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       STATISTICAL AND ANALYTICAL


           SUPPORT CONTRACTS


              FINAL REPORT


            (DELIVERABLE  4)
               VOLUME II
                                      U.S. Ervii oriental Protection Agency
                                      Region 111 Inloimation Resource
                                      Cente, tfPM52)
                                      841 Cuestnut Street
                                      Philadelphia, PA 19107        >
              Prepared for

     Water Quality Data Analysis
             Working Group
        (Chesapeake Bay Program)
           410 Severn Avenue
          Annapolis, MD  21405
              Prepared by

Martin  Marietta  Environmental Systems
            9200  Rumsey Road
          Columbia,  MD  21045
                                      February 1987
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                        TABLE  OF  CONTENTS


                                                          Page


VOLUME I

EXECUTIVE SUMMARY	      i i i


  I.  INTRODUCTION	."	      1-1


      A.  OBJECTIVES	      1-1


      B.  HISTORICAL  PERSPECTIVE	      1-2


 II.  DATA ANALYSIS APPROACH  AND RATIONALE	    II-l


      A.  RECOMMENDED ANALYSIS APPROACH	    II-l


      B.  LINKAGE WITH  HISTORICAL DATA		    11-12


III.  PRELIMINARY EVALUATION  OF  THE SAMPLING PROGRAM.   III-l


      A.  SYSTEMATIC  VS  RANDOM COLLECTION OF SAMPLES.   III-2


      B.  SPATIAL ALLOCATION  OF  SAMPLING EFFORT	   III-3


      C.  TEMPORAL ALLOCATION OF SAMPLING EFFORT	   III-5


      D.  VERTICAL SAMPLING ALLOCATION	.•.   III-6


      E.  VARIABLES MEASURED  AND MEASUREMENT
          TECHNIQUES	   III-ll


 IV.  DATABASE PROCESSING	    IV-1

  V.  CONCLUDING REMARKS	      V-l


 VI.  LITERATURE CITED	    VI-1


VOLUME II


APPENDIX A:  Deliverable 1


APPENDIX B:  Deliverable 2


APPENDIX C:  Deliverable 3


APPENDIX D:  Summary  of  Multivariate Statistical Methods
             Appropriate for  Chesapeake Bay Monitoring Data.
             Prepared by Dr.  Roger Green, University of Western
             Ontario, London, Ontario, N6A-5B7.


RP-825
RP-853
                               ill

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                              Martin Marietta Environmental Systems


                                    VOLUME II



                                   INTRODUCTION
    Under contract X-003321-02 (Monitoring Analysis Task) with
the Chesapeake Research Consortium, Martin Marietta Environmental
Systems was contracted to provide statistical and analytical
support to the Monitoring Subcommittee.  The analytical  and
statistical services contract involved the production of  three
interim deliverables and a Final Report.  The three interim
deliverables focused on data analysis and are presented  in
this volume as Appendices A-C.  Appendix D is a summary  of
multivariate statistical methods prepared by Dr. Roger Green  of
the University of Western Ontario for the Chesapeake Bay  Program
(CBP).  The contents of Appendices A-D are briefly summarized
below.
APPENDIX A:
       Deliverable 1;  Summary of data on relevant temporal  and
                       spatial scales

       Task A — Graphical and tabular characterization of the
                 data.

       Task B — Identification of a spatial aggregation  scheme
                 which captures the major patterns of spatial
                 variation in water quality and comparison of
                 the spatial aggregation scheme with the
                 segmentation scheme previously employed  by  the
                 CBP.

       Task C — Identification of a temporal aggregation scheme
                 which captures major patterns of seasonal and
                 interannual variation in water quality.

       Task D — Development and application of statistical
                 tests to examine the redundancy of information
                 in surface and above pycnocline samples, and
                 bottom and below pycnocline samples.

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                      Martin Marietta Environmental Systems


Appendix B:
       Deliverable 2;  Additional graphical displays of  the
                       data in support of CBP mission objectives,
                       Specifically graphics support for the
                       preparation of the water quality  section
                       of the "State-of-the-Bay Report."

       Task A — Isopleth mapping of dissolved oxygen concentra-
                 tions and salinity distributions during  summer
                 1984-1985

       Task B — Three dimentsional graphical displays of
                 selected water quality variables as a function
                 of latitude (station) and month (cruise)

       Task C — "Dilution plots" of selected water quality
                 variables versus salinity during spring and
                 summer 1984-1985

       Task D — Estimation of the volume of "anoxic" water  in
                 the mainstem Chesapeake Bay

       Task E — Time plots of dissolved oxygen at selected
                 monitoring stations

       Task F — Annual time series of selected water quality
                 variables averaged by season, Chesapeake Bay
                 segment, and depth strata.
Appendix C:


    •  Deliverable 3;  Review and evaluation of univariate
                       statistical analysis methods appropriate
                       for detection of temporal  trends  in  CBP.
                       water quality monitoring data.


Appendix D:


    •  Summary of multivariate statistical methods appropriate
       for CBP monitoring data.  Prepared by Dr.  Roger  Green
       for the Water Quality Data Analysis Working Group
       (Chesapeake Bay Program).
                               VI

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 I
 •                                    APPENDIX A

                     STATISTICAL AND ANALYTICAL SUPPORT CONTRACT
 ™                                  DELIVERABLE 1
 •                              (Chapters  II and III)
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                                  Martin Marietta Environmental Systems
                     APPENDIX  A;   DELIVERABLE  1


 II.   RESULTS..-	     II-l
       A.   TASKS B  AND C	     II-l
       B.   TASK A	     11-28
       C.   TASK D	     11-218
III.   LITERATURE CITED	   III-l
                                           A-3

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                          II.  RESULTS
                       A.  TASKS B AND C
Objectives


     To determine spatial patterns or groupings among
mainstem water quality monitoring stations which  capture
major regional variations in water quality (Task  B), and  to
develop temporal groupings among"cruises which can be used to
identify seasonal and interannual patterns in the data  (Task C)
Methods
     Principal components analysis  (PCA)  (Morrison  1976) was
used to determine spatial (over stations) and temporal  (over
cruises) aggregation schemes.  PCA was selected because  it
provides a method for parsimoniously reducing the large  amounts
of complex and correlated information contained in  a set of
water quality variables into a new set of uncorrelated vari-
ables (i.e., the principal components)-.

     Each principal component represents  a weighted linear
combination of the original water quality variables.  The
choice of weights (parameter estimates) are determined by an
eigenanalysis of the data covariance or correlation matrix.
This matrix summarizes the relationships  among the measured
values of the water quality variables.  From these  calculations,
each principal component is associated with an eigenvalue and
eigenvector extracted from the covariance or correlation
matrix.  The magnitude of the eigenvalue  is a measure of the
total system variation explained by that principal  component.
The coefficients associated with each eigenvector provide the
weights employed in the calculation of the principal component
(i.e., a linear combination of the variables).  Analysis results
are interpreted by examining the correlations between the
water quality variables and the new principal components (Cooley
and Lohnes 1971).  That is, the magnitude of the correlation
between the values of a principal component and a water quality
variable indicates the contribution of that variable to the
principal component.

     Principal components analyses were performed on surface
and bottom, surface only, and bottom only values of a subset
of the water quality variables measured.  All analyses were

                              II-l

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performed on the appropriate correlation matrix.  Table "II-l
summarizes the variables included in these analyses.  Multiple
PCA analyses were performed on various combinations of the
water quality variables to identify a spatial and temporal
aggregation scheme which was consistent with all of the vari-
ables as well as subsets of the variables.  Table II-2 shows
the combinations of the water quality variables upon which PCA
were performed.  To identify spatial groupings of stations,
station centroids were calculated by averaging principal compo-
nent values for each station (i.e., averaging component scores
over all cruises for each station).  Bivariate plots of the
first and second component station centroids were then used to
identify groups of similar stations.  To identify temporal
groupings of cruises, cruise specific centroids were calculated
by averaging component scores over all stations for each cruise.
Bivariate plots of the first and second component cruise cen-
troids were then used to define groups of similar cruises.

     We first identified spatial and temporal aggregation
schemes from the PCA applied to surface and bottom values of
all of the water quality variables (physical, nutrient, and
response variables).  To ensure that the proposed spatial and
temporal aggregation schemes derived from analysis of surface
and bottom values of all of the water quality variables was
also consistent with subsets of these variables, PCAs were
applied to the remaining combinations of water quality variables
shown in Table II-2.  Station and cruise centroids were calcu-
lated as described above and plotted using designations cor-
responding to the proposed spatial and temporal aggregation
scheme.
Results
                                                                       I
     Spatial Analysis


     The first two principal components from a PCA performed           '
on surface and bottom values of all of the water quality vari-
ables (PNR in Table II-2) accounted for 61 percent of total            I
variation in the data.  Salinity and water density were posi-          |
tively correlated with the first component (PCI), while nitrite/
nitrate concentrations were negatively correlated with PCI'.            .
Water temperature, bottom orthophosphorus, and ammonium concen-        1
trations were positively related to PC2.  Dissolved oxygen             '
concentrations and bottom chlorophyll-a concentrations were
negatively correlated with PC2.  Figure Il-la shows the station        1
centroids for the first and second principal components.               t
Shown on the axes are the original water quality variables
which were highly correlated with each principal component             i
                              II-2

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                     Martin Marietta Environmental Systems

     Principal component 1 represents a physical salinity/
density gradient among stations, as well as a nitrite/nitrate
gradient.  Stations with positive scores on PCI have higher
salinities and water densities  (sigma-t),  and low nitrite/
nitrate concentrations.  Stations with negative scores on PCI
have lower salinities and water densities, and relatively
higher nitrite/nitrate concentrations.

     Principal component 2 characterizes a gradient of dissolved
oxygen concentration, water temperature, and dissolved nutrients
in bottom waters.  Characteristics of stations with positive
scores on PC2 are relatively higher water  temperatures, low
dissolved oxygen concentrations, higher concentrations of
orthophosphorus, ammonium and nitrite/nitrate in bottom waters,
and lower concentrations of chlorpphyll-a  in the bottom layers.
Stations with negative scores on PC2 are distinguished by
generally lower water temperatures, higher dissolved oxygen
concentrations, and lower concentrations of dissolved nutrients
(NH4, PO4F, NO23) in bottom waters.

     Five regions were identified from this analysis of
Chesapeake Bay water quality data and are  indicated by region
specific symbols in Fig. Il-la.  Table II-3 lists the stations
included in each proposed region.

     These proposed regions were identified by a visual inspec-
tion of the clustering of station ce.ntroids.  The consistency
of the spatial aggregation scheme was further examined by
calculating station centroids for each seasonal period and
visually inspecting the resulting clustering of these centroids.
The seasonal periods employed in this analysis were identified
from the temporal aggregation scheme presented in the next
section.

     Figure II-2 compares the proposed spatial aggregation
scheme developed from this analysis with the previously pro-
posed segmentation scheme employed by the Chesapeake Bay
Program.  Notable features in the proposed scheme are the
division of the lower main Bay  into east and west subregions,
the separation of the Bay mouth stations, the lateral extension
of the northern main Bay region into the northern end of the
middle Bay region, the collapsing of the two northernmost CBP
segments into the Upper Bay/Susquehanna Flats region, and the
combination of CBP segments 4 and 5 into the middle Bay region.

     The consistency of the proposed segmentation scheme is
displayed in Figs. II-lb-g.  These figures show the station
centroids from PCA applied to the combinations of water quality
variables footnoted in Table II-2.  Similar results were ob-
tained for the remaining combinations of variables.  The pro-
posed spatial aggregation scheme determined from the PCA ap-
plied to surface and bottom values of all of the water quality
variables has been superimposed on the results.  For each PCA,

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                     Martin Marietta Environmental Systems

station centroids from the first two principal components are
shown with symbols corresponding to each of the proposed, spatial
groupings (regions).  These figures show that the proposed
spatial aggregation scheme determined from analysis of all of
the water quality variables is also applicable for the various
subsets of the water quality variables.


     Temporal Analysis


     Cruise centroids for the first two components from a
principal components analysis of surface and bottom values of
all of the water quality variables (PNR in Table II-2) are
shown in Fig. II-3a.  The first two components accounted for
61% of the variation observed in the data.  Table II-4 provides
a summary of cruise designations and associated sampling
periods used in Pig. II-3.  Principal component 1 is positively
correlated with salinity and water density.  Principal component
2 is positively correlated with water temperature and bottom
dissolved nutrient concentrations.

     Distinct seasonal patterns are evident in Fig. II-3a.
These seasonal groupings were identified by a visual inspection
of the clustering of cruise centroids.  Summer conditions are
app-rent in 1984 (cruises 1 through 7) and 1985 (cruises 22
through 27).  Winter conditions are represented by cruises 13
through 17. . Two transitional periods, fall 1984 (cruises 8
through 12) and spring 1985 (cruises  18 through 21), are also
apparent.

     Results of principal component analyses applied to the          •
selected subsets of water quality variables listed in Table
II-2 (Figs. II-3b-g) suggests that the proposed temporal aggre-      I
gation scheme is also generally applicable to subsets of vari-       |
ables.  Similar results were also obtained with the remaining
combinations of variables shown in Table II-2.                       •
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Table II-l.  Water quality  variables  (and their variable
             name designations)  included in Principal
             Components Analyses  of spatial and temporal
             patterns among Chesapeake Bay mainstem stations
             and cruises
Parameter
Class1
Surface
Bottom
Water temperature

Salinity

Water density

Orthophosphate

Ammonium

Nitrite and Nitrate

Chlorophyll-a

Dissolved oxygen

Conductivity

Silica

Total nitrogen

Total phosphorus

Station depth
  P

  P

  P

  N

  N

  N

  R

  R

  P

  N

  N

  N

  P
TEMP-S

SAL-S

SIGMA-S

PO4F-S

NH4-S

NO23-S

CHLA-S

DISOXY-S

COND-S

SI-S

TN-S

TP-S

DEPTH
TEMP-B

SAL-B


SIGMA-B

PO4F-B

NH4-B

N023-B


CHLA-B

DISOXY-B


COND-B

SI-B

TN-B

TP-B
*Variables included in this  analysis  were  classified into
 one of three categories:  P ซ physical, N ป nutrient,
 R * system response.
                           II-5

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Table II-2.
Combinations of water quality  variables  (from
Table II-l) upon which Principal Component
Analyses were performed  to determine  spatial
and temporal groupings of CBP  mainstem stations
and cruises
Analysis^3*
     Surface/Bottom Data
    Parameters
                       S & B
   P(b)


   R(b)

  PN


  PR


  NR
 PNR
          S & B

          S & B


          S & B

          S & B


          S & B


          S & B



          S & B
TEMP, SAL, SIGMA,
  PO4F, NH4,
  NO23, CHLA,
  DISOXY

P04Fr NH4, N023

SAL, TEMP, COND,
  DEPTH

CHLA, DISOXY

TEMP, SAL, P04F,
  NH4, N023

TEMP, SAL, CHLA,
  DISOXY

TN, TP, PO4F, NH4,
  N023, SI,
  DISOXY, CHLA

TEMP, SAL, SIGMA,
  PO4F, NH4,
  NO23, CHLA,
  DISOXY, TN, TP
         quality variables are  designated as  belonging to
   one of three categories (see Table  II-l):
     P ป physical variables
     N ป nutrient variables
     R * system response variables

  )Results from PCA are presented  in Figs.  II-l and II-3
   for the combinations of variables.
                           II-6

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          Table  II-2.  Continued
          Analysis^3)
Surface/Bottom Data
    Parameters
           PNR'
           PNR1
       B
TEMP, SAL,  SIGMA,
  PO4F, NH4,
  N023, CHLA,
  DISOXY, TN,t  TP

TEMP, SAL,  SIGMA,
  P04P, NH4,
  NO23, CHLA,
  DISOXY, TN,  TP
                                     II-7

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Table II-3.
     Proposed segmentation  of  mainstern water quality
     monitoring stations  based on a principal compo-
     nents analysis of  selected physical,  nutrient,
     and system response  variables.  Station desig-
     nations are historic Chesapeake Bay Program
     identifiers.
Region
     Designation
Monitoring Stations
 II
III
 IVa
 IVb
        Upper Bay/Susquehanna  Flats
Upper Main Bay
Middle Main Bay
Lower Main Bay-West
Lower Main Bay-East
        Bay Mouth
1.1
2.1, 2.2

3.1, 3.2, 3.3W,  3.3E,
3.3C
4.1W, 4.2W, 4.2E,  4.3W

4.1C, 4.IE, 4.2C,
4.3C, 4.3E, 4.4
5.1, 5.'2, 5.3, 5.4W
5.4

5.5
6.1, 6.2, 6.3, 6.4
7.3

7. IN, 7.la, 7.1,  7.2E,
7.2, 7.3E
8.1

7.4N, 7.4,
8. IE
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Table II-4.
Summary of Chesapeake  Bay  mainstem water
quality cruise designations and time periods
in which each cruise was conducted
 Cruise
                         Dates
    1
    2
    3
    4
    5
    6
    7
    8
    9
   10
   11
   12
   13
   14
   15
   16
   17
   18
   19
   20
   21
   22
   23
   24
   25
   26
   27
                     18-27  June 1984
                     10-14  July 1984
                     23-25  July 1984
                     6-7  August 1984
                     27-30  August 1984
                     10-19  September 1984
                     24-26  September 1984
                     8-10 October 1984
                     22-24  October 1984
                     5-16 November 1984
                     10-12  December 1984
                     14-24  January 1985
                     11-14  February 1985
                     4-6  March 1985
                     18-26  March 1985
                     8-11 April 1985
                     22-24  April 1985
                     6-8  May 1985
                     20-22  May 1985
                     3-5  June 1985
                     17-19  June 1985
                     8-10 July 1985
                     22-24  July 1985
                     6-8  August 1985
                     19-21  August 1985
                     9-11 September 1985
                     23-25  September 1985
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           Figure II-l,
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Results of a Principal Components Analysis
applied to various combinations of water quality
variables measured at CBP ma ins tern stations.
Shown are station centroids averaged over
cruises.  Each station centroid is identified
by symbols corresponding to the five spatial
groupings listed in Table II-3.   (square =
Region I, triangle ป Region II, star =
Region III, plus a Regions IVa and IVb,
diamond * Region 'V).  Water quality variables
that are highly correlated (|r( > 0.5) with each
of the first two principal components are shown
on the respective axes.  Unless otherwise noted,
all analyses were performed on surface and
bottom values.  The combinations of water
quality variables included in this figure are:
(a) TEMP, SAL, SIGMA, P04F, NH4 N023, CHLA, DISOXY
(b) PO4F, NH4, NO23
(c) CHLA, DISOXY
(d) TEMP, SAL, COND, DEPTH
(e) TEMP, SAL, SIGMA, P04F, NH4, N023, CHLA,
    DISOXY, TN, TP
(f) TEMP, SAL, SIGMA, PO4F, N023, CHLA, DISOXY,
    TN, TP (surface only)
(g) TEMP, SAL, SINGMA, P04F, NH4, N023, CHLA,
    DISOXY, TN, TP (bottom only)
                                     11-11

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                        Martin Marietta Environmental Systems
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Figure II-2,
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with the  previous spatial  segmentation  scheme
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                                11-20

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Results of a Principal Components Analysis
applied to various combinations of water  quali-
ty variables measured at CBP mainstem  stations.
Shown are cruise centroids averaged  over  sta-
tions.  Each cruise is identified by symbols
corresponding to the proposed  temporal groupings
identified on page II-4.   (square =  Summer
1984, triangle = Fall 1984, star = Winter
1985, plus ป Spring 1985, diamond a  summer
1985).  Water quality variables that are  highly
correlated (|r| > 0.5) with each of  the first
two principal components are shown on  the
respective axes.  Unless otherwise noted, all
analyses were performed on surface and bottom
values.  The combinations of water quality
variables included in this figure are:

(a) TEMP, SAL, SIGMA, P04F, NH4 N023,  CHLA, DISOXY

(b) P04F, NH4, N023

(C) CHLA, DISOXY

(d) TEMP, SAL, COND, DEPTH

(e) TEMP, SAL, SIGMA., P04F, NH4, N023,  CHLA,
    DISOXY, TN, TP

(f) TEMP, SAL, SIGMA, P04F, NO23, CHLA, DISOXY,
    TN, TP (surface only)

(g) TEMP, SAL, SIGMA, P04F, NH4, N023,  CHLA,
    DISOXY, TN, TP (bottom only)
                                     11-21

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                     Martin Marietta Environmental Systems

                           B.  TASK. A
     To provide succinct graphical and tabular summarizations
of water quality data collected at Chesapeake Bay mainstem
monitoring stations.
Methods
     Two methods were used to graphically display mainstem
water quality data.  The first method provides a longitudinal
"slice" down the Chesapeake Bay main channel.  For these graphs,
key nutrients, physical parameters, chlorophyll-a, and dissolved
oxygen concentrations were plotted against mainstem station
latitudes for selected cruises.  Sample depth relative to the
pycnocline (surface, above pycnocline, below pycnocline, and
bottom) are denoted.  Mainstem data collected in the eastern
and western Virginia mainstem are designated to allow compari-
son between these two regions.  The latitudes associated with
major tributaries are annotated, as well as the latitude asso-
ciated with the Maryland-Virginia border.

    •The second graphical method employed displays key nutrient
data, physical data, dissolved oxygen, and chlorophyll-a concen-
trations as a function of time for select stations (2.2, 3.2,
4.4, 6.3, 7.2, and 7.4).  Sample depth, where appropriate, is
indicated.

     Tabular data summaries were also prepared for key nutrient
and physical variables, and dissolved oxygen and chlorophyll-a
concentrations.  The temporal and spatial aggregation schemes
developed in Tasks B and C were used to collapse the 49 mainstem
x 27 cruise combinations into a more manageable number (i.e.,
a table consisting of 5 seasons X 6 regions).  Summary statis-
tics (mean, variance, coefficient of variation, minimum value,
maximum value, and number of observations) were calculated and
presented for 11 water quality variables for each of the 5
seasons and 6 regions.  Summary statistics were calculated for
the entire water column and, where appropriate, for layers of
the water column with respect to the pycnocline (above, within,
and below).

     To facilitate the production of additional summary graphics,
Martin Marietta Environmental Systems has transferred all
graphics software and corrected data sets used to generate
the plots presented under Task A to the CBP.
I
                            11-29

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                      Martin Marietta Environmental Systems


Results
    Longitudinal mainstem summary graphics  for  nitrite/nitrate,
orthophosphate, total nitrogen, total  phosphorus,  ammonium,
total suspended solids, secchi depth,  salinity,  water tempera-
ture, chlorophyll-a, and dissolved oxygen concentration are
presented in Figs. II-4 to_II-14.  Similarly, summary graphics
of the temporal distribution of these  parameters are presented
in Figs. 11-15 to 11-25.  Summary statistics  for the entire
water column are presented in Table  II-5, and summary statistics
for the same variables stratified by position relative to the
pycnocline are presented in Tables II-6  to  11-16.
                              11-30

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 I
                              Martin Marietta Environmental Systems
            Table  II-5.
Water column summary  statistics  for select
water quality variables  from CBP mainstern
stations.  Data are presented based on the
spatial  (see Table II-3)  and temporal (see
page II-4) aggregation schemes developed in
Tasks B and C.  The water quality variables
included in this table are:

(a) nitrite/nitrate

(b) orthophosphorus

(c) total nitrogen

(d) total phosphorus

(e) ammonium

(f) total suspended solids

(g) secchi depth

(h) salinity

(i) water temperature

(j) chlorophyll-a

(k) dissolved oxygen
I
             11-31

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Martin Marietta Environmental Systems

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                                            Martin Marietta Environmental Systems
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           11-43

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Martin Marietta Environmental Systems

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                   Martin Marietta Environmental Systems
Table II-6.
Summary statistics  for nitrite/nitrate by depth
strata (a ~ above  pycnocline, b — below
pycnocline) from CBP mainstem monitoring sta-
tions.  Data presentation is based on the
spatial (see Table  II-3)  and temporal (see page
II-4) aggregation schemes developed in Tasks B
and C.
                           11-45

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Martin Marietta Environmental Systems

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            J.J.-48

-------
                   Martin Marietta Environmental Systems
Table II-7.
Summary statistics  for orthophosphorus by depth
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tions.  Data  presentation is based on the
spatial (see  Table  II-3)  and temporal (see page
II-4) aggregation schemes developed in Tasks B
and C.
                           11-49

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Martin Marietta Environmental Systems

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           11-5:

-------
                   Martin Marietta Environmental Systems
Table II-8.
Summary statistics  for total nitrogen by depth
strata (a -- above  pycnocline, b ~ below
pycnocline) from CBP'mainstem monitoring sta-
tions.  Data presentation is based on the
spatial (see Table  II-3)  and temporal (see page
II-4) aggregation schemes developed in Tasks B
and C.
                           11-53

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Martin Marietta Environmental Systems  '

-------
                                           Martin Marietta Environmental Systems
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           11-56

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Table II-9.
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Martin Marietta Environmental Systems

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I
                               Martin Marietta Environmental Systems
           Table  11-10.
Summary statistics  for ammonium by depth
strata  (a —  above  pycnocline, b — below
pycnocline) from CBP mainstem monitoring
stations.  Data  presentation is based on
the spatial (see Table II-3) and temporal
(see page II-4)  aggregation schemes
developed in  Tasks  B and C.

-------
Martin Marietta Environmental Systems

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           11-64

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                               Martin Marietta Environmental Systems
I
           Table  11-11.
Summary statistics for total suspended solids
by depth strata  (a ~ above pycnocline, b  —
below pycnocline)  from CBP mainstem monitoring
stations.  Data  presentation is based on the
spatial (see Table II-3)  and temporal (see
page II-4) aggregation schemes developed in
Tasks B and C.
                                      11-65

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Martin Marietta Environmental Systems
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          Table 11-12.
Summary statistics  for salinity by depth strata
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developed in Tasks  B  and C.
                                     11-69

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Martin Marietta Environmental Systems

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                                                       11-73

-------
Martin Marietta Environmental Systems

-------
                    Martin Marietta Environmental Systems
Table 11-13.
Summary statistics  for water temperature by
depth strata  (a  —  above pycnocline, b —
below pycnocline c  — within pycnocline) from
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sentation is  based  on the spatial (see Table
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schemes developed in Tasfcis B and C.
                            11-75

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Martin Marietta Environmental Systems

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          11-77

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Martin Marietta Environmental Systems







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          11-78

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                                              11-79

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Martin Marietta Environmental Systems

-------
                              Martin Marietta Environmental Systems
          Table 11-14.
Summary statistics  for chlorophyll-a by depth
strata (a -- above  pycnocline, b — below
pycnocline) from CBP mainstem monitoring sta-
tions.  Data presentation is based on the
spatial (see Table  II-3)  and temporal (see page
II-4) aggregation schemes developed in Tasks B
and C.
I
             11-81

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Martin Marietta Environmental Systems

-------
                                          Martin Marietta Environmental Systems
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                                                      11-83

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Martin Marietta Environmental Systems
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           11-84

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                     Martin Marietta Environmental Systems
Table 11-15.
Summary statistics  for dissolved oxygen by
depth strata  (a  —  above pycnocline, b —- below
pycnocline, c  — within pycnocline) from CBP
maihstem monitoring stations.  Data presentation
is based on the  spatial (see Table II-3) and
temporal (see  page  II-4) aggregation schemes
developed  in Tasks  B and C.
                             11-85

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Martin Marietta Environmental Systems

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                                           Martin Marietta Environmental Systems
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Figure II-4.  Longitudinal  display of nitrite/nitrate
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       11-115

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Figure 11-10.  Longitudinal display of secchi depth  for GBP
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                          11-139

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            Figure 11-23.  Temporal display of water  temperature for
                           See Fig. 1-1 for  locations  of  the stations
                           The stations included  in  this  figure are:
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                           (a) Station CB2.2

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                                     11-243

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Martin Marietta Environmental Systems

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           Figure  11-24.   Temporal display of chlorophyll-a concen-
1                           trations for selected CBP mainstera stations
                           for  all cruises.  See Fig. 1-1 for locations
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   ^^      Figure 11-25.  Temporal  display of dissolved oxygen concen-
                           Itrations  for selected CBP mainstem stations
                           for  all cruises.  See Fig. 1-1 for locations
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                                       11-259

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                            C.   TASK  D


Objective


     To determine whether surface/above pycnocline  and  bottom/
below pycnocline samples provide  redundant information.


Methods•
     The upper and lower depths of  the pycnocline  for  each
station and cruise combination were identified  using the  method
employed by the CBP data generators (MD OEP  Scope  of Work
1984; VIMS Scope of Work Attachment B  1984;  ODU Scope  of  Work
1984).  Then, all nutrient, chlorophyll-a, and  total suspended
solids samples were classified into strata  (surface, above
pycnocline, below pycnocline, and bottom) as well  as into
depth ^strata  in the water column .(above or below the pycnocline)
Values were included  in the analysis only if a  complete water
column record was available (i.e.,  surface and  above pycnocline
samples; below pycnocline and bottom samples).

     The Wilcoxon signed ranks test, a nonparametric analog
of the student's matched-pair t test  (Conover 1980), was  used
to test the redundancy of information within each  depth strata
for nitrate/nitrite,  ammonium* total nitrogen,  total phosphorus,
orthophosphorus, total suspended solids, and chlorophyll-a
concentrations.  This test was chosen because of the possibility
of non-normally distributed data.   The null  hypothesis is that
the difference between paired concentrations within a  depth
strata (surface/above pycnocline; below pycnocline/bottom)  is
zero, i.e., the paired data are redundant.

     For each station and cruise combination, differences
"between surface/above pycnocline and below pycnocline/bottom
values were calculated for each of  the water quality variables.
These differences were then grouped according to the temporal
and spatial aggregation scheme developed in  Tasks  B and C
(i.e., 5 seasons x 6  regions).  For each spatial and temporal
group, these  calculated differences were further divided  into
groups corresponding  to each data generator.  The  two-sided
Wilcoxon signed ranked test was applied to the  surface/above
pycnocline and below pycnocline/bottom differences for each of
variables in  the spatial/temporal/data generator groups.  A
two-sided test was used since the direction  of  the differences
was not important.
                                      11-26'

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                      Martin Marietta Environmental Systems


    Summary tables of the results were prepared for surface/
above pycnocline and below pycnocline/bottom differences for
each nutrient variable, chlorophyll-a, and total suspended
solids for each spatial/temporal/data generator group.  Summary
statistics include the average difference between paired values,
the sample variance, the Wilcoxon test T statistic, the proba-
bility of observing a test statistic giventhe null hypothesis,
the observed range of sample differences, and the number of
observations use'd in the analysis.
Results
    Summaries of the above analyses are presented by data
generator in Tables 11-16 (OEP), 11-17 (VIMS), and 11-13  (ODU).

    In Tables 11-16 to 11-18, significant results are indicated
by the presence of an asterisk next to the probability values.
The sign of the test statistic T aids inthe interpretation of
the results.  In tests of the above pycnocline depth strata, a
positive sign indicates that surface concentrations are generally
greater than above pycnocline concentrtions, and a negative
sign indicates the opposite.  In below pycnocline depth strata
analyses a positive sign on the test statistic T indicates that
below pycnocline concentrations are generally greater than
bottom concentrations, a negative test statistic indicates the
opposite.  Significant results provide a clear indication that
two samples within a depth strata provide non-redundant
information on the vertical distributions of concentrations.

    Caution should be used in interpreting th enon-significant
rest results in Tables 11-16 to 11-18 if the variance of  the
calculated differences is large.  A large variance indicates
that the difference between surface/above pycnocline or below
pycnocline/bottm values are large for some station-cruise
combinations within the teraporal/spatial/data generator group.
Thus, even though no consistent pattern of concentration
differences may exist, (i.e., non-significant Wilcoxon signed
ranks test results), in some instances the surface/above
pycnocline or below pycnocline/bottom samples are providing non-
redundant information.
                              11-268

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I
                             Martin Marietta Environmental Systems
         Table 11-16.
Results of analyses evaluating the redundancy
of selected water quality measurements collected
by MD OEP.  Comparisons are made between:   1)
measurements taken at the surface and above  the
pycnocline and 2) measurements taken below  the
pycnocline and at the bottom.  The water quality
variables included in this table are:
(a) nitrite/nitrate — surface/above pycnocline
(b) nitrite/nitrate — below pycnocline/bottom
(c) ammonium — surface/above pycnocline
(d) ammonium — below pycnocline/bottom
(e) total nitrogen — surface/above pycnocline
(f) total nitrogen —below pycnocline/bottom
(g) total phosphorus — surface/above pycnocline
(h) total phosphorus — below pycnocline/bottom
(i) orthophosphorus — surface/above pycnocline
(j) orthophosphorus — below pycnocline/bottom
(k) total suspended solids -- surface/above
                              pycnocline
(1) total suspended solids — below pycnocline/
                              bottom
(m) chlorophyll-a — surface/above pycnocline
(n) chlorophyll-a — below pycnocline/bottom
                                     11-269

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Martin Marietta Environmental Systems

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-------
                               Martin Marietta Environmental Systems
          Table 11-17.
Results of analyses evaluating the  redundancy
of selected water quality measurements collected
by VIMS.  Comparisons are made between:   1)
measurements taken at the surface and above  the
pycnocline and 2) measurements taken below the
pycnocline and at the bottom.  The  water  quality
variables included in this table are:
                i
(a) nitrite/nitrate — surface/above pycnocline

(b) nitrite/nitrate — below pycnocline/bottom

(c) ammonium — surface/above pycnocline

(d) ammonium — below pycnocline/bottom

(e) total nitrogen — surface/above pycnocline

(f) total nitrogen —below pycnocline/bottom

(g) total phosphorus —i surface/above pycnocline

(h) total phosphorus — below pycnocline/bottom

(i) orthophosphorus — surface/above pycnocline

(j) orthophosphorus — below pycnocline/bottom

(k) total suspended solids — surface/above
                              pycnocline

(1) total suspended solids — below pycnocline/
                              bottom

(m) chlorophyll-a — surface/above  pycnocline

(n) chlorophyll-a — below pycnocline/bottom
I
              11-285

-------
Martin Marietta Environmental Systems

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                    Martin Marietta Environmental Systems
Table 11-18.
Results of analyses evaluating the  redundancy
of selected water quality measurements  collected
by ODU.  Comparisons are made between:   1)
measurements taken at the surface and above  the
pycnocline and 2) measurements taken below the
pycnocline and at the bottom.  The  water quality
variables included in this table are:
(a) nitrite/nitrate — surface/above pycnocline
(b) nitrite/nitrate — below pycnocline/bottom
(c) ammonium ~ surface/above pycnocline
(d) ammonium — below pycnocline/bottom
(e) total nitrogen — surface/above pycnocline
(f) total nitrogen —below pycnocline/bottom
(g)' total phosphorus — surface/above pycnocline
(h) total phosphorus — below pycnocline/bottom
(i) orthophosphorus — surface/above pycnocline
(j) orthophosphorus •— below pycnocline/bottom
(k) total suspended solids ~ surface/above
                              pycnocline
(1) total suspended solids — below pycnocline/
                              bottom
(m) chlorophyll-a — surface/above pycnocline
(n) chlorophyll-a — below pycnocline/bottom
                            TI-301

-------
Martin Marietta Environmental Systems

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Martin Marietta Environmental Systems
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 III.   LITERATURE CITED
 •        Conover, W.J.   1980.   Practical Nonparametric Statistics.  2nd
                ed.   New  York:   John Wiley and Sons.
           IGooley, W.R.,  and P.R.' Lohnes.  1971.   Multivariate Data
                Analysis.   New  York:  John Wiley  and  Sons.
I           Morrison,  D.R.   1976.   Multivariate Statitical Methods.  New
                York:   McGraw-Hill Book Co.

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f.                                    APPENDIX B
                    STATISTICAL AND ANALYTICAL SUPPORT CONTRACT:
                                   DELIVERABLE 2
 I                                (Chapters  II and III)
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                                 Martin Marietta Environmental Systems
                    APPENDIX B;   DELIVERABLE 2



 II.  RESULTS	     II-l


      A.   ISOPLETH MAPPING	     II-l


      B.   THREE  DIMENSIONAL GRAPHICS	     11-30


      C.   DILUTION PLOTS	     11-38


      D.   VOLUME OF ANOXIC WATER	     11-58


      E.   DURATION OF LOW DISSOLVED OXYGEN
           CONDITIONS	     11-59


      F.   HISTORICAL COMPARISON  OF WATER QUALITY....     11-65


III.  LITERATURE CITED	    III-l
                                         B-3

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                                Martin Marietta Environmental Systems
                          II.  RESULTS

                      A.  ISOPLETH MAPPING

Objective

    To display summertime dissolved oxygen concentration and
salinity as a function of latitude and depth.

Methods

    Isopleth maps of salinity and dissolved oxygen  concentration
for 25 selected mainstem stations (Table  II-l)  for  cruises
conducted in July and August of each year (1984:  cruises 2-5;
1985:  cruises 22-25) were prepared.  Isopleth  maps were
constructed for each cruise individually  and from data  averaged
over all July and August cruises.  Since  not all depths were
sampled on every cruise, summertime averages were calculated
for a given sampling depth if samples were collected  at that
depth for at least three of the four July/August cruises.

Results
    Figures II-l and II-2 show the isopleth maps of  salinity
for each July and August cruise of 1984 and 1985.  Figure  II-3
shows .the salinity isopleths averaged over July and  August
cruises.
    Isopleth maps of dissolved oxygen concentration  for  July
and August cruises of 1984 and 1985 are shown  in Figs. II-4 and
II-5.  Dissolved oxygen isopleth maps averaged over  July and
August cruises are shown in Fig. II-6.
                                        II-l
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                       Martin Marietta Environmental Systems
Table H-l.
Monitoring  Stations used for  isopleth mapping
of mainstern salinity and dissolved  oxygen
concentrations
Station
CB1.1
CB2.1
CB2.2
CB3.1
CB3.2
CB3.3C
CB4.1C
CB4.2C
CB4.3C
CB4.4
CB5.1
CBS. 2
CBS.3
CBS .4
CBS.5
CB7.1
CB6.1
CB7.1S
CB6.2
CB6.3
CB7.2
CB6.4
CB7.3E
CB7.3
CB7.4
                                II-3

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       Figure  II-l.   Isopleth maps of salinity  (ppt)  for CBP ma ins tern
                      stations for summer 1984 cruises.   Sample depth
                      (in meters) is indicated by RDEPTH and station
                      latitude is indicated by RLAT.   Cruises included
                      in this figure are:
                      (a)  Cruise 2  (7/10-14/84)
                      (b)  Cruise 3  (7/23-25/84)
                      (c)  Cruise 4  (8/6-7/84)
                      (d)  Cruise 5  (8/27-30/84)
                                         II-5
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Figure II-2.  Isopleth  maps of salinity (ppt) for CBP mainstern
              stations  for  summer 1985 cruises.  Sample depth
              (in meters)  is indicated by RDEPTH and station
              latitude  is  indicated by RLAT.  Cruises included
              in this figure are:
              (a)  Cruise  22  (7/8-10/85)
              (b)  Cruise  23  (7/22-24/85)
              (c)  Cruise  24  (8/6-8/85)
              (d)  Cruise  25  (8/19-21/85)
                                       11-11
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Figure II-3.  Isopleth maps of  average  summer (July/August)
              salinity (ppt) for  CBP  mainstern stations in 1984
              and 1985.  Sample depth (in meters)  is indicated
             .by RDEPTH and station latitude is indicated by
              RLAT.  Asterisks  (*) identify depths sampled on
              three cruises and periods (.) identify depths
              sampled on four cruises.   Plots included in this
              figure are:
                                                      <

              (a)  1984 Summer  (July/August) average salinity

              (b)  1985 Summer  (July/August) average salinity
                                       11-17

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         Figure I 1-4.   Isopleth maps of dissolved oxygen  concentration
                        (ml/L)  for CBP ma ins tern stations for 1984  summer
                        cruises.  Sample depth (in meters)  is  indicated by
                        RDEPTH  and station latitude is indicated by
                        RLAT.   Cruises included in this figure  are:

                        (a)  Cruise 2  (7/10-14/84)

                        (b)  Cruise 3  (7/23-25/84)

                        (c)  Cruise 4  (8/6-7/84)

                        (d)  Cruise 5  (8/27-30/84)
                                       11-21

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I          Figure II-5.  Isopleth maps of  dissolved oxygen concentration
                        (ml/L) for CBP mainstem stations for 1985 summer
                        cruises.  Sample  depth (in meters) is indicated
                        by RDEPTH and station  latitude is indicated by
m                      RLAT.  Cruises included in this figure are:
                        (a)  Cruise 22   (7/8-10/85)
ง  •                    (b)  Cruise 23   (7/22-24/85)
—                      (c)  Cruise 24   (8/6-8/85)
•                      (d)  Cruise 25   (8/19-21/85)
                                       11-27

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Figure II-6.  Isopleth maps of average  summer  (July/August)
              dissolved oxygen concentration  (ml/L)  for CBP
              mainstern stations  in  1984 and 1985.  Sample
              depth (in meters)  is  indicated by  RDEPTH and
              station latitude is indicated by RLAT.  Asterisks
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              cruises.  Plots included  in this figure are:

              (a)  1984 Summer (July/August) average dissolved
                   oxygen concentration

              (b)  1985 Summer (July/August) average dissolved
                   oxygen concentration
                                        11-33

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                 B.  THREE DIMENSIONAL GRAPHICS
Objective


     To display selected water quality variables as  a function
of latitude and month.
Methods
     Three dimensional plots were produced to display  surface
and bottom concentrations of selected nutrient variables/
active chlorophyll-a concentrations/ and transformed Secchi
depth as a function of latitude and month.  Secchi depth was
transformed by calculating the natural logarithm of the inverse
of Secchi depth (ln(l/Secchi depth)).  The inverse of  Secchi
depth was used because this results in high values of  the  trans-
formed Secchi depth being associated with poor water clarity
and low values of the transformed Secchi depth being associated
with clear water.  The natural logarithm of the inverse of
Secchi depth was used to reduce the range of values of the
inverse of Secchi depth/ allowing for better depiction in
graphical displays.  Selected nutrient variables included  in
these three dimensional graphical displays are surface and
bottom concentrations of orthophosphate/ ammonium, nitrate, the
sum of nitrite and nitrate/ total nitrogen/ and total phosphorus.
Stations included in these displays are identified in Table
II-l.  Water quality data from stations at similar latitudes
in the east and west channels of the Virginia mainstem (i.e.,
CBS.5 and CB7.1; CB6.1 and CB7.1S; CB6.3 and CB7.2; CB6.4 and
CB7.3E) were averaged for each sample date and depth.

     All data were linearly interpolated using the SAS G3GRID
software.  Two graphical displays were produced for each
variable.  The two displays were produced by rotating the
graph axes to. highlight either the temporal or spatial aspect
of the data.

     Detailed adjustment of the positioning of the axes on
these graphs to provide the "best view" of the data was not
performed by Martin Marietta Environmental Systems.  Based on
requests from the Analytical Working Group/ these three dimensional
graphical displays are intended to provide the Working Group
with an initial screening of the selected data.  The Chesapeake
Bay Program, in conjunction with Computer Sciences Corporation,
will use this information to produce camera-ready graphics for
the water quality section of the "State-of-the-Bay" Report.
                             11-37

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                      Martin Marietta Environmental Systems

Results
     Examples  of the three-dimensional graphics  produced  in
support of  the water quality  chapter of the  "State-of-the-Bay1
Report are  presented in Fig.  II-7.
                              11-38

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                                Martin (Marietta Environmental Systems
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f          Figure II-7.  Sample  three dimensional  displays  of selected
                        water quality variables as  a  function of latitude
                        and month.  Sample view included  in this figure
 _                      are:
1
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(a)   Temporal perspective of the sum of  the
     surface concentrations of nitrite and
     nitrate
                        (b)  Spatial perspective  of  bottom ammonium
—                           concentration  (Note  that  June 1984  data
fl                           are incorrect  as shown)
(c)   Temporal perspective of surface total
     phosphorus concentration
(d)   Temporal perspective of bottom active
     chlorophyll-a concentration
(e)   Spatial perspective of bottom dissolved
     oxygen concentration
                                       11-39

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Martin Marietta Environmental Systems

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                      Martin Marietta Environmental Systems


                       C.  DILUTION PLOTS


Objective


     To provide a graphical display of the  mixing  dynamics
(conservative/nonconservative) of selected  nutrient  variables.


Method


     Bivariate scatter plots of surface  total  nitrogen con-
centration and total  phosphorus concentration  as a function of
salinity were produced for selected cruises.   Mainstern stations
included in these graphs are listed in Table II-l.  Data points
in each plot are labeled with  the CBP station  designation, and
different symbols are used to differentiate between  Virginia
western channel and Virginia eastern channel stations.


Results


     Surface total nitrogen dilution plocs  for selected cruises
are presented in Fig. II-8.  Surface total  phosphorus concentration
dilution plots are presented in Fig. II-9.
                              11-45

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                               Martin Marietta Environmental Systems
         Figure II-8.
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Display of  total  nitrogen concentration
(mg/L) as a function of  salinity (ppt)  for
selected cruises.  CBP station names are used
to identify stations.  Maryland mainstern and
Virginia western  channel stations are represented
by triangles  (A)  and Virginia  eastern channel
stations are  represented by asterisks (*).
Cruises included  in this figure are:
(a)  Cruise 4   (8/6-7/84)
(b)  Cruise 15  (3/18-26/85)
(c)  Cruise 16  (4/8-11/85)
(d)  Cruise 18  (5/6-8/85)
(e)  Cruise 19  (5/20-22/85)
(f)  Cruise 20  (6/3-5/85)
(g)  Cruise 22  (7/8-10/85)
(h)  Cruise 24  (8/6-8/85)
                                       11-47

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Martin Marietta Environmental Systems

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Figure II-9.
Display of  total  phosphorus concentration
(mg/L) as a  function  of  salinity (ppt) for
selected cruises.   CBP station names are used
to identify  stations.  Maryland mainstern and
Virginia western  channel stations are represented
by triangles (A)  and Virginia eastern channel
stations are represented by asterisks (*).
Cruises included  in this figure are:
                                (7/10-14/84)


                                (8/6-7/84)


                                (3/18-26/85)
 (a)  Cruise 2


 (b)  Cruise 4


 (c)~  Cruise 15

 (d)  Cruise 16   (4/8-11/85)


 (e)  Cruise 18   (5/6-8/85)


 (f)  Cruise 19   (5/20-22/85)

 (g)  Cruise 20   (6/3-5/85)


 (h)  Cruise 22   (7/8-10/85)

 (i)  Cruise 24   (8/6-8/85)
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                                       11-57

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Martin Marietta Environmental Systems
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                                        11-67

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                      Martin Marietta Environmental Systems


                    D.   VOLUME OF ANOXIC WATER


Objective


    To estimate  the volume  of anoxic water  in  the mainstem of
Chesapeake Bay during  summer 1984 and summer  1985,  and to
compare these estimates with historical estimates.

    As discussed in the Final Report, this  task was not completed
as part of this  contract.
                              11-68

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         E.   DURATION  OF LOW DISSOLVED OXYGEN CONDITIONS



 Objective
     To  prepare  graphical displays of dissolved oxygen concen-
 tration over the  year at selected mainstem monitoring stations,
 and  visually compare these to historical graphical displays of
 dissolved  oxygen  data.
Methods
     To  date/  time  plots  of dissolved oxygen have been prepared
 for  one station  in the current CBP water quality monitoring
 program.   Based  on future  consultations with members of the
 Analytical Working Group,  additional stations may be identified,

     Dissolved oxygen  concentration between 18 and 21 meters at
•monitoring station CB5.1 was  plotted versus month.   Two sets of
 plots were prepared.   The  first set displays the average
 dissolved  oxygen concentration between 18 and 21 meters.  The
 second  set of bivariate  plots displays the dissolved oxygen
 concentration observed at  21  meters (which was sampled on every
 cruise).
          Results
    Plots  of  the  average  dissolved  oxygen concentration between
 18  and  21  meters  and  dissolved  oxygen concentration at 21 meters
 at  station CB5.1  were nearly  identidal.  the 21 meter dissolved
 oxygen  concentration  data are presented  in Fig. 11-10.  Dissolved
 oxygen  at  20  meters for  1936-1938 and 1970 at a station located
 in  the  same vicinity  as  CB5.1 is  shown in Fig. 11-11.
                                        11-69
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         Figure  11-10.  Dissolved  oxygen concentration  (ml/L) as a
                         function of  month at 21-meter depth at CBP
•                       mainstern  station C85.1.  Years  included in this
                         figure are:

                         (a)  1984
                         (b)  1985

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                                        11-71

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Martin Marietta Environmental Syrtems

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Figure 11-11.  Historical dissolved oxygen concentration (ml/L)
               observed in 1936-1938 and 1970 as  a function of
               month at 20-meter depth in the vicinity of
 •                       station CB5.1.   [From Fig. 4 in Officer  et al.
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                                        11-75

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Martin Marietta Environmental Systems
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                      Martin Marietta Environmental Systems


           F.  HISTORICAL COMPARISON OF WATER QUALITY


Objective


    To provide annual time series of selected water quality
variables by season/ Chesapeake Bay program segment/ and depth
level.


Methods


    Average values of selected water quality variables were
calculated for the current monitoring program (1984 and 1985)
and for data in the CBP historical mainstem database (1936
through 1981).  Water quality variables included total nitrogen,
total phosphorus, and dissolved oxygen  concentration.  Average
values were calculated for each year by CBP segment/ season,
and depth level in the water column (_ฃ  10 meters and > 10
meters).

    Methods of calculation were consistent with those employed
by the Chesapeake Bay Program (1983).  The following criteria
were used in calculating the average values for each year,
segment/ season, and depth level.  First/ monthly averages were
calculated only if three or more values were present for a
month. 'Second, seasonal averages were-calculated only if at
least two monthly means were available  for months in that
season.  Thus/ seasonal means are unweighted averages of
available monthly means.  Each table indicates the number of
observations that contribute to each calculated average.


Results


    Average values of dissolved oxygen, total nitrogen, and
total phosphorus concentration by year, season/ CB segment, and
depth level are presented in Table II-2/ II-3, and II-4,
respectively.

    After these tables were prepared, some problems in the CBP
historical database were discovered.  Therefore, the average
values shown in Tables II-2 -through II-4 should not be interpreted
for trends in Chesapeake Bay water quality.
                              11-78

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                Martin Marietta Environmental Systems
Table II-2.
Average values  of  dissolved oxygen
concentration  (mg/L)  from current and
historical mainstem data.  Results are
presented by year^,  segment, season, and
depth- strata.   Segments included in this
table are:

(a)  CBP Segment CB3

(b)  CBP Segment CB4

(c)  CBP Segment CBS
                         11-79

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Martin Marietta Environmental Systems

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Martin Marietta Environmental Systems







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Table II-3.
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(a)  CBP Segment CB3

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(c)  CBP Segment CBS
                              11-85

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Table II-4.
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                              11-91

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III. LITERATURE CITED
 I     EPA (U.S.  Environmental Protection Agency).   1983.   Chesapeake Bay:
             A profile of environmental change.  Appendix B.


 ™     Hires/  R.I./ E.D. Stroup, and R.C. Seitz.  1963.  Atlas  of
             the distribution of dissolved oxygen and pH in  Chesapeake
I             Bay 1949-1961.  Chesapeake Bay Institute Graphical
             Summary Report No. 3.


 ||     Officer/ C.B., R.B. Biggs/ J.L. Taft, L.E. Cronin/ M.A.  Tyler/
             and W.  R. Boynton.  1984.  Chesapeake Bay Anoxia:
 —          -Origin/ development/ and significance.  Science  223(4631):
 1           22-27.


I        SAS Institute/ Inc.  1985.  SAS/Graph Users Guide/ Version  5.
             Gary/  North Carolina:  SAS Institute/ Inc.


 B      Seitz/  R.C.   1971.  . Temperature and salinity distributions  in
             vertical sections along the longitudinal axis and across
             the entrance of the Chesapeake Bay (April 1968  to March 1969),
             Chesapeake Bay Institute Graphical Summary No.  5.
  *mr


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                                 Martin Marietta Environmental Systems
                                       APPENDIX C
                     STATISTICAL AND ANALTYICAL SUPPORT CONTRACT;

                                     DELIVERABLE 3

                         (Chapters I - VIII,  Plus Appendix A)
                                          C-l


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                  APPENDIX C:   DELIVERABLE  3


 I.  INTRODUCTION	     1-1

     A.  STATISTICAL  AND ANALTYICAL SERVICES
         CONTRACT	     1-1

     B .  REPORT ORGANIZATION	     1-2

II.  DETECTION OF TRENDS IN WATER QUALITY DATA......    II-l

     A.  DYNAMICS OF  WATER QUALITY	    II-l

     B.  METHODS FOR  TREND DETECTION	    II-2

     C.  UNIVARIATE AND MULTIVARIATE STATISTICAL
         METHODS	    11-5
              D.   TREND DETECTION AND FUNCTIONAL ANALYSIS...   II-5

 ป   •   III.  MAINSTEM CBP WATER QUALITY MONITORING
              PROGRAM	  III-l
              A.   SUMMARY OF DATA COLLECTION METHODS	  III-l


I
m       IV.  UNIVARIATE STATISTICAL METHODS FOR  TEMPORAL
•            TREN D  DETECTION	   IV-1
     B.  FORMULATION  OF  RESPONSE AND EXPLANATORY
         VARIABLES	   III-l
     A.  INTRODUCTION	    IV-1

     B.  BOX-JENKINS  INTERVENTION ANALYSIS	    IV-1

     C.  PARAMETRIC METHODS	    IV-6

     D.  DISTRIBUTION  FREE METHODS	    IV-12
                                       C-3

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                 TABLE OF CONTENTS (Continued)
                                                         Page
   V.  DEMONSTRATION OF  SELECTED METHODS	      V-l

       A.  INTRODUCTION.... c.. . .c	      V-l

       B.  CALVERT CLIFFS  AND CBP MONITORING DATA..      V-2

       C.  ANALYSIS OF CALVERT CLIFFS DATA	      V-2

       D.  ANALYSIS OF 1984/1985 CBP MONITORING
           DATA	      V-39

  VI.  ANALYSIS FRAMEWORK  FOR TEMPORAL TREND
       DETECTION IN CBP  WATER QUALITY MONITORING
       PROGRAM	c	     VI-1

       A.  INTRODUCTION	00	     VI-1

       B.  RESPONSE AND  EXPLANATORY VARIABLES	     VI-1

       C.  SPATIAL AND TEMPORAL SCALES OF ANALYSES.     VI-2

       D.  APPLICATION OF  METHODS FOR TEND DETECTION    VI-4

VIII.  LITERATURE CITED	   VIII-1

APPENDIX A

       BIBLIOGRAPHY	      A-l
                                                                      I
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                     I.   INTRODUCTION
    Under contract X-003321-02 with the Chesapeake Research
Consortium, Martin Marietta Environmental Systems provides
statistical and analytical support to the Monitoring Subcom-
mittee of the Chesapeake Bay Program.  Specific technical
activities for this contract are directed by the Water Quality
Data Analysis Working Group of the Monitoring subcommittee
(the Analytical Working Group).  This working group consists
of water quality experts from state and federal agencies and
universities.  The major focus of the analytical program is
the water quality data collected at the 40 mainstem stations
as part of the CBP monitoring program.

    The analytical and statistical support contract involves
the production of four deliverables.  The first three deliver-
ables which are working documents, focuse on graphical and
statistical analyses.  The final deliverable is a report that  •
synthesizes the information and results presented in Delivera-
bles 1-3.  This document is Deliverable 3.

    Deliverable 3 is a review of statistical methods appropriate
for detecting temporal trends in Chesapeake Bay water quality
data.  -Emphasis is placed on the interfacing of these general
statistical techniques with the specifics of the present water
quality monitoring program.  As part of Deliverable 3, an
analysis framework for temporal trend detection tailored to the
present monitoring program is presented.
                    B.  REPORT ORGANIZATION
    This document is organized as follows:

    •  Chapter II presents a general discussion of the dif-
       ficulties associated with the detection of trends  in
       water quality data and the methods available for de-
       tection of trends

    •  Chapter III focuses on specific issues associated  with
       trend detection applied to- water quality data from the
       present CBP water quality monitoring program, including
       the formulation of response and explanatory variables,
       the treatment of censored data, and the potential  problems
       with homogeneity of trends

    •  Chapter IV presents a brief review of univariate
       statistical methods for temporal trend detection
                                       1-1
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                                                 *

Chapter V.uses Chesapeake  Bay water quality data  to
demonstrate some of  the  methods described in Chapter  IV

Chapter VI presents  an analysis framework for temporal
trend detection in water quality data and describes how
the general statistical  methods can be applied to the
present CBP monitoring program.
                                                                 I
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         II.  DETECTION OF TRENDS IN WATER  QUALITY DATA


                 A.  DYNAMICS OF WATER QUALITY


     Detection of temporal trends in Chesapeake Bay water
quality data is difficult.   Water quality in Chesapeake  Bay
depends on interactions among physical, chemical, and  biolog-
ical processes that operate on a variety of temporal and spa-
tial scales.  The water quality at a given  location at any
given time is the net result of the superimposition of multiple
cycles with different periodicities, as well as the result of
episodic events and natural long-term trends.  For example,
periodic cycles that can affect water quality by causing spa-
tial and/or temporal variation include cycles operating  on
diurnal time scales or less (e.g., tides, solar radiation),
seasonal time scales (e.g., tributary flow, biotic migrations,
water temperature), and annual time scales  (e.g., long-term
cycles).  Episodic events that can cause spatial and/or  tem-
poral variation in water quality include extreme floods  and
events resulting from wind-induced flow dynamics, such as
destratification (mixing) of the water column, formation of
Langmuir circulation, and the introduction  of coastal  waters
into the Bay.  Water quality measurements over time typically
exhibit cycles due to the supe'rimposition of those cycles that
operate on time scales that are greater than the frequency of
measurements.  Cycles occuring on fine time scales compared
to the frequency of measurements and episodic events contribute
to "noise" surrounding the observed cycles  in water quality.
The goal of trend detection is to determine any trends in
water quality, given the complicated nature of the observed
cycles coupled with the high "noise" level  resulting from the
super-imposition of fine time scale cycles  and episodic  events.

     Trend detection is even more difficult for the CBP,
since the goal of the analyses is not only  to detect trends,
but to attribute these trends to specific causes (i.e.,  manage-
ment actions vs. natural trends).  To attribute detected trends
to a particular cause implies that any trends in the causative
factors are also measurable.  As with water quality, these
causative factors also exhibit spatial and  temporal variation
(e.g., point and nonpoint nutrient loadings to the Bay).

     Another difficulty with detection of trends in water
quality data is that there are usually only limited data avail-
able.  Limited data can result in the appearance of short-
term trends when, in fact,  no long-term trends exist.  Similarly,
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detection of long-term trends, when they do exist/  is difficult
with limited data.  Consider the following  hypothetical  example.
A six-year time series of summer DO concentrations  below the
pycnocline shows a "downward trend" because during  the first
three years tributary flow patterns (e.g. ,  low spring flows)
resulted in a poorly defined pycnocline and therefore relatively
high DO, whereas during the latter three years tributary flow
patterns (e.g., high spring flows) resulted in little vertical
mixing and therefore low DO concentrations.  However, analysis
of the magnitudes of spring flow over  the long term (e.g.,  50
years) do not show a year-to-year trend.  That is,  the pattern
observed in spring flow for the six years of data  (low flow in
the first three years and high flow in the  last three years)
is not part of a long-term "upward" trend in spring flow.
Thus, the short-term downward trend in DO is due to the  limited
amount of available data (6 years) being coincident with a
short term pattern in spring flow (i.e., three years of  low
spring flows followed by three years of high spring flows).
If 50 years of DO data were available, we would not find a
long-term trend in DO attributable to  a trend in spring  flows.


                B.  METHODS FOR TREND  DETECTION


     There are many methods available  for detection of trends
in water quality.  These methods range from anecdotal judgements
to tabular/graphical analyses  (e.g., Ahlgren 1978;  Edmondson
1972) to sophisticated statistical techniques (e.g., Lettenmaier
1976; McLeod et al. 1983).   Past analyses performed as part of
the CBP have included both tabular/graphical and simple  statis-
tical analyses.  Figure II-l is an example  of a graphic  used
to infer that there has been a trend of worsening hypoxia in
Chesapeake Bay.  Table II-l summarizes the  results  of correla-
tion analyses of water quality data with time (years), performed
as part of CBP, which have been used to show a trend of  in-
creasing nutrient concentrations in Chesapeake Bay.

     There are tradeoffs associated with the various methods for
trend detection.  Obviously, any conclusions based  on anecdotal
information are tenuous at best.  Tabular/graphical approaches
tend to be relatively simple and straightforward to apply,  but
interpretation of results tends to be  very  subjective.   Also,
while use of tabular/graphical approaches may appear to  require       •
few assumptions, this is usually because all of the assumptions       Q
required for conclusions to be drawn are not explicitly  stated.
Statistical methods tend to" have more  restrictive data require-       m
ments, but allow for more objective treatment and  interpretation      g
of complex situations.  When statistical analyses are used,
statements concerning trends are based on probabilistic  state-
ments associated with specific hypotheses,  and conclusions  are
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          O t_  ป-
          go  o

          if
            ฃ
            1
                                                                                                  o
                                                                                                  03
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                                                                            f*.
                                                                            o
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Table II-l. Summary of the significance of Pearson
correlation coefficients computed between
average values of water quality variables in
the summer for regions (segments) of Chesapeake
Bay and year. The significance

tion was tested
at alpha
3
+ ป increasing trend, - =

Segment
CB-1
CB-2
CB-3
CB-4
CB-5
CB-6
CB-7
CB-8
WT-1
WT-2
WT-3
WT-4
WT-5
WT-6
WT-7
WT-8
TF-1
RET-1
LE-1
TF-2
RET- 2
LE-2
TF-3
RET-3
LE-3
TF-4 '
RET-4
LE-4
TF-5
RET- 5
LE-5
(Reproduced from
TP IFF TN
000
+ 00
000
000
0 0







0 0

0 0
0
0 +

0
00-
000
000






0 0

0
Flemer
et
N03 NO2

+
0
0
0






0
0

0
0
+

+
0
0
0


0

0

0

0
0
+
0
0
0







0

0
0
0

0
0
0
0

0
0
•f
0

—


0.05
of each correla-
(0 =
decreasing
al.
NH3
+
0
0
0
0







-



0


0
0
0

0
0
0
0

0

0
1983b)
TKN
0
0
0
0
0







0



0


_
0
0

0
0
0
0

0

0
no trend,
trend) .

CHL-AU
+
0
0
0
0







0

+
0
0


0
0
0









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less susceptible to (although  not completely  independent  of)
the biases of individual  investigators.  Furthermore,  proper
application of statistical techniques  involves  the  explicit
statement of all assumptions,  as well  as tests  and  diagnostics
for how well these assumptions are met.  Thus,  if data permit,
statistical analysis for  trend is preferrable over  tabular/
graphical and anectodal techniques.

     As discussed above,  statistical techniques  have many advan-
tages over graphical/tabular approaches for trend detection.
However, statistical methods also carry with  them more extensive
data requirements.  Therefore, for statistical methods to be
useful for trend detection, methods must be reviewed for  their
appropriateness for application to available  data.  Feedback
between the monitoring program and data requirements for
statistical .analyses allows for aspects of the monitoring
program and the statistical methods to be modified  in  response  •
to each other's specific  needs and limitations.  This  is
especially important for  the CBP water quality monitoring
program in order to ensure that data being collected now, and
data collected in the future,  fulfill  the data  requirements
of those statistical methods likely to be used.

     In addition to providing  feedback on the design of the
ongoing CBP monitoring program, preparation of a statistical
analysis plan also provides a  review of potential approaches
for the analysis of historical data.


      C.  UNIVARIATE -AND  MULTIVARIATE  STATISTICAL METHODS
     Univariate methods are emphasized in this report.  This
is not intended to imply that multivariate methods  (analysis
of multiple response variables) are not appropriate for CBP
monitoring data.  A major role of multivariate methods would
be in exploratory data analysis in order to understand rela-
tionships among response variables, and it is likely that
multivariate methods would be applied prior to, or  in conjunc-
tion with, univariate methods.  In our discussion of the analy-
sis framework, we indicate several situations when multivariate
methods would be appropriate.  As part of the demonstration
of the methods using the 1984/1985 monitoring data, we use
results from a mult-ivariate analysis of the data in conjunction
with univariate tests for trend.
                  D.   TREND DETECTION AND FUNCTIONAL ANALYSIS
             We distinguish two types of statistical analyses applicable
        to the water quality data:   trend detection and functional analysis,

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In general, the same statistical methods can be used for trend
detection and for functional analysis.  However/ the objectives
of the two types of analyses differ.  The goal of functional
analysis is to remove variation in the response variable using
explanatory variables and then to interpret, in a mechanistic
context, the determined relationships.  The variation  in the
response variable removed by the explanatory variable  may or
may not be trend.  With trend detection, as applied to Chesapeake
Bay water quality data, the goal is to use explanatory variables
only to remove natural variation that could obscure detection
of trends.  This seemingly subtle difference has important
ramifications for the formulation and incorporation of explana-
tory variables into analyses.

     In summary, both functional analyses and trend detection
include explanatory variables to remove "noise" in the data
(e.g., variation among sampling stations at a given time).
However, functional analyses include other explanatory variables
to elucidate relationships between the response variable and
explanatory variables.  Trend detection only includes  explanatory
variables to remove "noise" in the data.  Results from trend
detection are not designed to provide information on the rela-
tionships among response and explanatory variables.
                      ป

     To illustrate, suppose we are analyzing average spring
chlorophyll-a concentrations in a region of Chesapeake Bay
over years.  An ANOVA approach could be appropriate for both
trend detection and functional analysis of these data.  Both
analyses would likely include station as a factor to remove
variation among sampling stations in the region.  However, it
is likely that a functionally oriented analysis would  also
involve incorporation of nutrient concentrations as explanatory
variables and would attempt to explain some of the variation
in chlorophyll-a by variation in nutrients.  This analysis may
show that high chlorophyll-a concentrations are associated
with high nutrient concentrations.  Such relationships may
exist regardless of whether chlorophyll-a and nutrient concen-
trations are trending in time.  The functional analysis approach
focuses on tests of hypotheses concerning the relationship
between chlorophyll-a and nutrient levels, rather than hypo-
theses about trends over time.

     To detect trends in spring chlorophyll-a concentrations          fe
that may be attributable to reduced nutrient loadings  and             I
lower overall nutrient concentrations, analyses should include
a time-related variable as an explanatory variable.  If a             •
trend in chlorophyll-a is detected/ then we would attempt to          g
relate the trends to various functionally oriented explanatory
variables (e.g., nutrient concentrations).  Inclusion  of nutri-
ent concentrations as an explanatory variable with the time-
related variable in initial analyses can create collinearity
problems among explanatory variables.  If there is a temporal


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trend in nutrient concentrations, the nutrient explanatory
variable would be confounded with the time-related  explanatory
variable, thereby greatly complicating  intrepretation  of  the
results.

     Clearly, there is a continuum between functional  analysis
and trend detection, and between functionally oriented explan-
atory variables and empirically oriented explanatory variables.
In this report we emphasize trend detection.  Therefore,  water
quality data are analyzed with minimal  inclusion of functionally
oriented explanatory variables on the basis  that many  of  the
candidate explanatory variables are likely to be related  to
pollution control management actions.  We view functionally
oriented analyses as useful for attempting to attribute
detected trends to explanatory variables.      -^

     Because of the emphasis on trend detection, several
statistical methods that are not especially  well suited for
temporal trend detection have been omitted from this review
(e.g., frequency domain analysis).  The omission of these
methods should not be interpreted as dismissal of these methods
for analysis of Chesapeake Bay water quality data.  Indeed,
several of these methods, as well as versions of methods  in-
cluded in this report, would be preferred for functionally
oriented analyses.  Subsequent work should include  a document
that is parallel to this one, but which emphasizes  statistical
analysis for temporal and spatial patterns in the water quality
data and the relationships among the water quality  variables.
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      III.  MAINSTEM CBP WATER QUALITY MONITORING PROGRAM


             A.  SUMMARY OF DATA COLLECTION METHODS


     Water quality is monitored at 21 raainstem  stations  in
Maryland and 19 mainstem stations in Virginia  (Fig.  III-l).
Virginia stations are sampled by the Virginia  Institute  of
Marine Sciences (VIMS) and Old Dominion University  (ODU).  All
Maryland stations are sampled by the Maryland Office of
Environmental Programs (OEP).  Sampling is conducted twice
monthly from March through October and once monthly  from November
through February/ resulting in a total of 20 cruises per year.
Each cruise covers, to the extent possible, the entire station
grid and takes about three days.  Table III-l  lists  the  water
quality variables measured at each station.

     The depths at which measurements are made  varies with
parameter and data generator (i.e., OEP, VIMS,  or ODU).  OEP
measures temperature, conductivity (salinity),  dissolved oxygen
concentration, and pH at 0.5 (surface), 1.0, 2.0, and 3.0 m
below the air-water interface.  Below 3.0' m depth, OEP measure-
ments are taken at 3.0-m intervals; however, if DO varies more
than 1.0 mg/L or conductivity varies more than  1000 micromho/cm
over any of the 3.0-m intervals, measurements are taken  every
1.0 m within that 3.0-m interval.  ODU and VIMS take measure-
ments of temperature, conductivity, DO, and pH  at 1.0 m  (sur-
face), and then at 2.0-m intervals thereafter.

     For all of the remaining variables shown  in Table III-l
except Secchi depth (i.e., nutrient variables,  chlorophyll-a,
and TSS), samples are taken at 0.5 m (surface)  and 1.0 m above
the bottom by OEP, and at 1.0 m (surface) and 1.0 m above the
bottom by VIMS and ODU.  In addition, at the Maryland stations
and a subset of nine mainstem stations in Virginia, two  samples
are taken above and below the pycnocline.


     B.  FORMULATION OF RESPONSE AND EXPLANATORY VARIABLES


Definition of Response Variables


     Univariate response variables can- be defined in several
ways.  Individual water quality variables measured as part of
the monitoring program can be directly used as  a response
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              CHESAPEAKE  BAY
               0  S  IO NAUTICAL MILES
Figure  III-l.
Locations of  GBP mainstem water quality monitoring
stations
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Table III-l.  Water quality variables  measured at various
              depths at each  of  the  CBP mainstern stations.
              Unless otherwise noted,  variables are measured
              by all of the data generators.
Temperature
Dissolved oxygen
Specific conductance  (or salinity)
pH
Secchi depth
Total Kjeldahl nitrogen (filtered, and unf.iltered) (a'b'c*
Nitrite plus nitrate(a'c)
Nitrite
Nitrate(b)
Ammonia (filtered)
Particulate organic nitrogen
Total dissolved nitrogen ^c
Total organic carbon^a'b'
Dissolved organic carbon
Particulate organic
Silicate (filtered)
Chlorophyll-a
                             /
                             v
(a)
(b)
(c)
VIMS
ODU
   OEP
*  1 June 1984-15 May 1985
** 16 May 1985-30 September  1985
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Table III-l.   Continued
Phaeophytin(b'c)
Total suspended solids
Total phosphorus
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variable (e.g., DO, chlorophyll-a), variables can be  integrated
over time and/or space (e.g., volume or duration of hypoxia),
or multiple water quality variables can be reduced to a univar-
iate response variable.  The latter can be achieved using
predefined water quality indices  (see Reed and McErlean 1979)
or multivariate statistical techniques.  An example of a
multivariate statistical technique would be the use of the
values of the first principal component from a Principal
Components Analysis (PCA) applied  to several water quality
variables.  Multivariate analyses  can be especially useful  for
defining univariate response variables, since they provide  an
objective means for capturing in a single measure the compli-
cated nature of the Bay's water quality.  A review of multivar-
iate techniques appropriate to CBP water quality monitoring
data is an area that deserves further investigation.

   .  For this report, we focus on  response variables defined as
individual water quality variables measured in the monitoring
program.  The methods are generally applicable to any response
variable.  Water quality variables that are likely candidates for
response variables are DO, chlorophyll-a, dissolved nitrogen and
phosphorus forms, total nitrogen and phosphorus, and total
suspended solids (TSS).  In addition, Secchi depth may be
treated as a response variable, although it is a special case
since only one measurement is associated with a station-cruise
combination.
Explanatory Variables


     Based on the earlier discussion, our general approach to  the
incorporation of explanatory variables for these response
variables is to include only empirically oriented explanatory
variables to remove "noise" from the data.  The only exception
would be analyses of DO trends, for which we would perform
analyses with and without the inclusion of a functionally oriented
explanatory variable that measures the intensity of stratifica-
tion, (e.g., some measure of the rate of change of salinity
with depth).

     It is important to realize that measures of the intensity
of stratification are also likely to be related to differences
in tributary flow.  Tributary flow may, in turn, be related to
nutrient loadings to the Bay, and nutrient loadings are one of
the major targets of management actions.  Care must therefore
be used to ensure that variation in DO removed by a measure of
the intensity of stratification are not trends in DO that are
perhaps attributable to reduced nutrient loadings from the
tributaries.
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     Information on wind direction, magnitude, and duration
may also be a useful explanatory variable, especially for
analyses during the fall season (for turnover events) and
perhaps for understanding "outlier" DO observations.


Censored Data


     For many of the dissolved forms of the nutrients measured
by the CBP monitoring program  (e.g., dissolved inorganic phos-
phorus, ammonia), substantial numbers of observations are
below detection concentrations, resulting in censored values.
These observations below detection can also affect data on
the particulate and total forms of the nutrients if calculation
of their concentrations involves the dissolved components.
The central issue with censored data is that concentrations
between zero and the detection limit exist, but these concen-
trations are not observable.

     If a substantial number of observations are below detection
concentrations and the overall range of observable values is
relatively narrow, problems can arise with a statistical analysis
for trend.  In such situations, censored variables can result
in data that cause violations of the data requirements and
assumptions underlying parameter estimation and hypothesis
testing for many of the statistical methods,  furthermore,
determination of trends in a censored response variable, or
use of a censored variable as an explanatory variable, can be
difficult because the unobservability of values between zero
and detection concentrations may cause an important portion of
the variation in the data to be masked.  Censored data is of
special concern to the CBP monitoring program.  Due to differ-
ences in laboratory measurement methods among the data genera-
tors (i.e., OEP, ODU, VIMS), detection limits vary spatially.
In addition, measurement methods have also changed over time
for some of the data generators causing temporal variation in
detection limits.  Variations  in detection limits with time
and space can introduce artificial temporal and spatial trends
in the data, thereby further confounding statistical determin-
ation of trends attributable to management actions.

     There are five general approaches for dealing with censored
data:                                                                 •

     (1) Categorize the censored variable.

     (2) Assign the same value to individual censored values          I
          (e.g., detection levels, zero).           .                  m
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     (3) Pool observations and assume a statistical distribution
         for the censored values in order to obtain less  biased
         estimates of distributional parameters of the  pooled
         data (e.g., mean, variance).

     (4) Pool observations, assume a statistical distribution  for
         all observations, and use regression or maximum  likelihood
         methods to obtain less biased estimates of distributional
         parameters of the pooled data.

     (5) Pool observations and use measures of central  tendency
         unaffected by censored data (e.g./ for < 50% of  the
         observations censored/ the sample median)

The first two approaches result in individual censored  observa-
tions being assigned to a specific value.  The latter three
approaches utilize pooled observations/ and thus do not result
in the assignment of values to individual censored observations.
The third and fourth approaches use statistical methods to
obtain less biased estimates of the distributional parameters
of the pooled observations (e.g./ mean and variance of  the
pooled data)/ and the fifth approach simply uses statistical
measures of central tendency unaffected by censored data.

     Examples of the approaches involving the assignment  of
values and the use of statistical methods (methods 2, 3,  and
4 above) were presented and compared using Monte Carlo  methods
in Gilliom and Helsel (1986) and Helsel and Gilliom (1986).
They found that the approach of assigning a specific value to
censored observations was inferior to several of the statisti-
cal approaches.   Furthermore, among the statistical methods,
the most robust method for estimating means and variances of
pooled observations was a regression method assuming a  lognorraal
distribution of observations (denoted LR in Gilliom and Helsel,
1986).   This method is implemented in the following manner:

     (1) Calculate the logarithm of uncensored pooled observa-
         tions (the response variable).

     (2) Calculate the probit of the ranks (standardized  to be
         between zero and one) of the all of the pooled observa-
         tions (the explanatory variable).

     (3) Apply linear regression to the above response  and
         explanatory variables.

     (4)" Calculate probit values for ranks from one to  the number
         of censored observations.

     (5) Use the regression model (step 2) to calculate values
         of the response variable corresponding to probit values
         generated in step 4.


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     (6) Estimate the mean and variance using the uncensored
         data combined with the regression generated values of
         the censored data.

     The categorization approach and the use of measures of
central tendency unaffected by censored values are alternatives
to the other three approaches, which attempt in some manner to
reconstruct values for the censored data.  Rather than  "filling
in" censored values, the idea behind categorization and unaf-
fected measures of central tendency is to deal directly with
the information available (i.e./ the censored value falls
somewhere between zero and the detection concentration).

     Use of a measure of central tendency requires that observa-
tions be pooled.  An alternative to the use of the mean as a
measure of central tendency is the sample median.  In fact, if
less than 50% of the pooled observations are censored,  then the
sample median is unaffected by the censored values.

     The categorization approach involves defining categories
for the censored variable based on some criterion.  Analysis
then proceeds using the categorized variable.  Any number of
categories can be defined, although to ensure sufficient numbers
of observations in each category usually two or three categories
are used.  For two categories, the sample median is commonly used
to define the "low" and "high" categories, since this results  in
equal numbers of observations in each category.  Provided less
than 50% of the observations are.censored, categorizing the
censored variable into two categories based on the median is
exact.  All values less than the median, some portion of which
are censored, are assigned to the "low" category, and all
values greater than the median are assigned to the "high"
category.  There is no need to attempt to distinguish among
the censored values.  Other criterion, as well as three or
more categories, can be implemented in a similar manner.  Care
must be used to ensure that categories are defined such that
all censored values are assigned to the same category.
     In summmary, in situations of heavily censored variables,
the feasible options are pooling data and defining "robust"
measures of central tendency (e.g., median), pooling data  and
implementing statistical techniques to obtain less biased
estimates of distributional parameters (e.g., means, variances),
and dealing with individual observations by categorizing the          |
censored variable.  We recommend the use of robust measures of        |
central tendency or categorization.  In Chapters V and VI  we
discuss how these various alternatives can be used with univar-       ป
iate methods for trend detection.                                     I
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        Homogeneity of Trend
     It is important in a monitoring program  as  large  as  the
CBP monitoring program (40 stations, 20 cruises/year,  multiple
depths) to perform analyses on temporal and spatial  scales
that allow for interpretation and generalization of  results.
On one hand, results of analyses of trends in water  quality at
individual stations, depths, or cruises are difficult  to  gen-
eralize to statements concerning trends in Chesapeake  Bay.   On
the other hand, if all of the data are included  in a single
analysis, interpretation of results is extremely difficult.
As discussed in more detail in Chapter VI, we advocate con-
ducting analyses for trend on seasonal-, regional-,  and depth
layer-specific bases.  Depth layers could be  defined as either
the water column or above and below pycnocline layers.

     In situations where trend analyses are being performed on
data grouped into season, region, and depth layer, the issue
of homogeneity of trend becomes important.  Improper aggrega-
tion of data into groups can cause trends in  some members of
the group to mask trends, in some, or even most,  of the other
members of the group.  Homogeneity of trend issues can involve
the consistency of trend in individual stations  grouped into a
region, in multiple cruises in a year grouped into a season,
and in multiple depth measurements grouped into  a depth layer.
For all of the methods described in Chapter IV,  homogeneity of
trend is an important consideration.
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                      IV.  UNIVARIATE STATISTICAL METHODS
                          FOR TEMPORAL TREND  DETECTION
                        A.  INTRODUCTION


     This Chapter presents a brief overview  of  selected
univariate methods for temporal trend detection.   These methods
were selected based on their appropriateness for  detecting trends
in water quality data.  Additional information  on these and other
methods for trend detection are presented  in the  bibliography
(Appendix A).  Some methods not included in  this  chapter may
also be appropriate for analysis of CBP water quality  data (e.g.,
two-sample tests).  As new methods become  available, or as
existing methods are deemed appropriate, these  can be  easily
incorporated into the analysis framework.

     Univariate statistical methods for trend detection have been
organized into the following general categories:

     •  Box-Jenkins Intervention analysis

     •  Parametric methods (GLM and linear logit  models)

     •  Distribution free (nonparametric)  methods.

In the next sections, we provide brief overviews  of methods in
each of these categories.


             B.  BOX-JENKINS INTERVENTION  ANALYSIS


Description


     Box-Jenkins time series analysis is an  empirical  approach
that entails the estimation of a model from  the data (see Box and
Jenkins 1976; McCleary and Hay 1980).  The observed time  series
of data is assumed to be a single realization from a stochastic
process.  The goal of Box-Jenkins modeling is to  estimate the
parameters of the underlying stochastic process that generated
the observed time series of observations.

     Box-Jenkins methods require stationary  time  series  (i.e.,
no trend or drift).  Differencing of a time  series  is  typically
used to obtain a stationary time series, and  analyses  proceed
using the differenced time series.


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                                                                       I
     The general form of Box-Jenkins models  is  an  ARIMA
(autoregressive integrated moving average) process,  . A*.IMA
models are specified with three parameters (ARIMA  (p,d,q)):

     •  p refers to the autoregressive structure  in  a model  (i.e.,    I
        the preceding p values of the response  variable are  used
        to predict the present value of  the  response variable).        —

     •  q refers the moving average component of  the model (i.e.,     ™
        the preceding q random shocks (or errors)  are used to
        predict the  present value of the response variable).         •

     •  d indicates the order of differencing that was applied to
        the response variable time series.                             •

Thus an ARIMA (2,1,2) model would be of  the  following form:

                          (l-e1B1-92B2)et                              I
              Yt - Yt_! = __—-_5—                            •


where:  .                         •                                      m

       Yt * response variable                                          •

        B * backshift operator (BnasXt-Yt_n)

        9 .ป moving average parameters

        $ * autoregressive parameters

       e^ = errors.

Specific Box-Jenkins models can consist  of either  an autore-
gressive component (AR), a moving average component  (MA),  or
both (ARIMA).

     The Box-Jenkins models described thus far  do  not take into
account any seasonal signals in the data (e.g., with the models
we can model DO in June as a function of DO  in  the preceding
April and May).  Seasonal ARIMA models allow for  inclusion of
additional seasonal signals into the model (e.gr  DO  in June  as
a function of DO in the previous June, as well  as  the preceding
April and May). ' Seasonal ARIMA models are denoted as ARIMA
(p,d,q)X(P,D,Q), where P, D, and Q refer to  the structure  of
the seasonal model.
Explanatory Variables


     Explanatory variables are  incorporated  into  Box-Jenkins
models by the use of transfer functions.   A  transfer function

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relates the present value of the  response  time  series to a linear
function of present and previous  values  of the  explanatory
variable.

Relevant Data Requirements


     Box-Jenkins methods generally  require a  long  time history of
data.  Observations are assumed to  be  from continuous variables.
Single observations of the response variable  are required (i.e.,
replicate values are not allowed)/  and these  observations must be
uniformly space in time with no missing  values.   Recently/
several methods have been proposed  for Box-Jenkins modeling of
data with missing values (e.g., see Lettenmaier 1980; Sturges
1983).


Assumptions


     The following assumptions are  required for parameter
estimation and hypothesis testing in Box-Jenkins models:       '

      (1) raean(et) = 0

      (2) var(et) * a  for all t  (homoscedasticity )
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      (3) cov(et /et+fc) * 0 for all t/k  (independence)

      (4) e^ are normally distributed.

Assumptions 1-4  imply a white noise process  for  the  errors  (et)


Model Building
     The Box-Jenkins approach involves three steps  to model
building:  model identification/ parameter  estimation/  and
diagnostic checking.

     •  Model Identification

        The autocorrelation function  (ACF)  and partial  autocor
        relation function (PACF) are  examined to assess the
        statioharity of the time series and to identify the
        general structure of the ARIMA model (i.e./ values for
        p/ d, and q).  When a transfer function is  included/
        the cross correlation function (CCF) between the  re-
        sponse and- explanatory time series  is examined  to
        determine the form of the transfer  function.
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     •  Parameter Estimation


        Initial parameter estimates for the identified model
        are first obtained directly from the ACF, PACF,  and,
        if appropriate, CCF.  An iterative maximum likelihood
        procedure is then used to obtain final estimates for
        the parameters.  Hypothesis testing of whether indi-
        vidual parameter estimates are significantly different
        from zero are performed using normal theory.  Insignif-
        icant parameters may be eliminated from the identified
        model.

     •  Diagnostic Checking

        Four diagnostics, based on the residuals  (observed  Yt
        minus predicted Yt), are typically examined to
        assess the adequacy of the determined model.  These
        diagnostics are plots of residuals versus time,  the
        ACF and PACF of the residuals, and tests of whether
        residuals are normally distributed (e.g., Kolmogorov-
        Smirnov test).  Residuals based on an "adequate"
        model would appear as a white noise process in these
        diagnostics.  If the residuals do not appear as  white
        noise, Box-Jenkins methods can be used to identify
        structure in the residuals, and this information can
        be utilized to modify the response time serie* model.
        Parameter estimation and diagnostic checking are then
        applied to the modified model.  This process continues
        until a model is obtained that results in white  noise
        residuals.


Trend Detection


     Box-Jenkins methods can be used for determining whether an
action (an "intervention") has had a significant effect  on  the
behavior of the response time series.  These analyses are termed
intervention analysis or ARIMA Impact assessment  (see McCleary
and Hay 1980; McDowall et al. 1980).  Intervention analysis
requires that the time of the onset of the intervening action be
known.  Furthermore, intervention analysis requires a long  time
history of both pre- and post-intervention data.  Four general
types of responses to the intervention can be detected (see
Figure IV-1).  These responses involve whether the response is a
permanent or temporary shift in the mean level of the time            •
series and whether the shift is abrupt or gradual.  In addition,      ||
for all of these of cases, the response may be coincident with
the onset of the intervention or may be delayed from the onset
of the intervention.
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                     DURATION
           PERMANENT
TEMPORARY
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        Figure  IV-1.
  Four general  types of responses to an  intervention
  that can be detected with  Box-Jenkins  intervention
  analyses (reproduced from  McCleary and Hay 1980)
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     Interventions are represented in Box-Jenkins models  as  a
special case of a transfer function  (i.e./ a step or pulse
transfer function).  The form of the intervention transfer
function can be specified a priori.  Model identification is
performed using the pre-intervention data/ and parameter

estimation is performed using both pre- and post-intervention
data.  Essentially/ the idea is to determine if incorporation  of
an intervention transfer function enables the model developed
from the pre-intervention data to also De applicable to
post-intervention data.  Conclusions concerning the effect of
the intervention are based on significance testing of  the
parameters associated with the intervention transfer function.


                     C. PARAMETRIC METHODS


General Linear Models (GLM)


     Description

                                     *
     GLM are a suite of methods that partition variation  in  a
continuous response variable to hypothesized sources (i.e./
explanatory variables).  The principal of GLM is that  the
response variable is a linear function of parameters
corresponding to terms for individual, or combinations of/
explanatory variables•(see Neter et  al. 1985).


     Explanatory Variables


     Explanatory variables can be categorical or continuous.
GLM that include only continuous explanatory variables are
called regression models.  GLM that  include only categorical
explanatory variables (which are termed factors) are ANOVA
models.  Models that include a mixture of categorical  and
continuous explanatory variables are known as ANCOVA models.

     ANOVA/ANCOVA models containing more than one factor  can be
characterized as crossed and/or nested designs.  To illustrate,
consider a response variable observed on two cruises in each year
for two years.  A crossed ANOVA model with cruise and  year as
factors would be appropriate if the  first cruise in each  year
has the same effect on the response  variable for year  1 as for
year 2.  In this situation, the ANOVA model would contain
terms corresponding to a cruise effect, year effect, and
interaction effect between cruise and year.
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               YJ.J  =  U + CRUISE i + YEARj + CRUISE x YEARji + e^

        where:

                    YJLJ = value of response variable for i   cruise
                          and jfc  year'
U * overall average of the response  variable

    the effect
    ith cruise
                CRUISED  =  the  effect on the response variable of the
                           th
                YEARj   ป  the  effect  on the response variable of the
                          jtn  year

       CRUISE x YEAR^j  a  the  interection effect on the response
                          variable due to the i^n cruise in the
                              year

                          error.

       A nested ANOVA model  with  cruise and year as factors would be
       appropriate  if the first cruise in year 1 was not related to the
       first cruise  in year  2, and  the second cruise in year 1 was not
       related to the second cruise in year 2.  That is, the model does
       not involve estimation of  the "effects" of a given cruise
       across years.  The nested  model would contain terms corresponding
       to year effects and cruise nested in year effects:
        where:
I
                   Y^J  ป  val
•                        itfi

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                           U + YEARj -I- CRUISE ( YEAR)
                             ue of response variable variable for
                              cruise and jtn year

                      U ป overall average of the response variable

                 YEAR.*  = the effect on the response variable of the
                     J    jtn year

         CRUISE(YEAR)\* * the effect on the response variable of the
    the effect on the response
    itn cruise in the jtn year
                    eij  * error.

             Factor  effects  in an ANOVA/ANCOVA model can be characterized
        as  fixed or  random effects.   Fixed effects imply conclusions are
        restricted to observed levels of factors.  Random effects imply
        that  inferences  will extend  to a population of factors levels
        (not  all of  which are observed).  We will be dealing with
        fixed effects models.


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     Relevant Data Requirements


     GLM requires that the response  variable  be  a continuous
variable.  In addition-, it is  preferrable  (although not
necessary) to have replicate values  of  the  response variable for
each combination of factors included in a model.
     Assumptions


     The following assumptions are  required  for GLM:

     (1) meanfe^) ป 0

     (2) var(ek) * c  for all k  (homoscedasticity)

     (3) covCefc/efc+in) a 0 for all k and  m (independence)

     (4) efc are normally distributed.

Note that these assumptions correspond to the assumptions
underlying Box-Jenkins methods.  For GLM, assumptions 1-3 are
required for parameter estimation and assumption 4  is required
for hypothesis testing.


     Parameter Estimation/Hypothesis Testing
     GLM base parameter estimation  on  ordinary least squares
methods and hypothesis testing on the  F-statistic.   Consider
the following two-factor crossed ANOVA model:
              Y ป U + Ai + Bj -i-

The following hypotheses are typically  evaluated:

      (1) Ho:  Aj. ป Bj = AxBij * 0  for all  i  and j

      (2) Ho:  AxB^j * 0 for all i  and j

      (3) Ho:  Ai - 0 for all i                                         •

      (4) Ho:  Bj ป 0 for all j.

Hypothesis 1 is a test of the significance of  the  overall             •
model.  Hypotheses 2-4 involve significance  tests  of interaction      *
and main effects.  Signficant interaction  effects  imply that
the effects of one factor are not  homogenous across levels of
another factor.  Evalaution of .hypotheses  concerning the main


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effect of a factor are difficult to  interpret  if  interactions
involving that factor are significant.   In  such situations,
separate analyses can be preformed at  each  level  of  one  of  the
factors involved in the interaction.

     In situations of interpretable  and  significant  main effects/
pairwise (for ANOVA and ANCOVA) and  multiple comparison  proce-
dures (for ANOVA) can be applied to  estimated  means  of the
response variable for each level of  the  signficant main  effect.
Examples of multiple range procedures  are Duncan's New Multiple
Range Test and Tukey's Studentized Range Test.


     Diagnostics


     Determination of model adequacy is  based  on  the analysis  of
residuals" (observed minus predicted  values  of  the response
variable).  Residuals can be examined  for their adherence to  the
assumptions made on the errors using graphical techniques and,
with sufficient data, hypothesis testing.   Examples  of statis-
tical tests include tests for homoscedasiticity (Bartlett
test; Hartley test), normality (X2 goodness-of-fit test), and
independence (Drubin-Watson test).   A  common remedy  for  hetero-
scedasticity is the use of transformations  applied to the
response variable.


     Trend Detection
     GLM can be used for trend dection by  inclusion of  a  time-
related factor or covariate into the model.  Use of a covariate
involves the specification of the functional form of the  trend
(e.g., a linear term for the time-related  covariate in  an
ANCOVA).  Use of a time-related factor does not require the
specification of the functional form of the trend.  Rather,
pairwise (for ANCOVA) or multiple comparison (for ANOVA)  tests
can be used to look for temporal patterns  in a significant
time-related factor.

     Detection of long-term trends with GLM can be difficult  in
data that exhibit short-term periodicities.  Two approaches to
dealing with short-term peridocities in the data are to include
a covariate (e.g., Lorda and Saila 1986) or factor (e.g., cruise
nested in year) in the model to account for these periodicities.
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Linear Logit Models
     Description
     Linear logit models and loglinear models deal with
categorical variables (see Feinberg 1980; SAS,  Inc.  1985).
These methods focus on the analysis of the number of  observa-
tions (counts) observed at various combinations of levels  of
variables.   Linear logit models are appropriate when a  distinc-
tion is made between response and explanatory variables.
Loglinear models are used to examine relationships   among
categorical variables, without distiguishing between  response
and explanatory variables.  In a general sens"e, loglinear
models can be viewed as the extension of contingency  table
analysis.  Most linear logit models can be recast in  terms of
a loglinear model.

     The form of linear logit models is analogous to  that  of
GLM; the major difference is the definition of  the response
variable.  Suppose a variable Y^ has three categories (i = •
low, medium/ high).  The response variable (Z)  for the linear
logit model would be the generalized logit function with two
possible outcomes:
                     log
PLOW

PHIGH

PMEDIUM

PHIGH
where:
          PLOW = proportion of low observations

       PMEDIUM * proportion of medium observations

         PHIGH a Proportion of high observations.

The calculation of PLOW' PMEDIUM' and PHIGH depends  On  the terms
in the model.  For example/ suppose we have observations  of Y^
for two cruises in each of two years.  A  nested  linear  logit
model with year/ and cruise nested in year/ as factors  would have
four cruise-year combinations.  For each  of these cruise-year
combinations, the proportions of low/ medium, and high  responses
would be determined.  These proportions would then be used in the
generalized logit function.
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             Explanatory Variables

 I-
   ^H       As with GLM, categorical explanatory variables can be
   1^^  included as factors (either crossed and/or nested) and continuous
        explanatory variables can be included as covariates (termed
        logistic regression).


             Relevant Data Requirements


 •          Linear logit models require non-zero counts for each level
        of the variable (Y^)  for each combination of factors defined
        •by the model.  When zero counts are present/ the standard
        procedure is to add a small constant value to each count.


 •           Assumptions


             •Linear logit models are based on the assumption of counts
        following a product multinomial distribution.


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     Parameter Estimation/Hypothesis Testing


     Parameter estimation in linear logit models can be performed
based on either maximum likelihood ox weighted'least squares
techniques.  Hypothesis testing is based on the X2 statistic.  As
with GLM, significance tests are performed on hypotheses
involving interaction and main effects, and interpretation of
significant main effects when interactions involving this factor
are present are difficult.  In situations of an interpretable
and significant main effect/ pairwise comparisons can be
performed on the estimates of the main effect parameters.


     Trend Detection


     Trend detection with linear logit models can be performed in
the same manner as with GLM.  A time-related explanatory variable
is incorporated into analyses either as a factor or covariate.
Pairwise comparisons applied to the parameter estimates of the
significant time-related factor can be then be used to discern
temporal trends.
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                 D.  DISTRIBUTION FREE METHODS


     Distribution free methods involve the analysis  of  scored
values of the response and explanatory variables.  Commonly used  .
score functions include the sign function and rank transforma-
tions.  We highlight four general distribution  free  methods
appropriate for trend detection:

     •  Cox-Stuart test for trend

     •  Kendall's Tau

     •  Spearman rank correlation

     -•  Friedman's two-way rank ANOVA.

Conover (1971) and Hollander and Wolfe (1973) provide excellent
discussions of distribution free statistical methods.   Note that
the use of rank transformed data with parametric methods  has
recently been proposed as a bridge between distribution free and
parametric analyses (see Conover and Iman.1981).


Cox-Stuart Test


     Description


     Cox-Stuart test is a test for trend and is based on  the sign
test.  Observations are grouped into pairs that are  equidistant
in time.  A sign test is performed on the differences between the
observations in each pair.  The test statistic  is the number of
positive differences.  To illustrate, suppose there  are 10
observations of a response variable ordered in  time  (ฅt,
tปl,10).  Cox-Stuart test would involve a sign  test  applied to
the five differences (*t"Yt+5' t * 1'5)-


     Explanatory Variables

     Explanatory variables cannot be directly incorporated
into the Cox-Stuart test for trend.  Two alternatives for             •
incorporating explanatory variables are the use of a linear           |
regression model (GLM) and an alignmnent procedure.  With linear
regression, the response variable is regressed on the explanatory     .
variable (or some function of the explanatory variable, e.g. .         •
ranks).  The Cox-Stuart test is then applied to the  time  ordered      ™
residuals from the regression model.
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     The data alignment procedure  is  analogous to the regression
approach in that residuals  are  analyzed  for temporal trend.  T.he
difference between the two  methods is that with the data align-
ment procedure, the explanatory variable is first categorized.
The response variable is then adjusted by the average value of
the response variable for each  category  of the explanatory
variable.  For example, with an explanatory variable with two
categories (low and high):



                      Yt-YLOW if Xt is low
      RESIDUALt          _
                                if  xt  is
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where:

             Yt s response variable

             Xt ป explanatory variable

                * average value  of response  variable for all
                  low Xt

          YHIGH ป average value  of response  variable for all
                  high Xt.

With sufficient data, any number of  categories,  as  well as any
number of explanatory variables, can be  incorporated.


     Relevant Data Requirements


     The Cox-Stuart test requires a  single value of the response
variable over time (i.e., replicate  values not allowed).   The
test only requires data to be sufficient to  allow calculation of
the signs of paired differences.  Thus,  at a minimum,  data must
be ordinal.  Note, the Cox-Stuart ignores tied values.


     Assumptions


     As is characteristic of distribution free methods, the
Cox-Stuart test does not require distributional  assumptions.
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     Hypothesis Testing


     Hypotheses testing with the Cox-Stuart statistic are  as
follows:

         Ho: Pr(YtYt+c) (no trend exists)

         Ha: Pr(YtYt+c) (trend exists)

where:

            c * number of paired observations.

The above test"is a two-tailed test; similar one-tailed  tests  can
be performed for detection of upward trend and downward  trend.
Critical values for the Cox-Stuart statistic can be obtained from
a binomial distribution table assuming equal probability of
positive and negative signs.  With large sample sizes, a
standardized Cox-Stuart statistic can be compared to normal
critical values.

Spearman Rank Correlation


     Description


     Spearman rank correlation is the distribution free  analog
to Pearson moment correlation.  Calculation of Spearman  rank
correlation is performed by computing the Pearson moment cor-
relation between rank transformed values of the two variables.
Spearman rank correlation measures the monotonic association
between two variables.  Use of Spearman rank correlation for
trend detection simply involves the correlation of a response
variable with a time.


     Explanatory Variables


     Direct incorporation of additional explanatory variables
into correlation analysis involves the use of partial Spearman
correlation coefficients.                                             •

     In addition, incorporation of explanatory variables can be
achieved using regression analysis or alignment procedures.
Analyses would proceed in the same manner as described for the
Cox-Stuart test (i.e., correlation of residuals with time).
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             Relevant Data Requirements
             Spearman rank correlation  requires  that the response
        variable be at least ordinal.
     Assumptions


     Spearman rank correlation does  not  require  any
distributional assumptions.


     Hypothesis Testing


     With Spearman rank correlation, the null  hypothesis is that
the two variables are independent.   For  trend  detection, this is
a test of whether the response and time-related  variables are
independent.  Tabled critical values of  Spearman rank  correlation
coefficients are available.  Approximate significance  tests of
partial Spearman rank correlation coefficients can be  performed
using crtitical values of Pearson correlation  coefficients
(Conover arid Iman 1981).


Kendall's Tau
     Description


     Kendall's Tau is a distribution free measure  of  concordance
between two variables.  Use of Kendall's Tau  in  trend detection
is based on time ordered observations of a response variable
(Yt/t*l,n).  In this situation, Kendall's Tau  involves the
sum of the signs of all unique pairwise comparisons.   First
the value of Kendall's statistic  (K) is calculated:

                         n-1  n
                     K -  I   I   sgn(Yi-Yi)
                         J-l i-j+1
        where:


            sgn(a)
              1 if a > 0
              0 if a = 0
             -1 if a < 0
        Kendall's Tau is then obtained by transforming K  to  a  value
        between -1 and 1.
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     Suppose we are examining year to year  trends  in  a  response
variable.  In situations when observations  are  available  for
multiple time periods within each year (e.g., monthly values),
two possible options are:

     •  Data can be averaged over months within each  year and
        Kendalls' Tau can be applied to these yearly  averages.

     •  Month-specific Kendall's Taus can be computed.  These
        values can then be combined into a  single  statement
        concerning trend, assuming independence (denoted  as
        S' in Hirsch et al. 1982) or accounting for possible
        dependence (Hirsch and Slack 1984).  Note  that  Van
        Belle and Hughes (1984) provide a X2-based test for
        determining homogeneity of trend for the different
        months.

     Explanatory Variables


     Incorporation of explanatory variables with Kendall's Tau
can be achieved using residuals from regression analysis  or  an
alignment procedure.
                                             ป


     Relevant Data Requirements


     Kendall's Tau requires the response variable  to  be at least
ordinal.  Note that Kendall's Tau is applied to single  values of
the response variable over time.


     Assumptions


     Kendall's Tau does not require any distributional  assump-
tions.


     Hypothesis Testing


     With Kendall's Tau, the null hypothesis is that  the  two         I
variables are independent.   For trend detection, this is  a
test of whether the response variable is independent  of time
order.    Tabled critical values of Kendall's Tau are  available.       •
For large sample sizes, Kendall's Tau can be standardized and         •
compared to normal critical values.
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Friedman's Two-Way Rank ANOVA



     Description


     Friedman's two-way rank ANOVA  is  a distribution free analog
to a two-way ANOVA without  interactions.   Analyses are performed
on rank transformed data.   The general model  is:

                     Y *  u  + a^  + bj + e^j

where:


        a^ * the effect on  the response varible of the
             ith level of factor a

        bj - the effect on  the response variable  of the
             jth level of factor b

       eij - error.

With Friedman's two-way rank ANOVA, one of the factors (a^) is
the effect of interest (treatment effect)  and the other factor
(bj) is considered a block  effect.  Observations  are ranked
within each level of the  block factor/ and ranks  are summed for
each level of the treatment factor.  Use of Friedman's two-way
rank ANOVA for trend detection involves defining  the effect of
interest as a time-related  factor.
     Explanatory Variables



     Friedman's two-way rank ANOVA  allows  for  a  design factor
(e.g., multiple stations) to be  incorporated as  the block factor,



     Relevant Data Requirements


     Friedman's two-way rank ANOVA  requires a  single observation
of the response variable for each combination  of levels of the
treatment and block factors.  The response variable must be
continuous.
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     Assumptions
     The errors are independent random  variables  from the same
continuous distribution.  Note that  a specific  form of the
distribution of the errors is not assumed.
     Hypothesis Testing


     The null hypothesis for Friedman's  two-way  rank ANOVA is that
treatments effects are zero (H0: a^  ป  0  for all  i;  Ha:  a^ ป 0 for
some i).  Sigificance testing for  this hypothesis is based on a
X2 statistic.  Trend detection with  Friedman's two-way  rank ANOVA
involves a time-related treatment  factor.   Rejection of the null
hypothesis (i.e./ treatment effects  are  not equal to zero)
implies that there are signigicant differences in the response
variable over time.  Multiple comparison procedures can be
applied to discern patterns in the significant time-related
treatment factor.  Alternatively,  a  test designed specifically
for trend detection is Page's test.  Page's test is based on an
ordered alternative hypothesis (Ha:  a^  ฃ  a2 <.•••<, an'  with at
least one inequality being strict).
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                     V.  DEMONSTRATION OF SELECTED METHODS
                                A.  INTRODUCTION
     In this Chapter/ we use historical time series data  at
four stations off of the Calvert Cliffs area of Chesapeake Bay
and data collected as part of the present CBP water quality
monitoring program to demonstrate some of the methods  described
above.  All analyses were performed using procedures in the
Statistical Analysis System (SAS)/ the database for CBP data.

     The primary objective of these analyses is to demonstrate
how the the statistical methods that were described in general
terms above can be applied to CBP water quality data.  In this
context/ the Calvert Cliffs data can be viewed as a "microcosmic"
version of CBP monitoring data (i.e./ both include data col-
lected at multiple depths and multiple stations during multiple
cruises in a year).

     Many of the distribution-free methods (as well as Box-
Jenkins methods) require a fairly long time series of  data.
At the present time/ only the first 18 months of data  from the
present monitoring program are available for analysis/ and these
are insufficient for application of the distribution free
methods.  Therefore/ the Calverts Cliffs data were assembled  from
a variety of sources so that the distribution free approaches
could, be demonstrated with actual Chesapeake Bay water quality
data.

     Parametric methods (GLM and linear logit models)  and most
of the distribution free methods are applied to at least  one of
the demonstration datasets.  The parametric method of  GLM is
applied to both Calvert Cliffs and present monitoring  data to
illustrate the versatility of the parametric methods.  The
application of GLM to the present monitoring data also provides a
good example of the use of results from multivariate techniques
in univariate analyses for trend.

     Box-Jenkins methods were not included in the demonstration
because a relevant "time of intervention" could not be defined at
Calvert Cliffs.  Furthermore/ because of the strict data  require-
ments associated with Box-Jenkins methods/ we do not forsee
their general application to data from the present CBP monitoring
program.

        Because the data used in analyses are limited/ we do
not rigorously assess the adherence to the assumptions under-
lying these methods.  Rather/ we emphasize implementation of

                              V-l
I

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                      Martin Marietta Environmental Systems

the methods to the data.  'Future analyses of CBP monitoring
data will involve more dataf thus allowing for more  rigorous
assessment of adherence to assumptions.


           B.  CALVERT CLIFFS AND CBP MONITORING DATA


     The data collection methods used for the present CBP
monitoring program were summarized in Chapter III.

     The Calvert Cliffs historical data consist of approxi-
mately monthly values of surface and bottom DO, salinity/  and
temperature for four stations off of Calvert Cliffs  for  1969 to
1986.  These data were assembled from a variety of sources.
Figure V-l shows the locations of the four stations  (labelled
A, B, C, and D) in Chesapeake Bay.  All four stations have a Dottorn
depth of approximately 30 ft (9 m).  Figures V-2 to  V-4  show
the time series data for variables used in analyses  (bottom DO,
bottom salinity/ and surface salinity) for each station.


             • C.  ANALYSIS OF CALVERT CLIFFS DATA


General Analysis Strategy
         t

     The objectives of the analysis were, first, to  detect year-
to-year trends in summertime bottom DO at Calvert-Cliffs and,
second/ to attempt to attribute any detected trends  to year-to-
year differences in the intensity of stratification.  We would
expect that in years of intense stratification (due  to flow
patterns in the tributaries/ lack of wind-induced mixing,  or
other reasons), that bottom DO would tend to be lower than in
years of the same conditions but with less intense stratifi-
cation.  The explanatory variable used as the measure of the
intensity of stratification was the difference between bottom
and surface salinity values.  Figures V-5 and V-6 show bottom
DO values and salinity differences (calculated as bottom minus
surface) for August of each year for each station.   However,
care must used in attempting to attribute trends in  summer DO
to salinity differences.  While proposed management  actions
will not directly affect the intensity of stratification,             •
salinity differences in the summer at Calvert Cliffs may be           I
correlated to flows in the Susquehanna River, and consequently
salinity differences may-be correlated with nutrient loadings.
Any trends in DO attributed to salinity differences  may  not be        I
due to stratification differences, but rather may be due to           •
differences in nutrient loadings.  Therefore, all methods  (GLM
and distribution free) were applied with and without salinity
difference as an explanatory variable.

                              V-2

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                              Martin Marietta Environmental Systems
       CHESAPEAKE  BAY
         0    5    10 NAUTlCAu MILES


I
Figure V-l.   Location  of the four  Calvert Cliffs stations  in
              Chesapeake Bay (A,  B,  C,  D) for  which historical
              data were assembled
                                    V-3

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                      Martin Marietta Environmental Systems

Distribution Free Methods


     A suite of analyses using distribution free methods  was
applied to the Calvert Cliffs data to attempt  to detect trends
in summer bottom DO and to attribute these trends to salinity
differences.  These analyses are organized into two groups
based on whether or not salinity differences were included  as
an explanatory variable.


     Analysis of DO without Salinity Differences


     Kendall's Tau and Spearman rank correlation coefficients
were computed between bottom DO and year for each station for
July/ for August/ and for the minimum DO value within  the year.
The results indicate that July and August values of bottom  DO
display similar trends/ and these trends are similar for  the four
stations (Table V-l).  We therefore decided to pursue  grouping
data over stations for August and for July and August  (which
we term "summer").  August DO values were examined/ since August
values showed the strongest downward trends in bottom  DO.
Additional analyses were not performed on minimum DO.

     Once data are grouped over stations and months (July and
August)/ two ways of collasping the data are possible.  -Data
can be averaged over stations and months and then analyzed/ or
all data can be analyzed but'with observations treated as
station-specific or month-specific values (i.e./'not averaged
over stations or months).

     We computed Kendall's Tau and Spearman rank correlation
coefficients between bottom DO and year/ with  DO averaged over
stations for August only and averaged over stations for the
summer (July and August).  The results show a  significant
downward trend in bottom DO averaged over all stations for
August and for the summer (Table V-2).

     An alternative to completely collapsing the data  by
averaging over stations and months is provided by Hirsh's
modification to Kendall's Tau and by the use of Friedman's
two-way rank ANOVA.  In this demonstration analysis/ Hirsch's     ^,
modification was used to compute Kendall's Tau for July and       ^   •
for August/ and then to combine these statistics into  a single        |
statement about trend for these months.  In practice,  before
applying Hirsch's modification to Kendall's Tau/ homogeneity          •
of trend needs to be established.  In this example/ the homo-         I
geneity of trend involving the four stations and the two  months
(July and August) was confirmed both graphically and with
Kendall's Tau and Spearman rank correlations applied to trends
                              V-18

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                                            Martin Marietta Environmental Systems
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                    Martin Marietta Environmental Systems
Table V-2.
Values and probability  levels  (p-values) of
Kendall's Tau and Spearman  rank correlation
coefficients computed between  bottom DO (aver-
aged over stations for  August  only) and year,
and between bottom DO  (averaged over stations
and averaged over July  and  August (Summer)) and
year
                               August
                                   Summer
Kendall's Tau

p-value  .
                   -0.52

                   0.003
-0.59

0.0006
Spearman rank

p-value
                   -0.70

                   0.001
-0.77

0.0002
                             V-20

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with time on a month-specific and station-specific  basis
(not all results shown).  For illustrative  purposes,  we applied
Van Belle's test to confirm the homogeneity of  trends in  July
and August bottom DO  (averaged over stations).  As  expected,
Van Belle's test showed that^the trends  in  July and August
bottom DO were consistent (xf * 0.44, ns  (p > 0.05)).
Therefore/ we computed Kendall's Taus for these two months
and used Hirsch's modification (Hirsch et al. 1982) to combine
the statistics.  The  result is that bottom  DO exhibits a
significant downward  trend for July and August  (S1  *  -0.44,
p < 0.01).

     Friedman's two-way rank ANOVA was applied  to summer  aver-
age (July and August) bottom DO to detect year-to-year differ-
ences, with station as the blocking variable.   Note  that we
have already established the homogeneity of trend among sta-
tions.  The results from the Friedman's test show that there
are significant year-to-year differences in average summer
bottom DO (Xj^ ป 96.5, p < 0.0001).  One option (not  shown)
for detection of trends with Friedman's test is to  apply  mul-
tiple comparison tests (e.g., Tukey's Studentized Range Test)
to determine any trends in ranked values of bottom  DO with
year.  Another option is to use the same two-way station-by-year
table of average summer bottom DO as with Friedman's  test, and
apply Page's test for monotonic trend.  Page's  test showed a
significant downward monotonic trend in summer  bottom DO  over
the four stations (Z ป -6.72, p < 0.001).
             Analysis of DO with Salinity Differences
     Having determined that summer bottom DO at Calvert Cliffs
exhibits significant downward trends, we next attemptea to  re-
late those trends to the salinity difference explanatory vari-
able.  As discussed in Chapter IV, explanatory variables cannot
easily be directly incorporated into distribution free methods.
We illustrate two different, though related, techniques for
incorporating explanatory variables into these methods.  A
third alternative (not demonstrated) is to perform approximate
significance tests on partial Spearman rank correlation coef-
ficients between DO and salinity differences, using significance
levels for Pearson correlation values.

     The first method is to use linear regression of DO on
salinity differences, and then to apply the distribution free
methods to detect temporal trends in the residuals (i.e.,
observed DO minus predicted DO from the regression model).
The second method for incorporating salinity differences as an
explanatory variables is the data alignment procedure.  Average
salinity differences are categorized into low and high categories
                                      v-21

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                      Martin Marietta Environmental Systems

based on the median value of salinity differences, and distri-
bution free methods are applied to time ordered residual DO
values calculated for high and low salinity difference years.
The advantage of the alignment procedure, as discussed in
Chapter IV, is that it allows for censored explanatory variables
to be easily incorporated into the distribution free methods.
For illustrative purposes, we apply the alignment procedure  to
salinity differences, even though salinity difference is an
uncensored variable.

     Kendall's Tau and Spearman rank correlation coefficients
were computed between bottom DO (averaged over all stations) and
year for August only and for July/August averages, after removing
the variation accounted for by bottom minus surface salinity
differences (i.e., intensity of stratification) using linear
regression and the alignment procedure.  For analyses applied
to August DO, salinity differences were computed for each
station and then the average "August" salinity difference was
calculated for each year.  For analyses applied to summer
(July/August) DO, bottom minus surface  salinity differences  were
computed for each station for each month, and the overall average
difference was calculated for each year.

     Within the linear regression approach, two alternative
regression models were investigated:  DO as a linear function
of salinity differences (DO ป BQ + B^'SALDIFF) and DO as an
exponential function or salinity differences (In (DO + 1J" *  B0
+ B]/SA{,DIFF, or equivalent!/, DO'ป Bo'exptBi'SALDIFF) - 1,
where B_ ป exp(BQ)).  The logarithm of  (DO + 1) is used to
avoid the problem of the logarithm of zero being undefined.
The exponential model is based on inspections of plots of
bottom August DO and summer DO values versus salinity differ-
ences, and the fact that, beyond some large value of salinity
difference, DO is bounded below by zero.

     The linear and exponential regression models were
significant for both August DO and summer (July/August) DO
(Table V-3).  Figure V-7 shows observed and predicted values
of August bottom DO for the linear and  exponential regression
models.  As expected/ the negative slope .coefficient (B]_)
indicates that the greater the difference between bottom and
surface salinities, the lower the value of bottom DO.

     We examined residual plots to determine if a more compli-
cated model of salinity differences was appropriate or if
there were obvious heteroscedasticity problems.  Figure V-8
shows examples of residual plots (in this case, residual DO
versus salinity differences) for August DO for the linear and
exponential models.  For both models, these plots do not indi-
cate any obvious'structure in DO residuals that could be ac-
counted for by a more complex model of  salinity differences.
Similar results were obtained for residual plots involving
summer bottom DO and salinity differences.

                              V-22

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Martin Marietta Environmental Systems


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         V-23

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                              Martin Marietta Environmental Systems
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     Conceptually, detection of temporal trends  in  these
residuals is based on the analysis of the residuals reordered
in time (years)/ rather than as a function of  salinity  differ-
ences.  (For August DO/ compare Figs. V-8 and  V-9.)   Table  V-4
shows the results of Spearman  rank correlation and  Kendall's
Tau applied to residual bottom DO values with  year  for  August
only and for July/August averages/ with residuals computed
from the linear and exponential regression models.   After
removing variation accounted for by salinity differences,
there is still a significant downward trend in residual August
DO values for both the linear  and exponential  regression models/
and for summer DO residuals with the linear regression  model.
However, there is not a significant trend in the summer DO
residuals using the exponential regression model.

     With the given information/ it is difficult to determine
which of the regression models is more appropriate.   Deciding
between the linear and exponential regression  models  is a model
building problem when using a  parametric method  (GLM).  The
original intent of the analysis was to use distribution free
methods to detect trends in DO.  The analyses  have  become a
functional analysis of the specific relationship between DO
and salinity differences.  The choice of the functional form
for the relationship between DO and salinity differences
affects conclusions concerning trends in DO.

     If salinity were a censored variable/ application  of the
alignment procedure would involve assigning salinity  differences
to high and low categories using the median value of  salini.ty
differences (1.74 for August only and 2.36 for summer).   We
elected to use two categories  to ensure that a sufficient number
of observations occurred in each category to allow  calculation
of the average DO for "high" and "low" salinity difference
years.  With additional data,  any number of categories  could
be defined, as well as any number of explanatory variables.
Average bottom DO was calculated for each salinity  difference
category for August only and for summer (July/August).  Each
observation of DO was then subtracted from the average  value
for the category corresponding to salinity difference for that
year.  Spearman rank correlation and Kendall's Tau  were calcu-
lated between these differences in DO and year (Table V-5).
After accounting for differences in salinity from year  to year
using the alignment procedure, we did not find a significant
downward trend in bottom August DO,' but we did for  summer
(July/August) DO.

     When analyses for trend detection include a functional
explanatory variable, collinearity can be a problem.  This  is
illustrated by the analysis described above.   Both  bottom DO
and salinity differences exhibit significant trends  in  time
(see Tables V-2 and V-6).  Salinity differences were  included
                                      V-28
I

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      Martin Marietta Environmental Systems
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Martin Marietta Environmental Systems
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            V-31

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                    Martin Marietta Environmental Systems
Table V-5.  Values  and  probability levels (p-values)  for
            Kendall's Tau and Spearman rank correlation
            coefficients applied to residual bottom DO
            values  and  year for August and for the summer.
            Residuals are computed from the alignment pro-
            cedure  based on two categories (high and  low)
            of  salinity differences.
                        August
                   Summer
Kendall's Tau

p-value


Spearman rank

p-value
-0.25

 0.14


-0.40

 0.10
-0.45

0.009


-0.60

0.008
                             V-32

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                    Martin Marietta Environmental Systems
Table V-6.
Values and probability  levels (p-values) for
Kendall's Tau and  Spearman rank correlation
coefficients computed between average August
salinity differences and year, and average
summer salinity differences and year
                        August
                              Summer
Kendall's Tau

p-value


Spearman rank

p-value
            0.45

            0.009


            0.60

            0.009
0.44

0.011


0.63

0.005
                          V-33
                                                         I

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                      Martin Marietta Environmental Systems
as a functional explanatory variable  in analyses  for  DO  trends
in three justifiable  ways  (linear and  exponential regression
models and alignment  procedure).  Significant  downward trends
in August bottom DO were detected with the  exponential regres-
sion model, whereas with the alignment procedure  there was no
significant trend in  August DO.  In contrast/  a significant
downward trend in summer DO was detected with  the alignment
procedure but not with the exponential regression model.   For
the linear regression model, significant downward trends  in DO
were detected for both August and summer DO.   Thus, because DO
and salinity are both trending in time, conclusions concerning
trends in DO become dependent on how  variation in DO  is  attri-
buted to variation in salinity differences.

     Based on the above application of distribution free
methods, we conclude  that summer bottom DO  at  Calvert Cliffs
exhibits a significant downward trend.  Some or all of this
downward trend in DO  can be attributed to an upward trend in
salinity differences  (or variables correlated  with salinity
differences).


Parametric Methods


     In the next section we demonstrate the parametric
approaches (GLM and linear logit models) using the 1984/1985
data from the present CBP monitoring program.  In this
section we apply the  parametric method of ANOVA/ANCOVA to
the Calvert Cliffs data.  Results are  organized into  two
groups based on whether salinity differences are  included
as an explanatory variable.


     Analysis of DO Without Salinity Differences


     An ANOVA was applied to bottom DO for July and August using
the four stations as  replicates.  July and August effects were
treated as nested within year effects:

            Yij - u + YEARi + MONTH(YEAR)j(i)  + ซij       (V-l)

where:

              u * overall mean DO

          YEARi • the effect on DO of  the ith  year
        MONTH (YEAR >.;,.• x  ป the effect on DO of the j
                   JU;    itR year
                                           itn month in the
                      year

                - error.
                                    V-34
I

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                      Martin Marietta Environmental Systems

Table V-7 shows the results from the ANOVA.  Both  the year and
the month nested in year effects are significant.  Although
not presented here, an assessment of whether assumptions
underlying ANOVA are met can be performed with  residual diag-
nostics and, if sufficient data are available,  with  statistical
tests.

     To illustrate how trends can be detected using  ANOVA, we
applied Duncan's Multiple Range Test and Tukey's Studentized
Range Test to the individual year averages  (Tables V-8 and V-9).
Results of Duncan's Multiple Range Test shows a general pattern
of later years having lower summer bottom DO values  than  earlier
years.  Average summer bottom DO values for 1982,  1984, 1985, and
1986 are significantly less than the summer average  bottom DO
values for 1969-1980.

     Results of Tukey's Studentized Range Test  also  show  a
similar general pattern of summer bottom DO values being  lower
in later years compared to earlier years (Table V-9).  Bottom
DO values for 1982, 1984, 1985, and 1986 are significantly
less than values for 1970-1980 (excluding 1975).  Also, bottom
DO values for 1984 and 1986 are significantly lower  than  values
for 1969-1981.
     Analysis of DO with Salinity Differences
     An .ANCOVA was applied to summer  (July and August)  bottom DO
values using stations as replicates and salinity difference  as
the covariate.  As with the ANOVA model, the ANCOVA model
included year and month nested in year effects.

            Yij ป u + YEARi + MONTH(YEAR)j( jj

                  + B'SALDIFF.
where:
              u
          YEAR
           13   wlj



overall mean DO

the effect on DO of the i1
                                              year
                                           th
MONTH(YEAR)iflN * the effect on DO of the j&" month  in  the
           ^  '   ith year

      SALDIFFij ป the salinity difference (bottom minus surface)
                  for the jth month in the  itft year

              B = parameter relating effect of salinity dif-
                 • ferences on DO

            ei'-i = error.
                                                                    .
                             V-35

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                    Martin Marietta Environmental Systems
Table V-7,
Source
Results of ANOVA applied to July  and  August
bottom DO values/ with year and cruise nested
in year effects  (* * p < 0.001)
    df
SS
MS
F-Value
Model            34
   Year          1-7
   Month(Year)   17
Error           100
              875.6
              455.5
              420.1
              191.3
            25.8
            26.8
            24.7
            1.91
            13.5*
            14.0*
            12.3*
                            V-36

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Martin Marietta Environmental Systems
Table V-8.




Results of
to ind
Means
Duncan1 s
ividual year
with
different (
Duncan


B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B





F
F
F








Grouping






D
D
D
D
D
D
D
D
D
D
D
0
D
D
D
D
D












A
A
A
A
A
A
A

E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E

G
G
G
G
G
G
G
G
G




C
c
C
c
c
c
c
c *
c
c
c
c
c
c
c
c
c














the same
alpha ป
Multiple Range
means of
letter
0.05, df
Mean
6.

5.

5.

5.

4.

4.

4.

4.

4.
*
4.

3.

3.

2.

1.

1.

0.

0.

0.
08

43

31

16

29

19

13

11

03

00

65

46

79

58

05

23

13

04
summer
are not
= 100 ,
N
8

8

8

8

8

8

4

8

7

8

8

8

8

8

8

8

4

8
Test applied
bottom DO.
significantly
MSE = 1.91)
Year
1973

1972

1970

1977

1974

1971

1979

1976

1980

1978

1969

1975

1981

1983

1985

1982

1986

1984
            V-37

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Martin Marietta Environmental Systems
Table

V-9.

Results of
to
Tukey 's
individual year
Means with
different
Tukey




B
B
• B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B

F
F
F
F
F
F
F
F
F
Grouping




















D
D
D
D
D
D
D
D
D






A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A


E
E
E
E
E
E
E














C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C








the same
(alpha ป
Studentized Range Test applied
means of
letter
0.05, df
Mean
6.

5.

5.

5.

4.

4.

4.

4.

4.

4.

3.

3.

2.

1.

1.

0.

0.

0.
08

43

31

16

29

19

13

11

03

00

65

46

79

58

05

23

13

04
summer
are not
• 100,
N
8

8

8

8

8

8

4

8

7

8

8

8

8

8

8

8

4

8
bottom DO.
significantly
MSE = 1.91)
Year
1973

1972

1970

1977

1974

1971

1979

1976

1980

1978

1969

1975

1981

1983

1985

1982

1986

1984
            V-33

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                       Martin Marietta Environmental Systems

     Table V-10 shows the results of the ANCOVA, and Table
V-ll shows the results of pairwise comparisons of adjusted
(least squares) yearly means of summer DO.  Year effects,
month nested in year effects, and salinity differences  are
significant.  Pairwise comparisons of least squares means of
DO, using an alpha * 0.05 for individual tests, show a  pattern
of latter years having lower DO than early years.  All  pairwise
comparisons of adjusted bottom DO for 1982-1986 with 1969-1980
show the DO concentrations in 1982-1986 to be significantly
lower than in 1969-1980 (except for 1983 vs. 1969 (p-value =
0.06), 1983 vs. 1980 (p-value * 0.07), and 1983 vs. 1986
(p-value * 0.06), which are marginal).

     The results obtained with the above ANOVA/ANCOVA analysis
of summer bottom DO at Calvert Cliffs are consistent with the
results obtained using the distribution free methods applied
to residuals from the linear regression model.  There is a
significant downward trend in summertime bottom DO at Calvert
Cliffs, and this trend is still significant after linearly
adjusting DC values for salinity differences (i.e., B'SALDIFF
in Eq. V-2).  Although not demonstrated, a more complicated
function of salinity differences could be incorporated  in the
ANCOVA (e.g., exponential).  These analyses would more  closely
parallel the distribution free analyses applied to residuals
from the exponential regression model.


         D.  ANALYSIS OF 1984/1985 CBP MONITORING DATA
     To further illustrate GLM and to illustrate  linear  logit
models, we used data collected in 1984 and 1985 as part  of  the
present CBP monitoring program.  We analyzed above pycnocline
layer chlorophyll-a and ammonia concentrations  (surface  and
above pycnocline values) from 9 stations  (Table V-12)  for the
summer periods defined for 1984 as cruises 1-7  (18 June  1984-
26 September 1984) and for 1985 as cruises 21-27  (17 June
1985-25 September 1985).  These stations  and cruises were
selected because, based on multivariate statistical analyses
(PCA) performed as part Deliverable 1, the stations can  be
considered to be located in a region of the Bay (.denoted Region
II in Fig. V-10) and the cruises can be considered to  belong
to summer periods.

     ANOVA was used with above pycnocline layer chlorophyll-a
concentrations.  Only about 2% of the above pycnocline layer
chlorophyll-a values measured at these 9  stations for  these
14 cruises were at the detection limit of 1 ug/L  (5 of 233
observations); thus censored values were  set to the detection
limit.  For ammonia, there were two detection limits:  0.02
mg N/L for summer 1984 cruises and 0.003  mg N/L for cruises in
                              V-39
                                                                    .

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                     Martin Marietta Environmental Systems
Table V-10,
Results of ANCOVA  applied to July and August
bottom DO values,  with year and cruise nested
in year effects and  salinity difference as
              the  covariate (* ป p < 0.005;
                                             **
                                    p < 0.001)
Source
    df
SS
MS
F-value
Model             35
   Year           17
   Month(Year)    17
   SALDIFF         1
Error             99
              890.3
              157.5
              295.5
               14.72
              176.6"
          15.4
           9.3
          17.4
          14.7
          1.78
          14.3**
          5.19**
          9.75*
          8.25*
                           V-40

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                    Martin Marietta Environmental Systems
Table V-ll.
Results of pairwise  comparisons of adjusted
(least squares) yearly means of summer DO.
Shown are the p-values associated with
HQ:Summer DO in year I * Summer DO in
year J.  Values are  adjusted based on the
ANCOVA model with  salinity differences as the
covariate.  Each individual comparison is
based on alpha = 0.05.   The values of I and J
denote years in time order (1 = 1969, 2 =
1970,..., 18 * 1986)
I/J
1
2
3
4
5
6
1
1
9
10
11
i:
13
14
13
1ซ
17
11
1
0.0605
0.35*7
0.0155
9.0084
0.3*33
0.3746
0.8891
0.0958
O.SI94
0.1462
0.9193
0.3309
0.0020
0.0595
0.0074
0.0004
0 0232
2
0.0605
0.3274
0.5969
0.3913
0.2992
0.4183
0.0763
0.8250
0.1328
0.9564
0.0695
0.0063
0.0001
0.0008
0.0002
0.0001
0.0009
3
0.3517
0.3274
fl'.l 275
0.0780
0.9627
0.962S
0.44S9
0.4440
0.6043
0.4278
0.3882
0.0601
0.0001
0.0060
0.0006
0.0001
0.0034
4
0.0155
0.5969
0.1275
0.7480
0.1147
0.1845
0.0231
0.4547
0.0413
0.7423
0.0203
0.0010
0.0001
0.0001
0.0001
0.0001
0.00.01
5
0.0014
0.3913
0.0780
0.7480
0.0659
0.1428
0.0095
0.2821
0.0220
0.5886
0.0100
0.0007
0.0001
0.0001
0.0001
0.0001
0.0002
6
0.3833
0.2992
0.9627
0.1147
0.0659
0.9299
0.4708
0.4113
0.6360
0.4120
0.4104
0.0685
0.0002
0.0081
0.0010
0.0001
0.0046
7
0.3746
0.4183
0.9628
0.1845
0.1428
0.9299
0.4833
0.3331
0.6071
0.4501
0.4122
0.0661
0.0001
0.0034
0.0001
0.0001
0.0011
8
0.8891
0.0763
0.4489
0.0231
0.0095
0.4708
0.4833
0*.1X96
0.7984
0.1906
0.9027
0.2837
0.0032
0.0609
0.0106
O.OOC .
0.0273
9
0.0958
0.8250
0.4440
0.4547
0.2821
0.4113
0.5331
0.1196
o!l978
0.8333
0.1077
0.0113
0.0001
0.0015
0.0003
0.0001
0.0015
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           Table V-ll.   Continued
 I
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 I
10
.6194
.1321
.6043
.0413
.0220
.6360
.6071
.7914
.1971
*
.2463
.7110
.1742
.0001
.0267
.0034
.0001
.0123
11
.1462 (
.9364 I
.4271 (
.7423 (
.3116 (
.4120 (
.4301 (
.1906 I
.1333
.2465
.
.1627
.0300
.0001
.0037
.0003
.0001
.0014 (
12
1.9193 (
1.0*93 (
1.3112 (
1.0203 (
1.0100 (
).4104 (
1.4122 (
).9027 (
.1077
.7110
.1627
.
.3417
.0036
.0747
.0116 (
.0007 <
j.0300 I
13
1.3309
).0063
1.0601
1.0010
1.0007
1.0615
1.0661
).2137
.0113
.1742
.0300
.3417
,
.0111
.3115
1.0446
).0012
1.0911
14
0.0020
0.0001
0.0001
0.0001
0.0001
0.0002
0.0001
0.0032
0.0001
0.0001
0.0001
0.0036
0.0111
.
0.1265
0.1179
0.9379
0.7141
15
0.0595 (
0.0001 (
0.0060 <
0.0001
0.0001 •
0.0011
0.0034
0.0609
0.0013
0.0267
0.0037
0.0747
0.3113
0.1265 (
(
0.2122
0.1179 (
0.3375 (
16
1.0074
1.0002
1.0006
.0001
.0001
.0010
.0001
.0101
.0003
.0034
.0003
.0116
.0446
1.1179
1.2122
.
1.7177
1.9325
17
0.0004
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0004
0.0001
0.0001
0.0001
0.0007
0.0012
0.9379
0.1179
0.7171
t
0.7586
18
0.0232
0.0009
0.00"34
0.0001
0.0002
0.0046
0 .0011
0.0273
0.0015
0.0123
0.0014
0.0300
0.0911
0.7148
0.3376
0.9325
0.758ซ
.
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Table V-12.
Stations  comprising Region  II  in the analysis
of 1984/1985 CBP monitoring data using
parametric  methods
                     CB 3.1

                     CB 3.2

                     CB 3.3C

                     CB 3.3E

                     CB 3.3W

                     CB 4.1W

                     CB 4.2E

                     CB 4.2W

                     CB 4.3W
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         Figure  V-10.
Grouping of CBP  monitoring stations  into regions
of similar water quality in Chesapeake Bay.
Groupings are  based on the results of  Principal
Components Analysis.   Region II  is denoted by
diagonal lines.
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summer 1985.  Assuming a detection limit of 0.02 mgN/L  for
both 1984 and 1985 results in approximately 26% of  the  above
pycnocline layer ammonia values being  "censored" (61 of  236
observations).  Therefore/ a linear logit model was used with
ammonia concentrations.


ANOVA with Chlorophyll-a


     ANOVA was applied to above pycnocline  layer chlorophyll-a
concentrations treating observations within the above pycnocline
layer as replicates.  Factors in the model were station/ year/
and cruise nested in year/ with appropriate two-way interactions.

             Yljk * u * STATIONi + YEARj + CRUISE(YEAR)k(j)

                    + STATION x YEARij + STATION x  CRUISE(YEAR)ik(j)

                    + eijk

where:

                u * overall average above pycnocline layer
                    chlorophyll-a concentration

         STATIONi ป the effect on chlorophyll-a of  the  i   station

            YEARj ป the effect on chlorophyll-a of  the  jth  year

 CRUISE(YEAR)]<(<) ป the effect on chlorophyll-a of  the  kth
                    cruise in the j    year

 STATION x YEARij = tne interaction effect  on  chlorophyll-a
                    between the itJl station and jtn year (in
                    addition to the station and year main effects)

STATION x
CRUISE( YEAR) ,-v ti\ * the interaction effect  on  chlorophyll-a
                    the interaction effect on chlorophyll
                    between the ith station and the ktn
                    cruise in the j"1 year (in addition  to
                    station and cruise nested in year effects)
             eijk
                    error.
     The results of the ANOVA are shown in Table V-13.  The
ANOVA model is not significant (Fii7,H5 z 1.13/ p = 0.26)/  and
station by year and station by cruise nested  in year interactions
are also not significant (p ป 0.94 and p = 0.41, respectively).
Therefore/ we opted to reapply the ANOVA model without  these
interaction terms.
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Table V-13
Results of ANOVA applied to
above pycnocline
layer chlorophyll-a concentration in Region II
for summer 1984 and 1985, with station, year,


Source
Model
Year
Cruise(
Station
Year x
Station
and cruise nested in year as
priate two-way interactions
df SS MS
117 108307.7 925.7
1 813.4 813.4
Year) 12 17542.9 1461.9
8 12037.0 1504.6
Station 8 2371.9 296.5
'x • 88 75542.4 858.4
factors and "appro-

F-value (p-value)
1.13 (0.25)
0.99 (0.32)
1.78 (0.06)
1.83 (0.08)
0.36 (0.94)
1.05 (0.41)
Cruise(Year)
Error
115 94352.7 820.5

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     ANOVA was applied to the above pycnocline layer
chlorophyll-a concentrations with station,  year,  and  cruise
nested in year effects, without two-way interactions.

        Yijk = U + STATION! + YEARj +  CRUISE(YEAR)]<(j)  +  eijj<

The results for this model.are shown in Table V-14.   The
station and cruise nested in year effects are marginally
significant (p ซ 0.07 and p * 0.05, respectively);  therefore we
could not treat stations and/or cruises as  replicates.  The
year effect is not significant (p ป 0.32).  The analyses  would
stop at this point since a nonsignificant year effect implies
there are no significant differences between 1984 and 1985
summertime above pycnocline layer chlorophyll-a concentrations.
We examined various residual plots to  assess homoscedasticity
and the possible need for transformations of chlorophyll-a
concentrations.  However, considering  the limited amount  of
data included in the analysis, there were no obvious  departures
from assumptions.  If a multiple years were included  in the
analysis and a significant year effect was  found, multiple
comparison tests would be used to detect any trends in
chlorophyll-a concentrations with years.


Linear Logit Model with Ammonia


     A linear logit model was used to  analyze trends  in above
pycnocline layer ammonia concentrations.  Ammonia concentrations
were classified into high and low categories based  on the
median of the ammonia values.  The remaining categorical  ex-
planatory variables were year and cruise nested in  year.  Note
that in this analysis with the linear  logit model we  treated
stations, as well observations within  the depth layer,  as
replicates rather than including a station  factor,  as in  the
ANOVA with chlorophyll-a.  This was deemed  necessary  for  the
linear logit analysis to ensure sufficient  numbers  of observa-
tions in each cell.  In addition, to avoid  cells with zero
counts, we added 0.5 to each cell.  The linear logit  model
thus consisted of year and cruise nested in year effects.
Table V-15 shows the the number of observations (plus 0.5)  in
each of the 28 cells (2 levels of ammonia x 14 cruises) defined
by the model.

          log Zij - K + YEARi + CRUISE(YEAR) j (i} +  e^j

where:

              Zjj * the ratio of the proportion of  low  ammonia
                    concentrations to  the proportion  of high
                    ammonia concentrations  for the  jtn  cruise
                    in the i   year (i.e.,  log Z^j  is the logit
                    response function)

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Table V-14.
Results of ANOVA applied  to  above pycnocline
chlorophyll-a concentration  in Region II
for summer 1984 and  1985,  with only station,
year, and cruise nested in year as factors
(i.e., no two-way  interactions)
Source
    df
SS
MS
F-value (p-value)
Model             21      30393.4   1447.3    1.77     (0.02)

   Year            1        813.4    813.4    1.00     (0.32)

   Cruise(Year)   12      17542.9   1461.9    1.79     (0.05)

   Station         8      12037.0   1504.6    1.84     (0.07)

Error           211     172267.0    816.4
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Table V-15.
     The number of observations (plus 0.5) in the  28
     cells defined by  the high and low categories  of
     above pycnocline  summer ammonia C9ncentrations
     and the seven summer cruises in 1984 and in 1985
                                  Ammonia Concentration
Year
Cruise
Low
High
 1984
  1

  2

  3

  4

  5

  6

  7
11.5

 9.5

11.5

 6.5

 1.5

 5.5

 0.5
 7.5

 9.5

 7.5

10.5

 3.5

13.5

18.5
 1985
 21

 22

 23

 24

 25

 26

 27
10.5

14.5

11.5

 8.5

14.5

17.5

 2.5
 8.5

 4.5

 7.5

10.5

 4.5

 1.5

16.5
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intercept

effect of i   year on the logarithm of  the
ratio of the proportions of low to high
ammonia concentrations
 CRUISE(YEAR)-(i)
                                   ,th
                                 ith year
               'ij
effect of the j   cruise in the
on the logarithm of the ratio of the pro-
portions of low to high ammonia concentra-
tions

error.
     The results for the above linear  logit model  are  shown in
Table V-16.  The cruise nested in year effect  is significant for
1985i and marginally significant .for 1984.  The year effect is
also significant.  Because there are only  two  years  in the
analysis/ we can interpret the significant year effect directly
from the negative sign of the 1984 year effect parameter (-0.59),
The negative sign implies that the proportion  of low ammonia
values above the pycnocline during the summer  of 1984  was
significantly lower than in the summer of  1985 (i.e.,  a greater
porportion of "high" ammonia concentrations were observed in
1984 compared to 1985).  If multiple years were included in
the analysis/ pairwise comparison of year-specific parameters
could be utilized to detect trends.  Similar results were also
obtained when the response function was simply the proportion
of low ammonia concentrations/ rather  than the logit function.
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Table V-16,
Results of the linear logit model  with the
logarithm of the ratio or  the  proportion of
"low" ammonia concentrations to  the proportion
of "high" ammonia concentrations as the
response variable, and year and  cruise nested
in year as categorical explanatory variables
as Ammonia concentrations  are  above pycnocline
layer values for the 1984  and  1985 summer
cruises
Source
           df
                         p-value
Intercept
Year
Cruise (Year
Cruise (Year
Error
  1984)
  1985)
1
1
6
6
0
 0.41
10.52
12.08
23.19
0.52
0.0012
0.06
0.0007
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      VI.  ANALYSIS FRAMEWORK FOR TEMPORAL TREND  DETECTION
            IN CBP WATER QUALITY MONITORING  PROGRAM


                        A.  INTRODUCTION


     In Chapters IV and V we described univariate statistical
methods for trend detection in water quality data, and  illus-
trated these methods using historical data at Calvert Cliffs
and data collected in 1984 and 1985 as part  of the present
CBP monitoring program.  In this chapter/ we present a  general
analysis -framework for temporal trend detection in water quality
data collected as part of the CBP monitoring program.   Many of
the ideas concerning trend detection in CBP  monitoring  data
presented in this chapter were discussed previously in  earlier
chapters and were implemented in the analysis of  the Calvert
Cliffs and CBP data presented in Chapter V.   For  all of the
methods, statements concerning trends based  on the data from
the present monitoring program are not possible until at least
5-10 years of data are available.

     We view this document as providing an initial foundation
for the analysis of temporal trends in Chesapeake Bay water
quality data.  A major role of this document is to focus debate
on trend detection of water quality in Chesapeake Bay on spe-
cific issues.  Statistical analysis of water quality data is
an iterative process/ and as more data become available and
the methods are applied to these data/ refinements and  improve-
ment of the proposed analysis scheme will be possible and
likely.

     The analysis framework focuses on data  as collected by the
CBP monitoring program and is organized into three major sections

     •  Response and explanatory variables

     •  Spatial and temporal scales of analysis

     •  Application of methods for trend detection


             B.  RESPONSE AND EXPLANATORY VARIABLES
            Initial  analyses  would focus on individual water quality
       variables  as  response  variables  (i.e./  DO/  chlorophyll-a,
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        dissolved and total nutrients, and TSS), although the analysis
        framework is applicable to any univariate response variable.

             Because the focus of this analysis framework is trend
        detection (rather than functional analysis)/ we would recommend
        not including functionally oriented explanatory variables in
        preliminary analyses.  The only exception would be for .analyses
        involving DO, for which we would recommend performing analyses
        with and without a measure of the intensity of stratification
        as an explanatory variable.  The measure of stratification
        would be computed on similar time and space scales as the DO
        response variable.  Once trends are detected in response -vari-
        ables/ we would attempt to relate these trends to trends in
        functionally oriented explanatory variables.

             Several of the water quality variables would serve as both
        response and functionally oriented explanatory variables.  For
        example/ nutrient concentrations can be analyzed for trends and
        can be used to explain trends in chlorophyll-a.  Similarly,
        tributary nutrient loadings to the Bay on a regional basis can
        be analyzed for trends and can be used to explain trends de-
        tected in other water quality variables.  Tributary data,
        measured as part of the present CBP monitoring program, would
        be used to compute the tributary nutrient loadings.  Additional
        sources of data for calculation of nutrient loadings are the
        USGS, the CBP nonpoint source model/ and ongoing water quality
        modeling activities.
                  C.  SPATIAL AND TEMPORAL SCALES OF ANALYSIS
     Analyses can be performed on a variety of spatial and
temporal scales.  Ideally/ at one extreme analyses would be
performed Bay-wide (all stations)/ for all cruises, and for
all depths (water column).  At the other extreme, the absolute
lowest bound to the generality of analyses, would be analysis
of data from a single depth at a specific station for a single
cruise each year (e.g. / a time series of minimum observed
bottom DO from among summer cruises at a station).  Examples
of temporal scales intermediate to these two extremes would be
investigation of trends by month (e.g., August DO), or cruises
grouped over multiple months (but not strictly months belonging
to seasons).

     As a compromise between practical considerations (likeli-
hood of detecting trends, interpretability of analyses) and
generality of conclusions spatially and temporally, we recommend
that the preferred level of analysis be regional- and seasonal-
specific for either the entire water, column or the above
pycnocline or below pycnocline depth layers.  On one hand,
detection of year-to-year trends based on a single depth at a

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single station for a single cruise per year are difficult  to
extrapolate to any meaningful statements about the Bay/ or
regions in the Bay.  On the other hand, as a first step in
analyses, detection of trends at a more general level  (e.g.,
annual instead of season-specific, Bay-wide instead of region-
specific) would be extremely difficult.  The issue of  homogeneity
of trend becomes even more complicated to assess.  For some
statistical methods, homogeneity of trend is required, whereas
for other methods interpretation of results when homogeneity
of trend does not hold is difficult.  In addition, many of the
water quality variables exhibit strong seasonal cycles, with
concentrations usually during one (or two) seasons serving as
an indicator of water quality (e.g. , summer DO, spring and
summer chlorophyll-a).  Similarly, the Bay exhibits strong
spatial variation in water quality, and these gradients in
water quality can be grossly grouped into distinct regions.
As with inclusion of all cruises, inclusion of multiple regions
in a single analysis would greatly complicate interpretation.
Also, we would a priori anticipate that these regions  would
likely exhibit different responses to reduced nutrient loadings.
Management actions are being implemented over time (years) in
different regions.  Also, because of the differences in the
water quality among these regions, even if management  actions
were implemented simultaneously throughout the Bay, these  regions
are likely to respond differently in time and magnitude to
reduced nutrients.      _

     Definition of the regions and seasons would be achieved
using multivariate analysis methods (e.g. , PCA, clustering
techniques).  Use of multivariate methods for defining temporal
and spatial groupings was illustrated as part of Deliverable  1.
PCA was applied to the 1984/1985 monitoring data to group  cruises
into seasons and stations into regions.  There are many other
multivariate methods, in addition to PCA, that would also  be
useful for defining spatial and temporal groupings in  Chesapeake
Bay.

     Whether data from the entire water column or data within a
depth layer (above pycnocline or below pycnocline) are analyzed
depends on the specific response variable.  Analyses on DO
would focus on the below pycnocline depth layer, whereas analyses
involving chlorophyll-a would probably focus on the above
pycnocline depth layer.  Detection of trends in nutrients  and
TSS could involve the entire water column or either of the
depth layers.
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                D.  APPLICATION OF  METHODS  FOR TREND DETECTION
 Distribution  Free  Methods


     Application of  the  distribution free methods, as described.
 in Chapter  IV,  to  the  present  monitoring data on a season-/
 regional-,  and  depth layer-specific basis is relatively
 straightforward.   For  all  of the  distribution free methods,
 homogeneity of  trend for the stations in a region, cruises
 in a season,  and observations  in  a depth layer would first be
 assessed.   This would .be achieved either with graphical
 i-nspection  or the  X2 approach  proposed by Van Belle and Hughes
 (1984).

     Cox-Stuart, Spearman  rank correlation, and Kendall's Tau
 would be applied to  values of  central tendency (mean or median)
 of the observations  in a region,  season, and depth layer over
 years (e.g.,  a  time  series of  annual average DO concentration
 below the pycnocline during the summer in some region).  Two
 alternatives  to this type  of analysis are provided by Hirsch's
 modification  to Kendall's  Tau  and Friedman's two-way ANOVA.
 Hirsch's modification  to Kendall's Tau would be used to allow
 individual  cruises within  a season to be explicitly included
.in the analysis.   Values of central tendency would be computed
 for each cruise for  data pooled over stations in a region.
 We would use  Friedman's  test to allow for individual stations
 to be included  in  the  analysis (i.e., a year-station model).
 In this case, values of  central tendency would be calculated
 for each station for data  pooled  over cruises in a season.
 Trends would  be detected using multiple comparisons tests and
 Page's test for monotonic  trend.

     For uncensored  or infrequently censored response variables,
 analyses would  be  performed on mean values (e.g., chlorophyll-a).
 For censored  response variables with less than 50% of the
 observations  censored, we  would use median values.   We do not
 forsee use  of distribution free methods for the analysis of
 response variables when  more than 50% of the observations are
 censored.

     Explanatory variables would  be incorprated via analysis of
 residuals,  and  for Spearman rank  correlation, by the calculation
 of partial  correlation coefficients with approximate significance
 tests based on  Pearson correlation tables.   For uncensored
 explanatory variables, residuals  would be obtained from regres-
 sion analysis of the response  variable on the explanatory
 variable.   For  heavily censored explanatory variables, the
 alignment procedure  based  on categorized values of the explan-
 atory variable  would.be  implemented.
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Parametric Methods
     GLM models would be implemented in much the•same way  as
linear logit models.  The difference would be that GLM would
be used for uncensored response variables and linear logit
models would be' used for heavily censored response variables.
Censored functionally oriented explanatory variables would be
incorporated into parametric methods by use of categorization.
For GLM, the categorized explanatory variable would be treated  .
as a factor.  For linear logit models/ the explanatory variable
would be treated as an additional categorical explanatory
variable.  Uncensored functionally oriented explanatory vari-
ables would be incorporated by using ANCOVA rather than ANOVA
for GLM, and either by categorization or use of  logistic models
for linear logit models.

     For all parametric approaches/ analyses would be performed
on a season-, region-, and depth layer-specific  basis.  In con-
trast to the distribution free and Box-Jenkins methods, the
parametric methods directly use the individual observations in
a region, season, and depth layer.  Also, homogeneity of trends
is assessed by examination of interaction effects.  The decision
becomes which of the sources of variation (stations in a region,
cruises in a season, or values within a depth layer) are treated
as factors and which are treated as "replicates."  We recommend
that multiple values in a depth layer be used as replicate
values and that the GLM and log-linear models be based on
station, year, and cruise nested in year factors with appro-
priate interactions:

   Yijk 3 U * STATIONi + YEARj + CRUISE (YEAR)fc( j)

          + STATION x YEARj.j + STATION x CRUISE(YEAR)ik(j)

          + eijk


where:

          response water quality variable for GLM

          generalized logit function for linear  logit models

Equation VI-1 would be modified appropriately to include
functionally oriented explanatory variables (either covariates,
or additional factors for GLM, or additional categorical explan-
atory variables for linear logit models).

     Figure VI-1 outlines a decision tree for interpretation
of the results from a GLM or linear logit model with terms for
the model defined as in Eq. VI-1.  The process consists of


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                     S  X Y
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      Redefine  station
      groupings into
        regions
                   S x C(Y)
                                                                    ns
       Trend?
 1. Option  of  eliminating S x Y from model.
 2. Option  of  eliminating S x C(Y) from model.
 3. Option  of  eliminating C(Y) from model.  (If options 2 and  3  are
   invoked, cruises in a season are treated as "replicates.")
 4. Option  of  eliminating S from model.  (If options 1, 2, and 4  are
   invoked, stations are treated as replicates."}
 5. Trend detection using pairwise or multiple comparison tests  on
   year-specific parameters.
Figure VI-1
Decision tree for interpretation of  the  results  from
the application of parametric methods  (GLM  and
linear logit models) for trend detection in CBP
water quality monitoring data.  Each step involves
the evaluation of the significance of  main  effects
or interaction terms.  The models are  based on
station, year, and cruise nested in  year factors  with
appropriate two-way  interactions (see  Eq. VI-1).
(* a significant, ns = not significant)
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sequential steps involving significance tests of terms corre-
sponding to interactions and- main effects.  A significant
Station x Year'interaction prevents further interpretation/
and implies that stations have been improperly grouped into
regions.  At several steps in the process/ certain combinations
of nonsignificant interaction and main effect terms allow  for
the option of simplifying the model or continuing with the
interpretation of the existing model.  The decision tree culmi-
nates in a significance test of the year effect.  If  the year
effect is significant/ trends can be detected by the  use of
pairwise or multiple comparison tests.


Box-Jenkins


     There are difficulties with the application of Box-Jenkins
intervention analysis to data from the present monitoring
program.  These difficulties center on the requirement of
Box-Jenkins methods for long time series of both pre- and
post-intervention data.  Because/ the initiation of the present
monitoring program in 1984 is coincided with the implementation
of some of the management actions/ little pre-intervention
information is available from the. monitoring data.  Furthermore,
it will take many years to collect sufficient post-intervention
time series to allow application of Box-Jenkins methods.   We
envision the best use of Box-Jenkins methods as being for  the
analysis for individual/ or small groups/ of stations, with
long time histories of regularly sampled data.  For example/
with an additional 5-10 years of data (i.e., post-intervention
information)/ Box-Jenkins methods can be applied to the Calvert
Cliffs time series presented in Chapter V.

     In addition to the requirement of long time series,
Box-Jenkins methods also require observations uniformly spaced  in
time throughout the year=  For Chesapeake Bay historical data,
this would likely involve monthly (January-December)  or seasonal
(winter/ spring/ summer/ fall) observations.

     DO, salinity/ and temperature are the likely variables for
which sufficient historical time series measurements  are
available to permit Box-Jenkins analysis.  Historical time series
of chlorophyll-a and nutrients are more difficult to  assemble
and interpret.  These variables are measured less regularly than
DO and, furthermore/ there have been major changes in the
analytical measurement techniques for these variables.

     Intensity, of stifatif ication would be incorporated into
analyses as an explanatory variable via a transfer function of
some measure of salinity differences with depth.  The other
explanatory variable would be the "intervention," with the
response defined as some combination of pulse or step and
gradual or immediate.

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             •Because DO and salinity  are  the likely variables for which
        sufficient  historical records exist for Box-Jenkins  analyses/
        •censored  variables are not an issue with the use  of  Box-Jenkins
  _     methods.

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VIII.  LITERATURE CITED
                                                                       I


                                                                       I


                                                                       I

Ahlgren, I.  1978.  Response of Lake Norrviken to reduced  nutri-
     ent loading.  Verh. Internat. Verein. Limnol.  20:846-850.         I

Box, G.E.P, and G.M. Jenkins.  1976.  Time Series Analysis:
     Forecasting and Control.  San Francisco:  Holden-Day.             I

Conover/ W.J.  1971.  Practical Nonparametric Statistics.
     New York:  John Wiley and Sons.                                   •

Conover, W.J., and R.I. Iman.  1981.  Rank transformations as  a
     bridge between parametric and nonparametric statistics.
     American Statistician 35: 124-133.                                I

Edmondson, W.T.  1972.  The present condition of Lake Washington.
     Verh. Internat. Verein. Limnol. 18:284-291.                       I

Feinberg, S.E.  1980.  The analysis of cross-classifieu  categorical
     data.  Cambridge, MA:  The MIT Press.                             i

Flemer, D.A., and others.  1983a.  Chesapeake Bay:  A profile  of
     environmental change.  U.S. Environmental Protection  Agency.   g^*

Flemer, D.A., and others.  1983b.  Chesapeake Bay:  A profile  of
     environmental change, Appendices.  U.S. Environmental
     Protection Agency.

Gilliom, R.J., and D.R. Helsel.  1986.  Estimation  of distribu-
     tional parameters for censored trace level water quality          I
     data:  1. Estimation techniques.  Water Resources Research       |
     22:135-146.

Heinle, D.R., C.F. D'Elia, J.L. Taft, J.S. Wilson,  M. Cole-Jones,      |
     A.B. Caplins, and E. Cronin.  1980.  Historical review of
     water quality and climatic data from Chesapeake Bay with          .
     emphasis on effects of enrichment.  Cheaspeake Research
     Consortium Publication No. 84.                                    '

Helsel, D.R., and R.J. Gilliom.  1986.  Estimation  of                  I
     distributional parameters for censored trace level  water          I
     quality data:  2. Verification and applications.  Water
     Resources Research 22:147-155.

Hirsch, R.M., and J.R. Slack.  1984.  A nonparametric trend test
     for seasonal data with serial dependence.  Water Resources
     Research 20:727-732.
         VIII-1

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                                         APPENDIX A
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                                        BIBLIOGRAPHY
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                                  BIBLIOGRAPHY
        Conover,  W.J.   1971.   Practical Nonparametric Statistics.
             New  York:   John Wiley and'Sons.
 I

 •            A.   DISTRIBUTION  FREE  (NONPARAMETRIC)  PROCEDURES




 I

       Conover, W.J., and R.L. Iman.   1981.   Rank, transformations
 •         as a bridge between  parametric and nonparametric
 •         statistics.  The American  Statistician 35(3):124-133.

I       Conover, W.J., and R.L. Iman.   1982.   Analysis of  covariance
            using the rank  transformation.   Biometrics '3t3 (3):715-724.

I       Cox, D.R., and A. Stuart.   1955.   Some quick tests for trend
            in location and dispersion.   Biometrika 42:80-95.

 _     Craig, C.D. ,  and G.D. Faulkenberry.   1979.   The application
 •          of ridit analysis to detect  trends in  visibility.
            Atmospheric Environment 13:1617-1622.

       Crawford, C.G., J.R. Slack, and R.M.  Hirsch.  1983.   Non-
            parametric tests for trends  in water quality  data using
            the Statistical Analysis System.   U.S.G.S. Openfile
 •          Report.  33-550.  Reston,  VA.

       Davis, J.C.   1973.   Statistics  and  Data Analysis  in  Biology.
 _          New York:  John Wiley and  Sons.

 ™     Farrell, R.   1980.   Methods for classifying changes  in
            environmental conditions.  VRI-EPA7.4-FR80-KR).
 I          Vector  Research Inc., Ann  Arbor,  MI.

       Frost, J., and R.T.  Clarke.  1973.   Use of  cross correlation
            •between hydrological time  series  to improve estimates
            of lag  one autoregressive  parameters.   Water  Resources
            Research 9(4):906-917.

•      Hirsch, R.M., and J.R. Slack.   1984.   A nonparametric trend
            test for seasonal data with  serial dependence.   Water
            Resources Research 20(6):727-732.

•      Hirsch, R.M., J.R. Slack,  and R.A.  Smith.   1982.   Techniques
            of trend analysis for monthly  water quality data.   Water
            Resources Research 18(1):107-112.
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Hollander, M./ and D.A. Wolfe.  1973.  Nonparametric  Statistical
     Methods.  New York:  John Wiley and Sons.

Konikow, L.F., and M. Person.  1985.  Assessment  of long-term
     salinity changes in an irrigated stream-aquifer  system.
     Water Resources Research 21(11):1611-1624.

Lehman, E.L.  1975.  Nonparametrics:  Statistical Methods
     Based on Ranks.  -San Francisco:  Holden-Day, Inc.

Noether, G.  1976.  Introduction to Statistics:   A Nonparametric
     Approach.  Boston:  Hougton Mifflin Co.

Nyblom, J.  1986.  Testing for deterministic  linear trend  in
     time series.  Journal of the American Statistical
     Association 81(394):545-549.

Puri, M.L., and P.K. Sen.  1971.  Nonparametric Methods
     in Multivariate Analysis.  New York:  John Wiley and  Sons.

Sen, P.K.  1968.  Estimates of the regression coefficient
     based on Kendall's Tau.  Journal of the  American Statistical
     Association 63(324):1379-1389.

Van Belle/ G., and J.P. Hughes.  1984.  Nonparametric tests
     for tren'4 in water quality.  Water Resources Research
     20(1):127-136.
         B.  GENERAL LINEAR MODELS/LINEAR LOGIT MODELS


Cunningham, R.B., and R. Morton.  1983.  A  statistical  method
     for the estimation of trend in salinity  in the River
     Murray.  Australian Journal of Soil Research  21:123-132.

Draper, N., and H. Smith.  1981.  Applied Regression Analysis.
     New York:  John Wiley and Sons.

Edwards, A.M.C., and J.B. Thornes.  1973.   Annual  cycle in
     river water quality:  A time series approach.  Water
     Resources Research 9(5):1286-1295.

Feinberg, S.E.  1981.  The Analysis of Cross-Classified
     Categorical Data.  Cambridge:  MIT Press.

Graybill, F.A.  1976.  Theory and Application of the Linear
     Model,  North Scituate, MA:  Duxbury Press.

Grizzle, J.E., C.F. Starmer, and G.G. Koch.   1969.  Analysis  of
     categorical data by linear models.  Biometrics 25:489-504.
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       Likens/ G.E., F.H. Borraan,  R.S.  Pierce/ J.S. Eaton/  and
            R.E. Munn.   1984.   Long-term trends in precipitation
            chemistry at Hubbard Brook/ New Hampshire.   Atmospheric
            Environment  18(12):2641-2647.

       Lorda/ E./ and S.B. Saila.   1986.   A statistical technique
            for  analysis of  environmental  data containing periodic
            variance components.   Ecological Modelling  32:59-69.

       Neter, J., W. Wasserman/ and M.H.  Kutner.   1985.  Applied
            Linear  Statistical  Models.   Homewood/  IL:   R.0.  Irwin,
            Inc.

       Oehlort,  G.W.  1984.   A  statistical analysis of  trends in the
            Hubbard Brook bulk  precipitation chemistry  database.
            Technical Report Mo. 260/  Series 2, Dept. of Stat. ,
            Princeton University.

       Smith/ R.V./ R.J. Stevens/  R.H.  Fox/ and C.E.  Gibson.   1982.
            Upward  trend in  nitrate concentrations in rivers
            discharging  into Lough Neagh for the period 1969-1979.
            Water Resources  Research 16:183-188.
                     C.   BOX-JENKINS  TIME SERIES ANALYSIS
       Barucos,  P.C./  K.W.  Hipel,  and.I.A.  McLeod.   1981.  Modeling
            hydrologic time series from the Arctic.   Water Resources
            Bulletin  17(3):414-422.

       Box,  G.E.P./ and G.M.  Jenkins.   1976.   Time  Series Analysis:
            Forecasting and Control.   San Francisco:   Holden-Day.

       Box/  G.E.P./ and G.C.  Tiao.  1975.  Intervention analysis with
            applications to economic  and  environmental problems.
            Journal of the  American Statistical Association 70(349):
            70-79.
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       IGlair/ T.A., and P.H. Whitfield.   1983.   Trends  in pH,
            calcium/ and  sulfate  of  rivers  in Atlantic  Canada.
            Limnology and Oceanography  28(1):160-165.
 I     D'Astous,  F./ and  F.W.  Hipel.  1979.  Analyzing  environmental
            time  series.  Journal of  the  Environmental  Engineering
 _          Division (ASCE)  105(EE5) -.979-972.
 *  .   Franzini/  L./ and  A.C.  Harvey.   1983.  Testing for deterministic
            trend and seasonal components in time  series  models.
            Biometrika 70(3):673-682.

       Fuller, F.C., and  C.P.  Tsokos. .1971.  Time series analysis of
            water pollution  data.  Biometrics 27:1017-1024.
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Hipel, K.W. , W.C. Lennox, T.E. Unny, and A.I. McLeod.   1975.
     Intervention analysis in water resources.  Water  Resources
     Research 11(6) :855-861.

Hipel, K.W. , A.I. McLeod, and W.C. Lennox.  1977.  Advances
     in Box-Jenkins modeling:  1.  Model Construction.  Water
     Resources Research 13(3) :567-575.

Hipel/ K.W. , A.I McLeod/ and W.C. Lennox.   1977.  Advances  in
     Box-Jenkins modeling:  2.  Applications.  Water Resources
     Research 13(3) :577-586.

Buck, P.M./ and G.J. Farguhar.  1974.  Water quality models
     using the Box-Jenkins method.  Journal of the Environmental
     Engineering Division (ASCE) 100(EE3) :773-752.

Lettenmaier, D.P.  1976.  Detection of trends in water quality
     data from records with dependent observations.  Water
     Resources Research 12( 5) :1037-1046.

Lettenmaier, D.Pc  1980.  Intervention analysis with missing
     data.  Water Resources Research 16(1) :159-171.

McCleary, R. , and R.A. Hoy.  1980.  Applied Time Series Analysis
     for the Social Sciences.  Beverly Hills:  Sage Publications.

McDowell/ D. , R. McCleary/ E. Meidinger, and R. Hay.   Interrupted
     Time -Series Analysis.  Beverly Hills:.  Sage Publications.

McLeod/ A.I.  1977.  Improved Box-Jenkins estimators.   Biometrika
     64(3):531-534.

McLeod, A. I., K.W. Hipel/ and F. Camacho.   1983.  Trend
     assessment of water quality time series.  Water Resource
     Bulletin 19(4) :537-547.

Mehta/ B.M. / R.C. Ahlent, and S.L. Yu.  1975.  Stochastic
     variation of water quality of the Passaic River.
     Water Resources Research 11(2) :300-308.

Muskens/ P.J.W.M. / and W.G.J. Hensgens.  1977.  Time series
     analysis on ammonia concentration and  lead values of .
     the river Rhine.  Water Research 11:509-515.

Pfeifer, P.E./ and S.J. Deutsch.  1980.  Identification and
     interpretation of first order space-time ARMA models.
     Techometr ics 22(3) ; 397-408 .

Rao, A.R. , and G. Padmanabhan.  1983.  Intervention analysis
     of total suspended particulate data from Chicago.
     Atmospheric Environment 17(3) :563-571.
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Reinsel, G., G.C. Tiao, M.N. Wang/ R. Lewis,  and  D.  Nychka.
     1981.  Statistical analysis of stratospheric ozone data
     for the detection of trends.  Atmospheric  Environment
     15(9):1569-1577.

Salas, J.O./ D.C. Boes/ and R.A. Smith.   1982.  Estimation
     of ARIMA models with seasonal parameters.  Water  Resources
     Research 18(4)-.1006-1010.

Salas/ J.O./ and J.T.B. Obeysekera.  1982.  ARMA  model identi-
     fication of hydrologic time series.  Water Resources
     Research 18(4):1011-1021.

Salas/ S.D./ G.Q. Tabios/ and P. Bantolini.   1985.   Approaches
     to multivariate modeling of water resources  time  series.
     Water Resources Bulletin 21(4):683-708.

Schaeffer/ D.F./ C. Conley/ and H. Chien.   1983.   Pollution
     source analyses using one-dimensional  timซ series.
     Environmental Management 7(5):421-426.

Sturges/ W.  1983.  On interpolating gappy  records  for time-
     series analysis.  Journal of Geophysical Research
     88(014)9736-9740..
                        D.   FREQUENCY  DOMAIN  ANALYSIS
Bloomfield, P./ G. Oehlent, M.L. Thompson/ and  S.  Zeger.
     1983.  A frequency domain analysis of trends  in  Dobson
     total ozone records.  Journal of Geophysical  Research
     88(C13):8512-8522.

Bloomfield/ P./ M.L. Thompson/ and S. Zeger.  A statistical
     analysis of Umkehr measurements of 32-46 km ozone.
     Journal of Applied Meteorology 21:1828-1877.
       Dowling/ J.M.   1974.  A  note  on  the  use  of  spectral analysis
_          to detect  leads and lags in annual  cycles of  water
•          quality.   Water Resources Research  10(2):343-344.

       Panofsky/ H.A./ and G.W.  Brier.   1968.   Some  applications
•          of statistics to meteorology.   Pennsylvania State
•          Uni V*ปT-S i t-v.
     University.


                   E.  TREND SURFACE ANALYSIS


Anderson/ P.M.  1970.  The uses and limitations of trend  surface
     analysis in studies of urban air pollution.  Atmospheric
     Environment 4:129-147.

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                       F.  MISCELLANEOUS
El-Shaaraw, A.H., ed.  1981.  Time Series Methods  in
     Hydrosciences.  New York:  Elsevier Science Publishing  Co.

Frzysko, M.  1983.  The discriminant analysis of multivariate
     time series.  IEEE Transactions on Information Theory
     29(4):612-614.

Gilliom, R.J.,-and D.R. Helsel.  1986.  Estimation of distri-
     butional parameters for censored trace  level  water  quality
     data:  1.  Estimation techniques.  Water Resources  Research
     22(2):135-146.

Gilliom, R.J., R.M. Hirsch, and E.J. Gilroy.  1984.  Effect
     of censoring trace-level water quality  data on trend-
     detection capability.  Environmental Science  and Technology
     18(7):530-535.

Glett, A.  1985.  Estimation for small normal data sets  with
     detection limits.  Environmental Science and  Technology
     19(12):1201-1206.

Green, R.H.  1979.  Sampling Design and Statistical Methods
     for Environmental Biologists.  New York:  John Wiley and
     Sons.

Helsel, D.R., and R.J. Gilliom.  1986.  Estimation of distri-
     butional parameters for censored trace  level  water  quality
     data:  2. Verification and applications.  Water Resources
     Research 22(2) 147-155.

Ott, W.R., and D.T. Mage.  1981.  Measuring  air quality  in-
     expensively at multiple locations by random sampling.
     Journal of the Air Pollution Control Association
     31(4):365-367.

Rasmussen, R.A., and M.A.K. Khalil.  1986.   Atmospheric
     trace gases:  Trends and distributions  over the last
     decade.  Science 232:1623-1624.

SAS Institute, Inc.  1985.  SAS Users Guide:  Statistics;
     Version 5 Edition.  Gary, NC:  SAS Institute, Inc.

Sokal, R.R., and F.J. Rohlf.  1969.  Biometry.  San Francisco:
     W.H. Freeman and Co.
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                  SUMMARY OF MULTIVARIATE STATISTICAL METHODS
 •               APPROPRIATE FOR CHESAPEAKE BAY MONITORING DATA
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                                  Prepared by
•                              Dr. Roger Green
                         University of Western Ontario
•                     .     London, Ontario, N6A-5B7
 M                                 Prepared for
                           Water Quality Data Analysis
                                  Working Group
 •                           (Chesapeake Bay Program)
 •                              410 Severn Avenue
                               Annapolis, MD  21405
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  ^B               Summary of multivariate statistical methods appropriate to
 •'                                Chesapeake Bay Program objectives

I       Introduction
            During my recent participation in the Data Analysis Workgroup I  suggested that ti
I       MMES evaluation of statistical methodology may be omitting some relevant multivariate
       methods - including some recent methods that are just entering aquatic/marine  ecology,
       I have been fortunate to have participated this summer in two- international  workshops
 I     which directly relate to this subject.  One was a NATO Advanced Research Workshop  on
       "Numerical Ecology", held during June in Roscoff, France, and the  other  was  an IOC/GEE
 •     (UNESCO) workshop on marine pollution methods held during August in Oslo, Norway.   (Th
       U.S. E.P.A. was a co-sponsor of the latter workshop)
 •          Here I will  attempt to summarize the current statistical  scene as it relates  to
       applied marine environmental  studies, with emphasis on (a)  monitoring applications,
       |(b) multivariate  methods, and (c) recent developments.  In  doing so I will borrow
       freely from those with whom I interacted at these two workshops, as well  as  from recen
       literature.  Since I can cite the literature more easily than  I  can cite  people with
       whom I interacted in free-wheeling workshop discussion,  I wish to  acknowledge  them now
       and to especially rcenticn John Field (Univ.  of Capetown)  and the other members of the
       irarine benthic working group  at the Roscoff workshop,  and  K.R.  Clarke (IMER, Plymouth
       U,K.) at the Oslo workshop.

       Multivariate analysis:
            Here multivariate analysis  is used  in  the  sense  of  >1  response variable, so that
       multiple regression  (with one Y)  is a  multi-variable method  but  not multivariate
       analysis.   There  are two  kinds of multivariate  analysis:  those which assume an a
       priori  structure  and are  used to  test  hypotheses  about multivariate responses,  and
       those which are used to  describe  vaguely  perceived  structure in multivariate
       observational  data.   The  former are usually  generalizations  of the univariate models we
       are  familiar with:   ANOVA,  regression, and so on.   All the  familiar univariate
       statistical  models based  on the assumptions  of  linear  additive effects and independent
       normally  distributed  errors have  multivariate equivalents and can be used for the same
       hypothesis-testing purposes.  The  response is no  longer confined to one dimension,  that
       is all.   The same assumptions apply.  The latter  ("structure-description") kind of
       multivariate analysis has  no  univariate equivalent, unless it is description  of the
       distribution of nb^'?*-v:itions  (e.g., mean, variance, skewness) on •  -•"-"!ซป v~--'-u1--

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These methods tend to fall into two broad classes, ordination  and  clustering, although!
there are exceptions.
                                                                                      I
Hypothesis-testing multivariate analysis methods
     If we have n observations on p variables, then we can  visualize  an n-by-p data    I
matrix (rows-by-columns).  If the n observations come from  g  sites, then the n rows are
subdivided into g sets of rows each containing n.  rows where  n=zn..  If p=l we have a  •
simple one-way ANOVA, and if p> 1 we have a simple one-way  MANOVA.  If p=Z then a      •
simple scatter plot of p, versus p-, with the observations  coded by site, will display.
the raw data.  For p>3 such a display is impossible, and any judgement about whether  J
and how - the groups (sites) are separated in the p>3 dimensional space is difficult
unless we can somehow reduce the dimensionality of the space.  We do  this (usually     |
following the formal MANOVA test of whether the sites are significantly separated in
the p>3 dimensional space) by applying canonical  discriminant analysis (CDA) to the   I
data, and thereby obtain a k< p dimensional representation  of the scatter - one which
gives us the best view of site separation.  We usually want a  k=2 representation s
that a 2-diir._nsional scatter plot is possible.
     MAfiOVA is easily done in more complicated designs.   In fact any  ANOVA design -
factorial, nested, mixed model - can be done for the multivariate (p>l) case.  For
example suppose an a priori structure corresponding to the  temporal and spatial
sampling design of the Chesapeake Bay program:  years with  times of year nested within
years,  and stations throughout the Say with different depths  sampled  at each station
time.  There is a logical ANOVA model  here, and with p> 1 chemical variables measured
it is also a logical MANOVA model.   We probably want to  identify (a)  redundant
variables (associated with major hydrological  phenomena  perhaps) and  (b) conservative
variables (not strongly associated with the among-years  effect in the design).  .Both of
these objectives could be accomplished by this analysis  methdti.
     With one or more continuous variables as  predictors  (e.g., salinity),  in addition
tc or instead of the design variables  (times,  stations,  depths), we can create
multivariate analysis of covariance or regression/correlation designs.  However with
the Chesapeake Bay program sampling design the implicit  ANOVA design  is too  logical  to
dismiss, and the use of a variable as  a covariate  in an  ANCOVA presumes that the
covariate is not logically correlated  with the design  effects  (e.g.,  that salinity
not logically correlated with depth,  station or time of year - not likely).   Therefore
an ANCOVA or regression approach,  multivariate or  otherwise, seems inappropriate here.
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      Good introductory references  to  such  hypothesis-testing multivariate methods are
 Cooley and Lohnes  (1962,  1971),  Harris  (1975)  and  Pimentel  (1978).  Somewhat more
 advanced treatments  are by  Morrison (1967)  and Finn  (1974).  Ecological examples are ii
 Green (1979)  and Legendre and  Legendre  (1983).

 Structure-description  multivariate analysis
      No attempt will be made here to  even  list, much less discuss, all ordination and
 clustering methods.  Ordination  can be  thought of  as an analysis of an unpartitioned
 n-by-p data matrix whereby  a new set  of variables, numbering less (usually much less)
 than  p,  is found which  optimally predicts  the  structure in the relationships among the
 original  p variables.   Ordination methods  differ from each other in (a) what is used as
 the optimality criterion, and  (b) how the  ordination algorithm "finds" the new axes
 which represent the  new variables.  Principal  components analysis (PCA) maximizes the
 amount of variation  accounted  for by  the new axes, and proceeds by way of an
 eigenanalysis on the p-by-p correlation (or covariance) matrix.  The new axes are
 uncorrelated.  PCA is elegant, simple to understand and easily done (one needs to
 calculate  a correlation matrix,  and to do an eigenanalysis of that symmetric matrix,
 and that's  it).  It does a fine  job of what it .does, but the new axes are rarely
 interpretable as simple environmental  factors "causing" the structure.  This bothers
 some  people a great deal.   Principal  coordinates analysis (PCoA)  starts with an n-by-n
 matrix of  "distances" (dis-similarities) among the samples, and then proceeds as in  the
 PCA.   The  same result as for the PCA can be obtained if the distance is defined
 appropriately.  There are two potential  advantages of PcoA over PCA, that the
 eigenanalysis is easier to do if n< p, and that one is  free to  choose  any of a  large
 number of possible  distance  measures,  using one appropriate to  the  data and  the
 objectives.  Thus PCoA is  more flexible, but less  clearly defined,  than PCA.  An
 ordination method called multi-dimensional  scaling (MDS)  finds  ~a  specified number  of
 new axes which preserve some relationship  among the among-sample  distances.   The
 procedure is an interactive  one and therefore more computationally  demanding  than  PCA
 or PCoA.  Non-metric MDS relies only on  the rank order  of the  among-sarcple distances.
 Finally, reciprocal  averaging (RA)  and correspondence analysis  (CA)  are ordinations  of
 contingency table count data,  in  whih  observations  and  variables  can be plotted  in the
 same space.  Various algorithms exist, but  an RA-CA type  solution can  be  obtained  by
doubly standardizing (both rows and columns)  an n-by-p  matrix,  and  then doing a  PCA  on
 it.

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                                                                                      I
                                                                             4     -•
      Clustering  can  be  thought of as an analysis on an unpartitioned n-by-p data matr-J
 whereby  a  partitioning  of the n samples into subsets of samples, numbering less
 (usually much  less)  than n,  is found such that the relationships among the subsets of I
 samples  optimally predict the structure in the relationships among the original n
 samples.   Clustering methods differ from each other in (a) what is used as the        I
 optimality criterion and (b) how the clustering algorithm "finds" the sample subsets,
 but clustering strategies are more diffuse than for ordination and less easily        I
 summarized in  terms  of  several specific methods.  Clustering methods are best defined*
 by a number of choices, often dichotomous choices, regarding strategy.  A method may bj
 hierarchical or  non-hierarchical.  If it is the former, then the algorithm may be     I
 agglomerative  (repeatedly fuses samples or groups of samples until all samples are in
 one group) or divisive  (repeatedly divides what is initially one group containing all |
 the samples).  The fusions or divisions are based, respectively, on some~n-by-n
                                                               2                      I
 distance matrix  or some p-by-p similarity (e.g., correlation, x ) matrix.  Divisive   I
 methods  are usually  monothetic (the next division is based on a statistic related to
 one variable), whereas  agglomerative methods are usua-lly polythetic (the next
 based on a statistic calculated from all the variables).  In hierarchical methods a
 particular linkage strategy  is chosen (i.e., how the samples or clusters of samples arj
 compared - nearest neighbour, farthest neighbour, average of cluster members, etc.)
 the clusters produced can be overlapping or discrete.   Usually hierarchical  methods   |
 produce  discrete clusters; overlapping clusters are usually produced by
 non-hierarchical methods.  Finally, a clustering method can be constrained by external
 criteria (e.g.,  that any cluster must contain samples  contiguous in space),  or it can
 be unconstrained and then evaluated by mapping the clusters and seeing whether they ar
 contiguous in space.  The most interesting recent developments in clustering methods
 have been methods that combine strategy choices that usually do not go together,  for
 example, polythetic  divisive clustering (e.g.  Lefkovitch 1976).
      It  should be emphasized that clustering and ordination are not methods  in
 competition with each other, although this has  been a  common attitude in the  past.   It
"is often a very  good strategy to do both,  and then plot the samples in the ordination
 space with cluster membership indicated by symbols.   Good introductory references  to
 ordination, clustering, and related methods are Dunn and Everitt (1982), Gauch  (198
 Legendre and Legendre (1983), and Pielou (1984).   All  are in an ecological,  or  at  1
 a  biological, context.
                                rlo3,rlv/ frOl  in^n a

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  I
 Aid be mentioned.  One is the use of rank cumulative species abundance plots,
 arhaps on a logarithmic scale, to represent community structure without giving an
 JBntity to each species.  Examples in marine benthic community studies, though
 iffaring somewhat in method and purpose, are by Sanders (1968) and Gray (1979).  The
 •ence of species identities has the potential advantage that patterns characteristic
 i  a certain condition (e.g., depth in the sea, or degree of pollution) can be
I   era!ized to zoogeographically different communities.  However there is the        v_
   advantage that one does not retain information associated with knowing what the
   fcies are (e.g., good indicator species).  A second class of methods selects a "best
   iable subset" from a larger set of response variables, the subset having the maximum
 predictive information about the larger set (maximum for any subset of that size).  The
 Aset will therefore tend to have low redundancy, which is to say that it will contain
 /ariables which have relatively-low correlations with each other.  The result of such
 M analysis is often similar to that of a PCA, except that each component will be
 represented by a single variable (which is an advantage when presenting the results  to
          audience).  The disadvantages are that the "components" will not be
  mpletely independent, and the amount of structure accounted for by a given number  of
  Imponents will not be quite as high as in the PCA.  Orlcci and Kenkel (1985) is a good
 ...fer'ence.  Ths third method, described by Hills (1969),  is simply to do an arithmetic
  Vobability plot of the normalized correlation coefficients in a correlation matrix.
  is display allows one to determine which of the correlations between species are the
I  in source of the correlation structure in the data.  Both the 2nd and 3rd methods  can
  so be applied to rank or presence/absence data, by simply changing the type of
correlation coefficient used so that it is appropriate  for the data.
I    There are some less  familiar  structure-description multivariate methods  which
are not commonly used in  ecology,  but could have valuable  applications in  marine
  tvironmental  studies.   One  is Procrustes  analysis  (Schonercann 1966,  Schonemann and
  rroll 1970,  Gower 1975)  in which the  correlation  structure of one set of variables
  Rn be fitted  directly  to  the structure of another  set,  resulting in  a description of
  e structure  which is  common to the sets  versus unique  to  each set.   I  know  of only
  e example in marine ecology (or  in  ecology at all), in which Fasham and  Foxton  (1979)
 it mid-Atlantic latitudinal  and depth-related spatial structure of biological
    ables  to the corresponding spatial  structure of physical-chemical  variables.   A
 umber of  useful  applications come  to mind.   For example,  spatial  structure  (among
stations  and depths)  in  Chesapeake  Bay  at  different times of the year,  or  in  different

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years, could be fitted t'o each other so as to produce a description of spatial
structure that is common to all times versus unique to each time.   Another method whic
can be used to compare structure-description matrices with each other is called
Individual Distance Scaling (INDSCAL).  There are no known marine  ecology examples.   I
                                                                                       I

                                                                                       I

                                                                                       I

                                                                                       J
the volume to come out of the NATO Roscoff workshop there will be a paper by de Leeuw
on this method.  Another more general, method which can be used to accomplish the same |
objective is called the Mantel test, which will be discussed by Soka'Y~in the same
volume.                                                                                I
     Correspondence analysis has been mentioned above as an ordination method
applicable to contingency table data.  It has been extended to the multi-dimensional   I
(> 2-way tables) case and is then referred to as multiple correspondence analysis
(MCA).  Gower's paper in the volume to come out of the NATO Roscoff workshop will be a I
key reference.  An obvious application would be to extend a "response variables by     '
stations" 2-way table to a "response variables by stations by times" 3-way table, use  j
the log linear model (Fienberg 1S70, McCullagh and Nelder 1983) to test for            I
non-independence of dimensions (just an extension of the G-statistic used for 2-way
tables), and then display the relationships in an ordination space produced by MCA.
Another approach to joint ordination of a response in a Iccations-by-times context is
called "3-way unfolding", which produces a true joint-space (as opposed to a
projection)  maximizing conservation of joint relationships among response variables,
locations and times.  There will  be a paper by Heiser in the NATO Roscoff workshop
volume on this method.
     There will be a paper by P.  Legendre in the Roscoff volume on "constrained
clustering"  which is a modification of commonly used cluster analysis methods,  whereby
only observations identified as contiguous in space (or time)  are allowed to join as
members of the same cluster.  This method might be used to objectively identify
environmentally homogeneous regions of the Bay, or to identify relatively homogeneous
periods of the year.  Another recent variant of cluster analysis is called "fuzzy sets
clustering"  (Bezdek 1981, and in  the NATO Roscoff workshop volume) which does  not
absolutely assign observations to clusters but rather assigns  probabilities  to
observation  membership in the clusters.   Thus if a cluster of  location-times was
identified as a low oxygen situation, the location-ti:;;es "adjacent" to their,  would have
probabilities, perhaps  interpretable as  relative severity indices, associated with  M
them.
     I do not have a final title  or publisher re the NATO Roscoff volume. T*a  ••'o

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 title was "Numerical  Ecology", but that is the title of an existing book (Legendre and
 Legendre 1983) by the workshop organizers and is therefore unlikely to be the title of
 the 'workshop volume.   I can provide this information as soon as I have it.
 References

 Bezdek, J.C.   1981.   Pattern recognition  with fuzzy objective function algorithms.
      Plenum,  New York.
 Cooley, W.W., and P.R.  Lohnes,  1962.   Multivariate  procedures for the  behavioral
      sciences.   Wiley,  New York.
 Cooley, W.W., and P.R.  Lohnes.   1971.  Multivariate data  analysis..  Wiley,  New York.
 Dunn,  G.,  and B.S. Everitt.   1982.  An introduction to  mathematical  taxonomy.
      Cambridge,  New  York.
 Fasham, M.J.R.,  and  P.  Foxton.   1979.  Zonal  distribution of  pelagic Decapoda
      (Crustacea)  in  the eastern North  Atlantic and  its  relation  to the physical
      oceanography.   J.  Exp.  Mar. Biol. Ecol.  37:  225-253.
 Fienberg,  S.E.   1970.   The  analysis of multidimensional contingency  tables.  Ecology
      51: 419-433.
 Finn,  J.P.  1974.  A general model for multivariate  analysis.  Holt, Rinehart and
     Winston, New York.
 Gauch,  H.G.,  Jr.  1982.  Multivariate  analysis in community ecology.   Cambridge, New
     York.
 Gower,  J.C.   1975. -Generalized Procrustes analysis.  Psychometrika  40: 33-52.
 Gray, J.S.  1979.  Pollution-induced changes  in populations.  Phil. Trans. Roy.  Soc.
     Lond. B.  286: 545-561.
 Green,  R.H.   1979.   Sampling design and statistical  methods for environmental
     biologists.  Wiley, New York.
 Harris, R.J.  1975.  A primer of multivariate statistics.   Academic, New York.
 Hills, M.  1969.  On looking at large correlation matrices.  Biometrika 56:  249-253.
 Legendre, L.,  and P.  Legendre.  1983.   Numerical  ecology.   Elsevier, New York.
 McCullagh, P., and J.A.  Nelder.   1933.   Generalized  linear models.  Chapman  and  Hall,
     New York.
Morrison, D.F.  1967.  Multivariate statistical  methods.  McGraw-Hill,  New York.
 Orloci, L., and N.C.  Kenkel.  1985.   Introduction to data  analysis.  Internat. Co-on.
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Pielou, E.G. • 1984.  The interpretation of ecological  data.   Wiley,  New York.
Pimentel, R.A.  1978.  Morphometrics:  the multivariate analysis  of biological  data.
     Kendall/Hunt, Dubuque, Iowa.                                                     I
Sanders, H.L.  1968.  Marine benthic diversity:  a  comparative study.   Amer.  Natur.  102:
     243-282.                                                                        I
Schonemann, P.H.   1966.  A generalized solution  of the orthogonal  Procrustes problem.
     Psychometrika 31:  1-10.      .                                                   •
Schonemann, P.H., and R.M. Carroll.   1970.   Fitting one matrix to  another under  choice!
     of a central dilation and a rigid motion.   Psychometrika 35:  245-255.


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                                            Martin Marietta Environmental Systems
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