CALCULATION OF THE FINAL ACUTE VALUE FOR
WATER QUALITY CRITERIA FOR AQUATIC ORGANISMS
RusselL J. Erickson
Center for Lake Superior Environmental Studies
University of Wisconsin-Superior
Superior, Wisconsin 54880
Charles E. Stephan
U. S, Environmental Protection Agency
Environmental Research Laboratory-Duluth
6201 Congdon Boulevard
DuluCh, Minnesota 55804
ENVIRONMENTAL RESEARCH LABORATORY-DULUTH
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
DULUTH, MINNESOTA 55804'
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DISCLAIMER
This report has been reviewed by Che Environmental Research
Laboratory-Duluth, U.S. Environmental Protection Agency, and approved for
publication. Mention of trade names or commercial products does not
constitute endorsement or recommendation for use.
AVAILABILITY NOTICE
This document is available to the public through Che National Technical
Information Service (NTIS), Springfield, Virginia 22161,
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ABSTRACT
The Final Acute Value (FAV) for a material, which is an integral part of
ehe procedure for deriving water quality criteria for aquatic organisms, LS
an estimate of th« fifth percentile of a statistical population represented
by the set of Mean Acute Values (MAV) available for the material, a MAV being
the concentration of the material that causes a specified level of acute
toxicity to aquatic organisms in some taxonotnic group. A new procedure for
calculating FAVs has been developed under the assumption that sets of MAVs
are random samples of such populations. Based on examination of available
sets of MAVs, it was inferred that FAV estination would be best served by
assuming that the populations have a log triangular distribution. Also,
because this or any other assumption will"likely not completely hold over the
entire range of data in all sets, it was judged that FAV estimation should be
based on a subset of the data near the fifth percentile. Based on
simulations, it was determined that a FAV for a set of MAVs would be best
calculated by (a) assigning each MAV a cumulative probability P^*R/(N+1)
(R»rank, N»nuraber of MAVs in the set), (b) fitting a line to Ln(MAV) versus
/Ffl using the four points with P^ nearest 0,05 and using the geometric
mean functional relationship to estimate slope, and (c) calculating the FAV
as the concentration corresponding to Pg*0,Q5 on this line. Major
modifications of this new procedure were found to result either in only minor
changes in FAVs or in FAVs at variance with the data. The old procedure for
calculation of FAVs was judged to have some theoretical and practical
shortcomings that make it less desirable than the new procedure, but FAVs by
the two procedures were generally similar. A procedure based on extreme
deviation from random sampling generally did not produce greatly different
FAVs.
i ii
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CONTENTS
Abstract • ...... 111
Tables v
Figures v
Acknowledgments , » vi
Introduction 1
Conclusions 13
Statement of Problem 16
Description of Percentile Estimation Methods 19
Selection of Distribution 26
Selection of Percentile Estimation Method and Subset Size . . 33
Application of Rscomnended Procedure and Alternatives to Data Sets . . 39
Discussion 50
References 53
Appendix I 54
IV
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TABLES
Number Pa|e
1. Example Sets of Species Mean Acute Values 4
2. Example Sets of Family Mean Acute Values 9
3. Average Goodness-of-Fits for In(SMAV) Data Sets 32
4. Average Goodness-of-Fits for In(FMAV) Data Sets 32
5. Means and Standard Deviations of Estimates of Fifth Percentile
(K<^~) by Various Methods, for 10,000 Samples from a Standard
Triangular Distribution 35
6. Mean True Cumulative Probabilities (£5) of Estimates of Fifth
Percentile (PCx^)) by Various Methods, for 10,000 Samples
from a Standard Triangular Distribution , 36
7. FAVs Calculated from SMAV Data Sets by Old Procedure, Recommended
New Procedure, and Various Modifications of Recommended New
Procedure 40
8. FAVs Calculated from FMAV Data Sets by Old Procedure, Recommended .
New Procedure, and Various Modifications of Recommended New
Procedure 42
FIGURES
1, Standard Probability Density Plots for Rectangular,
Triangular, Normal, and Biexponential Distributions ,,..,. 30
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ACKNOWLEDGMENTS
Charles Norwood helped design Chis project and Robert W. Andrew
performed some of the calculations.
VI
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INTRODUCTION
On November 28, L980, the U.S. Environmental Protection Agency published
"Guidelines for Deriving Water Quality Criteria for the Protection of Aquatic
Life and Its Uses" as Appendix B of an announcement of the availability of water
quality criteria documents (1). Calculation of the Final Acute Value (FAV) is
an important part of the process described in these Guidelines. A FAV is a
concentration of a material derived from an appropriate set of Mean Acute Values
(MAVs), a MAV being the concentration of the material that causes a specified
level of acute toxieity to an aquatic taxon in laboratory tests. The FAV is
defined to be lower than all except a small fraction of the MAVs that are
available for the material. The fraction was set at 0.05 (i.e., the FAV lies at
the fifth percentile of the MAVs) because other fractions resulted in FAVs that
were deemed too high or too low in comparison with the sets of MAVs from which
they were obtained. However, if the set contains a MAV for an important species
that is lower than the calculated FAV, the FAV is set equal to that MAV.
In order to be useful, the procedure for obtaining a FAV from a set of MAVs
must be objective so that different parties will obtain the same FAV from a set
of MAVs. The development of a reasonable mathematical framework for FAV
calculation was therefore necessary. In addition, it is desirable that the
rationale for Ch« calculation procedure be relatively easy to understand and
that the computations' be as simple as possible. Section IV.1-0 of the
Guidelines described a procedure for calculating a FAV from a suitable set of
Species Mean Acute Values (SMAVs). Because of criticism of this procedure, this
project was initiated to define the general problem of calculating a FAV, to
evaluate alternative procedures, and to recommend the most appropriate
1
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procedure- This project was not intended to evaluate che definition of the FAV
or the procedures for obtaining MAVs.
Development of an appropriate procedure for calculating FAVs requires che
availability of typical sets of SMAVs. Some of the water quality criteria
documents (1) contain such sets in Table 3 of the section on Aquacic Life
Toxicology. Twenty data sets for freshwater species and seventeen for saltwater
species were considered to be acceptable for the purposes of this project
because they contained SMAVs from at least eight families in a variety of
taxonowic and functional group*. Th«§« data s«tt (TabIt I) contain froo 8 co 45
SMAVs for a variety of organic and inorganic materials. Because ail acceptable
sets of SMAVs (that w*re available at the completion of this project in May,
1982) were used and because th*y include a diversity of species and materials,
this group of 37 data sets should be repr«s«nt*tive of the data sees from which
FAVs will be calculated.
There is son* concern that FAVs would be more appropriately bated on a
taxonomic level higher than species (e.g., family). Statistical analysis of
data sets similar to those in Table 1 has shown that differences between
families are usually greater, often by an order of magnitude or more, than
average differences within families (2). Therefore, if a set of SMAVs has a
disproportionate number of species from a sensitive or insensitive family, che
FAV might bs undesirably affected. For example, of che 29 SMAVs for zinc in
fresh water, six are from Salmonidae and are all among the twelve lowest SMAVs.
Resolution of which taxonomic level is most appropriate is not of concern here,
but because the definition of tht FAV might be so modified, Family Mean Acute
Values (FMAVs), the geometric mean of all the SMAVs available for a family, were
computed for all data sets in Table 1 and are reported in Table 2. Subsequent
2
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analysis will consider how the use of these two different taxonoraic levels might
affect recommendations about the procedure for calculating a FAV. This does
riot, however, constitute an endorsement of either species or family as che most
appropriate taxonoraic level.
This report will first define the problem of FAV calculation and then
discuss the general methods available for estimating percentiles. Next, the
example data sets in Tables 1 and 2 will be examined to determine an appropriate
statistical distribution to use in the FAV calculation procedure. Simulated
samples from the selected statistical distribution will then be used to
determine the procedure most appropriate for calculation of the FAV. Finally,
the procedure selected will be applied to the example data sets and the
significance of deviations from various assumption! of the procedure will be
evaluated,
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TABLE I. EXAMPLE SETS OF SPECIES MEAN ACUTE VALUES.'
COPPER
(FRESHWATER)
Rank SMAV
45
44
43
42
4!
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
260.
150.
148.
145.
117.
91.8
47.9
46.5
35.2
23.1
22.9
21 .8
20.1
18.9
14,4
10.1
8.4!
5.81
5.37
5.00
4.95
3.97
3.29
2.80
2.28
2.20
2.20
2.13
2.13
2.12
1 .99
1.83
1.68
1.42
1.34
1.23
1 .07
1.02
0.91
0.91
0.76
0.55
0.43
0.28
0.23
DOT
(FRESHWATER)
Rank SMAV
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
2!
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
!
1230.
362.
192.
175.
68.
67.
54.
48.
48.
40.
33.
25.
18.
17.
14.
12.
10.
9.3
8.5
8.0
7.3
7.8
7.3
5.0
4.9
4.3
4.0
3.9
3.5
3.2
3.0
3.0
2.6
2.4
.9
.9
.7
.7
.6
.4
1.1
0.36
CADMIUM
(SALTWATER)
Rank SMAV
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
50600.
50000.
21200.
21000.
19200.
12200.
10100.
6600.
5290.
4100.
3940.
3800.
3500.
3440.
2930.
2590.
2410.
1800.
1710.
1670.
1480.
1220.
1080.
760.
645.
320.
169.
144.
135.
78.
41.3
CADMIUM
(FRESHWATER)
Rank SMAV
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
138.
135.
134.
133.
125.
91.4
86.7
80.7
55.9
54.7
47.0
38.2
35.9
30.3
28.0
22.3
19.7
12.2
7.01
3.57
2.87
1.67
1 .1?
0,87
0.29
0.09
0.04
0.03
0.02
TOXAPHENE
(FRESHWATER)
Rank SMA¥
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
- 3
2
1
180.
28,
26.
24.
20.
15.
14.
14.
14.
13.
13.
12.
11.
10.
9.8
9.2
8.7
6.3
6.
4.2
4.1
4.
3.
3.
3.
2.5
2.3
2.
1.3
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\BL£ 1. Continysd
ZINC
(FRESHWATER)
Rank SMAV
29
28
27
26
23
24
23
22
2!
20
19
18
17
16
15
14
13
12
1!
10
9
a
7
6
5
4
3
2
1
2260.
1019.
732.
716.
708.
699.
331.
524.
413.
567.
315.
293.
285.
255.
172.
169.
92.8
82.6
81.4
64.9
57.9
37.6
49.3
42.0
26.2
23.1
21.2
9,09
9.89
ENORIN
(FRESHWATER)
Rank SMAV
28
27
26
23
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
!
352.
64.
60.
34.
32.
5.9
4.7
3.1
2.1
1.8
1.5
1.3
1.2
1.1
1.0
0.85
0.78
0.76
0.75
0.69
0.54
0.47
0.46
0.44
0.41
0.33
0.32
0.15
MERCURY
(SALTWATER)
Rank SMAV
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1680.
1260.
400.
315.
230.
223.
158.
116.
98.
98.
89.
84.
79.
70.
60.
50.
17.
14.
14.
14.
10.
7.6
6.6
5.6
4.8
3,5
ZINC
(SALTWATER)
Rank SMAV
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
70600.
50000.
39000,
24600.
9460.
8100.
6330.
4090.
3640.
3380.
2440.
2160.
1780.
1450.
1270.
1000.
950.
591.
498.
400.
321.
310.
290.
166.
UNDANE
(FRESHWATER)
Ran*. SMAV
22
21
20
19
18
17
16
15
14
13
12
It
10
9
8
7
6
5
4
3
2
1
676.
485.
460.
207.
141 .!
138.
90,
83.
68.
67.1
64.
55.6
49.
45.
44.
44.
40.
32.
32.
10.5
10.
2.
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TABLE 1. Contlnuad
COPPER
( SALTWATER )
Rank
22
21
20
19
18
17
16
15
14
13
12
1 1
10
9
8
7
6
5
4
3
2
1
SMAV
600.
560.
526.
487.
412.
364.
330.
181.
141.
138.
136.
129.
128.
124.
120.
86.
69.
52.
50.
39.
31.
28.
HEPTACHLOR
(SALTWATER)
Rank
19
18
17
io
15
14
13
12
11
10
9
a
7
6
5
4
3
2
1
SMAV
194.
138.
112.
is m
J J t
50.
32.
14.5
10.
8.
6.22
3.77
3.4
3.
3.
1.5
1.06
0.36
0.8
0.057
NICKEL
(FRESHWATER)
Rank
22
21
20
19
13
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
SMAV
2230.
2030.
1540.
1080.
1010.
730.
720.
665.
659.
627.
509.
507.
457.
440.
440.
401.
388.
302.
234.
208.
78.5
54.0
0 1 ELDS 1 N
(FRESHWATER)
Rank
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
\
SMAV
740.
620.
567.
250,
213.
130.
41.
39.
24. •
22»
20.
15.
10.8
8.1
8.
6.1
5.0
4.5
2.5
0 1 ELOR I N
(SALTWATER)
Rank
21
20
19
18
17
16
15
14
13
12
11
10
9
a
7
6
5
4
3
2
1
SMAV
50.0
34.0
31.2
23.0
19.7
18.0
14.2
10.8
10.0
8.9
S.6
7.0
6.0
5.0
5.0
4.5
3.5
2.3
1.5
0.9
0.7
LINQANE
(SALTWATER)
Rank
19
ia
17
16
15
14
13
12
11
10
9
a
7
6
5
4
3
2
1
SMAV
3680.
450.
103.9
66.0
60.0
56.0
47.
35.0
30.6
28.0
14.0
10.0
9,0
7.3
6.28
5.0
5.0
4.44
0.17
ALDRIN
(FRESHWATER!
RanK
21
20
19
18
17
16
15
14
13
12
It
10
9
8
7
6
5
4
3
2
1
SMAV
19000.
4900.
180.
143.
50.
45.9
42.
34.
32.
28.
27.
27.
21.
16.
10.
9.
8.
7.4
6.1
4.5
4.
CHROMIUM(VI)
(SALTWATER)
Rank
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
SMAV
105000.
93000.
91000.
57000.
32000.
30500.
22000.
17200.
15000.
10000.
7500.
6600.
6300.
4400,
4300.
3650.
3100.
2000.
2000.
ENDS IN
(SALTWATER)
Rank
2!
20
19
13
17
16
15
14
13
12
1 1
10
9
a
7
6
5
4
3
2
1
SMAV
14.2
12.
3.1
1.8
1 .7
1.2
1 .1
0.95
0.65
0.63
0.6
0.36
0.31
0.3
0.3
0.28
0.1
0.094^
0.05
0.048
0.037
CHROMIUM* M 1)
(FRESHWATER)
Rank
18
17
16
15
14
13
12
1!
10
9
3
7
6
5
4
3
2
1
SHAV
1075.
728.
633,
233.
224,
224.
224.
191 .
191 .
139.
161 .
138.
136.
132.
123.
113.
47.
33.4
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3LE 1. Continued
HEPTACHLOR
(FRESHWATER)
Rank
13
17
16
15
14
13
12
t!
10
9
8
1
6
5
4
3
2
1
SM*V
320.
148.
101.
81 .9
78.
61 .3
47.3
42.
29.
26.
24.
23.6
13.1
7.8
2.3
1.8
1 ,1
0.9
TOXAPHENE
(SALTWATER)
Rank
14
13
12
11
10
9
a
/
6
5
4
3
2
1
SMAV
1120.
324.
43.8
21.
16.
8.2
4.5
* ^
4.4
1.4
1.1
1.1
0.5
0.11
NICKEL
(SALTWATER)
Rank
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
SMAV
350000.
320000 .
150000.
49000.
47000.
25000.
17000.
9670.
7960.
6360.
2030.
1180. -
634.
600.
508.
310.
152.
CHROMlUM(Vi)
(FRESHWATER)
Rank
!4
13
12
11
10
9
8
7
6
5
4
3
2
1
SM*¥
195000.
134000.
120000.
69000.
59900.
59000.
45100.
304 QQ.
30000.
25000.
6800.
6400.
3100.
'67.
DOT
ALORIN
(SALTWATER)
Rank
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
SMAV
89.
7,9
7,0
6.0
4.0
3.9
2.0
1.8
1.6
1.1
1.0
0.68
0.6
0.53
0.4
0.38
0.14
CHLORDANE
(FRESHWATER)
Rank
14
13
12
11
10
9
3
7
6
5
4
3
2
1
SMAV
190.
82.
59.
58.
57,
56.
45.
40.
37.
26.
25.
15.
6.3
3.
(SALTWATER)
Rank
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
SMAV
100.0
36.0
33.0
33.0
25.0
17.0
13.0
12.0
9.0
8.0
7.2
6.0
5.6
5.0
4.1
1,5
SELEN 1 UM
(FRESHWATER)
Rank
13
12
11
10
9
8
7
6
5
4
3
2
1
SMAV
42400.
28500.
26100.
24100.
13600.
12600.
10200.
9000.
6500.
3870.
1460.
710.
340.
CYANIDE
(FRESHWATER!
Rank
15
14
13
12
1 1
10
9
8
7
6
5
4
3
2
1
SMAV
2326.
2240.
639.
431.
318.
167.
147.
137.
125.
125.
103.
102.
102.
83.
57.
SELENIUM
(SALTWATER)
Rank
13
12
11
10
9
3
7
6
5
4
3
2
1
SMAV
17348.
14651 .
9725.
7400.
4600.
4400.
3497,
1 740.
1200.
1040.
300.
600.
599.
-------
TABLE 1. Continued
ENOOSULFAN
(SALTWATER)
Rank
12
11
10
9
8
7
6
5
4
3
2
!
SMAV
730.
157.
7.6
1.31
0.83
0.76
0.38
0.30
0.14
o.to
0.09
0.04
ENOOSULFAN
(FRESHWATER)
Rank
10
9
8
7
6
5
4
3
2
1
SMAV
261.
88.
6.0
5.3
3.8
3.7
3.2
2.3
0.83
0.34
ARSENIC f€RCURYb SILVER SILVER
(FRESHWATER) (FRESHWATER) (FRESHWATER) (SALTWATER)
Rank
12
11
10
9
8
7
6
5
4
3
2
!
SMAV Rank SMAV Rank SMAV Rank SHAV
41760. 11 2000. 10 5.77 10 1400.
23130. 10 2000. 9 5,52 9 550.
26042. 9 2000. 8 4.11 8 500.
22040. 8 1000. 7 0.112 7 250.
18096. 7 784. 6 0.0230 6 210,
15660. 6 249. 5 0.015 5 36.
14964. 5 240. 4 0.014 4 33.
13340. 4 50. 3 0.0123 3 21.
5278. 3 20. 2 0.0121 2 20.
1348. 2 10. 1 0.00192 1 4.7
879. 1 5.
812. .
CHLORDANE
(SALTWATER)
Rank
&
7
6
5
4
3
2
1
SMAV
120.
17.5
16.9
11.8
6.4
6.2
4.8
0.4
8 TsHsn from Table 3 in the "Aquatic Life Toxicology" sections of the water quality criteria documents (1).
For the purposes of this project, the Species Maan Acut» Intercepts for several of tne metais in frash »o
ware considered to be Species Mean Acute Values. All SMAYs are In
13 Tha acute value for F axone 1 1 a c I y peat a should have been published originally as 20 yg/L, not 0.02 ^g/L (3).
-------
TABLE 2. EXAMPLE SETS CF FAMILY MEAN ACUTE VALUES.8
CADMIUM
(SALTWATER)
Rank
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
FMAV
37600.
21200.
19200.
1 1100.
6600.
5290.
3940.
3800.
3500.
3440.
3260.
2930.
2410.
1800.
1 710.
1670.
1480.
1220.
1080.
760.
645.
320.
156.
78.
75.
CADMIUM
(FRESHWATER)
Rank
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
FMAV
138.
133.
36.7
85.9
55.9
54.8
30.3
28.5
28.0
19.7
12.2
8.86
7.01
2.87
1.58
1.15
0.50
0.048
COPPER
(FRESHWATER)
Rank
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
a
7
6
5
4
3
2
1
FMAV
260.
150.
145.
117.
46.5
45.3
38.7
35.2
22.9
14.4
10.0
3.86
3.58
3.56
2.28
2.13
2.12
1.73
1 .42
1.34
0.99
0.76
0.30
ENDRIN
(FRESHWATER)
Rank
17
16
15
14
13
12
11
10
9
a
7
6
5
4
3
2
1
FMAV
109.
64.
60.
32.
4.7
4.3
1..80
1.50
1.30
1.0
0.95
0.85
0.66
0.65
0.49
0.48
0.44
MERCURY
(SALTWATER)
Rank
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
FMAV
1680.
1260.
400.
315.
230.
223.
158.
116.
98.
89.
84.
83.
79.
60.
50.
17.
14.
14.
12.
6.6
6.5
4.8
3.5
COPPER
(SALTWATER)
Rank
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
FMAV
600.
526.
487.
412.
330.
268.
212.
160.
138.
136.
129.
120.
69.
66.
40.
39.
28.
DOT
(FRESHWATER)
Rank
20
19
18
17
16
15
14
13
12
11
10
9
a
7
6
5
4
3
2
1
FMAV
1230.
92.
67.
54.
36.
33.
32.
25.
19.
17.5
10.
7.0
4.1
4.0
3.2
2.4
2.3
1.7
1.6
1.3
CHROMIUM(VI)
(SALTWATER)
Rank
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
FMAV
105000.
93000.
91000.
57000.
32000.
30500.
22000.
17200.
1 5000 .
10000.
7500.
6600.
6300.
4300.
3650.
2970.
2490.
Zl
NC
(SALTWATER)
Rank
20
19
18
17
16
15
14
13
12
It
10
9
8
7
6
5
4
3
2
1
FMAV
70600.
50000.
39000.
9460.
6330.
6330.
4090.
3640.
3380.
2440.
2160.
1 780.
1450.
1000.
543.
525.
400.
321.
310.
166.
0 1 ELDR 1 N
(SALTWATER)
. Rank
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
FMAV
34.0
31.2
23.0
19.7
13.0
16.7
14.2
7.6
7.0
6.0
5.0
4.5
2.3
1.5
0.9
0.7
-------
TABLE 2. Continued
ENDRIN
(SALTWATER)
Ran*
16
15
14
13
12
1 1
10
9
a
7
6
5
4
3
2
1
FMAV
14.2
12.
3.1
1.7
1.1
1.1
0.63
0.6
0.47
0.3
0.29
0.1
0.094
0.05
0.048
0.037
ALDRIN
(FRESHWATER)
Rank
14
13
12
11
to
9
a
7
6
5
4
3
2
1
FMAV
9650.
180.
143.
50.
27.5
27.
21.
20.
16.
Jo-
lt.
9.
3.
7.4
HEPTACHLOR
(SALTWATER)
Rank
16
15
14
13
12
1 1
10
9
8
7
6
5
4
3
2
t
DOT
FMAV
194.
188.
112.
55.
21.5
10.
a.
3.92
3.77
3.4
3.
3.
1.5
0.86
0.8
0.057
LINDANE
(SALTWATER)
Rank
16
15
14
13
12
1 1
10
9
8
7
6
5
4
3
2
1
FMAV
3680.
450.
66.0
56.0
55.9
47.
35.0
30.6
14.0
9.0
7.3
6.66
6.28
5.0
5.0
0.17
NICKEL
(SALTWATER)
Rank
14
13
12
11
10
9
a
7
6
J
4
3
2
1
FMAV
39.
7.9
7.0
6.0
4.0
2.0
1.6
1.4
0.87
0.&8
0.6
0.53
0.4
0.14
(SALTWATER)
Rank
14
13
12
11
10
9
a
7
6
;
4
3
2
1
FMAV
350000.
320000.
150000.
47000.
35000.
1 7000.
9670.
7960.
6360.
2080,
1180.
600.
366.
310.
NICKEL
(FRESHWATER)
Rank
16
15
14
13
12
11
10
9
a
7
6
5
4
3
2
1
FMAV
2230.
2030.
1540.
1080.
730.
720.
665.
627.
609.
457.
446.
440.
401 .
345.
234.
65.1
CHROMIUM< 111)
(FRESHWATER)
Rank
13
12
11
10
9
8
7
6
5
4
3
2
1
FMAV
885.
633.
224.
224.
211.
207.
153.
138.
136.
132.
123.
47.
33.4
ZINC
(FRESHWATER)
RanK
15
14
15
12
11
10
9
8
7
6
5
4
3
2
I
FH*¥
2260.
1019.
716.
708.
531 .
463.
315.
251 .
213.
161 .
136.
92.8
48.8
42.0
13.7
ALORIN
(SALTWATER)
Rank
13
12
11
10
9
8
7
6
5
4
3
2
1
FMAV
100.0
36.0
33.0
33.0
25.0
13.0
12.0
9.8
8.0
7.2
5.0
5.0
5.7
10
-------
'ABLE 2. Continued
TOXAPHENE
(SALTWATER)
Rank
13
12
11
10
9
8
7
6
5
4
3
2
1
FMAV
1 120.
824.
43.8
16.
9.6
3.2
4.5
4.4
1.4
1.1
1.1
0.5
0.11
HEPTACMLCfi
(FRESHWATER)
Rank
10
9
3
7
6
5
4
3
2
1
FMAV
180.
148.
58.6
37.0
29.5
24.8
7.8
2.8
1,8
1.0
SILVER
/ ^ » 1 Tt J
\ J/-IU ( *
Rank
10
9
8
7
6
5
4
3
2
1
A -ren %
FMAV
1400.
550.
500.
250.
210.
36.
33.
21.
20.
4.7
DIELDRIN
(FRESHWATER)
RanK
12
11
10
9
8
7
6
5
4
3
2
1
FMAV
740.
593,
191.
39.
30.
24.
20.
11.
8.
5.5
5.0
4.5
CHROMIUM(VI >
(FRESHWATER)
Rank
10
9
3
7
6
5
4
3
2
1
S
FMAV
1 62000.
71900.
63800.
59900.
30400.
30000.
25000.
6400.
4600.
67.
ILVER
K r Rc.ormn i en f
Rank
9
8
7
6
5
4
3
2
1
FMAV
5.77
5.52
4.11
0.112
0.0230
0.015
0.013
0.0123
0.00192
TOXAPHENE
(FRESHWATER)
Rank
12
11
10
9
3
7
6
5
4
3
2
1
FMAV
180.
28.
21.
20.
13.
12.0
8.0
5.8
4.7
3.5
2.6
1.3
SELENIUM
(FRESHWATER)
RanK
10
9
8
7
6
5
4
3
2
1
FMAV
42400.
28500.
24100.
13600.
12600.
9580.
6500.
6170.
1660.
340.
MERCURY
( FRESHWATER )
Rank
9
a
7
6
5
4
3
2
1
FMAV
2000.
2000.
2000.
1000.
784.
244.
32.
10.
5,
SELEN 1 UM
(SALTWATER)
RanK
12
11
10
9
8
7
6
5
4
3
2
1
FMAV
17348.
14651.
9725.
7400.
4600.
4400.
3497.
1200.
1180.
1040.
600.
599.
LINOANE
(FRESHWATER)
RanK
10
9
8
7
6
5
4
3
2
1
FMAV
532.
207.
138.
94.8
68.
53.1
52.9
22.4
22.
10.
ENOOSULFAN
(FRESHWATER)
Rank
9
8
7
6
5
4
3
2
1
FMAV
261.
38.
5.9
3.8
3.7
3.2
2.3
0.83
0.34
ENOOSULFAN
(SALTWATER)
RanK FHAV
11 730.
10 157.
9 3.16
3 0.33
7 0.76
6 0.38
5 0.30
4 0.14
3 0.10
2 0.09
1 0.04
CYANIDE
(FRESHWATER)
Rank FMAV
10 2326.
9 2240.
8 431 .
7 306.
6 199.
5 167.
4 125.
3 118.
2 83.
1 77.
ARSENIC
(FRESHWATER)
Rank FMAV
8 41760
7 29130.
6 22040.
5 20190.
4 13096,
3 1 4 1 30 .
2 1794.
1 379.
11
-------
TABLE 2. Continued
CHLOROANE
(FRESHWATER)
Rank
8
7
6
5
4
3
2
1
FMAV
190.
39.
58.
44.
32.
21.
15.
6.3
CHLOROANE
(SALTWATER)
Rank
3
7
6
5
4
3
2
!
FMAV
120.
17.5
16.9
11.8
6.4
6.2
4.8
0.4
a Calculated from the Sp«c!«s M§an Acuta Values In Table 1. All FMAVs ar« In \ig/L.
12
-------
CONCLUSIONS
1. Calculation of a FAV from a typically small set of MAVs requires chat che
set be considered a sample from a statistical population and chat che FAV
be considered an estimate of the fifth percentile of that population.
2, The set of MAVs must be assumed to have been obtained from the statistical
population by a specific sampling procedure; of reasonably simple sampling
procedures, an assumption of random sampling appears most consistent with
actual data selection,
3. Available sets of MAVs suggest that the statistical populations are highly
positively skewed and that estimation would be benefitted by logarithmic
transformation of MAVs.
4. Available sets of ln(HAV)s suggest that the statistical populations are
significantly and variably skewed and that FAV calculation should be based
on a subset of ln(MAV)s nearest the fifth percentile.
5. Available sets of ln(MAV)s suggest that FAV estimation is better served by
the assumption of a triangular distribution of the populations of ln(MAV)s
than by the assumption of a normal, rectangular, or biexponential
distribution.
6. Simulations using a triangular distribution indicate that 'parametric'
methods for percentile estimation and "graphical1 methods in which ranked
data are assigned cumulative probabilities PR*P(E(XR)) produce
undesired biases in the true cumulative probabilities corresponding to
fifth percentile estimates.
7. Simulations also indicate that a graphical method with (a) ranked data,
XR=»ln(MAV), assigned cumulative probabilities PR"R/(N+l), (b) ?g
transformed to its corresponding standard variate, ZR"/PR, and
13
-------
(c) a line fitted to % versus XR by the geometric mean functional
relationship produces the least bias among alternatives examined.
8. These simulations also suggest that it is appropriate to restrict the
calculation procedure to the four X^s with P^s nearest 0,05, because
(a) in the absence of skewness the precision of fifth percentile estimates
is little worsened by this and (b) in the presence of skewness this avoids
the introduction of substantial bias.
9. The old procedure used in the 11/28/80 Guidelines has some aspects which
are contraindicated either theoretically or empirically and the new
procedure described here should replace it; however, FAVs calculated from
example data sets by the two procedures usually do not differ by more than
a factor of 2.
10. Modifications of the recommended new procedure with respect to assumed
distribution, general percentile estimation method, and subset size (up to
half the set size) rarely cause FAVs calculated from example data sets to
vary by more than a factor of 2. Therefore, even if it is debatable
whether optimal decisions were made in developing the recommended new
procedure, it is unlikely that any alternative procedure, within reason,
would produce substantially different results.
II. Modification of the recommended new procedure to use the entire data set
often produced substantially different FAVs from example data sets, but
many of these FAVs were sufficiently at variance with the lowest MAVs in
the data sets to reject this modification.
12. Modification of the recommended new procedure to reflect an extreme
deviation from random sampling consistently produced higher FAVs from
example data sets, but the average increase was only about fifty percent;
14
-------
therefore, questions about the propriety of applying methods based on
random sampling to a system in which sampling is not strictly random
are probably not of great importance.
13. Recommendations about FAV calculation are the same whether HAVs are for
species or families.
15
-------
STATEMENT OF PROBLEM
A FAV is defined as an estimate of the concentration corresponding to the
fifth percentile of a suitable set of MAVs for a material; i.e., the FAV exceeds
five percent o£ the MAVs and is exceeded by ninety-five percent. Because the
number of species tested with any particular material is usually rather small,
most sets of MAVs will not have a datum which can reasonably be designated as
the fifth percentile; rather, the set of MAVs must be assumed to be a sample of
a population (in the statistical sense) that is large enough that a fifth
percentile is defined. For example, if resources permitted, the MAVs of a
material for many hundreds of aquatic taxa could be determined. Such a set of
MAVs could reasonably be considered to tvave a fifth percent lie that could be
obtained by inspection and is the type of statistical population of hfaich the
available sets of MAVs are assumed to be samples. Of course, the population of
MAVs would need to be determined using a mix of taxa that is acceptable to
toxicologists for calculating a FAV. The above assumption is inherent to the
definition of the FAV and any objection to it, or modification of it, was not a
subject of this project.
An additional assumption is necessary because any estimation method using a
sample from a population requires that the manner in which the sample was
obtained be adequately specified. The toxicity of a material is measured by
many independent investigators, who select test species based on a poorly
defined combination of tradition, convenience, happenstance, and intent to
diversify the mix of species. All available data meeting certain quality
standards (I) are incorporated into the aets of MAVs. This incorporation step
does not affect the nature of the sampling process, except that a FAV will not
be calculated unless the set of MAVs is of a minimum size (eight) and contains
16
-------
representatives of certain categories of species (1). It is not possible to
represent this process in a form suitable for applying appropriate exact
estimation methods. The issue then becomes what feasible description of
sampling (e.g., random, systematic) most closely approximates this process,
Random sampling was selected for the following reasons:
(1) Although they meet certain minimal diversity standards, the available sets
of MAVs vary markedly, and apparently haphazardly, in the species and
higher taxonomic levels they contain. Such variation is not compatible
vith entirely systematic sampling schemes and suggests that an appropriate
sampling assumption should contain a strong random element.
(2) Even where some elements of systematic sampling are evident in the
available sets of MAVs, a high correlation of toxicity with these elements
is usually not apparent. Without such a correlation, an assumption of
systematic sampling is not particularly needed because for practical pur-
poses it can be approximated by an assumption of random sampling,
In addition to being as much, or more, in accord with actual sampling
procedures than other tractable sampling assumptions, the assumption of random
sapling nay b* justified, in part, by noting chat, in general, deviations from
this assumption may occur without seriously compromising results. Methods based
on random sapling do not lose all ch*ir utility if it is not possible to
rigorously define * population and to formally conduct random sampling from it.
The population may even be somewhat hypothetical, being defined, in part by the
data selection process. Sampling may be nonrandom, but as long as the sampling
process has a low enough correlation with response, results under an assumption
of random sampling will not deviate by more than a certain amount from results
17
-------
under more appropriate assumptions. Consideration will be given below to what
errors would be introduced if random sampling were assumed for percentile
estimation when sampling is actually nonrandora.
Finally, if a procedure adopted for calculating FAVs results in criteria
chat are somehow independently validated, the procedure can be considered to be
entirely empirical and the assumptions become part of the definition of che FAV
needed to produce the desired criteria. This, however, is speculative and the
question remains as to whether the procedure developed here employs the most
appropriate assumptions and, if not, whether this has any substantial impact on
FAVs.
18
-------
DESCRIPTION OF PERCENTILE ESTIMATION METHODS
Method* for eseimacing, from random samples, a specified percentile of a
population can generally be placed into one of two categories. These categories
are presented here primarily to facilitate discussion and are not meant to imply
that methods in different categories do not have some important coramon features
or are not sometimes nearly equivalent. One notable feature of any method for
estimating percentiles is the need to make at least some distributional
assumptions about the population from which the sample was drawn.
For methods in the first category, the parameters in the general
mathematical equation for the assumed distribution are estimated from the sample
by mathematical procedures formulated to-produce estimates with desired
properties, such as being unbiased, having minimum variance, or having maximum
likelihood. The common formulas for estimation of mean and variance from a
sample from a normal population is an example of such a method. Once the
parameters are estimated, it is a simple matter to substitute then into the
general equation for the distribution and to estimate a desired percentile.
This category will be referred to as 'parametric methods'.
The second category of methods involves ranking the data in a sample and
then plotting the ranked data (X^) versus a cumulative probability (PR)
assigned to each rank (R). For calculation simplicity, plotting is usually on a
coordinate system for .which the cumulative form of the assumed distribution is a
straight line. A line is fitted to the plotted data by eye or by some
appropriate mathematical curve-fitting technique and the estimate of the desired
percentile is read off the plotted line or computed from the equation for the
line. This category will be referred to as 'graphical methods', although
explicit graphing is never strictly necessary. In most cases, graphical methods
19
-------
are not mathematically rigorous and do not produce the unbiased, maximum
likelihood, or minimum variance estimates that parametric methods are designed
co produce. This does not mean, however, that graphical methods will not
perform adequately in practice; in face, in some cases their performance is very
similar to that of parametric methods. Furthermore, for some cases suitable
parametric methods do not exist or are unreasonably cumbersome; graphical
methods thus might be a very useful alternative.
For both parametric and graphical methods, discussion here will be
restricted to a class of distributions which have only two parameters, these
being a location parameter and a scale parameter. By this it is meant that, for
each distribution type, there exists a standard distribution with standard
variate denoted 'z', such that for any distribution of this type wich variate
denoted 'x' there exists a location parameter 'L1 and a scale parameter 'S1 such
that x"L+S"z. For example, for the normal distribution, the mean and standard
deviation are usually used as the location and scale parameters, respectively,
and the standard normal variate (also called the "standard normal deviate' (4))
is then as usually tabulated.
Paratastric Method*
The only parametric method considered here will be a general one, termed
'best linear unbiased estimation' (5,6), which can be applied to any distribu-
tion characterized by location and scale parameters. This method is 'unbiased1
in that the parameter estimates will, on the average, equal the true population
parameter values. It is 'linear' in that the parameter estimates are linear
functions of the data. It is 'best1 in that the parameter estimates have the
lowest variances of all linear unbiased techniques. There may be nonlinear or
biased methods that have smaller variances, but, in general, the performance,
20
-------
with respect to bias and variance, of this technique cannot be much improved,
Parameter estimates (L, S) are obtained by minimizing the value of the matrix
expression:
, ^ /N , T - 1 , /^ A x
( c - L - S-)T'V K(c - L - S-z
where: N is the sample size;
jc is a (Nxl) matrix consisting of the ranked sample;
j^ is a (Nxl) matrix of the expected values of ranked standard variates
of random samples of size N from the assumed distribution;
V_ is the (NxN) varianee/eovariance matrix for ranked standard variates
of random samples of size N from the assumed distribution; and
T denotes matrix transposition.
/\ A A
This method has the additional advantage that Xp"L*S-zp is also the best
linear unbiased estimate for x_, the pc" percentile of the population.
This can be demonstrated in a variety of ways, but is mo»t obvious when it is
realized that any particular percentile could, quite legitimately, be designated
the location parameter.
This method also has the advantage that it can be applied to an arbitrary
sub sample of the data and still produce the best linear unbiased estimate that
can be obtained from that subsaaple. Applying the method to a subsample simply
requires eliminating, from matrices JK, _z_, and V^" , the elements referring to
data not in the desired sub sample. (Note: The calculation and inversion of V_ is
not affected by these deletions; rather, deletions are made after inversion.) A
notable property of such 'censoring1 of data is that, if the remaining data are
those nearest the percentiie of interest, the variance of the percentile
estimate is little worsened as the number of data used is reduced. This
suggests that using all the data, other than to determine ranks and define V_,
has relatively little utility in this kind of estimation.
The ability of this method to use only a subsample of the data has
particular significance when the distribution of the population from which the
21
-------
data are drawn ii not perfectly characterized. For example, when concerned with
che fifth percentile, deviations from the assumed distribution that are
restricted to the upper part of the distribution will impact the calculations
little if only the lowest few data in the sample are used. Even if the
distributional assumption is violated near the fifth percentile, the impact o£
this violation will be reduced as the number of data formally used in the above
equations ii reduced, as long as the data used are those nearest the percentile
of interest. Of course, the method still makes distributional assumptions about
both the data used and those not used and errors will arise if these assumptions
are incorrect, but as long as the distributional assumptions are not grossly
violated in the range of the selected subsample, theae errors will generally be
minimal. The question then arises as to the optimal sub sample size (n) , a small
size having the advantage of reducing the effects of deviation from the assumed
distribution and a large size having the advantage of reducing the variance of
estimates when the distributional assumptions are correct. The answer to this
question is specific to the problem of concern and will be considered below.
One troubling aspect of this methodology is, ironically, its lack of bias.
This is a problem because the lack of bias is in the variate rather than in the
cumulative probability; i.e., in repeated sampling, x_ will average x_, but
the true cumulative probability (P(xp)) corresponding to x_ will not average
p, unless cumulative probability is linearly related to variate, which ia only
true for rectangular distributions (simulated sampling from a variety of popula-
tions is presented below to demonstrate this point). Because che definition of
the PAV is based on protecting a certain percentage of a specified taxon, this
method is inappropriate; rather, a method that is unbiased with respect to the
22
-------
desired cumulative probability is desired. We art aware of no published ueehods
of this sort. It ii for this reason that graphic*! methods are now considered.
Graphical Methods
Graphical methods for examining cumulative distributions inherently have four
isaucs that must be resolved:
(I) Cuaulative Probabilities Assigned to Ranked Data
Formulae reported (4,7,8) for calculating the cumulative probability P^ to
assign to a datura XR with rank R in a sample of size N include R/*f,
-------
(d) ?a"P(S(%)) has, by definition, obvious theoretical foundations because
it denotes the cumulative probability corresponding to cha expected value
of XR. (S(XR) ii 4iso called 'rankit' (4)). this formula, is a
counterpart co PR"R/(H+l), differing by being based on the expected value
of ranked data rather than the true cumulative probabilities corresponding
co ranked data. Unlike PR*R/(H*l), its values are distribution-dependent,
Its use will also b« further explored below, but because it is based on the
expected value of the variate, it is anticipated that it will show the same
prob lea of bias as the parametric method above.
(2) Transformation of Axes
Thisis dictatedby the assumed distribution and by the reecriction adopted
her* chat che assumed distribution should produce a linear plot on the
selected axea. In general, it is the axis against which cumulative
probabilities are plotted that is transformed and the transformation is based
on the standard distribution of the assumed distribution; in fact, this can be
treated as a transform of PR to a corresponding standard variate ZR. In
such a case, the plot becomes one of a ZR assigned to etch rank versus the
observed datum X^. The slope dX/dZ is the scale parameter and the intercept
on the X axis is the location parameter.
(3) Subeaaple Site
This issue is identical to that discussed for the parametric method. A
later section vill consider how the sub sample size (n) can beec be determined,
baaed on simulations under various assumptions.
(4) Fitting a Line to PlottedData
Because the restriction of a linear plot has already been made, this issue
reduces to how to compute the slope of the line nose appropriate to the data.
Because, for any N, che 2R or PR assigned to a ranked datum is fixed, and
thus nay be an analogy to an independent variable, and because the line to be
fitted can be expressed as Xgl+S'Zg, it may be thought that the standard
I ease-squares regression formula with XR as the dependent variable and ZR
as the independent variable would b« chs preferred choice. As will be seen
below, this turns out to be the case when ?n"P(E(XR)) is used and when
ipi rather JChan P(Yp), is desired to be unbiased. Aa before, achieving an
unbiased P(Yp) is not amenable to exact techniques and an empirical approach
muse be used. To this end, three different, but simple, slope formulas were
considered (again, this approach is strictly empirical, employing these
formulas ae representing a range within which a reasonable slope night lie;
nothing is implied here about a theoretical justification for one or che other
formula and it is not implied chat this application meets the assumptions on
which any of the formulas are based):
(a) L3-X - standard bivariaee least-squares with Xg as che dependent variable
(residuals minimized in X-direccion):
24
-------
(b) LS-Z - standard bivariate least-squares with ZR as Che dependent variable
(residual* minimized in Z-direction):
a -
(c) GMFR - geofoetric mein functional relationship (residuals are minimized in
th« direction of ch« arithmetic reciprocal of the §lop«; this method
produces the geometric meen of the slopes by the two previous methods and
has seen some application in regression where both variables are in error
(9,10)):
•\
Whatever slope formula is used, the line always passe* throufh the
mean X& and the mean ZR. The location parameter estimate U, which is the
intercept on the X axis, is therefore
A
L
25
-------
SELECTION OF DISTRIBUTION
All methods for estimating the fifth percentile of a population from a
sample require at least some assumptions about the distributional
characteristics of the population. Few data sets from which an FAV will be
calculated will be large enough that such characteristics can be inferred from
the individual data set. However, the large number of sets available (Tables I
and 2) provides an opportunity for evaluating these characteristics and for
determining which characteristics can be reasonably applied to all data sets and
which parameters must be estimated individually from each set.
It is desirable to keep the number of unknown distributional parameters as
low as possible, not only because analysis becomes markedly more complicated as
the number of parameters increases, but also because data sets of the minimum
size (H»8) may be overly fitted if the number of parameters is not small. The
example data sets (Tables I and 2) vary widely in their means and coefficients
of variation. Therefore, at least two parameters, a location parameter (e.g.,
mean) and a scale parameter (e.g., standard deviation), are required.
Because these two parameters relate to the first and second moments of the
samples, an obvious third parameter to consider is skewness, which is related co
the third moment of the samples. Skewness is also strongly indicated by
inspection of the example data sets. A skewness measure (4), the normalized
third central moment, Was estimated for each example data set. All sets showed
positive skewness. The skewness was substantial enough to reject, at Che 0.10
level of significance, the hypothesis that the set was a random sample from a
normally distributed population for 35 of 3? SMAV sets and for 34 of 37 FMAV
sets; at the 0.01 level of significance, this hypothesis was rejected for 30
SMAV sets and 25 FMAV sets.
26
-------
Because of this strong positive skewness, a logarithmic (base e)
cran8formation was applied to each MAV, so that discussion will now relate to
che distribution of In(MAV). The skewness measure for each data set was
recomputed and the average skewness decreased from 2,39 for SMAVs and 2.08 for
FHAVs to 0.06 for SMAVs and 0.07 for FMAVs.
The small average skewness does not, however, mean that individual sets can
be considered to be samples from nonskewed populations. When the skewness
measures of individual data sets were tested under the same null hypothesis as
above, the hypothesis was rejected at the 0.10 significance level for 8 SMAV
sets and 7 FMAV seta and at the 0.01 signficance level for 3 of the SMAV sets
and 2 of the FMAV sets. Although this is substantially fewer than before
logarithmic transformation, it still indicates that skewness in some sets might
be too large to ignore. Furthermore, among the sets with significant skewness,
the skewness was sometimes positive and sometimes negative, indicating that the
populations these sets represent vary substantially in skewness.
Therefore, despite logarithmic transformation, skewness in the data must
still be dealt with by the methodology adopted for the estimation of the fifth
percentile. Two general approaches were considered for this. Firse.
distributions with a third parameter that affects skewness and which can be
estimated from a sample could be used. This approach greatly increases the
difficulty of parameter estimation and it is questionable whether the smaller
data sets reliably have enough information to make this effort appropriate or
worthwhile. The second approach is to limit the estimation method to a subset
of the data near the percentile of interest. By doing this, the effects of
having non-zero skewneas are markedly reduced and the location and scale
parameter estimates apply only locally, incorporating the effects of skewness at
27
-------
chat locality. This approach allows Che use of relacively simple estimation
methods and, as will be further discussed below, has very little impact on the
precision of fifth percentile estimates even if a population is not skewed. The
second approach will therefore be employed here,
Higher moments of the data sets were not directly examined because (a) the
decision to limit analysis to a subset of the data makes such an examination
complicated and (b) the effects of higher moments should be adequately accounted
for either by this limitation or by the examination of specific distributions
that follows,
Inference of distributional characteristics from the example data sets was
therefore limited to symmetric distributions with just location and scale
parameters to be estimated; furthermore, the most relevant information in the
data sets is that nearest the fifth percentile. The approach followed here was
to examine the fit of specific distributions to the example data sets. Four
distributions were considered:
(D R « c t an gu 1 a r Pi s t r i b u t ion
This was included as an extreme case because it assumes that the
relative frequency of ln(MAV)g remains constant between some lower and upper
limits, whereas theoretical considerations and inspection of the data sets
suggest that the frequency declines as the lower and upper lisits are
approached (i.e., very sensitive and very resistant taxa are rarer than
those with moderate sensitivity). The standard probability density function
for this distribution is:
.' f(z) - 1//TT ; -/T< z <
f ( z) - 0 ; z < -JT, z >
(2 ) Triangular Distribution
This was included because it is the simplest distribution that incor-
porates two basic properties that the frequency of ln(MAV)s should have: (a)
sensitivity should have lower and upper limits (no species succumbs to
infinitesimal concentrations of a material and no species tolerates infinite
concentrations) and (b) the frequency of ln(MAV)s should decline to zero as
28
-------
Che Haiti are approached (fewer species are near the limits than are near
the raidrange). The scandard probabiLicy density function for this distribu-
cion is:
f(z) - (1- z|)//6 ; -/6 < z < /6
f(z) - 0 ; z < -/6 , z > /6
( 3 ) Normal Distribution
This was included due Co its broad applicability and Co provide a
curved alcernacivt to the linear frequency trend of the triangular
d iscribucion ; chis curvicure causes relatively rare sensitive or resistant
taxa Co have somewhat more extreme ln(HAV)i (relative co the range of the
majority of the caxa with moderate sensitivity) than does the triangular
distribution. No lover or upper limits exist, but the frequency becomes so
small at reasonably moderate deviations from the mean chat Chis deficiency
is probably of limited consequence. The scandard probability density
function for chis distribution is:
f(z) - — e~z /2 ; -- < z < +-
(4) Biexponenti al Distribution
This was included as an extreme case in which the most sensitive and
resistanc taxa have greatly different ln(MAV)s than the majority of the taxa
with moderate sensicivicy. The scandard probability density function for
this distribution is:
f(2) .-L-.-^M . .„<,<+.
/2
Shapes of these distributions when chey have mean * 0 and standard deviacion « I
are displayed on Figure 1.
The best linear unbiased estimation method* discussed earlier was used to
estimate location and seal* parameters from each example data sec for each
combination of the four distributions above and four subset sizes (n*4, N/4,
N/2, and N, where N-daca set size; n also was required co be ac least 4, which
The matrix V_ for this method was calculated by exact integrals for all
distributions except normal, for which approximate formulas were used (11)
29
-------
1. Standard probability cUn«lty plot* for rectangular C
triangular (•-•-•), noraal ( ), and bIexponential (
distribution*.
O
0.60-
m
«0.40-
.>-V
„%,
-O
D
_Q
o
L.0.20
Q-
n n
—
h- S
/'
'*'*'
**< ''
mmA*****\' 1
/•': \x>
« .* \ *.
^ * ' \
f* * *. .
/f •' "' v*
x"/ / *• v\
.• t : : \x.
• / •' '-. ^ v.
/ / '*. v^
•*"
.
i 1 i 1 . 1 .
*
\
>• \
"• v,*\
* **^»x
| :X*T4rf'--«Li_.,..r
-3.0
-2.0 -1.0 0.0 1.0 2.0
Number of Standard Deviations from Mean
3.0
-------
was considered a ainimui number to use co test distribution fits), Frcra each
such estimation, Che expected value (E(X^)) of each ranked datum was estimated
as L+S '£(£3) i 'rfhere L is che location parameter estimate, S is the scale
parameter esciaata, and E(Z^) is ch* expected value of the datum of rank R in
snples of size N fcoa ch* asinawi standard discribucion. th« raeio:
(£.•., th« fraction of ch* varianct of ch* subiae not explained by fitting the
data co ch* distribution) was adopt wi as a o*aiur* of good n*aa-of- fit of ch* data
CO th* aasua*d distribution ov*r ch* sis* (R) of th« subs*C ua«d. Avtraf*
good n*ii-of- fits for all SMAV s*cs ar* r*port*d in Tabl* 3 and for FMAV sats ia
Table 4.
Th* criangular distribution was j«L*et*
-------
TABLE 3. AVERAGE GOODNESS-OP-FITS3 FOR In(SMAV) DATA SETS
ASSUMED DISTRIBUTION
RECTANGULAR
TRIANGULAR
NORMAL
BIEXPONENTIAL
a "Goodness-of-f It" Is the
not explained by fitting
N
0.169
0.082
0.081
O.L06
fraction
SUBSET SIZE (n)
•M/2 N/4b >4
0.150 0.144 0.135
0.099 0.114 0.118
0.104 0.134 0.137
0.230 0.235 0.206
of variance of 'n1 data points
data to assumed distribution; lower values
indicate better fits; values should be compared only within columns.
b n =» 4 when N <_ 16.
TABLE 4. AVERAGE GOODN!SS-QF-FITSa
ASSUMED DISTRIBUTION
RECTANGULAR
TRIANGULAR
NORMAL'
BISXPONEOTIAL
N
0.145
0.095
0.087
0.109
FOR In(FMAV) DATA SETS.
SUBSET SIZE (n)
N/2 N/4b 4 •
0.127 0.121 0.125
0.103 0.108 0.116
0.120 0.129 0.143
0.277 0.218 0.238
a "Goodness-of-f1t" Is the fraction of variance of 'n' data points
not explained by fitting data to assumed distribution; lower values
indicate better fits; values should be compared only within columns.
b n - 4 when N < 16.
32
-------
SELECTION OF PERCENTILE ESTIMATION METHOD AND SUBSET SIZE
Ten thousand computer-generated random samples from a standard triangular
distribution (location parameter " mean * 0; scale parameter =• standard devia-
tion =• 1) were used to estimate the fifth percentile of the distribution for
each combination of the following methods, sample sizes, and subsample sizes.
(I) Percentile Estimation Methods
Seven methods were examined. These included one parametric method (best
linear unbiased estimation) and six graphical methods (all possible
combinations of the two formulas for assigning cumulative probabilities
(P(S(XR)), E(P(XR)) and the three formulas for computing slope (LS-X,
LS-Z, GMFR)).
(2) Sample Sizes
Sample sizes (N) of 8, 15, and 30 were selected as being representative of
the minimum size, a moderate size, and a large size that are found in the
available sets of SMAVs and FMAv"s .
(3) Subsample Sizes
Subsample sizes (n) of the 4, N/4, N/2, and N points closest to the fifth
percentiie were considered; for N/4, an additional restriction of n>4 was
imposed; n"4 was considered to be the minimum reasonable size, a lesser
number making analysis too sensitive to a spurious datum.
From location and scale parameter estimates, the estimate of the fifth
percentile was calculated aa xg*L-l.675*S, -1,675 being the fifth percentiie
for the standard triangular distribution. The average xg over the 10,000
simulations was designated as xg and should equal -1.675 for methods unbiased
with respect to the variate. Because the parameters of the population from
which the samples were drawn are known, the true cumulative probability P(xg)
corresponding to each xg was calculated. The average P(x5) over the 10,000
simulations was designated Pg and should equal 0.050 for methods unbiased with
respect to cumulative probability. "x"g is tabulated in Table 5 and Pg is
33
-------
tabulated in Table 6. Table 5 also includes the standard deviations for '£5 in
order to indicate the relative precision of the various methods.
As expected, the best linear unbiased estimation method did produce an
essentially unbiased loj, as did the graphical method using P(E(X^)) co
assign cumulative probability and LS-X to calculate slope (Table 5), However,
it is bias in P<$ that is of paramount concern here. The best linear unbiased
estimation method and all graphical methods using P(E(X^)) were substantially
more biased than the graphical methods using E(P(XR)) (Table 6) and were
therefore dropped from consideration. In addition, the standard deviations of
^5 by the best linear unbiased method were usually no better than 10% less
than those of the graphical methods using E(P(Xa)) (Table 5), indicating that
the better precision of the best linear unbiased method is of little
consequence.
Although they did have lower biases than the other methods, none of the
graphical methods using E(P(X^)) to assign cumulative probability had an
unbiased Pj and the bias varied with n and N (Table 6). Furthermore, none of
the formulas for calculating slope had the lowest bias for ail combinations of n
and N. The zeotnetric mean functional relationship was selected as having the
lowest average bias over all combinations.
Selection of Che most appropriate subsample size required consideration of
the precision of xj (Table 5) for the selected percentile estimation method
(graphical method using E(P(X^)) to assign cumulative probability and GMFR to
calculate slope). For N»8, the standard deviation of x
-------
TABLE 5- ME^3 AND STANDARD DEVIATIONS3 OF ESTIMATES OF FIFTH PERCENTILE (x5)
BY VARIOUS METHODS, FOR 10,000 SAMPLES FROM A STANDARD TRIANGULAR
DISTRIBUTION.
N n
PARAMETRIC
METHOD
-- r,
LS-X
84-1
(0
8 -1
(0
15 4 -1
(0
8 -1
(0
15 -1
(0
30 4 -1
(0
8 -1
(0
15 -1
(0
30 -1
(0
,68
.57)
.68
.52)
.67
.39)
.67
.38)
.67
.36)
.67
.25)
.67
.25)
.67
.25)
.67
.24)
-1
(0
-1
(0
-1
(0
-1
(0
-1
(0
-1
(0
-I
(0
-1
(0
-1
(0
.68
.57)
.68
.53)
.67
.39)
.67
.39)
.67
.38)
.67
.25)
.67
.26)
.67
.26)
.67
.26)
n-p(E(xR):
LS-Z
-1
(0
-1
(0
-1
(0
-1
(0
-1
(0
-1
(0
-1
(0
-1
(0
-1
(0
.79
.61)
.81
.55)
.73
.40)
.75
.40)
.76
.39)
.69
.25)
.71
.26)
.72
. 26)
.72
.26)
\— — «.
-GRAPHICAL-
/
GMFR
-1
(0
•™ 1
(0
-1
(0
-1
(0
-" 1
(0
— 1
(0
-1
(0
-1
/ rs
w
-1
(0
.73
•59)
.74
.54)
.70
.39)
.71
.39)
.71
.38)
.68
.25)
.69
.26)
.70
"i r \
.io;
.70
.26)
rF
LS-X
-1
(0
-1
(0
-1
(0
-1
(0
-1
(0
-1
(0
-1
(0
-1
f n
\v
-1
(0
.83
.62)
.83
.56)
.77
.41)
.77
.41)
.77
.39)
.73
.26)
.73
.26)
.73
.27)
.73
.27)
l=E(P(XR))
LS-Z
-1.95
(0.67)
-1.98
(0.58)
-1.84
(0.44)
-1.85
(0.43)
-1.86
(0.40)
-1.75
(0.26)
-1.77
(0.27)
-1.77
(0.27)
-1.77
(0.27)
GMFR
-1.89
(0.64)
-1.90
(0.57)
-1.80
(0.42)
-1.81
(0.42)
-1.81
(0.39)
-1.74
(0.26)
-1.75
(0.27)
-1.75
t A 1 1 \
\\j-t-i /
-1.75
(0.27)
Standard deviations of estimates in parentheses.
35
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TABLE 6. MEAN TRUE CUMULATIVE PROBABILITIES (P ) OF ESTIMATES OF FIFTH
PERCENTILE (P(x5)) BY VARIOUS METHODS, FOR 10,000 SAMPLES
FROM A STANDARD TRIANGULAR DISTRIBUTION.
M n METHOD
PARAMETRIC GRAPHICAL
PR-P(E(XR)) PR=E(P(XR))
LS-X LS-Z GMFR LS-X LS-Z GMFR
8 4
8
15 4
8
15
30 4
8
15
30
0.076
0.072
0.063
0.062
0.061
0.055
0.055
0.055
0.055
0.076
0.073
0.063
0.063
0.062
0.056
0.056
0.056
0.056
0.066
0.058
0.057
0.054
0.052
0.054
0.051
0.051
0.051
0.071
0.065
0.060
- 0.058
0.057
0.055
0.053
0.053
0.053
0.063
0.057
0.053
0.053
0.052
0.048
0.049
0.050
0.050
0.054
0.044
0.047
0.044
0.042
0.046
0.044
0.044
0.044
0.058
0.050
0.050
0.049
0.048
0.047
0.047
0.047
0.047
36
-------
lowest at n*4 and did not vary among Che n by more Chan 3%, There is thus no
substantial, advantage, with respect to precision, to using n>4.
Another factor in the selection of the subsample size is the possibility of
skewness. Therefore, simulations were also conducted using a skewed triangular
distribution for which the mode was 20%, rather than 50%, of the distance from
the lower to the upper limit. For all sample sizes, using n*N resulted in
?5<0.02 and, for N-15 and N-30, using n-N/2 resulted in Pj being about 0.03,
Using n"4, however, resulted in ?5 being between 0.04 and 0.05 for all N.
Using n*4 also resulted in substantially lower standard deviations for X5 than
using n*N/2 and n*N,
Because for a nonskewed population the smallest subsample size considered
performed little, if any, worse than larger subsamples, and because the
possibility and consequences of skewness give good reason to restrict the
subsample size, a subsample size n™4 is recommended . Limited consideration was
given to a smaller subsaraple (n™2), but for N*8 this resulted in substantial
bias in Pg (Pg^T.OZ) and in a 202 increase in the standard deviation of
*5 ; also, the use of so few data markedly increases the sensitivity of results
to an occasional unusually low datum.
Consideration was also given to the possible effects of nonrandonj sampling
by determining the bias introduced into Pg if the method recommended above is
used when samples are not obtained randomly. Two nonrandom sampling schemes
ware investigated. First, samples were taken in an entirely systematic fashion
highly correlated with variate, data being uniformly distributed over percen-
tiles with the ic^ datum (Xj_) being set equal to x_. , where p^*(i-0. 5)/N.
With such a scheme, PCxg) equals 0.015, 0.024, and 0.034 for N equal to 8, 15,
37
-------
and 30, respectively. Although PCx;) is therefore substantially biased, this
is an extremely unrealistic depiction of the sampling and the_ biases are extreme
upper limits. Sampling schemes with a more realistic systematic component would
result in much lower biases.
The second nonrandotn sampling scheme used was stratified sampling in which
each member of the sample was assumed to be randomly sampled from a restricted
percentile range, The range for the ith datum of a sample was [(IOOS)(i-Q.5)/N]
* 25%; i.e., a fifty percentile range centered on the value used for the ic^
datum of the systematic sample discussed above. For low and high i, the range
was compressed so that percentiies were maintained between 0 and 100 and so
that, over the entire sample, each member of the population had an equal chance
of being drawn. Specifically, for low i, if a percentile was computed by the
above formula to be <0, its absolute value was used. This results in the ranges
for low i being narrower than the nominal 50Z (as small as 27Z for i*i and N»3Q)
and sampling within the ranges being somewhat skewed to low percentiies;
analogous compression and skewing occurred for high i. Because the recommended
procedure would heavily employ data with low i, the systematic component of this
procedure is therefore even greater than implied by the restriction of sampling
to fifty percentile ranges. Using this sampling scheme, ten-thousand samples of
size 8, 15, and 30 from a standard triangular distribution were computer
generated and FAVs were calculated from each sample by the procedure recommended
above based on random sampling. ?5 was 0.040, 0.042, and 0.044 for N equal to
8, 15, and 30, respectively. This small bias suggests that, even with a strong
systematic element in the actual sampling, an assumption of random sampling
performs well as long as there is also a substantial random element in the
sampling, or at least an element that is not correlated with variate.
38
-------
APPLICATION OF RECOMMENDED PROCEDURE AND ALTERNATIVES TO DATA SETS
In che previous sections all issues necessary for Che recommendacion of a
procedure for FAV calculation have been considered. The recommended new
procedure assumes the sec of ln(MAV)s is a random sample from a triangular
distribution with unknown location and scale parameters. The mechanics of che
procedure can be summarized as follows:
The ln(MAV)s are ranked and each assigned a cumulative probability
PR*R/(N+I), where R is the rankAand N the number of data in the aet. A
line of the form ln(MAV)«¥s/Pji+L is fit to th« four points with PR
nearest 0.05. (The square root of PR constitutes transformation to the
variate of a standard triangular distribution somewhat different than, but
equally valid to, that used above; it is used here because it is simpler to
calculate when PR<0.5). S, L, and FAV are calculated as follows:
i -
\
2/4
£ -
InFAV - Sno.05 + £
Example calculations ar* provided in Appendix I.
This procedure was applied to che example data sets in Tables I and 2. The FAVs
thus calculated for each data set are included in Tables 7 and 8, along with the
lov*st MAV as a reference.
Also included in Tablet 7 and 8 are PAVs calculated for each data set using
che old procedure presented in che November 28, 1980 version of the Guidelines
(1), This procedure can be described as follows:
The ln(MAV)s are ranked and assigned to fixed intervals with width - 0.25
and with the first interval starting at the low«st In(HAV). Each interval
is assigned a cumulative proportion P*Ro«x/N, where N is che number of
ln
-------
TABLE 7. FAVs CALCULATED FROM SMAV DATA SETS Bt OLD PROCEDURE, RECOMMENDED NEW PROCEDURE, AND VARIOUS MODIFICATIONS
OF RECOMMENDED NEW PROCEDURE,
MATERIAL
COPPER
DOT
CADMIUM
CADMIUM
TOXAPHENE
ZINC
ENORIN
MERCURY
ZINC
LINOANE
COPPER
NICKEL
DIELORIN
0 ALORIN
ENDRIN
HEPTACHLOR
DIELORIN
L1NDANE
CHROMIUM! VI)
CHROMIUMU II )
HEPTACHLOR
NICKEL
DOT
ALDRIN
CYANIDE
TOXAPHENE
CHROMIUM
-------
Table 7. Continued
MATERIAL
ARSENlCd II)
MERCURY
SILVER
SILVER
ENOOSULFAN
CHLQRDANE
WATER
FRESH
FRESH
FRESH
SALT
FRESH
SALT
N
12
11
10
10
10
8
LOWEST
SHAV
810
5.0
0.0019
4.7
0.340
0.400
OLD
PROCEDURE
440
3.7
0.0014
3.3
0.218
0.090
NEW
PROCEDURE
340
2,6
0.0014
3.3
0.183
0.200
r»=N/2
260
1.6
0.0017
3.9
0.214
0.200
n=N
620
2.9
0.0003»
2.8
0.1 25»
0.378
NATIONS OF
UNIFORM
OIST.
430
3.5
0.0019
4.4
0.258
0.352
RECOMMENDED NEW PROC
NORMAL SLOPE
DIST. CHANGE
310
2.3
0.0012
3.0
0.158
0.162
430
2.7
0.0018
3.7
0.186
0.278
•tuuRE — —
PARAM.
METHOD
560
3.6
0.0018
4.0
0.251
0.313
NONRANDOM
SAMPLING
820
5.2
0.0019
4.7
0.340
0.280
GEOMETRIC MEAN OF RATIOS OF FAV BY
MODIFIED PROCEDURE TO THAT BY
RECOMMENDED PROCEDURE:
NUMBER OF DATA SETS FOR WHICH FA¥
BY MODIFIED PROCEDURE DIFFERS FROM
THAT BY RECOMMENDED PROCEDURE BY
MORE THAN A FACTOR OF 1.4:
NUMBER OF DATA SETS FOR WHICH FAV
BY MODIFIED PROCEDURE DIFFERS FROM
THAT BY RECOMMENDED PROCEDURE BY
MORE THAN A FACTOR OF 2.0:
0.89 1.00 1.04 0.93 1.1 I
26
12
0.96 1.07 1.19 1.48
13
-------
Table 8, FAVs CALCULATED FROM FMAV DATA SETS BY OLD FttOCEOURE, RECOMMENDED NEW PROCEDURE, AND VARIOUS M30IFICATIOHS
Of RECOMMENDED MEW PROCEDURE.
MATERIAL
CADMIUM
COPPER
MERCURY
DOT
ZINC
CADMIUM
ENDRIN
COPPER
CHROMIUMiVi )
DIELDRIN
ENDRIN
HEPTACHLOR
LINOANE
NICKEL
ZINC
ALDRIN
DDT
NICKEL
CHROMtUMUm
ALDRIN
TOXAPHENE
OIELDRIN
TOXAPHENE
SELENIUM
ENDOSULFAN
HEPTACHLOR
CHROMIUM(VI)
SELENIUM
LINOANE
CYANIUE
SILVER
SILVER
WATER
SALT
FRESH
SALT
FRESH
SALT
FRESH
FRESH
SALT
SALT
SALT
SALT
SALT
SALT
FRESH
FRESH
FRESH
SALT
SALT
FRESH
SALT
SALT
FRESH
FRESH
SALT
SALT
FRESH
FRESH
FRESH
FRESH
FRESH
SALT
FRESH
N
25
23
23
20
20
18
17.
17
17
16
16
16
16
16
15
14
14
14
13
13
13
12
12
12
11
10
10
10
10
10
10
9
LOWEST
FMAV
n
0.30
3.5
1.30
166
0.048
0.44
28
2490
0.70
0.037
0.057
0.170
65
13.7
7.4
0.140
310
33
3.7
0.110
4.5
1.30
600
0.040
1.00
67
340
10.0
77
4.7
0.0019
OLD
PROCEDURE
45
0.34
3.7
1.22
166
0.038
0.33
27
1940
0.67
0.036
0.044
0.121
50
10.4
4.0
0.102
160
30
3.5
0.065
1.6
0.99
440
0.033
0.75
a
154
8.2
58
3.3
0.0011
NEW
PROCEDURE
70
0.38
3.8
1.28
182
0.058
0.40
25
2370
0.53
0.031
0.061
0.192
66
12.3
6.7
0.130
210
23
3.3
0.087
3.7
1.07
440
0.033
0.50
23
167
6.4
63
3.3
0.0013
n*N/2
79
0.45
2.5
0.98
146
0.086
0.38
23
2130
0.55
0.021
0.106
0.398
93
13.7
5.8
0.149
130
28
3.2
0.094
2.7
1.12
330
0.028
0.34
38
209
7.0
60
3.9
0.0013
MODIFICATIONS OF
n*N UNIFORM
OIST.
119
0.26
2.7*
0.55**
109*
0.234
0.09**
26
1550*
0.76
0.017*
0.117
0.395
122
19.1
1.1**
0.092
110*
33
2.3*
0.047
1.0**
0.87
320*
0.003**
0.52
243
513
7.0
25"
2.8
0.0003*
69
0.39
3.8
1.30
187
0.069
0.41
27
2440
0.58
0.032
0.076
0.248
75
14.1
6.9
0.149
240
27
3.5
0.112
3.9
1.24
490
0.039
0.68
56
267
8.1
68
4.4
0.0018
RECOMMENDED NEW PRC
NORMAL SLOPE
OIST. CHANGE
70
0.38
3.8
1.27
180
0.054
0.40
25
2340
0.51
0.030
0.055
0.170
62
11.5
6.6
0.122
200
21
3.2
0.077
3.6
1.00
420
0.031
0.44
16
136
5.8
61
3.0
0.0011
73
0.39
3.8
1.29
187
0.063
0.42
26
2370
0.55
0.032
0.073
0.272
70
12.7
6.8
0.138
220
24
3.4
0.095
3.8
1.08
470
0.034
0.53
31
181
6.9
64
3.7
0.0016
i/~*fTAl f3C ^ ___
PAR AM.
METHOD
97
0.42
4.0
1.37
192
0.075
0.44
28
2540
0.68
0.036
0.077
0.263
76
14.2
7.2
0.148
270
31
3.5
0.113
4.1
1.20
540
0.036
0.67
39
256
7.6
71
4.0
0.0017
NQt«ANDQM
SAMPLING
77
0.58
4.4
1.46
236
0.143
0.46
32
2680
0.77
0.041
0.143
0.578
103
19.3
7.5
0.182
320
36
3.9
0.147
4.6
1.42
600
0.042
1.00
67
340
10.0
77
4.7
0.0017
-------
Table 8. Continued
MATERIAL
MERCURY
ENDOSUCFAN
ARSENICH I 1)
CHLORDANE
CHLORDANE
WATER
FRESH
FRESH
FRESH
FRESH
SALT
N
9
9
8
a
a
LOWEST
FHAV
5.00
0.340
880
6.3
0.400
OLD
PROCEDURE
3.42
0.208
570
3.7
0.090
NEW
PROCEDURE
0.94
0.169
220
4.0
0.200
n=N/2
0.94
0.169
220
4.0
0.200
n*H
2.32
0.094»
730
4.6
0.378
;ATIONS TO RECOMMENDED NEW PRC
UNIFORM NORMAL SLOPE
DIST. DIST. CHANGE
1.79
0.248
410
5.3
0.352
0.73
0.144
170
3.6
0.162
1.12
0.172
250
4.0
0.278
PARAM.
METHOD
1.89
0.234
450
4.6
0.313
NOtttANOOM
SAMPLING
4.76
0.319
790
5.6
0.280
GEOMETRIC MEAN OF RATIOS OF FAV BY
MODIFIED PROCEDURE TO THAT BY
RECOMMENDED PROCEDURE:
NUMBER OF DATA SETS FOR WHICH FA¥
BY MODIFIED PROCEDURE DIFFERS FROM
THAT BY RECOMMENDED PROCEDURE BY
MORE THAN A FACTOR OF 1.4:
NUMBER OF DATA SETS FOR WHICH FAV
BY MODIFIED PROCEDURE DIFFERS FROM
THAT BY RECOMMENDED PROCEDURE BY
MORE THAN A FACTOR OF 2.0:
0.91
1.00 1.01
0.89 1.23 0.92 1.08 1.25 1.58
29
13
17
-------
interval with the lowest P if no interval has a P less Chan 0.05) is desig-
nated Interval A. The next highest nonempty interval is designated Interval
B. The FAV is then computed as ln(FAV)=VA+(VB-VA)/(PB-PA)-(0.05-PA).
Criticisms of this procedure include:
(1) The formula PaR/N is positively biased as discussed earlier and chus
results in negative bias in the FAV.
(2) The positive bias in the cumulative proportions is increased by using the
maximum rank in an interval rather than the average rank, when there is more
than one In(MAV) in the interval.
(3) Often only one In(MAV) is in Interval A or Interval 1 or both, making the
method quite sensitive to data variation.
(4) A linear relationship of P versus V is assumed, which is equivalent to
assuming a rectangular distribution; as discussed earlier this is
contraind icated by the available data sets.
(5) The use of intervals is meant to causa pooling of ln(MAV)s which are
indistinguishable; the interval width of 0.25 was selected because it is a
typical value for the standard deviation of replicate acute toxicity tests;
this value may not be appropriate for all species and materials and is
strictly appropriate only when ln(MAV)s are based on only one toxicity test.
More importantly, this pooling method works effectively only for Interval A,
because tht starting point for Interval B is fixed by that for Interval A
and therefore does not necessarily, properly oool data in the vicinity of
Interval B. In any event, this pooling serves no useful purpose except to
prevent the slope for interpolations and extrapolations from being
inappropriately calculated based on two identical, or nearly identical,
ln(MAV)s, a purpose which can be better served by routinely using more
points to assess data trends.
(6) The use of intervals containing variable numbers of ln(MAV)s makes the
method sensitive to minor changes in the data set which may move ln(MAV)s
into or out of intervals; this sensitivity can be quite marked and can even
be anomalous. For example, in the SMAV data set for heptachlor in fresh
water (Table 1), Interval A would contain the lowest two SMAVs and Interval
8 would consist of the third lowest SMAV, However, if the lowest SMAV was
just 6Z lower (for Example, because a new toxicity test for that species
lowered th« mean slightly), the two lowest SMAVs would be sufficiently
separated to be in separate intervals, which wauid become the new, and
markedly different, Intervals A and B. The calculated FAV would change from
0.52 to 0.83, a change that not only is much larger than the change in the
SMAV that caused it (60% versus 6%), but also is in the opposite direction
to the change in the S>1AV.
44
-------
These criticisms are sufficient to warrant the replacement of this old procedure
with the new procedure recomnended above. However, the two procedures generally
do not produce markedly different results (Tables 7 and 8). On the average,
FAVs calculated using the old procedure are only HZ lower than those calculated
using the new procedure for the SMAV data sets and 9Z lower for the FMAV data
sets. Individual FAVs were within a factor of 1.4 for over 75Z of che FMAV data
sets and over 85Z of the SMAV data sets and within a factor of 2.0 for over 85X
of the FMAV sets and 94% of SMAV data sets. Where differences are greater than
twofold, comparison of the FAVs with the data sets does not clearly indicate
that one or the other of these procedures results in more questionable FAVs.
The consequences of modifying the major features of the recommended new
procedure were also explored to determine how sensitive FAVs are to such
changes. If the sensitivity is low, any objection to compromises or approxima-
tions used in arriving at the recommended new procedure are largely irrelevant,
because more exact analysis or different compromises (within reason) would have
little effect in practice. If sensitivity is high, the basis for the recommended
new procedure becomes more critical and further examination is warranted.
Three features of che recotn"l6nded new procedure were modified co sngn the
range over which they could reasonably be varied. The assumed distribution was
changed to rectangular and to normal. The subset size (n) was changed to N/2
and N. The pcrcentile estimation method was changed to the graphical method
with slope formula LS-X, but still with P^^E(P(X^)), and to the parametric
method (best linear unbiased estimate). The FAVs for these modifications are
included in Tables 7 and 8. Also included in these tables are (a) the geometric
mean of the ratios of the FAV by each modified procedure to that by the
recommended procedure, (b) the number of data sets for which the FAV by each
45
-------
modified procedure differs by more than a faccor of 1.4 from that by che
recommended procedure, and (c) che number of data sets for which the FAV by each
modified procedure differs by more than a factor of 2.0 from that by the
recommended procedure.
Changes in the assumed distribution, in the percentile estimation method,
and in the subset size to N/2 had only minor effects on results. The geometric
means of the ratios of the FAVs by these modifications to that by the
recommended procedure were close to 1.0 (0.92-1.25), For individual data sets,
FAVs by these modifications differed by more than a factor of 1.4 from the FAVs
by the recommended new procedure for no more than 20% of the data sets and by
more than a factor of 2.0 for no more than 6t of the data sets.
Modification of subset size to n"N, however, caused major changes. The
geometric means of the ratios of the FAV by this modification to that by the
recommended procedure were close to 1,0 (0.93 for SMAVs, 0.89 for iFMAVs), but
individual FAVs changed by more than a factor of 1.4 for over 70% of the SMAV
daca sets and for nearly SOX of the FMAV data sets and by more than a factor of
2.0 for about one-third of both the SMAV and FMAV data sets. Such differences
do not demonstrate, per se, thac this isodified procedure is less appropriate
than the recotnaended new procedure, but it does raise such a suspicion. In
particular, when n"N, an unusually large number of sets have a FAV that is well
below both the lowest MAV and the FAV calculated by the recommended new
procedure. Using the location and scale parameter estimates from the modified
procedure with n="N, the fiducial probability that the lowest MAV could be so
high was evaluated for each data set; where this probability is <0.20 a single
asterisk is placed next to the FAV for n"N and where the probability is <0.10 a
double asterisk is used. The frequency of these marked entries suggests that
46
-------
using n"H is in fact inappropriate. This is directly rtLated to the existence
of statistically significant skewness in the data sets, positive skewness
resulting in inappropriately low FAVs when n"N and negative skewness resulting
In high FAVs,
A related observation that also contraindicates the use of n*N is that,
when compared to the recommended new procedure and the diverse modifications
already mentioned, th* modification with n*N both frequently produces the lowest
FAV (for 20 SMAV data sets and 19 FMAV data sets) and frequantly produces the
highest FAV (for 14 SMAV data sets and 12 FMAV data sets). Such frequent
occupation of both extremes is again due to the variable skewness of the sets
and is indicative of the error of assuming all data are equally useful in
estimating low percentiles when distributional assumptions are not completely
met. Furthermore, th«s« problem* with using n"S art not restricted to the
modification with n"{f already discussed. Graphical methods with the other slope
formulas and other distributional assumptions (e.g., normal) were tested using
n*N with similar results. Likewise, the best linear unbiased method using n"N
and assuming a normal distribution showed similar problems. (This latter method
is equivalent to the simple approach of calculating a sample mean and unbiased
standard deviation (12) and estimating the fifth parcentile as lying 1.645
standard deviations below the m«an, -1.645 being th« fifth percentile of a
standard normal distribution.)
Another advantage of not using n"N is that certain semiquantitative data
can be used. Acute tests on some materials with some species produce 'greater
than' values because concentrations high enough to cause effects were not, or
could not be used (because of solubility or time constraints). Regardless of
the reason, because such data are usually for resistant species, they can
47
-------
usually be used if n-4, but, if n-N, either they must be excluded, thereby
biasing the data set, or additional acute tests must be conducted, thereby
increasing costs. Finally, it should be noted that the use of na4 does not
constitute "not using all the data1, because ail data are used in setting ranks
and cumulative probabilities and thus in selecting which four daca will be used
explicitly in final calculations; rather, the use of n-4 is more properly
interpreted as a simple scheme of giving greater weight to those MAVs which
provide the most information about the fifth percentile.
The consequences of nonrandom sampling can also be partly addressed here.
Assuming that the available data sets somehow resulted from the strictly
systematic sampling scheme discussed earlier, FAVs were calculated by assigning
cumulative probabilities P^»(R-0.5)/N to the ranked ln(MAV)s and interpolating
between the two data with P^ nearest 0.05 (or extrapolating using the lowest
two points if N<10), the interpolation being based on the assumption of a
triangular distribution. The results of this exercise are included in the last
columns of Tables 7 and 8 and indicate that higher FAVs are produced than by the
recommended new procedure, but the differences average only about a factor of
1,5 for SMAV sets and 1.6 for FMAV sets and are less than a factor of 1.4 for
65Z of the SMAV sets and 55% of the FMAV sets. Considering that this alterna-
tive sampling scheme is so extreme, and chat therefore a scheme with a more
realistic systematic component would produce results much nearer those obtained
by the recommended new procedure, this further suggests that the issue of the
sampling assumption is not of great importance.
A final point that should be emphasized is that, whether SMAVs or FMAVs are
used, the same conclusions are reached regarding the appropriate attributes of
48
-------
Che procedure for estimating fifth percent!Lea. Also, although it does not bear
on Che recommendations made here, it is interesting to note that FAVs calculated
from SMAVs are similar to those calculated from FMAVs, The geometric mean of
the ratios of the FAV computed from FMAVs to that computed from SMAVs, by the
recommended procedure, was 1.04. The two FAVs differ by a factor of 1.4 for
only 12 sets, by a factor of 2.0 for only 6 sets, and by more than a factor o£
2.8 for no sets.
49
-------
DISCUSSION
The recommended new procedure uses linear extrapolation or interpolation to
estimate the fifth percentile of a statistical population of mean acute values
(MAVs) from which the available MAVs are assumed to have been randomly obtained.
The available MAVg are ranked from low to high and the cumulative probability
for each is calculated as PR*R/(N+l), where R * rank and N » number of MAVs in
the sec, Extrapolation or interpolation is based on an assumed linear
relationship between>/P^ and In(MAV), and uses only the four points with P^
closest to 0.05 because this subset provides the most useful information
concerning the fifth percentile.
The bases for the new procedure are mostly mathematical, with some input
from toxicological and practical considerations. The FAV, however, is basically
a eoxicological value, and the acceptability of any calculation procedure to
toxicologists will be based on the acceptability of che resulting FAVs. Most
aquatic toxicologists will judge the acceptability of an FAV by comparing it
with che lowest MAVs in the data set, and thus it is quite appropriate that the
four MAVs with estimated cumulative probabilities closest to 0,05 be given the
most weight in calculating the FAV"; in fact, Cue reconsn ended procedure is
largely a formalization of the way one would obtain a FAV by 'eyeballing' the
data.
An important property of the new procedure is that the resulting FAV is not
very sensitive to modifications in the procedure or slight changes in the data
set. A variety of calculation procedures all produced FAVs that were quite
similar for most data sets. In addition, the recommended new procedure rarely
produced either the highest or lowest of the FAVs obtained with the procedures
examined. Another important property of a procedure is its performance with
50
-------
data sets which contain apparent discontinuities in the Lower tail, For
heptachlor, lindane, and chlordane in salt water and chromium(VI) in fresh
water, the lowest SMAV is at least a factor of 10 lower than the second lowest
SMAV (Table 1). In addition, for these four and cadmium in fresh water, the
lowest FMAV is at least a factor of 10 lower than the second lowest FMAV (Table
2). Of chese, only chromium(VI) in fresh water has a very large range of FAVs
in Tables 7 and 8. Even though the two lowest MAVs are far apart, most of chese
daca sets seem to provide adequate information about the FAV because similar
FAVs were obtained using a variety of procedures. These examples support the
idea that the best approach to take toward calculating the FAV is to select a
procedure that is best on the average and then use it with ail data sets, except
possibly in extraordinary cases.
An unfortunate aspect of the methodology for calculating the FAV is the
necessity of extrapolating to estimate the 0.05 cumulative probability for small
data sets; if extrapolations become too great, the FAVs will be suspect. For
only 5 of the 74 data sets is the FAV more than a factor of 2 lower than the
lowest MAV. Thus, for the available data sets this procedure rarely
extrapolates much below the lowest value in the data set.
Overall, the recommended new procedure is the best of the procedures
examined, regardless of whether the FAV is calculated from SMAVs or FHAVs. It
is a straightforward procedure for interpolation or extrapolation based on
fitting a line to the most useful points. It produces results similar to and
usually intermediate to those obtained by other reasonable procedures. In
addition, the calculations are relatively easy to perform with the aid of a hand
calculator, as described in Appendix 1. The major weakness of this procedure is
51
-------
that ie as«uai«» the same degree of tailing for all daca sets. Fortunately, for
most data sets the FAV is not very dependent on the assumed degree of calling
and deviation from the assumed intermediate degree of tailing is not too
critical. Other procedures wauld suffer as much or more from the same or other
weaknesses.
52
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REFERENCES
1, U.S. EPA. 1980. Federal Register. 45:79318-79379. November 28.
2. Javitz, H. and J. Skurnick. 1980. Analyses of relaCtonships among
various aquatic toxicity test: results. Final Report, Task 9, EPA
Contract 68-01-3887.
3. Kelt, M. 1981, Letter to J. H. McCormtck. U.S. EPA, Duluch, MN.
4. Sokal, R.R. and F.J. Rohlf. 1969. Biometry^ Freeman, San Francisco.
5. Lloyd, E.H. 1952. Least squares estimation of location and scale
parameters using order statistics. Biometrika, 39:88-95.
6. Johnson, N.L. and S. Kotz. 1970. Distributions in statistics: Continuous
unvariate distributions - 1. Wiley, New York.
7. Box, G.E.P., W.G. Hunter, and J.S. Hunter. 1979. Statistics for
Experimenters. Wiley, New York.
8. Cunnane, C. 1978. Unbiased plotting positions - a review. J_. Hydro 1.
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9. Sprsnt, P, and G.R, Dolby. 1980. Response to query. Biometrics,
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10. Barker, P., Y.C. Soh, and R.J. Evans. 1988. Properties of the geometric
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Cliffs, New Jersey.
53
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APPEHDIX 1
A.
General Instructions for Recotsnended flew Procedure for FAV Calculat
ion .
1. Based on data ate size (N), determine four ranks (R) with cumulative
probabilities (?R»R/(N+O) closest to 0.05; for N<60, this will be
R-l through 4; for 60
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C. Example Computer Program in BASIC Language for Calculating che FAV
10 REM THIS PROGRAM CALCULATES THE FAV WHEN THERE ARE LESS THAN
20 REM 59 MAVS IN THE DATA SET.
30 X-0
40 X2-0
50 Y-0
60 Y2-0
70 PRINT "HOW MANY MAVS ARE IN THE DATA SET?"
80 INPUT N
90 PRINT "WHAT ARE THE FOUR LOWEST MAVS?"
100 FOR R-l TO 4
110 INPUT V
120 X-X+LOGCV)
130 X2-X2+(LOG(V))*(LOG(V))
140 P-R/CN+1)
150 Y2-Y2+P
160 Y-Y+SQR(P)
170 NEXT R
180 S-SQR((X2-X*X/4)/(Y2-Y*Y/4))
190 L-(X-S*Y)/4
200 A-S*SQR(0.05)+L
210 F-EXP(A)
220 PRINT "FAV « "F
230 END
D. Example Printout from Program
HOW MANY MAVS ARE IN THE DATA SET?
? 8
WHAT ARE THE FOUR LOWEST MAVS?
? 6.4
? 6.2
? 4.8
? .4
FAV » 0.1998
55
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