Review and Evaluation of
Dendrochronological Methods
Synthesis and Integration Report Number 10

St	United Statu?
Wrl A Environmental Protection
Agency

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Review and Evaluation of
Dendrochronological Methods
Synthesis and Integration Report Number 10
October, 1987
Prepared By:
William G. Warren
Research Associate
Department of Forest Management
Oregon State University
Corvallis, OR 97331
Prepared For:
Synthesis and Integration Project
Forest Response Program
US EPA Environmental Research Laboratory - Corvallis
200 SW 35th St.
Corvallis, OR 97333
Notice
This document is an internal report. It has neither been peer
reviewed nor approved by the U.S. Environmental Protection
Agency. It is being circulated for comment on technical merit and
policy implications. Do not release. Do not quote or cite.

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Review and Evaluation of Dendrochronological Methods.
General Concepts
There is no question that the history of a tree's response
to its environment is embodied in its sequence of tree rings.
There are, however, a multitude of environmental factors that are
acting simultaneously, both natural and anthropogenic. Natural
factors include climate, pests and pathogens, and, in some
instances, fire; examples of anthropogenic factors are thinning
and fertilization and, possibly, atmospheric pollution. Cook
(1985b) formally describes the situation in terms of the so-
called linear aggregate model, an earlier version of which was
given by Graybill (1982)? specifically
Rt = Gt + Ct + 5D1, + 6D2t + 6Pt + E(
where Rt is the observed ring-width series measured along a
single radius, the subscript t referring to time, i.e. year t.
Gt is the growth trend associated with increasing age and
size of the tree,
Ct is the climatically-related growth variation common to a
stand of trees,
Dl, is the variation due to endogenous disturbances, that is
those that affect a small subset of trees in a stand,
D2t is the variation due to natural exogenous disturbances,
that is those that have a standwide impact on radial growth,
Pt is the variation due to anthropogenic pollutants that
again have a standwide impact, and
Et is a random component, unique to each tree or radius.
The 6 associated with each Dlt, D2t and Pt is a binary
indicator of the presence (6 = 1) or absence (5 = 0) of that
component for a particular year or group of years.
Conceptually there is little, if any, difference between
this and Sundberg's (1974) seemingly neglected representation
Ar(t) = Mt)r,(t) [1 - 0(t) ]C(t)
in which Ar(t) corresponds to Rt, K(t) is an individual growth
level function (cf. Gt) , ri(t) is a climatic factor which "for
fixed t can probably be regarded as common to all trees in a
region, or at most as having a slight and smooth spatial trend"
(cf. ct) . The [1 - 0(t)] introduces the effect of external
agents with 0(t) si (and may be negative) and 0(t) ¦ 0 for
t < t0 (cf. Pt) . Finally g(t) is a random process and "with
satisfactory accuracy we may probably assume that log{?(t)) is a
stationary Gaussian process centered to have expectation zero,
with independence between realizations for not very closely
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adjacent trees" (cf. Et) .
The linear aggregate model is, perhaps, the more_sophis-
ticated in that it separates out the effects of endogenous and
exogenous disturbances which Sundberg more or less embodies in
6(t). Sundberg's representation, however, preceded the linear
aggregate model by approximately a decade. The other noticeable
difference is that Sundberg deliberately chose a multiplicative
model, in contrast to an additive model, although there is
evidence to the effect that the linear additivity of the
components in the latter was intended to simplify the exposition
and not to imply any structural relationship. Any formal
development should not, therefore, involve linear additivity as a
necessary component.
The challenging problem is, thus, to isolate any pollution
signal from the various other synchronous components.
It seems reasonable to address this problem through an
extension of the methodology developed for dendrochronology and
its outgrowths, namely dendroclimatology and dendroecology.
Dendrochronology has its roots in antiquity. The
relationship between time and the growth of trees dates back more
than two millennia to Theophrastus, the student and successor to
Aristotle. That rings of trees are annual in nature was well
understood by a number of botanists, foresters and astronomers in
Europe and America in the 18th and 19th centuries; indeed, some
understood the principles of crossdating (Studhalter, 1956).
Fritts (1976), quoting Dean (1978) observes that the "French
naturalists Duhamel and Buffon in 1737 discovered that a
conspicuous frost-damaged ring occurred 29 rings in from the bark
on each of several newly felled trees. Other investigators
confirmed their observation and this feature was subsequently
used as a marker for the 1709 ring. Crossdating based on relative
ring widths was independently recognized in 1827 by A.C. Twining
in Connecticut, in 1838 by the mathematician Charles Babbage in
England, in 1859 by Jacob Keuchler in Texas, and in 1904 by A.E.
Douglass in Arizona". While it was Twining who first grasped the
full significance and potential of crossdating, it was the work
of Douglass that led to dendrochronology being recognized as a
legitimate and continuing field of scientific investigation.
Dendrochonology in its purest form, as implied by the word
itself, deals with the use of tree rings to date particular
events, such as otherwise unrecorded geomorphological phenomena,
for example earthquakes, landslides and floods. Examples of this
type are provided by Druce (1966), Helley and LaMarche (1973),
and LaMarche and Wallace (1972). The incidence of forest fires
has likewise been dated, e.g. Dieterich (1980).
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The greatest interest, perhaps, has been in climatic
phenomena; in relation to the above it may be noted that it was
the excessively cold winter of 1708-09, one that was beyond
living memory, that destroyed Charles XII*s army in its invasion
of Russia. One particular area of interest comes from the
suggestion that the blocking of radiant energy from the sun by
dust in the atmosphere, resulting from major volcanic erruptions
could cause a reduction in temperature which would be reflected
in tree ring widths of the period. While the so-called "year
without a summer" ,1816, (Francis 1976) followed the erruption of
Tambora in 1815 and appears to be reflected in the ring widths of
white spruce growing near Great Whale River (Cri Lake), Quebec,
Canada (Parker et al. 1981), there appears to be no universal
association between major volcanic erruptions and tree growth.
Thus, while LaMarche and Hirschboek (1984) report several
instances of frost-ring events in conjunction with volcanic
erruptions, they also note various instances of frost rings in
years with no known volcanic activity, and instances of major
erruptions for which no frost rings have been associated. In
reporting his own study, Parker (1985) concluded that "No
immediately-apparent relationship was detected on a continent-
wide scale. ... A possible response is suggested for trees from
sites near the tree line in northern Quebec and northern
Manitoba". The phrase "near the tree line in northern Quebec and
northern Manitoba" should be borne in mind in what follows
concerning the relationship between stress and sensitivity.
Dendoclimatology goes further than the dating of extreme
climatic events, and much of the work in this area has been
devoted to climatic reconstruction. For example, Schweingruber et
al. (1978) have demonstrated that excellent reconstructions of
past climate could be produced using series of ring-width and
wood-density measurements from subalpine firs. Also Hughes et al.
(1984) found that maximum latewood density and ring width
measurements from the Scottish Highlands could be used to
reconstruct July-August temperature for Edinburgh, the period
studied being from 1721 to 1975. (See below for futher discussion
on the role of tree-ring wood density in dendrochronology,
dendroclimatology etc.) However not all such studies have been so
successful.
A thorough description of dendrochronology and dendro-
climatology has been given by Fritts (1976) who himself has been
a major contributor to the methodology and application. Much of
the work of the past decade has been founded on this definitive
text as well as the various papers of its author. As will be
demonstrated below, this "standard" methodology has its
limitations which have, perhaps, come more into focus with the
current necessity to assess the effect of atmospheric pollution
on forest growth. Accordingly, recent years have seen the
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introduction of some alternative methodological approaches, for
example Warren (1980), Monserud (1986), Visser and Molenaar
(1987), Van Deusen (1987), and Cook (1987b).
There are several well defined and generally recognized
steps in a dendrochronologically-based study directed to
establishing an effect of atmospheric pollution, namely
crossdating, standardization (the removal of the natural growth
trend), identification and removal of the effects of disturbances
unrelated to pollution, and estimation of the effects of climatic
variation. Each of these steps will now be critically examined.
Crossdating.
It should go without saying that any dendrochronological
study is dependent on the accurate dating of each ring. This may
not seem like a problem if the date of felling, or of the
collection of an increment core, of a tree is known; one should
simply be able to count, ring-by-ring, back from the bark to
identify the year in which each ring was laid down. This would be
the case if it were not for the possible occurrence of missing
and/or false rings.
Tree rings are identified by the visible transition from the
higher density wood laid down towards the end of the one growing
season (the latewood) to the lower density wood laid down at the
beginning of the subsequent growing season (the earlywood). There
is, however, no sharp transition from earlywood to latewood.
False rings tend to arise when, because of some late-season
climatic phenomenon, there is a transition from latewood back to
earlywood within the growing season; thus two (or even more)
rings may appear within the one year. On the other hand,
sometimes during a year of extreme climate, or as a result of
some disturbance, the tree may not form a ring on all portions of
the stem; the ring is then said to be partial, locally absent or
missing along certain stem segments or radii. Again climatic
conditions may be such that wood laid down in a growing season
never attains sufficient density to be identified as latewood; in
this case the ring is not so much missing as inseparable from the
following ring. Fritts (1976) observes that "A surprising number
of absent sets or false rings occur, even for sites that are
relatively temperate".
Further, Fritts writes "... ring widths can be crossdated
only if one or more environmental factor becomes critically
limiting, persists sufficiently long, and acts over a wide enough
geographic area to cause ring widths or other features to vary
the same way in many trees. The principle implies that the
narrower rings provide more precise information on limiting
climatic conditions than do wider rings. During years when rings
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are generally wide, factors may become limiting to different
degrees in each tree, depending on its locale, ecological
position in the site, and a great variety of nonclim^iic factors.
As a result considerable variation in growth patterns may occur
from tree to tree. If the growth of a tree is never limited by
some climatic or environmental condition, there will be no
information on climate in the widths of the rings and they will
not crossdate". In a later section he states "If climate is
highly limiting to growth, all replicated samples within and
among trees will show approximately the same ringwidth variation
and the rings will be easy to crossdate. ... If climate is not
highly limiting, variations in factors of the site may cause
marked differences in ring sizes between trees, and differences
in growth on opposite sides of the same trees may result from
variations in the structure of the forest stand, lean of the
stem, and competition from neighboring trees. A greater amount of
replication may be necessary to achieve a reliable chronology
from such a site". In a later chapter it is observed that "No
specimen is selected for climatic analysis if there is any
question as to the dating, even if the problem involves only one
ring".
These statements imply that crossdating is best achieved
with trees that are growing under somewhat, but sublethally,
stressed conditions and that reliable crossdating would be
difficult, if not impossible, for unstressed trees. This is not
necessarily the case. Sequences that exhibit a wide variation in
ring width from year to year are commonly referred to as
"sensitive"; such are observed in trees from, for example, the
dry interior of British Columbia or near the northern tree line
in the Northest Territories. On the other hand, sequences that
exhibit little variation in ring width are referred to as
"complacent"; trees from the moist British Columbia coast, for
example, generally fall into this category. From the traditional
dendrochronological point of view sensitive series are the most
useful since, if there are no missing or false rings, these are
the most easily crossdated. Unfortunately, it is with senstive
series that missing and false rings are the more likely to arise.
Thus, complacent series, from trees for which the date of felling
or sample extraction is known, present fewer problems in
crossdating. Complacent series carry a weaker climatic signal and
are thus of less value for climatic reconstruction or the dating
of historic, or prehistoric, phenomena; the last-mentioned may
well be impossible. Conversely it may be that, because of the
weaker climatic signal, such samples may provide a clearer
indication, or measure, of any effects of pollution. Most
existing chronologies have, however, been built from samples
selected on the basis of the traditional objectives of
dendrochronology and dendroclimatology and, indeed, not
surprisingly, these principles have maintained themselves in the
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collection of tree-ring data when the objective has been the
isolation of possible pollution effects. In short, the analysis
of tree-ring data collected under conventional dendro-
chronological principles is unlikely to be efficient for the
detection of a pollution signal. Further, if sampling is
restricted to sensitive series, how representative of the
population would such trees be? Is it possible that pollutant
would have a greater effect on otherwise stressed trees than on
unstressed trees, or indeed have little or no effect on the
latter? Any restriction imposed on sample collection may well
lead to biased estimates of the effects of pollution.
With respect to the actual crossdating operation, Fritts
(1976) observes that "Crossdating, as practiced in North America,
is a time-consuming process in that it requires both tedious
examination of the specimens by eye, an operation that can be
facilitated by certain graphical techniques, and computer
analysis". He also notes that "The crossdating obtained in most
North American studies by professional dendrochronologists
involves visual examination of every specimen. The Laboratory of
Tree-Ring Research requires that a routine dating check be
performed by a second person, and if serious discrepancies are
found, the dates for the entire set must be verified by yet
another worker". He concludes that "It is highly likely that some
of the tedium involved in the visual crossdating operation will
be increasingly facilitated by computer analysis. However, at
present there is no fool-proof shortcut to the procedure of
careful visual comparison". Many recent studies now employ the
computer program COFECHA (Holmes, 1983; Holmes et al., 1986)
which calculates the correlation coefficients between a single
series and a master tree series. As with much modern-day
analysis, there is a temptation to eliminate the tedium by total
reliance on a piece of computer software. The consensus of
experienced dendrochronologists is, however, that the proper use
of COHECHA is for verification after visual crossdating; that is,
the software is not a replacement for personal skill and
experience. Indeed the linear correlation between two ring-width
sequences, progressively shifted against each other, not
uncommonly reveals more than one distinct, and roughly equal,
maxima. In such circumstance an automated process could be quite
erroneous.
Standardization
In its simplest form standardization involves the smooth
tracking of the natural growth trend of ring width on age,
followed by the division of the actual ring width by the smoothed
estimates to form a sequence of relative ring width indices.
Ideally such indices could be looked on as random variables
distributed about unity with constant variance. The ratio of the
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actual and smoothed widths is taken rather than their difference
since the raw ring-width series are, in general, heteroscedastic,
i.e. the variance of the ring widths in a relatievly .small time
period is a function of the mean ring width for the period.
Indeed there tends to be a linear relationship between the mean
and the standard deviation of ring width which, in turn, implies
that the ratio will have stable variance.
Early attempts at standardization relied on linear or
modified negative exponential functions to model the natural
growth trend, i.e
Gt - P0 + Dit
and
Gt = k + a exp(-pt), respectively.
Note that (Jj would generally be constrained to be < 0. Both
these functions are monotonic and thus cannot accomodate the
common pattern of increasing ring widths during the first several
years of growth followed by the decreasing ring widths of
subsequent years. In the fitting of these functions it is
necessary, therefore, to remove from the analysis all data, prior
to a subjectively chosen point. Since, from geometical argument,
some form of exponential decline in ring width is, ultimately, to
be expected, other more flexible exponential forms have been
considered, e.g.
Gt = atp exp(-ct),	(Siren, 1961; Warren,1980)
Gj = ata_,|f'xexp[-(t/P)B], (Yang et al., 1979)
Gt = a[l + Pexp(*yt)]k, (Van Deusen, pers. com.)
The last mentioned embodies the modified negative expontial (k =
1) and the logistic growth curve (k = -1) as special cases. In
addition, the power function Gt = at~p has been employed by
Kuusela and Kilkki (1963).
These functions are either monotonic or possess a single
maximum and, thus, are unable to track any departures from trend
caused by endogenous or exogenous disturbances, including change
in competitive status. Their applicability is thus generally
limited to undisturbed trees in open canopy stands where inter-
tree competition is not a factor. This has often led dendro-
chronologists to avoid sampling closed-canopy stands; such action
would, however, prohibit a study of the interaction between
pollution effects and competition.
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The earliest, and simplest, method of avoiding this
limitation was to represent the growth trend as a polynomial,
G, = pQ + Pjt + P2t2 + . . . + pptp
with the order of the polynomial unknown and dictated by the
usual forward inclusion stepwise procedure of fitting. Whereas
there may be some physical justification for the exponential-type
functions, the same cannot be said of the polynomial; to describe
the latter as a "model" is to stretch the meaning of the word.
While the exponential-type models may be too simplistic the
polynomial is too data dependent; it is also known to suffer from
potentially severe end-fitting problems and poor local goodness
of fit (Cook and Peters, 1981).
A rather different approach, advocated by, in particular,
Parker (1970), is the application a digitial filter. Specifically
Gt = £ wi^t-i i
i=-n
rt
2 w, = 1, and, for i ~ 0,w, = w_r
l=-n
This type digital filter is simply a centrally weighted moving
average of the actual ring widths. The number, 2n+l, and value of
the weights, wt, depends on which frequencies of variation are
to be retained and which are to be eliminated (filtered out).
Thus, three weights in the proportion 1:2:1 will block out
rapidly changing variations and pass slowly changing variations;
such is refered to as a low-pass filter. Conversely one may have
a high-pass filter, i.e. one that blocks frequencies with
relatively long wave lengths (low frequencies) and transmits the
high frequencies (short wave lengths). Both kinds may be used
simultaneously. It is clear that, for application to tree-ring
widths, there is little, if any, basis for selecting the weights
and, thus, it would be difficult to justify the choice of any
particular filter. Different filters will lead to different index
values, although once a filter has been prescribed the procedure
becomes totally data dependent; it can not therefore be properly
described as model-based.
A somewhat similar procedure is cubic-spline smoothing
introduced by Cook and Peters (1981). In this, cubic curves are
fitted to subsets of points and pieced together under the
restriction that the first derivative be continuous. This is
again a totally data-dependent, rather than model-based,
technique.
A rather different approach is that taken by Barefoot et al.
(1974), namely exponential weighted smoothing, in which an
estimate of each point in a series is a weighted sum of all
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observations preceding the point, with heaviest weight being
given to the most recent observations. Specifically
Gt = Rt + [(l-ot)/a]Rt
where
Rt = aRt + (1—0£) Rt_j
and
Rt = a(Rt - r,_,) +(i-a) Rt.j	(Brown, 1959)
The current average, Rt, is, thus, taken as a weighted average of
the current observation, R(, and the previous average, Rt_j.
<\>
Likewise, the current trend, Rt/ is taken as a weighted average
of the current trend, Rt - Rt_,, and the previous trend, Rt_j. The
smoothed current value, Gt, is then estimated as the current
average plus an adjustment for trend. The average and estimate
for the first year may be taken as the observed value, and the
trend as zero. [Note: the versions given in Barefoot (1974) and
Cook (1987a) both contain apparent typographical errors].
The quantity a is the weighting factor that determines the
degree of smoothing; Barefoot et al. chose a = 0.2 which allowed
the previous 10 - 15 years data to influence the current year's
estimate. Thus, in contrast to the symmetrical digital filter,
the exponential smoother is a one-sided filter that relies on
only the current and prior values for its estimate; the estimates
thus have the desirable feature of evolving in time in a manner
that parallels the growth of the tree. The resultant series is
nevertheless dependent on a, the choice of which is again
omewhat arbitrary, and once a is selected, the procedure is
likewise totally data dependent.
Another method of trend removal is obtained from simply
differencing the successive observations. If the ring-width
series is taken as a random walk with a determininstic drift then
Rt = Rt_i + et + 8
where et is a serially uncorrelated random variable and 6 is
the deterministic drift of the process. Then the average of the
VRt = Rt - Rt_, provides a measure of the drift. In practice a
logarithmic transformation is usually applied, i.e. one works
with the log(Rt) rather than the Rt. This is tantamount to
assuming that Rt = P exp(-kt)(1 + €t) .
A disadvantage of this procedure is that no direct estimate
of the growth trend component, Gtf is obtained. However, as
with other standardization processes, it might provide a
stationary series of random variables to which autoregressive -
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moving average modeling (e.g. Box and Jenkins, 1970, 1976) can
be applied [see the following section on chronology development].
On the other hand Monserud (1986) points out that a f-irst order
autoregressive model with $ = 1.0 [defined in the following
section] is assumed to be the underlying model if first
differences are to produce a stationary series, and that such
strong nonstationarity is unlikely to be found in tree growth,
for the factors causing the changing growth trend over time are
fairly well understood and can largely be explained by changes in
tree size and stand density. Taking first differences thus seems
an unnecessarily strong assumption for producing stationary
series, and Monserud believes that fitting a growth trend and
then standardizing seems more reasonable.
Warren (1980) attempted to bridge the gap between overly
simplistic models and totally data-driven smoothing by the
introduction of a compound increment function. His approach has
not received much attention, probably because of a perceived
awkwardness in estimating the parameters, although Monserud
(1986) notes, with respect to four chronologies for which he was
forced to use a ninth degree polynomial for removal of the growth
trend, that "the additive [i.e. compound] version of Warren's
(1980) original model would likely perform very well in
describing such complicated series".
Before closing this section mention should be made of the
very different method employed by the Russians Shiyatov and
Mazepa (1987) . It is based on generating smooth curves from the
maximal and mininal possible ring widths for each series as a
function of time; this leads to a so-called "corridor". Currently
it appears to be a purely graphical and subjective technique with
undefined mathematical properties.
Chronology Development
On the completion of standardization for each tree from a
particular site, the next step in the conventional process is to
form a site chronology by averaging the index values on a year-
by-year basis. Such a series should then contain a measure of the
climatic component, Ct, plus any endogenous and exogenous
disturbances that have not been removed along with the natural
growth trend. If standardization has been accomplished by fitting
(non-compound) exponential-type (or linear) functions, the
effects of any endogenous or exogenous disturbances will also
remain in the individual series.
The effects of non-synchronous endogenous disturbances
should, of course, be reduced by the averaging process. A more
effective means of minimizing these effects seems likely achieved
by the use of robust estimates such as the biweight mean
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(Mosteller and Tukey, 1977). Cook (1987a) gives some indication
of the quantitative improvement that can be achieved in this
manner and concludes that "the results indicate the high level of
outlier contamination in closed-canopy forest tree-ring data that
will corrupt the common climate signal if left unattended".
With the exception of Cook (1987b) no attention appears to
have been given to the forming of the principal components, yet
this seems to be a situation in which principal components
logically might be expected to be useful. Suppose that the
majority of the standarized series (from a given site) carry a
common signal (plus random noise) but a few series are aberrant
in their pattern due to undetected or unallowed for endogenous
disturbances. The first principal component should then tend to
give equal weight to those series with the common signal but
little, or inconsistent weight, to the remaining series. Indeed,
if the remaining series also carried a common, albeit different,
signal, this should show up in the second principal component.
Thus, the computation of the principal components opens up the
possibility of demarcating subgroups of trees that respond
similarly within the subgroup but differently between subgroups.
By definition, the first principal component should maximize the
signal to noise ratio. Any need to exclude, subjectively, series
perceived to contain a low common signal would be obviated.
Whether the principal component would have any advantage over the
biweight estimate is an open question since, presumably, the
latter allocates its weights on a year-by-year basis. This could
result in a series being heavily weighted in one year while
receiving little weight in another. This aspect merits further
investigation.
Because of the likely existence of persistence effects, the
application of autoregressive (AR) or autoregressive-moving
average (ARMA) methodology would seem appropriate, and possibly
mandatory in the case of standardization by the fitting of (non-
compound) exponential-type functions or where the detrending has
been done by the method of differencing.
An autoregressive model for the index values may be written
as
=	+ $2*»-2 * * * $p*l"P + ef
Thus the response at time t is dependent on the responses of the
p previous years? the model is said to be of order p and written
AR(p). There is thus a carry over effect of previous years plus a
random perturbation.
Complementary to this is a moving average model (this
should not be confused with the moving average, or running mean,
filter used to remove certain frequencies for curve smoothing?
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see above). It may be written
and said to be of order q, i.e. MA(q). It represents a linear
combination of the current and the q prior random perturbations.
The two may be combined into an autogressive-moving average
model of order p,q, i.e. ARMA(p,q).
Differences in tree size or position within a stand can
produce differences in either the speed of degree of response to
an environmental "shock". Thus "to remove the effects of
unwanted, disturbance-related transience on the common signal
among trees, the tree-ring indices can be modelled and
"prewhitened" by AR(p) and ARMA(p,q) processes before the mean
value function is computed. The order of the process can be
determined at the time of estimation using the Akiake Information
Criterion (Akiake, 1974)" (Cook, 1987a). The biweight mean is
recommended. The state-of-the-art means of accomplishing this is
generally looked on as being the autoregressive standardization
program, ARSTAN, of Cook (1985a) [see also Cook and Holmes, 1985;
Holmes et al. 1986].
Monserud (1986) reports that, in 31 out of the 33 series he
examined, a satisfactory representation was given by ARMA(1,1)
models. With AR(p) models a satisfactory representation was
obtained with p = 1, 2, 3 and 4 in 3, 15, 5 and 10 cases
respectively. Cook (1987a) reports that he found AR(1) - AR(3)
models satisfactory in most cases with his studies of eastern
North American conifer and hardwood chronologies.
From the ARMA(p,q) parmeters, $ , 6j, one may estimate
the et and hence the (robust) means et, which may be taken
as an initial estimate of the climate component, Cr Cook
(1987) points out that this estimate is incomplete in that there
still may exist a persistence component. An autorgressive model
may again be applied and, hence, a final estimate of Ct
generated. [There are complications in employing an ARMA model at
this stage; these are explored in Guiot (1987)].
Disturbance Effects
Once the climate component, which is assumed to be common to
all trees in a site, has been estimated, it is possible to
estimate the effects of any endogenous disturbances, that were
not taken out in the standardization process, on the individual
trees. [Any effects of exogenous disturbances that remain are
common to the trees of a site and should be embodied in the
estimate of the climate component]. Removal of the common
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component from an individual tree series should leave the effect
of the disturbance(s) plus random noise.
In the terms of Cook (1987a) a disturbance pulse~can be
thought of as an autocorrelated form of Et in the sense that
Dlt represents a persistent departure from the common signal.
However Dt differs from Et in that, while Et is always
present to some extent, the random shock leading to the creation
of Dlt has a defined arrival time and an impact on radial
growth that eventually disappears from the record (cf. Warren,
1980). These properties suggest that an endogenous disturbance
may be modelled by means of intervention analysis (Box and Tiao,
1975), cf. Warren (1983). This assumes that the times of the
interventions are known and, unfortunately, for trees within
forest stands, this is rarely, if ever, the case. The problem
can, however, be addressed by "intervention detection" (e.g.
Reilly, 1984) which is essentially an a posteriori search
performed by undertaking intervenion analysis at successive
points in the sequence until an identifiable effect is located.
However, from the viewpoint of detecting a pollution-related
signal, it is the removal, rather than the estimation, of the
effects of endogenous effects that is important.
Intervention analysis and intervention detection may also be
appropriate tools for separating the effect of exogenous
disturbances, D2t, from the climate component, Ct. The date
of exogenous disturbances, such as fire, fertilization, or
epidemics of defoliating insects, will often be known, or much
more readily inferred from the data than the effects of
endogenous disturbances. It is also necessary that this component
be removed before one can attempt to relate the climate effect to
meteorological measurements.
Effect of Climate
To this point the objective has been to obtain an estimate
of the climate component, Ct, free from the natural growth
trend and any endogenous or exogenous disturbances, including
atmospheric pollution. If this component can be accurately
described in terms of meteorological measurements, after allowing
for the effects of age and other disturbances, it may be possible
to predict ring widths of trees suspected of being affected by
atmospheric pollution. Consistent overestimation (or under-
estimation) would then reasonably suggest a negative (or
positive) effect of pollution and, indeed, provide a measure of
that effect.
Most attempts at relating the estimated climatic signal to
meteorological variables have followed Fritts et al. (1971), or
Ashby and Fritts (1972) (also Fritts, 1976). The dependent
13

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variable is the site chronology obtained from the indexed
series, or, of course, the estimated Ct. The weather variables
are taken as the monthly mean temperatures and total
precipitation for January through September of the associated
growth year and for May through September of the previous year,
that is 14 temperatures and 14 precipitations for a total of 28
potential predictor variables. Mean ring-width indices from l to
3 years previously have sometimes been included, primarily when
the persistence effect has not otherwise been accomodated. There
may thus be up to 31 potential predictor variables.
To avoid the problems of multicollinearity the principal
components of the weather variables then replace the actual
monthly values. The number of potential predictors is sometimes
then reduced by omitting those components which account for a
negligible amount of the variability amongst the predictors. In
this way Ashby and Fritts (1972) reduced the number of weather-
related variables to 20. A forward-selection stepwise-regression
procedure is then employed, inclusion being based on the
sequential F statistic being > 1. This seems a very liberal
criterion since, under the null hypothesis, the expected value of
the, F statistic is d/(d-2) where d is the number of degrees of
freedom of the denominator. More stringent criteria (e.g. F > 2)
have sometimes been used. The resulting multiple coefficient of
determination, R2, is usually computed and its significance
judged by reference to standard statistical tables.
A drawback here, that seems not to be generally recognized,
is that the probability values obtained from standard tables
are not applicable in the case of a "steered" regression, of
which forward selection is a particular form. In this, one is not
picking a potential predictor, or set of predictors, at random,
for which the standard tables would be applicable, but the
predictor, or subset, with the maximum R2. Thus the critical
values of a test of the null hypothesis, at any specified level,
must exceed those given in the standard tables. In the limited
number of cases where appropriate values have been determined,
the difference between the proper value and that given by
standard tables is, in general, substantial. (See for example,
Draper et al., 1971; Pope and Webster, 1972; Rencher and Pun,
1980).
Unfortunately the problem is not well documented in the
statistical literature, possibly because of the inherent
theoretical difficulties. The few explicit results that are
available have been, for the most part, generated by Monte Carlo
methods. There are vast number of possible combinations of the
number of data points, the number of potential predictors and the
number of these selected (let alone the correlation structure of
the predictor variables, in the general case). Also the time for
14

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computation becomes substantial so it is not surprising that
only a small segment of the possible space has been covered.
Reliance on conventionally computed probabilities or tabled
critical values seems to have led to undue confidence in the
resulting regressions to predict the effect of weather variables.
McLenahen and Dochinger (1985), for example, present, inter alia,
the number of principal components entered under a forward-
selection F > 1 criterion and the resulting values of the
multiple R2, for all combinations of 5 sites and two periods
(1901-1930 and 1931-1978); all 28 principal components were,
however, considered. Interpolation and modest extrapolation of
the Rencher and Pun (1980) tables show that, while all ten values
of R2 exceed their expected values under the null hypothesis,
only one exceeds the 5%-level critical value, and it only
marginally so (see Appendix). There is thus serious doubt whether
the derived regressions have any real predictability. Indeed, in
relation to their expected values, the pattern of results for the
two periods is remarkably similar, seemingly invalidating the
authors' conclusion that tree growth in the more recent period
had come under strong influence of factors unrelated to climate.
Ashby and Fritts (1972) included 12 out of 2 0 orthogonal
(principal-component) predictors. These accounted for 59% of the
variability in the ring-width index chronology. The series length
was 55 years. This combination of values is too far removed from
the Rencher and Pun tables for extrapolation to be reliable
(Rechner and Pun provide a formula for approximate interpolation
but state that extrapolation with it may be risky and needs
further investigation). It would appear, however, that
significance at the 5% level would be, at best, marginally
attained.
Cropper (1982) took an actual 32-year chronology and
regressed it against regional climatic data. He did the same with
a computer-generated independent chronology (i.e one generated
from random numbers without regard for the climatic data, so that
only chance associations with a climatic data set could be
expected). He reported that 88% of the variance of the actual
chronology was accounted for exclusively by climate and 15
climatic variables were deemed significant at the 5% level.
Slightly less variance was accounted for in the case of the
artificial chronology but 16 variables were included as
significant. The analyses were run in exactly the same way
following Fritts et al. (1971). Cropper concluded that "there is
the potential for the response-function techniques to be
misleading by attributing significance to non-significant
variables".
15

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Notwithstanding the significance, real or otherwise, of the
resulting regression, validation of the chosen regression is to
be recommended. The general approach here is to divide the total
series into two parts, one from which the regression parameters
are estimated, the other to which the resulting regression is
applied and its performance evaluated. It is assumed that the
"verification" set is not affected by some undetected
disturbance, such as pollution; thus the time period up to, say
1940 may be used for estimation, 1940-1960 for verification, and
the years since 1960 omitted. The difficulty lies in choosing a
meaningful measure of the regression's applicability in the
verification period. The so called "reduction of error" (RE)
statistic has been employed (Kutzbach and Guetter, 1980; Fritts
et al., 1979). Specifically
RE = 1 - i(yt - y^VKy, - yc)2
where the summation is over the verification set, the y( and
£ are the actual and estimated values, while y^ is the
mean value from the estimation (or calibration) set. Although the
properties of this statistic do not appear to have been studied,
positive values are regarded as as indicitive of satisfactory
performance. Unfortunately, it is possible to obtain, from the
estimation sets, a collection of multiple R2 values that appear
to be in good agreement with the null distribution (after
allowance for forward selection), and thus imply little or no
predictability, along with a preponderance of positive values of
RE from the verification sets. This seems pardoxical, but it
should be noted that RE may be positive because £(y - y()2 is
small, which would be the case if the verification data were
closely tracked by the regression estimates, and/or because
£(Y, " Yc)2 is large. The latter may result from the
climate variables in the verification period being atypical in
relation to those experienced in the calibration period. This is
an aspect that deserves further study.
There is another problem. The majority of trees for which
dendrochronological records are available are those which have
survived to the present, or at least recent, time. Thus trees
that may have succumbed to some environmental factor will be
excluded from the record. A reccurrence of that factor may then
show up as a decline simply because it cannot been fairly
represented in the regressions.
Finally there is the choice of potential predictors.
Calendar months are an artificial division of the year which are
not, in general, in synchronization with biophysical
developments. It would seem to make very little difference to a
plant if 2 inches of rain fell on 31 March or on 1 April, but
this could have a noticeable effect on regressions based on
16

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monthly values. The forming of principal components, which is
first and foremost and arithmetical artifact, is unlikely to
remedy this. Next, from the physiological point of view, it is
difficult to see how additive linear models of precipitation and
temperature can give anything other than, at best, a very crude
predictor. The use of the Palmer drought index, as a measure of
moisture availability, as advocated by some authors, would seem
to be a step in the right direction, but not on a calendar month
basis. Further, there is no doubt a differential temperature
response at any given moisture level, provided the latter is
adequate, but what constitutes an adequate moisture level is,
itself, likely a function of temperature. Further, if the
moisture level is adequate, why should additional moisture result
in any response? May there not be a threshhold temperature below
which growth will be impeded whatever the moisture availability?
A more sophisticated physiological approach to relating growth to
climate seems necessary.
Newer Concepts
The most recent development in the analysis of tree-ring
data has been the introduction of the Kalman filter. The method
stems from Kalman (1960) who presented a system for updating and
predicting that is based on the space-state formulation of a
linear dynamic model. It encompasses ARMA models (Box and Jenkins
1970, 1976), standard multiple regression, and regression models
with time-varying parameters (Harvey, 1981). The methodology has
application in other areas but it relevance to dendrochronology
seems not to have been recognized prior to Visser and Molenaar
(1986, 1987) and Van Deusen (1987). It provides a means of simul-
taneously reducing a number of series to a single chronology and
generating climate-based predictions. The climate parameters can
be allowed to vary over time to provide a test of the hypothesis
that conditions in the past are similar to conditions at present.
The essence of the Kalman filter is in two equations, the
"observation" or "measurement" equation,
+ Xt
and the "transition" or "system" equation,
a = G.a w .
in which the state variables a( evolve over time according to
a first-order Markov process. In a dendrochronological
application y is an nt x 1 vector of tree-ring data at
time t, F, is an nt x p matrix of known quantities (e.g. the
values of climate variables) , at is a p x 1 vector of
17

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underlying state parameters, Gt is a fixed p x p matrix,
v and wt are, respectively, nt x 1 and p x 1 vectors
of residuals with zero expectation of covariance matrices Vt
and Wt. Van Deusen(1987) and Visser and Molenaar (1987) provide
examples.
This seem to provide the closest thing to a unified
mathematical framework for analyzing tree-ring series; indeed,
Visser and Molenaar (1987) note the possibility of incorporating
the estimation of growth curves and the effect of weather
variables in the one model. Nevertheless the approach is not
without limitations.
Firstly there is the need to specify, explicitly or
implicitly, a form for the natural growth function; Van Deusen's
(1987) standarization by differencing the logarithms of
successive ring widths is a case of implicit definition.
Secondly, the first-order Markov assumption for the
transition equations is rather strong; put simply, it means that
the situation in the current year is determined, apart from a
random perturbation, solely by the situation in the immediately
preceding year.
Finally, the covariance matrices, Vt and Wt, have to be
known in advance. In general this is not the case and their
"estimation" will involve some element of subjectivity.
It may well be that departures from these assumptions have
little impact on the inferences that result; this possibility
merits investigation.
In addition, there is nothing in this approach that would
resolve the criticism concerning the choice of climate variables;
in short, monthly temperatures and precipitations are no less
inappropriate under this formulation as under any other.
If these and other such difficulties prove inconsequential,
or can be satisfactorily resolved, the amalgamation of
intervention analysis and Kalman filtering appears to have
considerable potential as an aid to answering the current
concerns about the existence and magnitude of any effect of
atmospheric pollution on tree growth. This potential is
highlighted by the apparent success of this mix elsewhere. Thus,
the abstract of the paper "Using the Kalman filter to include
intervention analysis in starmax models" presented by D. K.
Blough, University of Arizona, at the 1987 joint meetings of the
American Statistical Association reads as follows:
"The Kalman-filter approach has been usd in fitting ARIMA
18

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models. This paper uses the multivariate extension of this
approach to model spatial time series which include periodic
interventions whose effects damp out exponentially. Included in
the development of these models will be previously studied
techniques for the inclusion of missing values, aggregate values,
non-linear transformation of the data, covariates and spatial
relationships. An example will be presented in which these models
are applied to the study of boll worm moth counts in cotton
fields over time. The amount of irrigation water in the field is
the covariate and the periodic application of insecticide is the
intervention. Maximum likelihood estimates of the model
. parameters are obtained using a quasi-Newton routine in the GAUSS
programming language on a personal computer."
Wood Density
The statement that the history of a tree's response to its
environment is embodied in its sequence of tree rings is not
meant to apply solely to ring widths. To determine the amount of
woody material actually produced one must consider also the
density of the wood, or the ring mass, the latter being defined
as the product of ring density and ring width. As noted above,
the density is not uniformly distributed across the ring but
normally increases steadily from the low density earlywood to the
high density latewood.
The ability to examine the intra-ring density profile
essentially began with the pioneering work of Polge (1963) in the
application of X-ray densitometry to tree-ring research. Prior to
this some use had been made of beta radiation (Cameron et al.
1959) but it is now generally conceded that, for most purposes,
the X-ray method is superior. Soft gamma rays have also been
employed (Woods and Lawhon, 1974) as well as photogrammetric
methods (Green, 1964, 1965; Elliott and Brook, 1967).
Such techniques permit the measurement not only of the ring
width but also the complete intra-ring density profile. From this
profile one can derive such summary statistics as the proportion
of latewood (and, of course, the widths of the earlywood and
latewood), the average and maximum density of the latewood, the
average and minimum density of the earlywood and, of course, the
whole ring average density. The location within the ring of the
maximum and minimum density also can be recorded. Further any
deviations from the normal profile can be identified. Since the
wood cells are laid down sequentially during the growing season,
such deviations must be in response to some more or less
synchronous environmental phenomena.
Indeed, Fritts (Chapter 2 1976) writes "Considerable
attention has been given in this chapter to the variations that
19

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occur in cell structure within the annual ring. It has been
stated that these variations can be attributed to particular
characteristics of the growing season and the associated
climates. While such small-scale variations in cell size have
little net effect on ring widths, they are an important source of
information that can be measured by densitometric analyses. The
close association between cell size, wood density, and factors of
the environment point to a tremendous amount of information that
must be present in the rings and which should be extractable by
measuring density variations along ring widths. If these new data
on density can be successfully related to and calibrated with
environmental data, the simultaneous analysis of density and ring
width may provide significantly more information on past climate
than is possible using only the widths of rings".
The addition of density to ring width can aid in the
troublesome operation of crossdating. Parker (1967, 1970) reports
the dating of high latitude spruce and larch samples that may not
have been possible from ring widths alone.
Nevertheless, even during the past two decades, most dendro-
chronological studies have been based solely on ring widths.
This, perhaps, reflects the forester's preoccupation with volume
rather than, and sometimes at the expense of, wood quality.
Several facilities for X-ray densitometric measurement of
increment cores exist in North America, principally the
Laboratory for Tree-Ring Reseach at the University of Arizona,
the Lamont-Doherty Geological Observatory, Palisades NY, and
Forintek Canada Corp., Vancouver BC, but only at the last
mentioned does it appear than any extensive densitometric
chronologies have been constructed. Other North American
facilities include the Oak Ridge National Laboratory, Oregon
State University, the University of Kentucky, and Laval
University, Quebec; these, however, have to date made little or
no contibution to the subject. Outside North America the
principal facilities appear to be at the Centre National de
Recherches Forestieres, Champenoux, France, the Swiss Federal
Forestry Research Institute, Birmensdorf, the Oxford Forestry
Institute, England, and the Forestry Research Institute, Rotorua,
New Zealand. Notable conributions to the application of
densitometry to tree rings have been made by workers at all these
four installations.
The influence of environemntal factors can be reflected in
one of more of ring width, and the various density measures
mentioned above. Conkey (1984a,b) reported that several stands of
high elevation red spruce in Maine sampled in 1977 showed no real
decline in ring-width values in recent decades relative to the
200-to-300-year record of growth, but the series of maximum wood
densities from the same trees displayed a flattening of high-
20

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frequency variation and a lower mean value since 1962. This led
her to suggest that wood density may provide an "earl^ warning
signal" of growth decline (Conkey, 1987).
Bodner (pers. comm.) examined the density profiles of trees
in the neighborhood of a point source of pollution that became
active in 1979. He found that, beginning with that date, there
was a clear decrease in maximum density coupled with an increase
in minimum density. An almost identical pattern was observed by
Parker (pers. comm.) with material obtained under similar
circumstances.
Any one of the above density measures can be analyzed in a
manner parallel to ring width, but it would seem appropriate to
handle all variables simultaneously, i.e. construct multivariate
analogs of all or any of the above approaches and, in particular,
intervention analysis and Kalman filtering. There is,
accordingly, a substantial, and seemingly relevant, area for
research that has not, as yet, been adequately exploited.
Other Concerns
For practical reasons most tree-ring analysis has been
performed on increment cores obtained at breast height. It seems
possible, however, that environmental factors would be better
refected in cores taken close to the base of the live crown.
Sample collection of such would, however, be relatively
expensive. Further, not only is the base of the live crown
sometimes difficult to define but also its position advances up
the bole as the tree ages. Some studies based on cores, or discs,
take at various positions along the bole, nevertheless, have
been, carried out, e.g. Parker et al. (1976).
Most samples have come from individual trees, mainly
dominant or codominant if not open-grown, without regard to their
location relative to neighboring trees. Thus, while one may be
able to relate individual tree growth to environmental factors,
the data provide no information on stand behavior. For example,
one may conjecture that, because of genetic variability, some
trees could be more adversely affected by pollution than others.
The more affected ultimately would provide less competition to
any less affected neighbors which, in turn, should respond
positively to the reduced competition. This is a possible
explanation for the behavior of a set of cores presented by
Schweingruber (1987). Of these, some show an abrupt and roughly
simultaneous decrease in ring width, others show no change while
in others there is a suggestion of a synchronous increase. Not
surprisingly, therefore, Schweingruber appears to advocate
looking at trees individually rather that automatically
generating site chronologies. It also highlights the need for
21

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direct evidence of the magnitude of competitive effects on tree
ring measures.
While the temporal variation of ring-width series has
received a great deal of attention, there has been little formal
investigation of their spatial variation. By this is meant, not
necessarily the relative location of sample trees within a site
but, more particularly, the inter-site variation. Ord and Derr
(1987) give an illustration of how the some of the relatively
simple techniques given in, for example Cliff and Ord (1981) may
be utilized, informatively, in this context. Note that Blough,
quoted above, included a spatial component in his study.
Conclusion
Notwithstanding the problems, tree rings provide the best
hope for determining if there is an effect of atmospheric
pollution and, if so, how its magnitude relates to levels of
deposition. The traditional methods of dendrochronology and
dendroclimatology are, however, not well suited to these new
objectives. In this respect, while great credit must be given to
Fritts and his coworkers for their development of much of the
conventional methodology, the uncritical adoption of that
methodology may have done more harm than good. In some instances
it has clearly led to a confidence in interpretations that, in
actuality, cannot be justified.
Alternative methodological ideas have recently found their
way into this arena. Hands-on experience with these techniques in
this context is to date very limited, and it is too early to
determine whether they will provide the hoped-for panacea.
Further, intra-ring density must also assume a role at least
equal to that of ring width.
22

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Appendix
Tree Ring Response of White Oak to Climate and Air Pollution
near the Ohio Valley: A Comment.
William G. Warren
In the study of the response of tree rings of white oak to
climate and air pollution near the Ohio Valley by McClenahen and
Dochinger (1985) the conclusions appear to be very much dependent
on the values of the coefficient of determination, R2, obtained
on fitting response functions in the manner of Fritts (1976).
Specifically, chronologies of standardized ring-width indices
were regressed on the principal components of monthly
precipitation and mean temperature for January through September
of the growth year and May through September of the preceding
year - a total of 14 months or 28 potential predictor variables.
The authors also considered additional variables, namely the
growth of each of the preceding three years; regressions were
performed with these variables included and excluded. A forward
selection procedure was used with variables included as long as
the F ratio for the next component to enter was greater than
unity. Response functions were obtained for the years 1901-1930,
1931-1978 and 1900-1978. When prior growth was not included the
number of principal components that entered ranged from 8 to 23
and was most commonly from 12 to 16. Our focus will be on these
cases.
Although McClenahen and Dochinger did not make any claims of
formal statistical significance, it is clear that any inferences
must be dependent on the results departing from what could be
reasonably expected under the null hypothesis, namely the
hypothesis that there is no association between the growth (as
measured by the standardized ring-width indices) and the climatic
factors, or the principal components thereof.
An often unappreciated fact is that, in a forward selection
procedure, the null distribution of R2 (of F) is not the standard
distribution as widely tabled. One is not then dealing with a
random value of R2 but with the maximum value of a set of random,
but not mutually independent, values of R2, so that the expected
value and the critical values for an alpha level test (alpha
small, commonly 5%) of the null hypothesis are, in general,
substantially greater than the tabled values. (See e.g. Draper et
al. 1971, Pope and Webster 1972).
The same applies to the multiple R2 obtained by a forward
selection procedure. Because of the intrinsic theoretical
difficulty little work on this topic has appeared. Rencher and
Pun (1980) nevertheless provide Monte Carlo estimates of the
expected value and 95th percentile of R2 for a limited number of
combinations of sample size, n, number of potential predictors,
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k, and the number of variables selected, p. They assume that the
regression variables are uncorrelated as is the case when the
principal components of the set of potential predictors are used.
Their results are based on 500 to 2000 realizations.
Interpolation and modest extrapolation from the Rencher and
Pun table gives
Table 1.

2
4
P
6
8
10
30: E(R2)
= .296
.461
.570
.651
.716
R. 95
= .430
.606
.716
.784
.839
48: R(R2)
= .188
.295
.370
.426
.471
R. 95
= .275
.405
.493
.568
.620
For the regressions with prior growth excluded McClenahen
and Dochinger give
Table
2




1
2
Site
3
4
5
1901-30: n = 30: p =
23
15
19
21
19
R2 =
.972
869
.965
.985
.960
1931-78: n = 48: p =
8
10
13
15
12
H
CN
«
.529
484
.639
.576
.609
The values of Tables 1 and 2 are plotted in Figs. 1 and 2.
Although the Rencher and Pun Tables do not extend beyond p = 10
it is clear in Fig. 2 that, while all the R2 values obtained by
McClenahen and Dochinger fall above their expected values (i.e.
their theoretical means) they all fall below the 95th percentile,
i.e. are less than the critical value for a 5% level test and
would commonly be declared "not significant".
In Fig. 1 there is virtually an additional point since when
p = ]c = n-1, R2 = 1.0 (assuming that an intercept is fitted). The
dotted lines between p = 10 and p = 28 are conjectural but one
can be confident that while all the McClenahen and Dochinger
values of R2 again fall above their expected values, with one
possible exception, they again all fall below the 95th
percentile. If, indeed, the R2 for Site 4 exceeds the 95th
percentile it does not do so by much.
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A similar pattern arises for 1900-1978 but is not presented
because of the uncertainty in extrapolating the Rencher and Pun
table to n = 79.
Thus, for both periods, 1901-30 and 1931-78, the pattern of
results does not seem too far removed from what would be expected
under the null hypothesis. In only one out of ten cases is an R2
possibly significant at the 5% level and there is about a one in
three chance of such spurious significance arising in ten trials.
Thus, notwithstanding the fact that all R2 values are somewhat
greater their expected values, there is little that can be
construed as definitive evidence of a relationship between ring
width and the climatic factors as measured.
More importantly, perhaps, is the fact that the pattern of
the R2 values in relation to what would be expected appears to
differ negligibly between the two periods.
The choice of F > 1.0 as a selection criterion in forward
inclusion seems rather foolhardy. Since the expected value of an
F statistic is d/(d-2), where d is the number of degrees of
freedom of the denominator, one must expect that a goodly
proportion of variables with no predictive ability would be
included.
In summary, therefore, it seems legitimately questionable
whether the climatic variables in the McClenahen and Dochinger
study have any predictive ability and, thus, whether inferences
based on the resulting response functions have any validity.
References
Draper, N.L., Guttman, I. and Kanemasu, H. 1971. The distribution
of certain regression statistics. Biometrika 58:295-298.
Fritts, H.C. 1976. Tree Rings and Climate. Academic Press, New
York.
McClenahen, J.R. and Dochinger, L.S. 1985. Tree ring response of
white oak to climate and air pollution near the Ohio River
Valley. J. Environ. Qual. 14:274-280.
Pope, P.T. and Webster, J.T. 1972. The use of an F-statistic in
stepwise regression procedure. Technometrics 14:327-340.
Rencher, A.C. and Pun, F.C. 1980. Inflation of R2 in best subset
regression. Technometrics 22:49-53.
iii

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Fig. 1 1901-1930 No Previous Growth
1.1
1
0.9
- 95th percentile
0:8
0.7
0.6
Expected
0.5
0.4
0.3
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

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Fig. 2 1931—1978 No Previous Growth
0.8
0.7
95th percentile
0.6
0.5
0.4
Expected
0.3
0.2
0.1
0
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

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