Measuring Compositional Change along GradientsAuthor(s): Mark V. Wilson and C. L. MohlerSource: Vegetatio, Vol. 54, No. 3 (Nov. 11, 1983), pp. 129-141Published by: SpringerStable URL: http://www.jstor.org/stable/20146004 .Accessed: 04/03/2011 11:22

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Measuring compositional change along gradients

Mark V. Wilson1 & C. L. Mohler2'* 1 Environmental Studies Program, University of California, Santa Barbara, CA 93106, U.S.A.

2 Ecology and Systematicsf Cornell University, Ithaca, NY 14853, U.S.A.

Keywords: Beta diversity, Gradient analysis, Multivariate methods, Niche, Ordination, Siskiyou Mts.,

Species distributions, Species turnover, Succession

Abstract

A new procedure for measuring compositional change along gradients is proposed. Given a matrix of

species-by-samples and an initial ordering of samples on an axis, the 'gradient reseating' method calculates 1)

gradient length (beta diversity), 2) rates of species turnover as a function of position on the gradient, and 3) an

ecologically meaningful spacing of samples along the gradient. A new unit of beta diversity, the gleason, is

proposed. Gradient rescaling is evaluated with both simulated and field data and is shown to perform well

under many ecological conditions. Applications to the study of succession, phenology, and niche relations are

briefly discussed.

Introduction

The observation that community composition varies along environmental gradients is basic to

much work in community ecology. Gradients not

only form the context for many field studies (see Whittaker 1967, 1972) but figure heavily in much

theoretical work on niche relations (Hutchinson, 1958; McNaughton & Wolf, 1970; Whittaker et al,

1973). Despite the large role of compositional gra dients, or coenoclines as they are sometimes called

(Whittaker, 1967), statistical methods for the anal

ysis of continuous compositional variation remain rather primitive.

We address three related problems. First, one

may want to know the biological length of some

coenocline, which is to say, the total amount of

compositional change associated with the gradient. Such a measure is particularly useful when compar

ing two or more gradients (Whittaker, 1960; Peet,

1978). For example, if Janzen (1976) is correct in

believing that mountain passes of given elevation are biologically 'higher' in the tropics, then the amount of compositional change from sea level to

the pass should be greatest at tropical latitudes and decline toward the poles. Whittaker (1960, 1965, 1972) refers to compositional differentiation along

gradients as beta diversity, in contrast to the alpha diversity or richness of a single sample. He and

subsequent authors have measured beta diversity in units of half-changes (HC), analogous to the half life of radioactive elements. Roughly speaking, a

coenocline of 1 HC has endpoints which are 50%

similar, a coenocline of 2 HCs has endpoints which are each 50% similar to the midpoint, etc. However,

measurement of coenocline length in half-changes is complicated by apparently random composition al variation among replicate samples, henceforth

referred to as 'noise', by the non-linear relation

* The authors thank E. W. Beals, R. Furnas, P. L. Marks, R. K.

Peet, O. D. Sholes and the late R. H. Whittaker for helpful comments. This work was supported in part by a National

Science Foundation grant to Robert H. Whittaker of the Section of Ecology and Systematics at Cornell University, and in part by

Mclntire-Stennis Grant No. 183-7551 and a grant from the

National Park Service, both to Peter L. Marks of the Section of

Ecology and Systematics at Cornell University.

Vegetatio 54, 129-141 (1983). ? Dr W. Junk Publishers, The Hague. Printed in the Netherlands.

130

between similarity measures and sample separation

along the gradient (Gauch, 1973), and by the pres ence of other, unmeasured gradients. To our

knowledge, previous workers have not attempted to test the accuracy of Whittaker's procedure for

determining beta diversity in half-changes, and al

ternative procedures are few.

Second, one may want to compare rates of spe

cies turnover at various points on a gradient, for

example, to define objectively ecotones between

community types (Whittaker, 1960; Beals, 1969) or

to locate periods of rapid change during a succes

sional or phenological sequence. Turnover rate

may be expressed as beta diversity per unit gradient as measured over an infinitesimal length of gra dient. (Conversely, beta diversity is turnover rate

integrated over the gradient (Bratton, 1975).) Whit

taker, Beals & Bratton all estimated compositional turnover using the measures percentage similarity

(PS) or percentage difference (PD=100-PS)

(Goodall, 1978). This approach may lead to prob lems because: 1) random variation in composition contributes to PD; 2) non-monotonic change in

species abundance may cause PD to underestimate

compositional change; and 3) spacing of samples

along the gradient reflects the investigator's percep tion of the environment rather than that of the

species which compose the coenocline.

Third, one may want a natural, species-defined

spacing of samples. In general, any ordination pro

duces a species-based but undefined spacing; in this

paper we consider in particular the spacing defined

explicitly by a constant species turnover rate along the gradient. When turnover rate is constant the

separation between samples is an expression of

sample distinctiveness, that can be used in weight

ing niche metrics (Colwell & Futuyma, 1971; Pie

lou, 1972). Furthermore, if habitat axes are rescaled

in beta diversity units, breadth and overlap meas

ures for different systems can be compared and

multi-dimensional niche metrics can be computed

directly rather than by the addition or multiplica tion of several one-dimensional measures (cf. Le

vins, 1968; Pianka, 1973; Cody, 1974). We believe the technique presented here, gra

dient rescaling, resolves the problems of measuring

compositional change, rates of species turnover,

and axis scale. The gradient rescaling approach is

treated in the following sequence. First, we give the

rationale of using species turnover as a basis for

gradient analysis and develop the computational procedures for measuring beta diversity in half

changes. Second, we describe our evaluation of the

technique using simulated data sets. Third, we illus trate the use of the technique with field data. Final

ly, we discuss and compare methods and mention some applications of our approach to the study of

compositional change along gradients.

Gradient rescaling

Rationale

The relationship between beta diversity, rate of

species turnover, and gradient scale may be ex

pressed as:

b

?=S R(x)?x (1) a

where ? is the beta diversity of the compositional gradient between points a and b, R is the species turnover rate and x are values along X, the initial

scaling of the gradient. Field measurements of en

vironmental factors along the gradient X are, of

course, in a scale chosen by the investigator (e.g.,

elevation in meters or acidity in pH units). The

choice of scale must be arbitrary, reflecting the

investigator's notion of what is easy, traditional or

interesting to record. Since the investigator's scale of measurement does not necessarily reflect how

organisms respond to changes in environmental

conditions, several authors have proposed rescaling gradient axes using species distributions as an

'ecoassay' (e.g. Colwell & Futuyma, 1971) of the environmental gradient. Gauch et al (1974) and Ihm & Groenewoud (1975) proposed scaling axes

such that species distributions most closely ap proximate Gaussian curves. Austin (1976) dis counted the generality of Gaussian curves in nature

and used the more flexible beta distribution of sta

tistics in an algorithm similar to that of Gauch et al

(1974). In contrast to these curve-fitting ap

proaches, Hill & Gauch (1980) scaled reciprocal

averaging ordination axes so that, on the average,

the species in each sample have distribution curves

of unit standard deviation. Our approach is based on the view that composi

tional turnover is the essence of community gra dients, and that environmental change is ecological

131

ly significant primarily to the extent that it influen

ces the relative abundances of species. Accordingly, we scale community gradients so that the rate of

compositional change is constant throughout. In

this scaling, each portion of the gradient is ecologi

cally equivalent to all other portions. This scaling allows the biological length of the compositional

gradient and the rate of species turnover along

physically-defined gradients (e.g., elevation), to be

accurately calculated. Our approach starts with the

relationship between gradient scale, rate of turn

over and beta diversity shown in equation (1). There are three phases. We first find a new scaling,

X', such that R(x') ? k, for all x\ where k is a

constant. That is, we find a new set of relative

sample positions such that the rate of species turn

over along the gradient is constant. Second, we

calculate the compositional length or beta diversity of the gradient as:

? = k - (b

- a)

where b - a is the distance between endpoints in

terms of the scaling X'. The value of ? is independ ent of the initial scaling. Finally, R*(x), the rate of

compositional change along the original gradient, is computed by using the original scaling to trans

form R(x'). Details of this procedure are presented below.

Early in the development of gradient rescaling we

devised a measure of beta diversity which is an

alternative to the half-change. This we name the

gleason, in honor of H. A. Gleason, who first point ed out the importance of continuous compositional change along environmental gradients (Gleason,

1926). One gleason (G) is the amount of composi tional turnover which would occur if all changes

were concentrated into a single species whose

abundance changed 100%. For example, a 1 G

coenocline might consist of three species, two of

which change from 50% to 0% relative abundance as the third species changes from 0% to 100% rela

tive abundance. There is no general function which

can interconvert gleasons and half-changes. In the

following sections we develop computational

procedures for both measures.

Computations

We start with field observations of species abun

dances within a sequence of samples, X, along an

ecological gradient. This gradient can be either a

direct environmental gradient or an indirect, com

positional gradient derived by ordination. We con

sider two measures of species turnover or composi

tional change: half-changes and gleasons. Compu tations for both measures are performed by a

FORTRAN program, GRADBETA, which is

available with documentation from Cornell Ecol

ogy Programs, Ecology and Systematics, Cornell

University, Ithaca, NY 14853 (Wilson & Mohler,

1980). For either gleasons or half-changes species im

portance values should be standardized so that

sample totals are equal. After this standardization, measures of compositional change are measures of

proportional turnover. In this way, the influence of

trends in total biomass are minimized in the inter

pretation of compositional turnover.

Field data always include variation due to sample error or the effects of chance and uninvestigated

ecological factors. Such noise causes the observed

rate of compositional change to be higher than the

change attributable to species distributions along

recognized gradients. Noise is often ignored in eco

logial analyses, but this practice may lead to misin

terpretation of ecological patterns. A convenient index of noise is internal associa

tion (I A), the expected similarity of replicate sam

ples. IA decreases as the average noise level of data

increases. Any index of compositional similarity can be used to estimate IA; our technique uses

percentage similarity (Whittaker, 1975; Goodall,

1978). Replicate samples are equivalent to samples

whose separation along the gradient is zero. There

fore, in a plot of sample percentage similarity of

each pair of samples versus the gradient separation of each pair, an estimate of IA may be obtained by

extrapolating percentage similarity to zero separa

tion (Whittaker, 1960). Whittaker accomplished this by fitting a straight line to the central portion of

a plot of the logarithm of percentage similarity versus gradient separation. We make two refine

ments. First, extrapolation is accomplished by line

ar least-squares regression instead of by a hand-fit

ted line. Second, instead of an arbitrary spacing between pairs of samples, we use the ecologically

more meaningful gradient distance derived by the

gleasons or half-changes procedure.

132

Half-change s

Whittaker (1960) defined the half-change as the

ecological distance between two samples with sim

ilarity of species composition 50% that of two repli cate samples. The Whittaker method for measuring the beta diversity of an entire gradient is based on

the posited relationship:

u? ., l\o%(lA)-\og(PS(a,b))\ HC(a, b) ? -

log2

where PS(a,b) is the percentage similarity of sam

ples a and b, I A is the expected similarity of repli cate samples, and HC(a,b) is the beta diversity of

the gradient segment from a to b, in half-changes. This half-change formula implies two basic as

sumptions. The first assumption is that sample sim

ilarity is negatively exponentially related to ecolog ical distance as measured in half-changes, so that as

ecological distance increases the rate at which sam

ple similarity decreases is constant (cf. Gauch,

1973). Whittaker (1960, 1972) presents data sets

that support this assumption. The calculations of

half-changes in our technique are also based on this

exponential relationship. The second assumption is

that samples are evenly spaced along the ecological

gradient. This assumption, which is not met by most field data, is avoided by our technique.

In outline our method is as follows. First, an

index of compositional turnover rate is computed for each sample point on the gradient and the

among-sample variance in this index is determined.

The turnover index used in the procedure is basical

HC(a,b) ly the average of-for all samples b in

Ax(a,b) some vicinity of a, where HC(a,b) is the number of

half-changes between samples a and b9 and Ax(a,b) is the gradient distance between a and b. The ratio

HC(a,b) -is the slope of a vector in a 2-space

Ax(a,b) whose axes are gradient distance and composition

al distance (in half-changes). The appropriate aver

age for the ratios is thus the slope of the vector sum.

Second, using a path-of-steepest-descent algorithm

(Berington, 1969), the sample positions are adjust ed iteratively until the variance in turnover rate

has reached a local minimum. This produces the new scaling, X'. Third, the final turnover rate and new sample positions are used to compute a set of

corrected turnover rates for the original scaling and a set of

pascal beta diversity values corresponding

to the intervals around each sample. Total beta

diversity is then computed as the sum of the partial beta diversities. See Wilson & Mohler (1980) for

details of the numerical application of these proce dures.

To compute the partial and total beta diversity values and the turnover rates with respect to the

original gradient we first define an interval, /, about

each of the final sample positions:

/(/) = (x'(2)-*'0))/2 j=l

(*'(/+l)-x'(/-l))/2 j=2,...,n-\ (2)

(x'(n)-x'(n-\))ll j=n

Partial beta diversities, BHC(j), are then computed as:

BHCU) = R(xV))-iU)

Thus the beta diversity associated with a sample is

simply the turnover rate at that point on the re

scaled gradient times the gradient interval associated

with the sample. Total beta diversity, ?//^, of the

gradient is then the sum of the partial beta diversi

ties. If var R = ? exactly, then R(x'(j)) = k for ally,

and ?HC = k [x'(n)

- x'(l)].

To compute the turnover rates for the original

scaling of the gradient we divide the above partial beta diversities by the intervals for the original scal

ing. These corrected rates, R*(x(j))9 are:

1BHC(\) (x(2)-x(\))

y=i

2BHr(j) /?*(*(/)) =

<-?m-y = 2,.. ., n - 1 (3)

2BHC(n)

(x(n)-x(n~ 1)) j = n

Equation (3) gives rates of compositional turnover at the position on the gradient of each sample.

Gleasons

A second natural measure of compositional

change is the amount that species importance changes from one point on a gradient to another,

summed over all species. The unit of change meas

ured in this manner we call the gleason. Rate of

change in terms of gleasons is defined as:

133

R(x) = X ?X

= (l/d*)2|d?,.|

where f? is the expected abundance of species i as a

function of gradient position, x. Substituting R(x) into equation (1) yields

b

G(a,b) = SZ\dYi\ (4) a i

where G(a,b) is the beta diversity in gleasons of the

gradient interval between points a and b.

When a and b are near each other, the integral in

(4) may be approximated as:

G(a,b) = X\AYfab)\ i

where A?fab) = Y ?(b)

- Y ?a). When the two

points are farther apart, however, some species may

rise and fall in abundance within the interval. In this

case 2 \Affab)\ does not describe the full change

in the abundance of species and thus G(a,b)>

X\AYfab)\. i In practice, the expected change in importance of

species /, A?z, is estimated from observations at

discrete samples along the gradient. To minimize

the underestimate of G caused by non-monotonic

species response, G is first computed for pairs of

adjacent samples and then the beta diversity of

longer intervals is obtained by summing these small

elements. In general, species show some random

variation in abundance not related to the gradient. Thus, the compositional change from one sample to

the next which is attributable to the gradient may be

viewed as the total difference in species abundance

in the two samples minus that component of the

difference which is due to noise. That is,

G(j,j+l) =

X\AY?jJ+ Dlob8crvcd-2|Alrl</J+ Dlnoisc

where A Yt(j,j + 1) is the difference in abundance of

species / in samplesy andy + 1 and the samples are

numbered consecutively with respect to the gra dient. Since we use species abundance values, Yi9 that have been standardized so that sample totals are equal, percentage similarity (PS) can be calcu lated as PS(j,j + 1) = 1 - y2X | Y{(j,j + 1) |. This leads to:

21 Y?UJ+ Dlobserved = 2-2 PS{jJ+ 1).

Moreover, since j andy'+ 1 are nearby on the gra

dient it is reasonable to assume that

2| Ay/(/j+ Dl ?oise = 2-2(//l), i

and thus G(j,j+ 1) can be estimated as

G{j,j+\) = 2[IA-PS{jj+\)l

The beta diversity of the whole gradient in gleasons,

?G, is then

?G = \ G(j,j +1) = 2W2 [IA

- PSUJ + 1)].

.7=1 7=1

Rescaled sample positions, x\ are derived from

knowledge that the rate of species turnover along the rescaled axis is constant:

GUJ+i) R(x') =-?-=k

or, equivalently,

*V+ D - (r) UA -

PSUJ+ 1)] + x'U). (5) k

To compute x(j) we first set the beginning of the axis

equal to zero, x(l) = 0, or some other convenient

value, and choose an arbitrary value for k. In practice we set k ? 1 since this causes x'(n) -x'(\) =

?G, and thus sample positions are in units of gleasons. The

sample positions, x'{j) for j = 2, 3,..., n, are then

computed recursively using equation (5). The partial beta diversity, BG(j), attributed to the

neighborhood around sample y is calculated as,

( G(U2)/2 j=\ BGU)= [GU-Uj)+GUJ+W2j

= 29...,n-l

{ G(n -

1, n)/2 j ? n

Note that XBG(j) =

?G. Corrected rates of change, j

measured as gleasons per original gradient unit, are

then calculated as

BrU) **(*(/))= -TT 7=l,2,...,/i,

where the intervals, /(/'), of neighborhoods around the

samples, are defined by equation (2). Thus, ?G, X' and R* can be calculated simply and

directly, without the iteration step required by the

half-changes procedure.

134

Evaluation

Gradient rescaling was developed for the analysis of ecological communities. In the field it is often

difficult to determine how many samples should be

taken and how well samples have been allocated to

different segments of ecological gradients. Sam

pling schemes and characteristics of the study sys tem also influence the noise level of data. Any suc

cessful method of community analysis, including

gradient rescaling, should be robust under a variety of field conditions. Although we have applied the

procedure to many field data sets the following evaluation relies primarily on simulated data, be cause only with a simulation can the true values of

estimated parameters be specified for purposes of

comparison.

Evaluation methods

There were four steps to the evaluation proce

dures. First, we constructed simulated data sets of

species distributions along gradients, both with and

without noise. Second, using a priori knowledge of

species distributions from the simulations, we cal

culated exact values of the rescaled sample posi tions, the corrected rates of change, and beta diver

sity. Third, we 'sampled' the simulated data at

different points along the gradient, and analyzed the resulting data using our gradient rescaling tech

nique. Fourth, the results of the gradient rescaling of the sampled data were compared to the true

values derived from the original simulations.

Gradients of species distributions were simulated

using a modification of CEP-1 of Gauch & Whit

taker (1972). This program constructs species dis

tributions as Gaussian curves along a gradient.

Species curves were drawn at random from a wide

range of peak abundance and breadths of distribu

tion, and were placed randomly along the gra dients. Modifications included a noise algorithm that, for each species distribution curve, produced noise proportional to abundance, and a routine

that added species until total abundance along the

gradient was constant. Gradients were simulated

with different ecological lengths and with different

noise levels. For our main experiment, 18 gradients were simulated in a factorial design, as follows:

three ecological lengths (short, medium and long;

approximately 1.5, 3.0 and 6.0 half-changes), two

noise levels (I A values of 90% and 70%), and three

replicates. For each of the 18 gradients the corre

sponding noise-free data set represented the underly

ing ecological conditions of the gradient. For our evaluation of the effects of sampling

intensity nine gradients were simulated: three eco

logical lengths (approximately 0.7, 3.0 and 10.0

half-changes) with three replicates each. These nine

gradients were simulated in noise-free and moder

ate noise versions (IA values of 100% and 80%). Each of the simulated gradients was 'sampled' by

computer at certain locations along the gradient to

mimic extremes of sample placement that could be

encountered in field work. In the main experiment

(Fig. 1), 15 samples were placed in four schemes:

evenly spaced, grouped at both ends, grouped in the

center, and grouped at one end only. These schemes

were the basis for the true or underlying sample

spacings. In each analysis by gradient rescaling this

sample scheme information was treated as un

known, just as the true relationships among sam

ples would be unknown in a field study. Thus, the

naive view that samples were evenly spread along the gradient was assumed for the sake of evaluating the technique. In the intensity experiment the inter

vals between samples was constant but the intensity was set at 3, 8, 15, 29 or 57 samples per gradient. The data on species abundance within samples were

then analyzed by our gradient rescaling technique,

using the parameter values and procedures recom

mended in Wilson & Mohler (1980). Results from

these analyses of the simulated field data were then

compared with the true values as determined from

knowledge of the noise-free species distributions.

I | 9 9 9 9 9 9 ? ? ?

| I | 9 9 ?

|W HI ? ?? 9 9 9 9

Fig. I. Sample placement schemes along the simulated gradients for analysis by gradient rescaling. I - evenly spaced, II - clus

tered at both ends, III - clustered at the center, IV - clustered at

one end. Each scheme has 15 samples.

135

Evaluation results

Gradient rescaling gave accurate estimations of

beta diversity under most combinations of gradient

length and noise level. Table 1 presents the average

proportional error of estimation, for both half

changes and gleasons, when sampling was from 15

evenly spaced locations along the simulated gra dient. Gleason estimations are nearly 100% accu

rate for the noise-free gradients. In contrast, the

estimation of half-changes for the low-noise gra

dients was better than for the noise-free gradients.

Very poor estimations in gleasons or half-change occurred only for the short gradients with high noise. Evidently the signal, in this case the patterns of species turnover, was swamped by the noise, the

random variations in species abundances. Without

noise, estimation for gleasons was better than for

half-changes, but with either low or high noise es

timation was better for half-changes. These con

trasting results may reflect the development of glea sons from theoretical considerations of composi tional gradients compared to the lineage of half

changes as an empirical technique. The ability to estimate beta diversity is affected

by the accuracy of assumptions concerning under

lying sample spacing. Table 1 also shows accuracy of estimation of beta diversity by the gradient re

scaling technique when the underlying sample posi tions followed three extreme distributions (see

Fig. 1). Patterns of estimation accuracy of both

half-changes and gleasons followed those seen for

evenly spaced samples, with best estimations of

gleasons for noise-free gradients and for longer

gradients, and best estimations of half-changes with

low noise gradients (Table 1). Overall, differences

among the four underlying sample distributions did not greatly influence estimation of beta diversity as

either half-changes or gleasons. That is, poor

knowledge of true sample spacing would not nor

mally be an impediment to the accurate estimation of beta diversity by the technique of gradient rescal

ing. A possible exception to this general robustness

involved data sets with noisy samples clustered to

ward the center of short gradients. Again, the signal to noise ratio is low for gradients with high noise.

Center-clustered sampling schemes are also least

useful in ordination (Mohler, 1981) and for extract

ing information about species distributions

(Mohler, 1983). Table 2 reports the ability of gradient rescaling to

recover the three types of underlying sample distri

Table 1. Accuracy of estimation of beta diversity by the gradient rescaling technique. The index is the average proportional error:

average of | estimation-true | / true. Short, medium and long refer to the relative lengths of the simulated gradients, approx. 1.5,3.0, and

6.0 half-changes, resp. Noise levels correspond to internal associations of 100%, 90%, and 70%. Sampling was at 15 locations in four

distributions: evenly spaced, end-clustered, center-clustered, and clustered at one end (see Fig. 1).

Gradient

length

Noise

level

Index of accuracy

Half-changes

Sample distribution

Even Ends Center One end

Gleasons

Sample distribution

Even Ends Center One end

short

medium

long

none (1)*

none(h)* low

high

none (1)* none (h)* low

high

none (1)* none (h)* low

high

0.14

0.09

0.09

0.47

0.05

0.05

0.02

0.21

0.03

0.03

0.02

0.09

0.15

0.09

0.06

0.09

0.07

0.07

0.02

0.25

0.03

0.04

0.02

0.11

0.14

0.10

0.32

0.66

0.02

0.03

0.02

0.34

0.02

0.02

0.04

0.07

0.14

0.09

0.09

0.22

0.06

0.05

0.01

0.26

0.03

0.02

0.03

0.07

0.00

0.01

0.41

0.83

0.01

0.00

0.22

0.19

0.02

0.03

0.06

0.16

0.00

0.01

0.61

0.63

0.02

0.01

0.21

0.26

0.04

0.06

0.09

0.33

0.00

0.01

0.39

0.85

0.01

0.01

0.23

0.47

0.04

0.03

0.07

0.11

0.00

0.01

0.55

0.65

0.02

0.01

0.27

0.09

0.04

0.05

0.08

0.12

* The first noise-free gradient was used to simulate the low noise gradient; the second was used for the high noise gradient.

136

Table 2. Ability of gradient rescaling to recover three types of underlying sample spacings: end-clustered, center-clustered, and clustered

at one end (see Fig. 1). A low ratio of convergence means good recovery, a high ratio means bad recovery. The simulated gradients are

the same as in Table 1. See text for more explanation.

Gradient

length

Noise

level

Ratio of convergence

Half-changes

Sample distribution

Ends Center One end

Gleasons

Sample distributions

Ends Center One end

short

medium

long

none (1)* none (h)* low

high

none (1)* none (h)* low

high

none (1)* none (h)* low

high

0.15

0.17

0.56

1.59

0.18

0.10

0.32

1.02

0.08

0.07

0.14

0.48

0.33

0.22

0.69

1.48

0.23

0.36

0.28

1.18

0.15

0.16

0.27

0.41

0.10

0.10

0.39

0.67

0.12

0.08

0.17

0.54

0.08

0.05

0.08

0.19

0.01

0.01

0.79

1.47

0.03

0.02

0.50

0.92

0.04

0.07

0.32

0.75

0.02

0.02

0.64

1.03

0.05

0.01

0.38

0.91

0.07

0.07

0.28

0.71

0.01

0.01

0.91

0.88

0.01

0.01

0.45

0.96

0.03

0.04

0.25

0.67

* The first noise-free gradient was used to simulate the low noise gradient; the second was used for the high noise gradient.

butions discussed, given the initial assumption that

samples are evenly spaced along the gradient. Suc

cessful recovery, or convergence, was measured as

the ratio of: 1) the mean absolute difference be

tween locations of the samples after rescaling and

their true, underlying positions to 2) the mean abso

lute difference between locations of the distorted

sample positions and the true positions. Thus, a

ratio of convergence of 0.0 indicates that true sam

ple positions have been perfectly revealed by gra dient rescaling; a ratio of 1.0 indicates the gradient

rescaling did not offer any advantage over the

naive, even spacing. As with estimating beta diver

sity, the gleasons procedure did well on the noise

free gradient of all lengths and with all underlying

sample distributions. Sample positions were recov

ered less well with the low and high noise gradients,

although ratios of convergence were lower for low

noise and longer gradients. Performance on the

short, noisy gradients was unsatisfactory; in these

cases gradient rescaling produced little or no im

provement on the initial even spacing. Underlying

sample distribution seemed to have little overall

effect on the ratio of convergence with the gleasons

procedure.

The half-changes procedure on the noise-free

gradients adequately recovered the true sample po sitions, but still did not do as well as did the glea sons procedure. With all low noise gradients and the high noise, long gradients, however, the half

changes did well or very well at converging to the

underlying sample distribution. Convergence was better with the longer gradients. In contrast to the

results with gleasons, the particular form of the

underlying sample distribution did make a signifi cant difference in the ability of the half-changes

procedure to recover sample positions. In general,

the distribution with samples clustered at one end had the best ratios of convergence, especially for the

high noise gradients. The number of samples collected to represent a

study system should have some effect on the success

of analysis. Many workers attempt to collect as

many samples as can reasonably be accomplished.

For estimating beta diversity, however, interme

diate sampling (n = 15) gave the best overall results

(Table 3). The sparsest sampling (n = 3) did the

worst at estimating beta diversity and intensive

sampling (n = 57) was not advantageous. With noi

sy gradients best estimation of beta diversity was

accomplished by sampling short gradients less in

tensively and by sampling long gradients more in

137

Table 3. Accuracy of estimation of beta diversity by the gradient

rescaling technique with different numbers of samples. The

accuracy index is the average proportional error: average of

estimate-true/true. Very short, medium and very long refer to

the relative lengths of the simulated gradients (approx. 0.7, 3.0,

and 10.0 half-changes, resp.). Noise levels correspond to

internal associations of 100% and 80%. For each sampling

intensity samples were spaced evenly.

Gradient

length

Noise

level

n Index of accuracy

Number of samples

3 15 29 57

very short

medium

very long

very short

very long

none

moderate

none

moderate

none

moderate

none

moderate

0.22

0.20

0.10

0.37

Half-changes

0.18

0.31

0.19

0.51

0.19

0.69

Gleasons

0.02

0.72

0.01

0.85

0.01

0.68

none

moderate

0.75

0.75

0.24

0.24

0.15

0.23

0.16

0.37

0.20

0.67

3 0.20 0.15 0.14 0.13 0.18

3 0.36 0.13 0.12 0.32 0.34

3 0.19 0.13 0.08 0.10 0.14

3 0.24 0.15 0.10 0.10 0.12

0.01

1.13

medium none 3 0.35 0.12 0.07 0.01 0.17

moderate 3 0.45 0.15 0.26 0.37 0.89

0.28

0.83

tensively, indicating an optimal sampling intensity of perhaps 3 samples per beta diversity unit.

Of the gradient characteristics varied in the eva

luation - beta diversity, noise level, and sample

spacing -

noise level had the largest effect on the

performance of gradient rescaling, indicating the

importance in field studies of using sampling tech

niques and methods of data summarization that

minimize noise. Gradient length was not an impor

tant factor except when high noise levels occurred

in short gradients; this combination produced poor results. All but extremes of sampling intensity re

sulted in good performances by gradient rescaling. From these evaluation results we conclude that beta

diversity and underlying sample positions can be

accurately measured by the technique of gradient

rescaling.

Comparisons with other methods

During this study we converted Whittaker's

(1960) original method of measuring beta diversity to a more objective linear regression technique (see

Computations), which successfully captured the ra

tionale of the original (R. H. Whittaker, personal communication). Like our technique of gradient

rescaling, Whittaker's method measures beta diver

sity, but unlike gradient rescaling it was not de

signed to give information on proper sample posi tions with respect to the gradient or on rates of

ecological change at particular sample positions. Our formalization of Whittaker's method was ap

plied to each of the simulated gradients discussed

above, and the beta diversity results compared to

the performance of the half-changes and gleasons

procedures of gradient rescaling. As expected, Whittaker's method performed best with those gra dients in which the samples were evenly spaced and

less well with distorted sample distributions. In

general, the half-changes procedure of gradient res

caling was superior to Whittaker's method except when sampling intensity was very high. Whittaker's

method usually did better than the gleasons proce dure at estimating beta diversity for gradients with

noise but did appreciably less well than the gleasons

procedure with noise-free gradients. In gradient rescaling, the effects of errors of esti

mation of internal association are compounded be cause the half-changes and gleasons procedures both calculate overall turnover by accumulating individual turnover values between samples. This

compounding of errors is responsible for the re

duced accuracy of gradient rescaling at the highest

sampling intensity (Table 3). Whittaker's method, in contrast, examines only overall turnover and

should be less sensitive to errors in estimating I A.

Despite this sensitivity of gradient rescaling, the

half-changes procedure, in particular, calculates

the true total turnover better than does Whittaker's method when sample spacing is not uniform.

Detrended correspondence analysis (DCA; Hill

& Gauch, 1980) is a recently developed ordination

technique, derived from reciprocal averaging (Hill, 1973). Some of the issues which we have addressed

in this paper -

calculating gradient length and re

scaling gradients - are also dealt with by DCA. In a

subsequent paper we will evaluate the performance of DCA in these two areas, and will compare DCA with our gradient rescaling technique.

138

Field example

We applied the gradient rescaling technique to

field data on the distribution of tree species along an elevation gradient on quartz diorite substrate in

the Siskiyou Mountains of Oregon, U.S.A. (Whit taker, 1960). The data were densities of stems great er than 1 cm dbh (diameter at breast height) within 6 elevation belts, which were centered at 610 m, 915 m, 1220 m, 1 525 m, 1800 m, and 2 025 m. Thir

ty-seven tree species were recorded.

These data of tree densities within elevation belts were analyzed using the gradient rescaling proce dures described above. Values for beta diversity, rescaled sample positions and corrected rates of

ecological change were obtained for both gleason and half-change units. Results for gleasons and

half-changes closely paralleled one another and on

ly the half-change results will be reported here.

When measured by our gradient rescaling me

thod the Siskiyou elevation gradient has a beta

diversity of 4.0 HC. In contrast, Whittaker's (1960)

original methods yield beta diversity values of ca

6.9 HC, an apparently large overestimation. Spe cies turnover of 4.0 HC indicates a significant amount of ecological change over the elevation

change of 1415 m. The comparison of beta diversity

along the Siskiyou elevation gradient with that of

other systems must await the availability of accu

rate beta diversity figures from the analyses of other

community gradients. The interpretation of species distribution curves

depends on the scaling used for the gradient. Gra

dient rescaling of the Siskiyou data produced small but important shifts in sample positions (Table 4). Shifts for samples 3 and 4 were the largest, both

approximately 10% of the total gradient length. The mean absolute displacement of the half-change scale from the elevation scale was 6.5%. In Figure 2 the density of stems of Abies concolor and Pseudo

tsuga menziesii, the most abundant tree species in

Table 4. Positions of samples along the Siskiyou Mountains

elevation gradient with respect to elevation (m) and, after

rescaling, with respect to half-changes. The mean absolute

displacements for the two sets of sample positions is 6.5%.

Elevation Sample positions belt _

Elevation Half- Elevation Half

(m) changes (%) changes

(%)

1 610 0.0 0.0 0.0

2 915 0.68 21.6 16.8

3 1220 1.37 43.1 33.8

4 1525 2.20 64.7 54.4

5 1800 3.34 84.1 82.6

6 2025 4.05 100.0 100.0

Sample positions, half-changes (-)

o m

1000 H

750 H

B 500 H

250 H

610 915 1220 1525

Elevation belts, midpoints (m) ( )

1800 2025

Fig. U.S

2. Distribution curves o? Abies concolor and Pseudo?suga menziesii along an elevation gradient in the Siskiyou Mountains, Oregon,

A., with respect to meters of elevation (?) and to half-changes (?).

139

the Siskiyou data set, are plotted against both ele

vation and half-changes. The curves against eleva

tion are skewed to higher elevations; the curve of

Abies concolor against half-changes is more sym metrical and slightly more broad. Niche breadth of

Abies concolor, measured as the standard deviation

of the distribution curve, is 13% wider with the

half-change scaling than with the elevation scaling.

Perhaps the most interesting and significant re

sults of this analysis of the Siskiyou elevation gra dient are the calculated rates of eological change

through the elevation belts (Fig. 3). Rates of change range between 2.2//C/1000 m and 3.7 HC/ 1000 m. Rates are generally higher through the

higher elevation belts, indicating that environmen

tal change with elevation is more biologically signif icant at higher elevations than at lower. The grea test rate of change is through the 1 800 m elevation

belt. This elevation corresponds to the broad sub

alpine-montane border as indicated by Whittaker

( 1960, p. 303). Our gradient rescaling technique has

given a quantitative corroboration for Whittaker's

field observations.

Discussion

We expect the technique of gradient rescaling

presented here will improve a variety of investiga tions into the nature of biotic communities. Gra dient rescaling furnishes accurate and robust esti

mates of beta diversity, which can be used in the

comparison of ecosystems from different geograph ical or biological regions. Rates of ecological change computed in gradient rescaling can be used to indicate the presence (or absence) of ecotones, either in space or through time. Gradient rescaling should be helpful also in analyses of species distri butions. Specifically, measures of niche breadth and overlap can now be calculated with respect to axes rescaled to constant rates of species turnover,

in effect weighting for the ecological separation between samples. Because breadth would then be

measured in objective and universal units of beta

diversity, direct comparisons of species distribu tions among systems are possible.

Although gradient rescaling is particularly well suited to the analysis of vegetation (e.g., Marks &

Harcombe, 1981; Wilson, 1982), the technique is relevant to research in other ecological systems as

well. Trends in faunal composition through space (e.g., James, 1971; Terborgh, 1971; Cody, 1975), animal niche relations along resource axes (e.g.,

Pianka, 1973; Cody, 1974), and patterns of com

munity change through time, including phenology of flowering and pollination (e.g., Mosquin, 1971;

to o

Il 2.0 O ^

o I

610 915 1800 1220 1525

Elevation belts, midpoints (m)

Fig. 3. Rate of ecological change through 6 elevation belts in the Siskiyou Mountains, Oregon, U.S.A.

2025

140

Reader, 1975; Bratton, 1976) and succession after

disturbance (Austin, 1977; Christensen & Peet,

1983) may all be examined with the tools of gra dient rescaling.

Some differences were apparent in the use of

gleasons and half-changes as units of beta diversity. Gleasons as units of compositional change have

several appealing advantages. The measure itself

(equation 4) is simple and straightforward. Derived

directly from a notion of species turnover rates,

gleasons are well suited for examining rates of

compositional change along gradients. Using glea sons has the disadvantage of retaining only infor

mation of the similarity of samples adjacent on the

gradient; using half-changes as units of composi tional change utilizes information about the sim

ilarities of some non-adjacent samples as well. In

general, the half-changes procedure should be su

perior when moderate or high levels of field noise are present.

Gradient rescaling can be applied to data in

which several factors are important in determining

species abundances. Each important axis must be

identified, using multivariate techniques, such as

factor analysis or ordination. Then data dimen

sionality is reduced by taking the average abun

dance of each species along segments of single axes, one axis at a time. This averaging is analogous to

marginal distributions of multivariate statistics.

Gradient rescaling can then be applied to each re

duced axis, and results combined into a multi-di

mensional representation. The ecological lengths, or beta diversities, of individual axes, for example, would indicate the relative importance of those axes to overall ecological structure. As with most

approaches to multivariate data, care must be taken

to identify axes of variation that are mutually or

thogonal.

Gradient rescaling is not useful for the analysis of

all forms of community data. Several assumptions and prerequisites are necessary for its proper use.

Samples must be orderable along axes of composi tional or environmental variation. These axes can

be obtained either by direct observations (e.g.,

along gradients of elevation, soil nutrient status,

etc.), or by indirect analysis (e.g., by ordination). When a gradient structure is inappropriate

- for

example, when biogeographical phenomena (e.g.,

migration) or historical effects (e.g., disturbance by

fire) create discontinuities in species distributions -

methods of analysis other than gradient rescaling must be employed.

Samples must also be of sufficient size and number to adequately represent gradient condi tions. Multi-dimensional data require larger numbers of samples. Poor sampling procedures can

contribute to high levels of noise and significantly reduce the accuracy of gradient rescaling results. In

particular, when field noise is high and gradient length is short, gradient rescaling (and probably any other technique) is not successful. Because the

technique performs best at intermediate sampling intensities, it is often possible to reduce noise for

purposes of the analysis by pooling replicate sam

ples. For most studies with adequate sampling along major gradients of environmental or compo sitional variation, gradient rescaling can be a pow erful tool for ecological analysis.

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Accepted 30.3.1983.