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Community similarity and the analysis of multispecies environmental data: A unified statistical approach
Wat. Res. Vol. 24, No. 4, pp. 507-514, 1990 Printed in Great Britain. All fights reserved
0043-1354/90 $3.00+0.00 Copyright © 1990Pergamon Pr-~,splc
COMMUNITY SIMILARITY A N D THE ANALYSIS OF MULTISPECIES ENVIRONMENTAL DATA: A UNIFIED STATISTICAL APPROACH ERIC P. S ~ r n * , KURT W. PorcrAscni" and Jorn~ CAIRNS JR~/ University Center for Environmental and Hazardous Materials Studies, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A. (First received May 1989; accepted in revised form November 1989) Ala~raet--The number of species or variables in some designed environmental studies is too large for analysis using standard inferential statistics. To analyze the type of data, a two-stage procedure and follow-up methods are described. First, data are reduced using a measure of similarity or distance. Then, a permutation procedure is used to make inferences. Assuming the hypothesis is rejected, several follow-up analyses are presented as tools in understanding the causes of rejection. For example, to further understand treatment effects, a randomization based, multiple comparison procedure may be used; to better understand what differences the similarity or distance measure describes, a method based on removal of a species is given. The methods are illustrated on a study of the effects of zinc on the periphyton community in the New River, Va, U.S.A. Key words---similarity indices, statistical tests, permutation tests, microcosm experiments, multispecies toxicity tests, environmental monitoring
INTRODUCTION Biological monitoring studies and multispecies toxicity tests involving measurements on a large number of species (or other variables) are difficult to analyze. Biological concerns are many, ranging from the loss of important species to changes in the abundance, biomass or biovolume of important species, to changes in the composition and diversity of groups of species. Although a number of researchers have recommended multivariate methods for detecting the changes associated with differences in locations or levels of a toxicant, most studies cannot use these methods for inference because the number of replicates is not adequate and it is unreasonable to obtain adequate replication. For example, it is not unreasonable for a researcher to observe as many as 100 different species in a study. To apply multivariate analysis of variance, one would need over 100 replicate samples (otherwise the degrees of freedom will be less than the number of variables and the pooled covariance matrix will be singular). Furthermore, it is unlikely that the assumptions of M A N O V A would be realized even if that many samples were obtained. Because of the large number of species typically absent (i.e. having zero abundance or biomass) in a sample at a given impacted site, yet present at other
*Department of Statistics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A. i'Present address: Department of Biology, University of Northern Iowa, Cedar Falls, IA 50614, U.S.A. :~Author to whom all correspondence should be addressed.
unimpacted sites, the normality assumption cannot be met. While transformations may help in dealing with skewed data, the problems caused by a large number of zeros cannot be remedied by a simple transformation. Therefore, some alternate methods are needed for the analysis of this type data. A common approach to summarization and interpretation of the above data is through a measure of community composition (Boyle et al., 1984; Washington, 1984; Perkins, 1983). While many researchers use a measure of diversity, measures of similarity are becoming increasingly popular. These measures allow comparisons within and among treatments and may be more sensitive than diversity (Boyle et al., 1984; Pontasch and Brusven, 1988). This paper focuses on the analysis of biological data arising from multispecies studies through the use of measures of community similarity. Of interest are three basic questions concerning these studies (1) are there differences due to the locations or treatments? (2) which species are primarily involved in the differences? and (3) which locations are different? The primary inferential method proposed is based on permutation or randomization procedures (Pitman, 1937). Such procedures were proposed for use in monitoring studies using diversity measures by van Belle and Fisher (1977) and Bell et al. (1981). Here, the methods are based on comparing similarities between samples from like and unlike sites or from replicate samples receiving different treatments. Similarities or measures of distance between species are commonly used to compare sites. However, most comparisons tend to be graphical and not based on 507
508
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an inferential procedure. The p e r m u t a t i o n m e t h o d s presented here are c o m p l e m e n t e d by graphical a n d s u m m a r y statistics to aid interpreting the test results. METHODS
For simplicity, the methods are discussed assuming a completely randomized design with a single factor of interest. For example, the interest may be in the effects of different concentrations of a single chemical on growth in microcosm experiments. Each microcosm would receive only one treatment and each treatment would be applied to several replicate microcosms. Observations, for example, on the biomass of individual species would then be taken at some suitable time. An alternate example would involve perhaps two sites, one impacted and a second unimpacted site that acts as a control. At each site, several samples are taken. Assuming that the samples could be viewed as statistical replicates, this might be viewed as a completely randomized design with two treatments. The first step in the randomization analysis is the summarization of data vectors through the use of similarity (or distance) measures. A similarity measure S~ describes the degree of relatedness between the species for two replicates (from the same or different treatment), i, and j. There exists a number of measures and a number of research papers describing the merits and demerits of various measures (see for example Washington, 1984 or Pontasch and Brusven, 1988). The object here is not to enter into that debate but to describe methodology that is useful after the measure is chosen. However, the choice of the measure is very important and determines the interpretation of the statistical hypothesis. Some guidelines on the choice of appropriate measures are found in discussions in Sneath and Sokal (1973), Hellawell (1978), Lamont and Grant (1979), Hajdu (1981) and others. A simplified summarization of the similarity measures categorizes them into three groups that are related to types of changes in community structure. First, if presence-absence data are used, the focus is on loss of species associated with the pollutant. Measures such as Jacard's coefficient S,j = a/(a + b + c)
(1)
where a is the number of species present in both replicates b is the number present in replicate i only and c is the number present in replicate j only, or the simple matching coefficient S,j = (a + d)/(a + b + c + d)
(2)
where d is the number absent in both replicate i and j, are useful for detecting changes in the occurrence of species. These two measures do not measure similarity the same way and one would not always expect the conclusions to be the same for an analysis using one or the other measure. The first measure is more useful when interest is in comparing one replicate with another, taking into account the total number of species in both replicates. The second measure takes into account the total number of species in the entire study. Hence, the second measure is more useful in describing effects in situations where loss of species is the primary effect; the first measure would be more useful when there is a change in the species composition, but the total number is relatively constant. Loss of species is, of course, not the only type of change that may occur in an ecosystem. Mild pollution may cause global decreases in the abundance of species, or some intolerant species may decline in abundance while others increase, or the relative abundances of species may change. In the first situation, measures that are based on quantitative or absolute measurements are to be preferred. Some measures include Euclidian distance
SMITH
et al.
where Xa and X~ refer perhaps to the biomass or biovolume of species k for replicates i and .L or a version of the Minkowski metric D,/= ~ l X~ - X~ I-
(4)
For relative changes, proportional abundances may be used, and a measure such as Bhattacharyya's (1946) measure of similarity S!~ = ~ (P~ p~)l .'
(5)
or the proportional similarity measure S,j = ~ min(P,, P~ )
(6)
may be useful. Here P~ and P~ refer to the proportion of species k for replicates i and j, respectively. Alternatively, if proportions are used with biomass data, a commonly used measure (Sullivan, 1978) is Stander's (1970) measure, which is more generally the cosine of the angle between two vectors 2
)
l/2
or
The second step in the randomization analysis is the test of treatment differences based on permutations of the matrix of similarity or distance measures. The simple data and calculations in Table 1 represent data on two treatments, a control and a dosage of a toxicant. Assume that there were five species studied and three replicate samples for each treatment. Below the data matrix is the matrix of similarity coefficients, using the cosine measure. Note the obvious structure in the matrix of coefficients. There are two groups of high similarities and a single block of small coefficients. The large coefficients represent the "within" similarities, that is, the similarity between the replicates receiving the same treatment. The block of small coefficients is the "between" similarities. These coefficients measure the similarity between samples receiving different treatments. If there are no treatment effects, the between similarities should have roughly the same values as the within similarities. If there are differences, the between similarities should be lower (less similar) than the within values. The permutation test compares the between and within similarities assuming that there is no difference due to the treatment. If there are large differences between the treatments, the test will usually indicate these differences. To test for differences, a statistic is needed to summarize the differences. Recognizing the similarity of the above situation with the analysis of variance method, a possible statistic (Good, 1982) is L = B~ if"
(8)
where B is the mean between similarity and if" is the mean within similarity (computational formulas are given in the appendix). Good (1982) suggests using L as a measure of separation. Since T, the total similarity, is a fixed number for a given similarity matrix, the sums B or W may also be considered for testing purposes (see for example Mielke et aL, 1981; Good, 1982; Zimmerman et al., 1985) in the completely randomized design. The permutation test requires computing the measure L for the data as collected. This value can be called L(data). Because the assumption is that there is no effect due to the pollutant, one or more of the control replicates can be switched with the same number of treatment replicates without a major change in the value of the statistic. If, however, there is a difference between treatments, switching data should cause a relatively large change in the value of
Analysis of multispecies experiments
509
Table 1. Hypothetical data from three replicate sites for two treatments (possibly sites above and below a source of pollution) and five species. Similarity matrix using cosine measure. Triangles contain within similarities while rectangle contains the between similarities. All possible similarity estimates for all possible permutations of the data are presented below the similarities Treatment
Rep.
Spl
Sp2
Sp3
Sp4
Sp5
1 1 1
1 2 3
10 12 18
5 2 9
8 9 4
2 5 1
I 0 2
2 2 2
4 5 6
5 3 4
7 4 8
9 6 2
15 12 16
5 9 8
Pep.
1
3
4
5
6
0.908
0.685
0.556
0.486
0.717
0.586
0.506
0.515
0.413
0.444
Estimated similarities
1.00
2 "",~955
1.00"'~.836 I.O0"X,,,J
1.00 " ~ , , ~ 4 6
0.925
l OO
1.01) Permutations Permutation
Trmt 1
Trmt 2
Total
Within
Between
I 2 3 4 5 6 7 8 9 l0 ll 12 13 14 15 16 17 18 19 20
1, 2, 3 1, 2, 4 1, 2, 5 1, 4, 3 l, 4, 3 1, 5, 3 1, 6, 3 4, 2, 3 5, 2, 3 6, 2, 3 4, 5, 1 l, 4, 6 I, 5, 6 4, 2, 5 4, 2, 6 5, 2, 6 4, 5, 3 4, 6, 3 6, 5, 3 4, 5, 6
4, 5, 6 3, 5, 6 4, 3, 6 4, 5, 3 2, 5, 6 4, 2, 6 4, 5, 2 1, 5, 6 4, 1, 6 4, 5, 1 6, 2, 3 2, 3, 6 4, 2, 3 1, 5, 3 5, l, 3 4, 1, 3 1, 2, 6 1, 5, 2 4, 1, 2 1, 2, 3
10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42
5.51 4.15 3.98 3.82 4.14 4.02 4.08 4.05 3.93 3.97 3.97 3.93 4.05 4.08 4.02 4.14 3.82 3.98 4.15 5.51
4.91 6.27 6.44 6.60 6.28 6.40 6.34 6.37 6.49 6.45 6.45 6.49 6.37 6.34 6.40 6.28 6.60 6.44 6.27 4.91
the statistic L. This value of L can be referred to as L(permute). Testing for differences in location requires a great number o f these data switches or permutations (say 1000) and the original value L(data) must be compared with the permuted values. If L(data) is more extreme than, for example, 95% o f the L(permute) values, then the null hypothesis of no differences between treatments would be rejected. Two comments are noteworthy here. First, in some cases (small samples), there are only a small n u m b e r of permutations possible. A total o f N observations at k levels with n, replicates per level gives N!/(n:n2!... nk!) different permutations. O f these permutations, only N!//-/[(i!)"m~!] (where m~ is the n u m b e r of groups with i replicates) are unique. W h e n this number is small, all possible permutations o f the data (Berry and Mielke, 1984) should be computed. Second, as in the analysis of variance, the test is a one-sided test. However, the rejection side depends on the statistic and measure that are used. For example, if ~r/Wis used, the null hypothesis is rejected if the observed value o f the statistic L ( d a t a ) = ~r/fl~ is large relative to the permuted values if a distance measure is used. Because distance measures are large when treatments are separated, a treatment effect would m e a n that L(data) should be large relative to the permuted values (the between treatment distance in the data
will be large relative to the within treatment distance). If a similarity measure is used, rejection occurs if the observed Br/I~ is small relative to the permuted values. Since the within treatment observations should be close to each other, the within similarities should be large. The between similarities should be small. Hence, relatively small values of L(data) are expected. Alternatively, if ~r or if'is used as the statistic and a similarity measure is used, then rejection occurs for small values of B or large values o f if'. If a distance measure is used, then the null hypothesis is rejected for large values of B or small values of if'. In Table 1, all 20 possible combinations o f the six replicates are divided into two groups of three, and the resulting totals are given for within treatment and between treatment. The total similarity remains fixed at 10.42, and there are 10 unique values, not 20. For example, if group 1 consists o f replicates 1, 2 and 3 and group 2 o f 4, 5 and 6, the within and between total similarities are the same as when group 1 consists o f replicates 4, 5 and 6 and group 2 consists o f 1, 2 and 3. Although the total of the between similarities for the observed data, 4.91, is less than all the other values, the associated significance level for the test of no difference is only 2/20 = 1/10. Within this small n u m b e r o f replicates, a strong statement cannot be made about the possible differences using the permutation procedure (or any
510
ERIC P. SMITH et al.
other procedure without making additional uncheckable assumptions). Thus, although the mean between similarity is smaller than all permuted values, the hypothesis that there is a treatment effect is not rejected at significance levels below 0.10. When the number of replicates is large, a large number of permutations (say 1000) can be computed. A critical value can then be chosen for a particular ~ level by ordering the permuted statistics and selecting the 100(1-~) percentile. This would then be compared with the observed value to test the null hypothesis.
Follow-up analyses o f replicates Assuming rejection of the null hypothesis of no treatment or location differences, follow-up analyses of treatments will determine which ones differ and why. There are a number of analyses that could be used to determine which treatments differ (assuming there are more than two) and which species are the important species for indicating the differences. It may also be of interest to look for possible odd data values that may influence the analysis. A number of procedures could be used; only a few are discussed here. In particular, interest in this discussion is in methods that use similarity or distance information, as they are consistent with the previously discussed overall test. Multidimensional scaling procedures (Gower, 1966) are a set of graphical procedures useful for comparing the treatments. While there are many multidimensional scaling approaches, we have found the one based on metric scaling to be quite useful. The objective of this set of procedures is to find a set of x and y data points that, when graphed, represent the location of the samples in a two-dimensional space (a reduced species space). The point (x, y), or treatment scores, are chosen so that the visual (Euclidian) distance between the points is approximately the same as the distance between the samples as measured by the chosen index. If a similarity measure is used, the points well separated would have low similarity. Graphs of the sample sites show groups of samples, for possible ordering of the sites, or odd data values. Thus, this method is somewhat similar to principal components analysis in that the dimensionality is reduced and scores plotted to visualize patterns. However, instead of using a covariance or correlation matrix (which would suggest Euclidian distance), a similarity or distance matrix is reduced. A better approach for assessing separation, one that may be used for inference, is to compare the mean similarities for the different sites using randomization. Two approaches may be used. One approach is to carry out more permutation tests, looking at the locations in a pairwise fashion, analogous to multiple comparison procedures (Foutz et al., 1985). Foutz et al. (1985) suggest forming comparisons of interest and carrying out the permutation test on the vector of comparisons. For example, if there are four groups, there are six pairwise comparisons of interest •12/#]2, ~ o j l ~ , ~14/1~r14,~23/~2], ~24/~'24 and B ~ / f f ' ~ , where ~ r and ~1 are the mean between and mean within similarity for groups i and j, respectively. At each step of the permutation procedure, a vector of length six is generated. After all vectors are generated, the critical values should be calculated. This is relatively easy to accomplish for the overall test by sorting the permuted values and obtaining the upper or lower percentile associated with the significance level. However, with multiple comparisons, the sorting is not of single values but vectors, and the problem oferror rates associated with the overall test and the pairwise comparisons also arises. Foutz et al. (1985) indicate that these problems can be solved by the proper sorting of the permuted vectors. If there are R pairwise comparisons and N is the number of permutations, choose a significance level • for experimentwise and comparisonwise error rates so that K = ~N is divisible by R. For example, if there are 6 multiple comparisons and 1000 permutations, • of 0.06 gives ~N of 60 which is divisible by 6 (so, K = 60, N = 1000 and R = 6). For the
overall test, the 1000 permutations could be sorted and the 60th smallest (or largest depending on the statistic and measure) value chosen for use as a critical value for the test. Sorting on the elements of the vectors produces the critical values for each comparison for use in the multiple comparison procedure. Error rates are maintained by: (1) selecting an element of the vector corresponding to a comparison--select a random number for 1 to R (the total number of comparisons); sort on this element (for example, if the second element is selected, all vectors would be ordered by the second element); (2) put this vector aside; choose another element; sort the remaining N-I vectors on this element; (3) repeat this process a total of K times with each element being sorted K / R times. In the above example, K = 60 and R = 6 . Each element would be used in the sorting process 10 times. The vector obtained in the last sort on each element contains the critical value for that element. There will be R (6 in the example) critical values, one for each comparison. The tests on the comparisons can be carried out by comparing the observed comparison with the critical value. Further details of the procedure and some extensions are presented in Foutz et al. (1985). This approach is only useful when there are an adequate number of replicates for each treatment level. Another follow-up analysis that may be of interest is confidence intervals for the mean between similarity. As the estimated similarities are not independent, standard methods cannot be used as in the usual analysis of variance (i.e. based on the normal distribution). Two approaches that work well for this problem are based on the bootstrap method and the jackknife method. As in the multiple comparison problem, these methods require reasonable sample sizes to work well. Some details are given in Smith (1985) and Smith et al. (1986).
Follow-up analyses of species Species as well as the treatments must be analyzed because changes in the species can indicate which species are reacting to the treatment as well as being informative on why there are differences between the treatments. As different measures of similarity may lead to different conclusions about changes due to the treatment, it is important to have methods that use the measure of interest. There are, of course, a number of possible methods useful for evaluating the influence o f individual species. The interest here, however, is not only in changes in individual species but which species are primarily responsible for the differences indicated by the test. As the test is strongly dependent on the measure of similarity, using the contribution (or importance) of the species to the overall statistic is proposed. If a measure is additive (for example, Bhattacharyya's measure), the contribution may be directly computed. However, most measures are not additive, so a method based on the effect of removing a species on the similarity is proposed. Let B'_ i be the mean of the between similarities with species i removed. Then INFi = 100(B- , -/~)//~
(9)
measures the percentage relative influence of species i on the mean between similarity. Large positive values indicate a species whose removal greatly increases the between similarity (i.e. makes the data more similar). These species usually have quite different proportional abundances between treatments. Species with large negative values decrease the between similarity when removed and represent species that show little change over most treatment levels but contribute to the between similarity.
Example As an example, consider the data set (Table 2) from an experiment on the effect of zinc on the periphyton community in the New River at Glen Lyn, Va. Twelve artificial streams received river water and one of four zinc treatments
Analysis of multispccivs experiments
511
Table 2. Data (biovolume in cells/era 2) on I 1 dominant species from a study on the effects of zinc on the a l o e community in the New River, influence measures and mean similarities Species Zn
Rep.
I
2
3
4
5
0.0
1 2 3 1 2 3 1 2 3 1 2 3
175 134 77 44 18 49 Ill 59 29 81 98 66
1745 931 393 1738 241 716 846 386 482 862 953 794
642 0 393
408 412 323 59 55 66 89 40 51 85 80 14
21 53 48 564 137 874 7 3 0 44 17 0
0.05 0.50 1.0
Cosine Bhatt.
0.06 -0.00
4.18 -0.76
14
11 16 44 22 29 65 53 37 0.47 0.87
6
7
730 596 705 1923 1103 1891 7 18 14 40 26 51
Influence 0.19 2.33 0.87 3.66
0.69 0.83
8
0 0 3 0 0 0 163 3 44 694 561 185
204 134 230 876 427 924 5133 802 995 8457 14,012 10,028
0.02 0.59
44.73 13.00
9
10
29 187 141 222 200 208 326 155 111 102 205 14
379 245 300 490 159 216 1515 594 1084 98 989 37
2293 1070 3613 5615 3725 7099 12,940 9489 20,116 1013 5740 1121
-0.01 -0.58
-15.09 -6.01
0.06 -0.12
11
Matrix of mean similarities using the cosine index Zn 0.00 0.05 0.50 1.00
0.00 0.872
0.05 0.882 0.982
0.50 0.824 0.933 0.967
1.00 0.265 0.322 0.365 0.973
Matrix of mean similarities using Bhattacharyya's measure Zn 0.00 0.05 0.50 1.00
0.00 0.923
0.05 0.901 0.982
(0.0, 0.05, 0.5 or 1.0 mg/l). Artificial substrates were placed in the streams and removed at a n u m b e r o f times throughout the experiment. Thus, the experimental design is a two factor repeated measures experiment (split-plot design but without blocking) (Milliken and Johnson, 1984). As the experiment was designed to obtain a significant time-zinc interaction, the individual times were analyzed separately. A detailed analysis o f the full data set has been presented elsewhere (Genter et al., 1987). Table 2 gives the data for day 20 o f the experiment and some s u m m a r y measures using Bhattcharyya's measure and the cosine measure. The permutation test indicated significant differences between the four treatments for both measures (no mean between similarities were lower than the observed). Figure 1 shows the separation between the replicates using both measures. With the cosine measure, the low zinc treatments are close together, while the highest
0.50 0.798 0.870 0.966
1.00 0.591 0.644 0.692 0.967
dose of zinc separates well from the low doses. The means o f the similarities in Table 2 suggest a relatively high degree o f similarity for replicates within a treament (the diagonal elements) while the between means show a decreasing similarity with increasing zinc concentration. Note that this is not clearly displayed in Fig. 1 due to the horseshoe effect (Kendall, 1971). Variance estimates suggest a high degree of variability for replicates using the cosine index. Confidence intervals and the multiple comparison procedure applied to the between means suggest overlappings between the 0.0, 0.05 and 0.50 treatments. The plot for Bhattacharyya's measure differs slightly from that of the cosine measure. In particular, the 0.00 treatment group (control) is closer to the 0.05 group while the 0.50 group is more distant. In both cases, one of the replicates from the control group overlaps with the 0.05 group of replicates. The reasons for these differences are
b. B h a t t a c h a r y y a
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Fig. I. Plots of samples using the first two axes from multidimensional sealing using (a) the cosine measure and (b) Bhattacharyya's measure. W.R. 2 4 / 4 - - H
0.6
512
Emc P. SMITH et al.
discussed below. Variance estimates for mean similarities using Bhattacharyya's measure are much lower, and, hence, confidence intervals are narrower. Confidence intervals and the multiple comparison procedure on the between means suggest no differences between the 0.00 and 0.05 groups but differences between all other pairs. The influence measures indicate some interesting relationships between the data and the similarity statistics. The data indicate that certain cell densities tend to be quite large. Easily observed differences are for species 5 and 7. However, these species have negligible importance to the overall similarities. The influence measures point to species 8 and 11 as the species that determine magnitude of B. This peculiarity is due to the high abundances of species 8 and 11. Understanding the relationship between the data, influence and the differences between treatments, require considering the proportions for each replicate (for Bhattacharyya's measures) and proportions adjusted for the square root of the sum of squared proportions (for the cosine measure) that are presented in Table 3. Species 11 dominates the cosine measure for treatments 0.0, 0.05 and 0.50. Only the last treatment alters this species dominance. The large negative influence indicates the dominance of the species for most of the treatments. Species that have a strong decreasing effect on between similarity are species 8, 6 and 2. Species 8 increases relative to others in replicates of the 1.0 treatment, while species 2 and 6 are diminished by increasing zinc. Differences in Fig. 1 between replicates for the 0.0 and 0.05 treatment are due to species 2, while species 6 contributes to differences between the replicates for the 0.05 and 0.50 treatment, and species 8 and 11 affect separation of the 0.50 and 1.0 treatments. Thus, the influence measures highlight the four types of changes in the species that are associated with the levels of the treatment and indicate the relative importance of each of the levels of the treatment. The interpretation based on influence using Bhattacharyya's measure is slightly different and may be
Table 3, Adjusted proportions
P~
understood by studying the unadjusted proportions in Table 3. Although species 11 dominates, the magnitude of that domination is diminished especially for the first group of replicates. Also diminished is the influence of species 2. While high for the cosine measure, the influence is now small and negative. With the exception of two replicates in the first treatment, the proportions for this species are relatively small. Thus, the cosine measure gives more weight to changes in dominant species, while Bhattacharyya's measure is not as influenced by the dominant species. Primary differences using Bhattacharyya's measure are indicated by the proportions for species 6, 8 and I 1. These species all have large influence. Species 6 again will lead to a separation between replicates for low (0.0, 0.05) and high (0.05, 1.0) zinc treatments in Fig. 1. Species 8 and I 1 contribute to the separation between replicates of the 1.0 zinc treatment and the other treatments. Bhattacharyya's measure indicates the replicates for the two low treatments are much closer together than suggested by the cosine measure. DISCUSSION This p a p e r has presented m e t h o d s for the analysis o f mulfispecies data. T h e m e t h o d s allow a researcher to a n s w e r three i m p o r t a n t questions a b o u t the data. First, are there any overall differences due to treatments? Second, if there are differences, which levels o f the t r e a t m e n t are associated with the differences? Third, which species are associated with differences? T h e m e t h o d s are unified in that they involve the use o f similarity o r distance measures. Thus, for example, the species that are d e t e r m i n e d to be i m p o r t a n t in the third analysis are the species that are influencing the
P~ and proportions (P~) for Glen Lyn data. Boxes indicate where primary changes occur
(a) Adjusted proportions Species Zn
oo
Rep. 2 3
1
2
4
5
0.000 0.105
0.252 0.086
0.032 0.013
006 00 0.082 0.021
3
6
03 07
0.05
I 2 3
0.007 0.005 0.007
0.276 0,061 0.096
0.002 0.003 0.002
0.009 0.014 0.009
0.090 0.035 0.117
0.50
! 2 3
0.008 0.006 0.00I
0.060 0.040 0.024
0.003 0.002 0.001
0.006 0.004 0.003
0.001 0.000 0.000
1.00
1 2 3
0.010 0.006 0.007
0.100 0.063 0.079
0.008 0.004 0.004
0.010 0.005 0.001
0.005 0.001 0.000
10
II
0.115 0.038
0.150 0.080
0.6551 0.961
0.000 0.000 0.000
0.035 0.051 0.028
0.078 0.041 0.029
0.892 0.949 0.947
0.012 0.000 0.002
0.023 0.016 0.006
0.108 0.062 0.054
0.922 0.994 0.997
00
7
8
00
0.000 0.001
~ ~
9
0 0
0.081 0.037 0.018
~ ~
0.012 0.013 0.001
0.011 0.065 0.004
0.000 0.000 0.001 0.000 0.000 0.000
0.031 0.036 0.037 i 10.076 10.070 10077
0.004 0.050 0.023 0.019 0.033 0.017
0,057 0.065 0.048
0.346 0.284 0.580
0.042 0.026 0.018
0.486 0.613 0.588
0.008 0.000 0.002
0.242 0.069 i0.043
0.015 0.013 0.005
0.072 0.051 0.047
0.611 O.820 0.876
0.060
~
0.009
(b) Proportions 0.00
l
0.05
2 3 I 2 3
0.026 0.036 0.013 0.004 0.003 0.004
0.263 0.247 0.063 0.150 0.040 0.059
0.097 0.000 0.063 0.001 0.002 0.00t
0.062 0.110 0.052 0.005 0.009 0.006
0.003 0.014 0.008 0.049 0.023 0.073
0.50
1 2 3
0.005 0.005 0.001
0.040 0.033 0.021
0.002 0.002 0.001
0.004 0.004 0.002
0.000 0.000 0.000
1.00
I 2 3
0.007 0.004 0.005
0.075 0.042 0.064
0.006 0.002 0.003
0.007 0.004 0.001
0.004 0.001 0.000
10.110[
J
0.025
0.009
0.008 0.044
0.015
0.001
0.003
Analysis of muitispecies experiments
similarity or distance measure and are associated with the treatment differences in the first and second analyses. The techniques discussed above provide researchers with a powerful set of techniques for the
analysis of multispecies data. The methods presented are only a few of the possible techniques available for analyzing community data. However, these methods are oriented to the analysis of data using some measure of similarity. A number of other techniques, such as principal components, discriminant analysis, and detrended correspondence analysis (Greenacre, 1984; Gauch, 1982) are dependent of Euclidian (or weighted Euclidian) type measures of distance. If interest is in Euclidian (or weighted Euclidian) distance, then these methods, which are available on some computer packages, should provide useful results. One method of interest is Gabriel's biplot analysis (ter Braak, 1983), which allows graphical displays of both the species and the replicates. One drawback to the permutation approach is that the size of the difference between the mean between and the mean within similarity may be statistically significant but not biologically important. If all between similarities are only 0.01 less than the smallest within similarity, the degree of significance using the permutation test is the same as if the difference were 0.50. This problem is shared with many nonparametric methods and similar problems occur in classical statistical methods, but should not commonly occur. The magnitude of the between and within similarities as well as the significance of the test should be considered. An approach based on the size of the similarities is available; however, it requires additional assumptions. Boyle et al. (1984) present a similar approach to the overall test of differences, using a t-test to compare the similarities at two locations. This test procedure ignores the dependence between estimated similarities and may result in inflated error rates. A procedure that does account for dependence is discussed by Dyer (1978). This procedure involves the use of linear models and assumes normality of the similarities and homogeneity of variance. The examples discussed above are for relatively simple designs. For example, we did not test directly for a time effect or interaction between time and treatment in the Glen Lyn study. Analyses of more complex designs are possible but not as straightforward due to the use of repeated measurements on the same experimental unit over time (Carter et al., 1982). Acknowledgements--This research was partially supported by an appointment to the U.S. Department of Energy Faculty Research Participation program administered by Oak Ridge Associated Universities to E. P. Smith while he was at the Environmental Sciences Division, Oak Ridge National Laboratory (operated by Martin Marietta Energy Systems Inc., under contract DE-AC05-84OR21400 with the U.S. Department of Energy), Oak Ridge, TN
513
37831-6036, U.S.A. Publication No. 3405, Environmental Sciences Division, ORNL. Comments by I. J. Good, K. Rose and A. Brenkert greatly improved a previous version of the manuscript. REFERENCES
Bell C. B., Conquest L. L., Pyke R. and Smith E. P. (1981) Some nonparametric statistics for monitoring water quality using benthic species counts. In Environmetrics 81: Selected Papers, pp. 100-121. Society of Industrial and Applied Mathematicians, Philadelphia. van Belle G. and Fisher L. (1977) Monitoring the environment for ecological change. J. Wat. Pollut. Control Fed. 49, 1671-1679. Berry K. J. and Mielke P. W. (1984) Computation of exact probabilities for multiresponse permutation procedures (MRPP). Commun. statist, simul. Computat. 13, 417-432. Bhattacharyya A. (1946) On a measure of divergence between two multinomial populations. Sankya 7, 401-406. Boyle T. P., Sebaugh J. and Robinson-Wilson E. (1984) A hierarchical approach to the measurement of changes in community induced by environmental stress. J. envir. Test. Eval. 12, 241-245. ter Braak C. T. F. (1983) Principal component biplots and alpha and beta diversity. Ecology 64, 454-463 Carter R. L., Morris R. and Blashfield R. K. (1982) Clustering two-dimensional profiles: a comparative study. Technical Report 175, Department of Statistics. University of Florida, Gainesville. Dyer D. P. (1978) An analysis of species dissimilarity using multiple environmental variables. Ecology 59, 117-125. Foutz R. V., Jensen D. R. and Andersen G. W. (1985) Multiple comparisons in the randomization analysis of designed experiments. Biometrics 41, 29-38. Gauch H. G. (1982) Multivariate Analysis in Community Ecology. Cambridge University Press, Cambridge. Genter R. B., Cherry D. S., Smith E. P. and Cairns J. Jr (1987) Algal-periphyton population and community changes from zinc stress in stream mesocosms. Hydrobiologia 152, 261-275. Good I. J. (1982) An index of separateness of clusters and a permutation test for its significance. J. statist, comp. Simul. 15, 81-84. Gower J. C. (1966) Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika 53, 325-338. Greenacre M. J. (1984) Theory and Applications of Correspondence Analysis. Academic Press, London. Hajdu L. J. (1981) Graphical comparison of resemblance measures in phytosociology. Vegetatio 48, 47-59. Hellawell J. M. (1978) Biological Surveillance of Rivers: A Biological Monitoring Handbook. Water Research Centre, Stevenage, England. Kendall D. G. (1971) Seriation from abundance matrices. In Mathematics in Archaeological and Historical Sciences (Edited by Hodson F. R., Kendall D. G. and Tautu P. A. P.), pp. 215-252, Edinburgh University Press, Edinburgh. Lamont B. and Grant K. J. (1979) A comparison of twenty measures of site dissimilarity. In Multivariate Methods in Ecological Work (Edited by Orloci L., Rao C. R. and Stiteler W. M.), pp. 101-126. International Cooperative Publishing House, Fairland, Md. Mielke P. W. (1979) On the asymptotic non-normality of null distributions of MRPP statistics. Commun. statist. theor. Meth. A8, 1541-1550. Mielke P. W., Berry K. J., Brockwell P. J. and Williams J. S. (1981) A class of nonparametric procedures based on multiresponse permutation procedures. Biometrika 68, 720-724.
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APPENDIX
Milliken G. A. and Johnson D. E. (1984) of
Messy
Data,
Volume
1: Designed
Analysis Experiments.
Wadsworth, London. Perkins J. L. (1983) Bioassay evaluation of diversity and community indexes. J. Wat. Pollut. Control Fed. 55, 522-530. Pitman J. (1937) Significance tests which may be applied to samples from any populations. J. R. Statist. Soc., Set. B 4, 119-130. Pontasch K. W. and Brusven M. A. (1988) Diversity and community comparison indices assessing macroinvertebrate recovery following a gasoline spill. Wat. Res. 22, 619-626. Smith E. P. (1985) Estimating the reliability of diet overlap measures. Envir. Biol. Fish. 13, 125-138. Smith E. P., Genter R. and Cairns J. Jr (1986) Confidence intervals for the similarity between algal communities. Hydrobiologia 139, 237-247. Sneath P. H. A. and Sokal R. R. (1973) Numerical Taxonomy: The Principles and Practices of Numerical Classifi cation. Freeman, San Francisco.
Stander M. J. (1970) Diversity and similarity of benthic fauna off Oregon. M.Sc. thesis, Oregon State University, Corvallis, Ore. Sullivan M. J. (1978) Diatom community structure" taxonomic and statistical analysis of a Mississippi salt marsh. J. Phycol. 14, 468-475. Washington H. G. (1984) Diversity, biotic and similarity indices: a review with special relevance to aquatic ecosystems. Wat. Res. 18, 653-694. Zimmerman G. M., Goetz H. and Mielke P. W. (1985) Use of an improved statistical method for group comparisons to study five effects of prairie fire. Ecology 66, 606-611.
Computational Details
Assume there are g groups and let S,j be the similarity between replicates i and j. Indexing on the replicates is i = 1, 2. . . . , N where N is the total number of replicates. Further, N = n t + n: + • • - + ng where nk is the number of replicates in group k, and N k = n I + n, + . . • + n k . For group k the total within similarity is
w,=
Yk
Nk
E
Es,,.
i-Nk
14-1 ! > i
Let T= V
S,~
,~1
!>J
be the total similarity. Then B = T - IV. The mean within similarity for group i is • ,= w
,, \2 I
To compute the overall mean similarity, one may use
, . -1
However, Mielke (1979) indicates that
if* = E n , ~ / N is a better estimate if the n, are not equal. The mean between similarity is given by [
g \i=lj>i
]
0043-1354/90 $3.00+0.00 Copyright © 1990Pergamon Pr-~,splc
COMMUNITY SIMILARITY A N D THE ANALYSIS OF MULTISPECIES ENVIRONMENTAL DATA: A UNIFIED STATISTICAL APPROACH ERIC P. S ~ r n * , KURT W. PorcrAscni" and Jorn~ CAIRNS JR~/ University Center for Environmental and Hazardous Materials Studies, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A. (First received May 1989; accepted in revised form November 1989) Ala~raet--The number of species or variables in some designed environmental studies is too large for analysis using standard inferential statistics. To analyze the type of data, a two-stage procedure and follow-up methods are described. First, data are reduced using a measure of similarity or distance. Then, a permutation procedure is used to make inferences. Assuming the hypothesis is rejected, several follow-up analyses are presented as tools in understanding the causes of rejection. For example, to further understand treatment effects, a randomization based, multiple comparison procedure may be used; to better understand what differences the similarity or distance measure describes, a method based on removal of a species is given. The methods are illustrated on a study of the effects of zinc on the periphyton community in the New River, Va, U.S.A. Key words---similarity indices, statistical tests, permutation tests, microcosm experiments, multispecies toxicity tests, environmental monitoring
INTRODUCTION Biological monitoring studies and multispecies toxicity tests involving measurements on a large number of species (or other variables) are difficult to analyze. Biological concerns are many, ranging from the loss of important species to changes in the abundance, biomass or biovolume of important species, to changes in the composition and diversity of groups of species. Although a number of researchers have recommended multivariate methods for detecting the changes associated with differences in locations or levels of a toxicant, most studies cannot use these methods for inference because the number of replicates is not adequate and it is unreasonable to obtain adequate replication. For example, it is not unreasonable for a researcher to observe as many as 100 different species in a study. To apply multivariate analysis of variance, one would need over 100 replicate samples (otherwise the degrees of freedom will be less than the number of variables and the pooled covariance matrix will be singular). Furthermore, it is unlikely that the assumptions of M A N O V A would be realized even if that many samples were obtained. Because of the large number of species typically absent (i.e. having zero abundance or biomass) in a sample at a given impacted site, yet present at other
*Department of Statistics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A. i'Present address: Department of Biology, University of Northern Iowa, Cedar Falls, IA 50614, U.S.A. :~Author to whom all correspondence should be addressed.
unimpacted sites, the normality assumption cannot be met. While transformations may help in dealing with skewed data, the problems caused by a large number of zeros cannot be remedied by a simple transformation. Therefore, some alternate methods are needed for the analysis of this type data. A common approach to summarization and interpretation of the above data is through a measure of community composition (Boyle et al., 1984; Washington, 1984; Perkins, 1983). While many researchers use a measure of diversity, measures of similarity are becoming increasingly popular. These measures allow comparisons within and among treatments and may be more sensitive than diversity (Boyle et al., 1984; Pontasch and Brusven, 1988). This paper focuses on the analysis of biological data arising from multispecies studies through the use of measures of community similarity. Of interest are three basic questions concerning these studies (1) are there differences due to the locations or treatments? (2) which species are primarily involved in the differences? and (3) which locations are different? The primary inferential method proposed is based on permutation or randomization procedures (Pitman, 1937). Such procedures were proposed for use in monitoring studies using diversity measures by van Belle and Fisher (1977) and Bell et al. (1981). Here, the methods are based on comparing similarities between samples from like and unlike sites or from replicate samples receiving different treatments. Similarities or measures of distance between species are commonly used to compare sites. However, most comparisons tend to be graphical and not based on 507
508
EPic P.
an inferential procedure. The p e r m u t a t i o n m e t h o d s presented here are c o m p l e m e n t e d by graphical a n d s u m m a r y statistics to aid interpreting the test results. METHODS
For simplicity, the methods are discussed assuming a completely randomized design with a single factor of interest. For example, the interest may be in the effects of different concentrations of a single chemical on growth in microcosm experiments. Each microcosm would receive only one treatment and each treatment would be applied to several replicate microcosms. Observations, for example, on the biomass of individual species would then be taken at some suitable time. An alternate example would involve perhaps two sites, one impacted and a second unimpacted site that acts as a control. At each site, several samples are taken. Assuming that the samples could be viewed as statistical replicates, this might be viewed as a completely randomized design with two treatments. The first step in the randomization analysis is the summarization of data vectors through the use of similarity (or distance) measures. A similarity measure S~ describes the degree of relatedness between the species for two replicates (from the same or different treatment), i, and j. There exists a number of measures and a number of research papers describing the merits and demerits of various measures (see for example Washington, 1984 or Pontasch and Brusven, 1988). The object here is not to enter into that debate but to describe methodology that is useful after the measure is chosen. However, the choice of the measure is very important and determines the interpretation of the statistical hypothesis. Some guidelines on the choice of appropriate measures are found in discussions in Sneath and Sokal (1973), Hellawell (1978), Lamont and Grant (1979), Hajdu (1981) and others. A simplified summarization of the similarity measures categorizes them into three groups that are related to types of changes in community structure. First, if presence-absence data are used, the focus is on loss of species associated with the pollutant. Measures such as Jacard's coefficient S,j = a/(a + b + c)
(1)
where a is the number of species present in both replicates b is the number present in replicate i only and c is the number present in replicate j only, or the simple matching coefficient S,j = (a + d)/(a + b + c + d)
(2)
where d is the number absent in both replicate i and j, are useful for detecting changes in the occurrence of species. These two measures do not measure similarity the same way and one would not always expect the conclusions to be the same for an analysis using one or the other measure. The first measure is more useful when interest is in comparing one replicate with another, taking into account the total number of species in both replicates. The second measure takes into account the total number of species in the entire study. Hence, the second measure is more useful in describing effects in situations where loss of species is the primary effect; the first measure would be more useful when there is a change in the species composition, but the total number is relatively constant. Loss of species is, of course, not the only type of change that may occur in an ecosystem. Mild pollution may cause global decreases in the abundance of species, or some intolerant species may decline in abundance while others increase, or the relative abundances of species may change. In the first situation, measures that are based on quantitative or absolute measurements are to be preferred. Some measures include Euclidian distance
SMITH
et al.
where Xa and X~ refer perhaps to the biomass or biovolume of species k for replicates i and .L or a version of the Minkowski metric D,/= ~ l X~ - X~ I-
(4)
For relative changes, proportional abundances may be used, and a measure such as Bhattacharyya's (1946) measure of similarity S!~ = ~ (P~ p~)l .'
(5)
or the proportional similarity measure S,j = ~ min(P,, P~ )
(6)
may be useful. Here P~ and P~ refer to the proportion of species k for replicates i and j, respectively. Alternatively, if proportions are used with biomass data, a commonly used measure (Sullivan, 1978) is Stander's (1970) measure, which is more generally the cosine of the angle between two vectors 2
)
l/2
or
The second step in the randomization analysis is the test of treatment differences based on permutations of the matrix of similarity or distance measures. The simple data and calculations in Table 1 represent data on two treatments, a control and a dosage of a toxicant. Assume that there were five species studied and three replicate samples for each treatment. Below the data matrix is the matrix of similarity coefficients, using the cosine measure. Note the obvious structure in the matrix of coefficients. There are two groups of high similarities and a single block of small coefficients. The large coefficients represent the "within" similarities, that is, the similarity between the replicates receiving the same treatment. The block of small coefficients is the "between" similarities. These coefficients measure the similarity between samples receiving different treatments. If there are no treatment effects, the between similarities should have roughly the same values as the within similarities. If there are differences, the between similarities should be lower (less similar) than the within values. The permutation test compares the between and within similarities assuming that there is no difference due to the treatment. If there are large differences between the treatments, the test will usually indicate these differences. To test for differences, a statistic is needed to summarize the differences. Recognizing the similarity of the above situation with the analysis of variance method, a possible statistic (Good, 1982) is L = B~ if"
(8)
where B is the mean between similarity and if" is the mean within similarity (computational formulas are given in the appendix). Good (1982) suggests using L as a measure of separation. Since T, the total similarity, is a fixed number for a given similarity matrix, the sums B or W may also be considered for testing purposes (see for example Mielke et aL, 1981; Good, 1982; Zimmerman et al., 1985) in the completely randomized design. The permutation test requires computing the measure L for the data as collected. This value can be called L(data). Because the assumption is that there is no effect due to the pollutant, one or more of the control replicates can be switched with the same number of treatment replicates without a major change in the value of the statistic. If, however, there is a difference between treatments, switching data should cause a relatively large change in the value of
Analysis of multispecies experiments
509
Table 1. Hypothetical data from three replicate sites for two treatments (possibly sites above and below a source of pollution) and five species. Similarity matrix using cosine measure. Triangles contain within similarities while rectangle contains the between similarities. All possible similarity estimates for all possible permutations of the data are presented below the similarities Treatment
Rep.
Spl
Sp2
Sp3
Sp4
Sp5
1 1 1
1 2 3
10 12 18
5 2 9
8 9 4
2 5 1
I 0 2
2 2 2
4 5 6
5 3 4
7 4 8
9 6 2
15 12 16
5 9 8
Pep.
1
3
4
5
6
0.908
0.685
0.556
0.486
0.717
0.586
0.506
0.515
0.413
0.444
Estimated similarities
1.00
2 "",~955
1.00"'~.836 I.O0"X,,,J
1.00 " ~ , , ~ 4 6
0.925
l OO
1.01) Permutations Permutation
Trmt 1
Trmt 2
Total
Within
Between
I 2 3 4 5 6 7 8 9 l0 ll 12 13 14 15 16 17 18 19 20
1, 2, 3 1, 2, 4 1, 2, 5 1, 4, 3 l, 4, 3 1, 5, 3 1, 6, 3 4, 2, 3 5, 2, 3 6, 2, 3 4, 5, 1 l, 4, 6 I, 5, 6 4, 2, 5 4, 2, 6 5, 2, 6 4, 5, 3 4, 6, 3 6, 5, 3 4, 5, 6
4, 5, 6 3, 5, 6 4, 3, 6 4, 5, 3 2, 5, 6 4, 2, 6 4, 5, 2 1, 5, 6 4, 1, 6 4, 5, 1 6, 2, 3 2, 3, 6 4, 2, 3 1, 5, 3 5, l, 3 4, 1, 3 1, 2, 6 1, 5, 2 4, 1, 2 1, 2, 3
10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42 10.42
5.51 4.15 3.98 3.82 4.14 4.02 4.08 4.05 3.93 3.97 3.97 3.93 4.05 4.08 4.02 4.14 3.82 3.98 4.15 5.51
4.91 6.27 6.44 6.60 6.28 6.40 6.34 6.37 6.49 6.45 6.45 6.49 6.37 6.34 6.40 6.28 6.60 6.44 6.27 4.91
the statistic L. This value of L can be referred to as L(permute). Testing for differences in location requires a great number o f these data switches or permutations (say 1000) and the original value L(data) must be compared with the permuted values. If L(data) is more extreme than, for example, 95% o f the L(permute) values, then the null hypothesis of no differences between treatments would be rejected. Two comments are noteworthy here. First, in some cases (small samples), there are only a small n u m b e r of permutations possible. A total o f N observations at k levels with n, replicates per level gives N!/(n:n2!... nk!) different permutations. O f these permutations, only N!//-/[(i!)"m~!] (where m~ is the n u m b e r of groups with i replicates) are unique. W h e n this number is small, all possible permutations o f the data (Berry and Mielke, 1984) should be computed. Second, as in the analysis of variance, the test is a one-sided test. However, the rejection side depends on the statistic and measure that are used. For example, if ~r/Wis used, the null hypothesis is rejected if the observed value o f the statistic L ( d a t a ) = ~r/fl~ is large relative to the permuted values if a distance measure is used. Because distance measures are large when treatments are separated, a treatment effect would m e a n that L(data) should be large relative to the permuted values (the between treatment distance in the data
will be large relative to the within treatment distance). If a similarity measure is used, rejection occurs if the observed Br/I~ is small relative to the permuted values. Since the within treatment observations should be close to each other, the within similarities should be large. The between similarities should be small. Hence, relatively small values of L(data) are expected. Alternatively, if ~r or if'is used as the statistic and a similarity measure is used, then rejection occurs for small values of B or large values o f if'. If a distance measure is used, then the null hypothesis is rejected for large values of B or small values of if'. In Table 1, all 20 possible combinations o f the six replicates are divided into two groups of three, and the resulting totals are given for within treatment and between treatment. The total similarity remains fixed at 10.42, and there are 10 unique values, not 20. For example, if group 1 consists o f replicates 1, 2 and 3 and group 2 o f 4, 5 and 6, the within and between total similarities are the same as when group 1 consists o f replicates 4, 5 and 6 and group 2 consists o f 1, 2 and 3. Although the total of the between similarities for the observed data, 4.91, is less than all the other values, the associated significance level for the test of no difference is only 2/20 = 1/10. Within this small n u m b e r o f replicates, a strong statement cannot be made about the possible differences using the permutation procedure (or any
510
ERIC P. SMITH et al.
other procedure without making additional uncheckable assumptions). Thus, although the mean between similarity is smaller than all permuted values, the hypothesis that there is a treatment effect is not rejected at significance levels below 0.10. When the number of replicates is large, a large number of permutations (say 1000) can be computed. A critical value can then be chosen for a particular ~ level by ordering the permuted statistics and selecting the 100(1-~) percentile. This would then be compared with the observed value to test the null hypothesis.
Follow-up analyses o f replicates Assuming rejection of the null hypothesis of no treatment or location differences, follow-up analyses of treatments will determine which ones differ and why. There are a number of analyses that could be used to determine which treatments differ (assuming there are more than two) and which species are the important species for indicating the differences. It may also be of interest to look for possible odd data values that may influence the analysis. A number of procedures could be used; only a few are discussed here. In particular, interest in this discussion is in methods that use similarity or distance information, as they are consistent with the previously discussed overall test. Multidimensional scaling procedures (Gower, 1966) are a set of graphical procedures useful for comparing the treatments. While there are many multidimensional scaling approaches, we have found the one based on metric scaling to be quite useful. The objective of this set of procedures is to find a set of x and y data points that, when graphed, represent the location of the samples in a two-dimensional space (a reduced species space). The point (x, y), or treatment scores, are chosen so that the visual (Euclidian) distance between the points is approximately the same as the distance between the samples as measured by the chosen index. If a similarity measure is used, the points well separated would have low similarity. Graphs of the sample sites show groups of samples, for possible ordering of the sites, or odd data values. Thus, this method is somewhat similar to principal components analysis in that the dimensionality is reduced and scores plotted to visualize patterns. However, instead of using a covariance or correlation matrix (which would suggest Euclidian distance), a similarity or distance matrix is reduced. A better approach for assessing separation, one that may be used for inference, is to compare the mean similarities for the different sites using randomization. Two approaches may be used. One approach is to carry out more permutation tests, looking at the locations in a pairwise fashion, analogous to multiple comparison procedures (Foutz et al., 1985). Foutz et al. (1985) suggest forming comparisons of interest and carrying out the permutation test on the vector of comparisons. For example, if there are four groups, there are six pairwise comparisons of interest •12/#]2, ~ o j l ~ , ~14/1~r14,~23/~2], ~24/~'24 and B ~ / f f ' ~ , where ~ r and ~1 are the mean between and mean within similarity for groups i and j, respectively. At each step of the permutation procedure, a vector of length six is generated. After all vectors are generated, the critical values should be calculated. This is relatively easy to accomplish for the overall test by sorting the permuted values and obtaining the upper or lower percentile associated with the significance level. However, with multiple comparisons, the sorting is not of single values but vectors, and the problem oferror rates associated with the overall test and the pairwise comparisons also arises. Foutz et al. (1985) indicate that these problems can be solved by the proper sorting of the permuted vectors. If there are R pairwise comparisons and N is the number of permutations, choose a significance level • for experimentwise and comparisonwise error rates so that K = ~N is divisible by R. For example, if there are 6 multiple comparisons and 1000 permutations, • of 0.06 gives ~N of 60 which is divisible by 6 (so, K = 60, N = 1000 and R = 6). For the
overall test, the 1000 permutations could be sorted and the 60th smallest (or largest depending on the statistic and measure) value chosen for use as a critical value for the test. Sorting on the elements of the vectors produces the critical values for each comparison for use in the multiple comparison procedure. Error rates are maintained by: (1) selecting an element of the vector corresponding to a comparison--select a random number for 1 to R (the total number of comparisons); sort on this element (for example, if the second element is selected, all vectors would be ordered by the second element); (2) put this vector aside; choose another element; sort the remaining N-I vectors on this element; (3) repeat this process a total of K times with each element being sorted K / R times. In the above example, K = 60 and R = 6 . Each element would be used in the sorting process 10 times. The vector obtained in the last sort on each element contains the critical value for that element. There will be R (6 in the example) critical values, one for each comparison. The tests on the comparisons can be carried out by comparing the observed comparison with the critical value. Further details of the procedure and some extensions are presented in Foutz et al. (1985). This approach is only useful when there are an adequate number of replicates for each treatment level. Another follow-up analysis that may be of interest is confidence intervals for the mean between similarity. As the estimated similarities are not independent, standard methods cannot be used as in the usual analysis of variance (i.e. based on the normal distribution). Two approaches that work well for this problem are based on the bootstrap method and the jackknife method. As in the multiple comparison problem, these methods require reasonable sample sizes to work well. Some details are given in Smith (1985) and Smith et al. (1986).
Follow-up analyses of species Species as well as the treatments must be analyzed because changes in the species can indicate which species are reacting to the treatment as well as being informative on why there are differences between the treatments. As different measures of similarity may lead to different conclusions about changes due to the treatment, it is important to have methods that use the measure of interest. There are, of course, a number of possible methods useful for evaluating the influence o f individual species. The interest here, however, is not only in changes in individual species but which species are primarily responsible for the differences indicated by the test. As the test is strongly dependent on the measure of similarity, using the contribution (or importance) of the species to the overall statistic is proposed. If a measure is additive (for example, Bhattacharyya's measure), the contribution may be directly computed. However, most measures are not additive, so a method based on the effect of removing a species on the similarity is proposed. Let B'_ i be the mean of the between similarities with species i removed. Then INFi = 100(B- , -/~)//~
(9)
measures the percentage relative influence of species i on the mean between similarity. Large positive values indicate a species whose removal greatly increases the between similarity (i.e. makes the data more similar). These species usually have quite different proportional abundances between treatments. Species with large negative values decrease the between similarity when removed and represent species that show little change over most treatment levels but contribute to the between similarity.
Example As an example, consider the data set (Table 2) from an experiment on the effect of zinc on the periphyton community in the New River at Glen Lyn, Va. Twelve artificial streams received river water and one of four zinc treatments
Analysis of multispccivs experiments
511
Table 2. Data (biovolume in cells/era 2) on I 1 dominant species from a study on the effects of zinc on the a l o e community in the New River, influence measures and mean similarities Species Zn
Rep.
I
2
3
4
5
0.0
1 2 3 1 2 3 1 2 3 1 2 3
175 134 77 44 18 49 Ill 59 29 81 98 66
1745 931 393 1738 241 716 846 386 482 862 953 794
642 0 393
408 412 323 59 55 66 89 40 51 85 80 14
21 53 48 564 137 874 7 3 0 44 17 0
0.05 0.50 1.0
Cosine Bhatt.
0.06 -0.00
4.18 -0.76
14
11 16 44 22 29 65 53 37 0.47 0.87
6
7
730 596 705 1923 1103 1891 7 18 14 40 26 51
Influence 0.19 2.33 0.87 3.66
0.69 0.83
8
0 0 3 0 0 0 163 3 44 694 561 185
204 134 230 876 427 924 5133 802 995 8457 14,012 10,028
0.02 0.59
44.73 13.00
9
10
29 187 141 222 200 208 326 155 111 102 205 14
379 245 300 490 159 216 1515 594 1084 98 989 37
2293 1070 3613 5615 3725 7099 12,940 9489 20,116 1013 5740 1121
-0.01 -0.58
-15.09 -6.01
0.06 -0.12
11
Matrix of mean similarities using the cosine index Zn 0.00 0.05 0.50 1.00
0.00 0.872
0.05 0.882 0.982
0.50 0.824 0.933 0.967
1.00 0.265 0.322 0.365 0.973
Matrix of mean similarities using Bhattacharyya's measure Zn 0.00 0.05 0.50 1.00
0.00 0.923
0.05 0.901 0.982
(0.0, 0.05, 0.5 or 1.0 mg/l). Artificial substrates were placed in the streams and removed at a n u m b e r o f times throughout the experiment. Thus, the experimental design is a two factor repeated measures experiment (split-plot design but without blocking) (Milliken and Johnson, 1984). As the experiment was designed to obtain a significant time-zinc interaction, the individual times were analyzed separately. A detailed analysis o f the full data set has been presented elsewhere (Genter et al., 1987). Table 2 gives the data for day 20 o f the experiment and some s u m m a r y measures using Bhattcharyya's measure and the cosine measure. The permutation test indicated significant differences between the four treatments for both measures (no mean between similarities were lower than the observed). Figure 1 shows the separation between the replicates using both measures. With the cosine measure, the low zinc treatments are close together, while the highest
0.50 0.798 0.870 0.966
1.00 0.591 0.644 0.692 0.967
dose of zinc separates well from the low doses. The means o f the similarities in Table 2 suggest a relatively high degree o f similarity for replicates within a treament (the diagonal elements) while the between means show a decreasing similarity with increasing zinc concentration. Note that this is not clearly displayed in Fig. 1 due to the horseshoe effect (Kendall, 1971). Variance estimates suggest a high degree of variability for replicates using the cosine index. Confidence intervals and the multiple comparison procedure applied to the between means suggest overlappings between the 0.0, 0.05 and 0.50 treatments. The plot for Bhattacharyya's measure differs slightly from that of the cosine measure. In particular, the 0.00 treatment group (control) is closer to the 0.05 group while the 0.50 group is more distant. In both cases, one of the replicates from the control group overlaps with the 0.05 group of replicates. The reasons for these differences are
b. B h a t t a c h a r y y a
a, cosine 0.5"
0.4[]
~
•',
[]
X
0.2£ • v
A m
0.0
4L
A
I
N
.o.0 2
._w
._w -o.2-
x
-0.5'
A M ÷
-1.0
-0.6
m
[] zn-O.O zn-O.05 zn-0.50 zn-1.00
+
-0.6'
-0.2
0.0
axis 2
. 0.2
.
. 0.4
.
.
0.8 -0.4
[] & . +
+ +
+ +
-
-0.4
-0.4"
4-
-0.2
0.0
0.2
ZnlO.O0 zn=0.05 zn=0.50 zn-l.0
0.4
axis 2
Fig. I. Plots of samples using the first two axes from multidimensional sealing using (a) the cosine measure and (b) Bhattacharyya's measure. W.R. 2 4 / 4 - - H
0.6
512
Emc P. SMITH et al.
discussed below. Variance estimates for mean similarities using Bhattacharyya's measure are much lower, and, hence, confidence intervals are narrower. Confidence intervals and the multiple comparison procedure on the between means suggest no differences between the 0.00 and 0.05 groups but differences between all other pairs. The influence measures indicate some interesting relationships between the data and the similarity statistics. The data indicate that certain cell densities tend to be quite large. Easily observed differences are for species 5 and 7. However, these species have negligible importance to the overall similarities. The influence measures point to species 8 and 11 as the species that determine magnitude of B. This peculiarity is due to the high abundances of species 8 and 11. Understanding the relationship between the data, influence and the differences between treatments, require considering the proportions for each replicate (for Bhattacharyya's measures) and proportions adjusted for the square root of the sum of squared proportions (for the cosine measure) that are presented in Table 3. Species 11 dominates the cosine measure for treatments 0.0, 0.05 and 0.50. Only the last treatment alters this species dominance. The large negative influence indicates the dominance of the species for most of the treatments. Species that have a strong decreasing effect on between similarity are species 8, 6 and 2. Species 8 increases relative to others in replicates of the 1.0 treatment, while species 2 and 6 are diminished by increasing zinc. Differences in Fig. 1 between replicates for the 0.0 and 0.05 treatment are due to species 2, while species 6 contributes to differences between the replicates for the 0.05 and 0.50 treatment, and species 8 and 11 affect separation of the 0.50 and 1.0 treatments. Thus, the influence measures highlight the four types of changes in the species that are associated with the levels of the treatment and indicate the relative importance of each of the levels of the treatment. The interpretation based on influence using Bhattacharyya's measure is slightly different and may be
Table 3, Adjusted proportions
P~
understood by studying the unadjusted proportions in Table 3. Although species 11 dominates, the magnitude of that domination is diminished especially for the first group of replicates. Also diminished is the influence of species 2. While high for the cosine measure, the influence is now small and negative. With the exception of two replicates in the first treatment, the proportions for this species are relatively small. Thus, the cosine measure gives more weight to changes in dominant species, while Bhattacharyya's measure is not as influenced by the dominant species. Primary differences using Bhattacharyya's measure are indicated by the proportions for species 6, 8 and I 1. These species all have large influence. Species 6 again will lead to a separation between replicates for low (0.0, 0.05) and high (0.05, 1.0) zinc treatments in Fig. 1. Species 8 and I 1 contribute to the separation between replicates of the 1.0 zinc treatment and the other treatments. Bhattacharyya's measure indicates the replicates for the two low treatments are much closer together than suggested by the cosine measure. DISCUSSION This p a p e r has presented m e t h o d s for the analysis o f mulfispecies data. T h e m e t h o d s allow a researcher to a n s w e r three i m p o r t a n t questions a b o u t the data. First, are there any overall differences due to treatments? Second, if there are differences, which levels o f the t r e a t m e n t are associated with the differences? Third, which species are associated with differences? T h e m e t h o d s are unified in that they involve the use o f similarity o r distance measures. Thus, for example, the species that are d e t e r m i n e d to be i m p o r t a n t in the third analysis are the species that are influencing the
P~ and proportions (P~) for Glen Lyn data. Boxes indicate where primary changes occur
(a) Adjusted proportions Species Zn
oo
Rep. 2 3
1
2
4
5
0.000 0.105
0.252 0.086
0.032 0.013
006 00 0.082 0.021
3
6
03 07
0.05
I 2 3
0.007 0.005 0.007
0.276 0,061 0.096
0.002 0.003 0.002
0.009 0.014 0.009
0.090 0.035 0.117
0.50
! 2 3
0.008 0.006 0.00I
0.060 0.040 0.024
0.003 0.002 0.001
0.006 0.004 0.003
0.001 0.000 0.000
1.00
1 2 3
0.010 0.006 0.007
0.100 0.063 0.079
0.008 0.004 0.004
0.010 0.005 0.001
0.005 0.001 0.000
10
II
0.115 0.038
0.150 0.080
0.6551 0.961
0.000 0.000 0.000
0.035 0.051 0.028
0.078 0.041 0.029
0.892 0.949 0.947
0.012 0.000 0.002
0.023 0.016 0.006
0.108 0.062 0.054
0.922 0.994 0.997
00
7
8
00
0.000 0.001
~ ~
9
0 0
0.081 0.037 0.018
~ ~
0.012 0.013 0.001
0.011 0.065 0.004
0.000 0.000 0.001 0.000 0.000 0.000
0.031 0.036 0.037 i 10.076 10.070 10077
0.004 0.050 0.023 0.019 0.033 0.017
0,057 0.065 0.048
0.346 0.284 0.580
0.042 0.026 0.018
0.486 0.613 0.588
0.008 0.000 0.002
0.242 0.069 i0.043
0.015 0.013 0.005
0.072 0.051 0.047
0.611 O.820 0.876
0.060
~
0.009
(b) Proportions 0.00
l
0.05
2 3 I 2 3
0.026 0.036 0.013 0.004 0.003 0.004
0.263 0.247 0.063 0.150 0.040 0.059
0.097 0.000 0.063 0.001 0.002 0.00t
0.062 0.110 0.052 0.005 0.009 0.006
0.003 0.014 0.008 0.049 0.023 0.073
0.50
1 2 3
0.005 0.005 0.001
0.040 0.033 0.021
0.002 0.002 0.001
0.004 0.004 0.002
0.000 0.000 0.000
1.00
I 2 3
0.007 0.004 0.005
0.075 0.042 0.064
0.006 0.002 0.003
0.007 0.004 0.001
0.004 0.001 0.000
10.110[
J
0.025
0.009
0.008 0.044
0.015
0.001
0.003
Analysis of muitispecies experiments
similarity or distance measure and are associated with the treatment differences in the first and second analyses. The techniques discussed above provide researchers with a powerful set of techniques for the
analysis of multispecies data. The methods presented are only a few of the possible techniques available for analyzing community data. However, these methods are oriented to the analysis of data using some measure of similarity. A number of other techniques, such as principal components, discriminant analysis, and detrended correspondence analysis (Greenacre, 1984; Gauch, 1982) are dependent of Euclidian (or weighted Euclidian) type measures of distance. If interest is in Euclidian (or weighted Euclidian) distance, then these methods, which are available on some computer packages, should provide useful results. One method of interest is Gabriel's biplot analysis (ter Braak, 1983), which allows graphical displays of both the species and the replicates. One drawback to the permutation approach is that the size of the difference between the mean between and the mean within similarity may be statistically significant but not biologically important. If all between similarities are only 0.01 less than the smallest within similarity, the degree of significance using the permutation test is the same as if the difference were 0.50. This problem is shared with many nonparametric methods and similar problems occur in classical statistical methods, but should not commonly occur. The magnitude of the between and within similarities as well as the significance of the test should be considered. An approach based on the size of the similarities is available; however, it requires additional assumptions. Boyle et al. (1984) present a similar approach to the overall test of differences, using a t-test to compare the similarities at two locations. This test procedure ignores the dependence between estimated similarities and may result in inflated error rates. A procedure that does account for dependence is discussed by Dyer (1978). This procedure involves the use of linear models and assumes normality of the similarities and homogeneity of variance. The examples discussed above are for relatively simple designs. For example, we did not test directly for a time effect or interaction between time and treatment in the Glen Lyn study. Analyses of more complex designs are possible but not as straightforward due to the use of repeated measurements on the same experimental unit over time (Carter et al., 1982). Acknowledgements--This research was partially supported by an appointment to the U.S. Department of Energy Faculty Research Participation program administered by Oak Ridge Associated Universities to E. P. Smith while he was at the Environmental Sciences Division, Oak Ridge National Laboratory (operated by Martin Marietta Energy Systems Inc., under contract DE-AC05-84OR21400 with the U.S. Department of Energy), Oak Ridge, TN
513
37831-6036, U.S.A. Publication No. 3405, Environmental Sciences Division, ORNL. Comments by I. J. Good, K. Rose and A. Brenkert greatly improved a previous version of the manuscript. REFERENCES
Bell C. B., Conquest L. L., Pyke R. and Smith E. P. (1981) Some nonparametric statistics for monitoring water quality using benthic species counts. In Environmetrics 81: Selected Papers, pp. 100-121. Society of Industrial and Applied Mathematicians, Philadelphia. van Belle G. and Fisher L. (1977) Monitoring the environment for ecological change. J. Wat. Pollut. Control Fed. 49, 1671-1679. Berry K. J. and Mielke P. W. (1984) Computation of exact probabilities for multiresponse permutation procedures (MRPP). Commun. statist, simul. Computat. 13, 417-432. Bhattacharyya A. (1946) On a measure of divergence between two multinomial populations. Sankya 7, 401-406. Boyle T. P., Sebaugh J. and Robinson-Wilson E. (1984) A hierarchical approach to the measurement of changes in community induced by environmental stress. J. envir. Test. Eval. 12, 241-245. ter Braak C. T. F. (1983) Principal component biplots and alpha and beta diversity. Ecology 64, 454-463 Carter R. L., Morris R. and Blashfield R. K. (1982) Clustering two-dimensional profiles: a comparative study. Technical Report 175, Department of Statistics. University of Florida, Gainesville. Dyer D. P. (1978) An analysis of species dissimilarity using multiple environmental variables. Ecology 59, 117-125. Foutz R. V., Jensen D. R. and Andersen G. W. (1985) Multiple comparisons in the randomization analysis of designed experiments. Biometrics 41, 29-38. Gauch H. G. (1982) Multivariate Analysis in Community Ecology. Cambridge University Press, Cambridge. Genter R. B., Cherry D. S., Smith E. P. and Cairns J. Jr (1987) Algal-periphyton population and community changes from zinc stress in stream mesocosms. Hydrobiologia 152, 261-275. Good I. J. (1982) An index of separateness of clusters and a permutation test for its significance. J. statist, comp. Simul. 15, 81-84. Gower J. C. (1966) Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika 53, 325-338. Greenacre M. J. (1984) Theory and Applications of Correspondence Analysis. Academic Press, London. Hajdu L. J. (1981) Graphical comparison of resemblance measures in phytosociology. Vegetatio 48, 47-59. Hellawell J. M. (1978) Biological Surveillance of Rivers: A Biological Monitoring Handbook. Water Research Centre, Stevenage, England. Kendall D. G. (1971) Seriation from abundance matrices. In Mathematics in Archaeological and Historical Sciences (Edited by Hodson F. R., Kendall D. G. and Tautu P. A. P.), pp. 215-252, Edinburgh University Press, Edinburgh. Lamont B. and Grant K. J. (1979) A comparison of twenty measures of site dissimilarity. In Multivariate Methods in Ecological Work (Edited by Orloci L., Rao C. R. and Stiteler W. M.), pp. 101-126. International Cooperative Publishing House, Fairland, Md. Mielke P. W. (1979) On the asymptotic non-normality of null distributions of MRPP statistics. Commun. statist. theor. Meth. A8, 1541-1550. Mielke P. W., Berry K. J., Brockwell P. J. and Williams J. S. (1981) A class of nonparametric procedures based on multiresponse permutation procedures. Biometrika 68, 720-724.
514
ERIC P. SMITH et al.
APPENDIX
Milliken G. A. and Johnson D. E. (1984) of
Messy
Data,
Volume
1: Designed
Analysis Experiments.
Wadsworth, London. Perkins J. L. (1983) Bioassay evaluation of diversity and community indexes. J. Wat. Pollut. Control Fed. 55, 522-530. Pitman J. (1937) Significance tests which may be applied to samples from any populations. J. R. Statist. Soc., Set. B 4, 119-130. Pontasch K. W. and Brusven M. A. (1988) Diversity and community comparison indices assessing macroinvertebrate recovery following a gasoline spill. Wat. Res. 22, 619-626. Smith E. P. (1985) Estimating the reliability of diet overlap measures. Envir. Biol. Fish. 13, 125-138. Smith E. P., Genter R. and Cairns J. Jr (1986) Confidence intervals for the similarity between algal communities. Hydrobiologia 139, 237-247. Sneath P. H. A. and Sokal R. R. (1973) Numerical Taxonomy: The Principles and Practices of Numerical Classifi cation. Freeman, San Francisco.
Stander M. J. (1970) Diversity and similarity of benthic fauna off Oregon. M.Sc. thesis, Oregon State University, Corvallis, Ore. Sullivan M. J. (1978) Diatom community structure" taxonomic and statistical analysis of a Mississippi salt marsh. J. Phycol. 14, 468-475. Washington H. G. (1984) Diversity, biotic and similarity indices: a review with special relevance to aquatic ecosystems. Wat. Res. 18, 653-694. Zimmerman G. M., Goetz H. and Mielke P. W. (1985) Use of an improved statistical method for group comparisons to study five effects of prairie fire. Ecology 66, 606-611.
Computational Details
Assume there are g groups and let S,j be the similarity between replicates i and j. Indexing on the replicates is i = 1, 2. . . . , N where N is the total number of replicates. Further, N = n t + n: + • • - + ng where nk is the number of replicates in group k, and N k = n I + n, + . . • + n k . For group k the total within similarity is
w,=
Yk
Nk
E
Es,,.
i-Nk
14-1 ! > i
Let T= V
S,~
,~1
!>J
be the total similarity. Then B = T - IV. The mean within similarity for group i is • ,= w
,, \2 I
To compute the overall mean similarity, one may use
, . -1
However, Mielke (1979) indicates that
if* = E n , ~ / N is a better estimate if the n, are not equal. The mean between similarity is given by [
g \i=lj>i
]