Ecological Archives XXX-XXX-XX

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1 Ecoloical Archives XXX-XXX-XX Marti J. Anderson, and Daniel C. I. Walsh. PERMAOVA, AOSIM and the Mantel test in the face of heteroeneous dispersions: What null hypothesis are you testin? Appendix A. Description of statistical tests and related methods. Let Y be an p matrix of i =,..., multivariate observations (rows) by k =,..., p variables (columns). Let D = {d } be a square symmetric matrix of distances (or dissimilarities) between all pairs of observations i =,..., and j =,..., with diaonal elements d = 0 i = j. For example, the Euclidean distance is: d ( E) p k ( y y ) (A.) ik jk In ecoloy, typically, some other resemblance may be calculated for non-neative count data, such as the Bray-Curtis measure: d ( BC ) p k p k y ( y ik ik y y jk jk ) (A.) which ranes from 0 to, and is also often expressed as a percent similarity: s 00 ( d ( BC ) ( BC ) ). Another commonly-used measure, calculated on presence-absence data and directly interpretable as the proportion of unshared species, is the Jaccard measure: d ( J ) p k ( y p k ik ( y ik ) ( y ) ( y jk jk ) ) ( y ik y jk ) (A.3) where () is an indicator function such that

2 0 if x 0 ( x) if x 0 ext, suppose the observations belon a priori to =,..., roups, with sample sizes n, n,..., n and n. Let X be an ( ) matrix of full rank containin orthoonal contrasts amon the roups. We can construct an projection matrix for this roup structure accordin to the classical linear least-squares solutions to the Gauss-Markov normal equations (e.., Plackett 949) as: H X[ XX] X (A.4) As outlined in McArdle and Anderson (00), this can be used to obtain a partitionin of the multivariate variability inherent in matrix D, by relyin on the followin transformation, due to Gower (966) and hihlihted for this purpose oriinally by McArdle (99). Let matrix A consist of elements a d, which, after centerin on its rows and columns, ives a matrix directly interpretable as sums of squares and cross products (SSCP). amely, matrix G of elements: a ai a j a (A.5) where j, a j i a and a i j a i a a. Indeed, supposin each variable (C) is centered on its mean to yield the centered data matrix Y c with elements { y } (namely, where ( C) for each column variable j, we have y ( y y j ) and y j i y ) and we have used Euclidean distances (E) d, then matrix G is equivalent to the outer product G Y Y. (A.6) c c The total sum of squares is obtained as the trace (sum of diaonal elements) of this matrix. If Euclidean distance is used, this is equivalent to the trace of the inner product SSCP for Y; i.e.

3 3 where tr indicates the trace of a matrix. If tr G] tr[ Y Y] tr[ YY ] (A.7) [ c c c c (BC) d, ( J ) d or some other resemblance measure is used to construct D, however, then the relationship between G and Y is not so straihtforward. ow the one-way PERMAOVA test-statistic (McArdle and Anderson 00) is easily obtained throuh a partitionin of the G matrix to yield a pseudo-f statistic: F pseudo tr[ HG]/ v tr[( I H) G]/ v (A.8) where v ( ), v ( ) and I is an identity matrix with ones alon the diaonal and zeros elsewhere. ote that this equation is equivalent to equation (4) in McArdle and Anderson (00) because of the idempotency of matrix H (i.e., HH = H) and the fact that tr[hgh] = tr[hhg]. ow, with just one variable (p =) and Euclidean distances, the value of pseudo-f is precisely equal to the oriinal univariate F ratio (Snedecor 934) used in classical analysis of variance. For the one-way case, a p value is calculated for PERMAOVA by a random reorderin (permutation or randomization) of the observation rows of Y relative to the fixed ordered list of n + n +..., n labels for the roups (Edinton 995, Manly 006). This is equivalent to a random simultaneous re-orderin of the rows and columns of matrix D, which maintains the inter-point structure in the multivariate space, but chanes the roup label with which each point is associated (Anderson 00b). If the desin is balanced, then all observations have an equal chance of fallin into any particular roup. If the desin is unbalanced, then this is not true; however, the structure of the existin imbalance in the number of replicates per roup is maintained under randomization and all re-orderins of the observations relative to this structure are equally likely. The test-statistic is re-calculated for each randomization ( F ( ), say)

4 4 and a distribution of ( ) F is thereby enerated under a null hypothesis of no differences amon the roups, conditional on the observed data. A random subset of all possible re-orderins can be used for accurate inference (Hope 968). A p value is calculated as the proportion of obtained under randomization that are reater than or equal to the observed value of pseudo-f. ote also that (A.8) can be calculated directly from sums of squared distances (or dissimilarities) in matrix D as described in Anderson (00a); namely, ( ) F F pseudo ( SS SS ) / v SS / v T W (A.9) W where SS T is the sum of squared inter-point dissimilarities divided by the number of points: ( ) SST d / i j( i) (A.0) and SS W is the sum of squared inter-point dissimilarities within each roup divided by the number of observations within that roup, and then summed across all roups: SS W ( ) i j( i) d / n (A.) Here and in what follows, is an indicator such that = if sample units i and j are in the same roup, or else = 0. ote also thattr[g ] SST. Leendre and Anderson (999, see Theorem in Appendix B therein) have shown the equivalence of (A.) with the sum-ofsquared distances to roup centroids in the case of Euclidean distances. A eometric F statistic constructed usin sums of squared Euclidean distances to centroids within and between roups was described as a possible multivariate randomization test by Edinton (995, pp. 88 9). Pillar and Orlóci (996) had also suested the use of a related test-statistic, Q SS SS ), B ( T W which, in the specific case of a one-way AOVA model only, is monotonic on the pseudo-f

5 5 statistic iven in (A.9) under permutation, as the derees of freedom (v and v ), and also SS T will all remain constant for any random re-orderin of the data, so identical p values will be obtained. It may be noted here that the PERMAOVA test-statistic has the advantae of bein constructed as a pivotal test-statistic (i.e., F pseudo calculated from a Euclidean distance matrix for variable is equivalent to the classical univariate F statistic), so should not be affected adversely by the presence of nuisance parameters and can be easily extended to multi-way desins. It is also clearly not restricted to the use of the Euclidean distance. ote also, however, that the construction of the test effectively relies holistically on sums of squared distances within (and between) roups, without any reard whatsoever for the particular direction of those distances within the multivariate space, which distinuishes it from the classical MAOVA test statistics. ext, the AOSIM statistic of Clarke (993) is easily described as a function of the ranks of matrix D. There will be M ( ) / inter-point distance values d within the upper-trianular (or, equivalently, the lower-trianular) portion of matrix D (excludin the diaonal); namely, for i =,..., ( ) and j = (i + ),...,. Let the values d be replaced by the rank order of their values, r, where the lowest value of d is iven a value of r = and the hihest value of d is iven a value of r = M. The AOSIM test-statistic (Clarke 993) is then iven by: ( rb rw ) R (A.) M / where r W is the averae of the ranked dissimilarities between observations within the same roup:

6 6 r W ( ) i j( i) r n ( n ) / (A.3) and r B is the averae of the ranked dissimilarities between observations in different roups: r B ( ) M i j( i) ( ) r n ( n ) / (A.4) A p value is obtained for the one-way case in the same way for AOSIM as for PERMAOVA, usin random re-orderins of the observations relative to the roup structure and calculatin a distribution of ( ) R provide a p value for the test. aainst which the value of R for the oriinal orderin is then compared to The Mantel test was first described as a test of association between two resemblance matrices (Mantel 967, Mantel and Valand 970). For a iven set of observations, suppose there are two resemblance matrices; for example, the first miht be dissimilarities based on species data while the second miht be eoraphic distances. A cross-product (or Pearson or Spearman correlation coefficient) is calculated between the matched paired values in the two matrices and this is compared with the distribution of the same under random re-orderin of the oriinal observations for one of the two matrices. The Mantel test may also be used for a oodness-of-fit test between a matrix of resemblances and a model matrix (Leendre and Leendre 998, see pp ). For example, to model the roup structure as in AOVA, the model matrix may have zeros in place of the between-roup distances and ones in place of the within-roup distances. In other words, the model matrix consists of the indicators, as defined in equation (A.) above. A cross-

7 7 product between the sub-diaonal elements (as these matrices are symmetric) then simply ives the sum of the within-roup dissimilarities, ( ) z (A.5) (,0) d i j( i) ote that the value of z (,0) will decrease with increasin deree of clumpin within roups, so the p-value for the test usin this statistic must be calculated as the proportion of values of z ( ) (,0) that are less than or equal to the observed value of z (,0). For one-way desins, other arbitrary contrast coefficients can be used in the indicator model matrix to distinuish the within-roup versus the between-roup dissimilarities, yet would yield the same result. For example, consider the use of (,+) rather than (0,) to ive: z ( ) (, ) ( ) i j( i) ( ) d d (A.6) i j( i) As the sum of all the dissimilarities in the sub-diaonal matrix of D is a constant, (A.6) will yield a cross-product that is monotonic with (A.5) under permutation, so will result in equivalent p values for the Mantel test. Furthermore, Leendre and Leendre (998, p. 56) demonstrated the clear relationship between the Mantel test and AOSIM. Specifically, in the model matrix, let the code for within-roup resemblances be: c W n ( n ( M / ) ) / (A.7) and the code for the between-roup resemblances be: c B M n ( n ( M / ) ) / (A.8)

8 8 then the Mantel cross-product statistic z yields a test statistic with an equivalent structure to the R-statistic of AOSIM, but it is calculated on the averaes of the between-roup and withinroup dissimilarity values themselves, rather than on the averaes of their ranks, namely: z ( c W, c ) B ( db dw ) (A.9) M / The use of (A.9) will yield an equivalent p-value for the one-way model as the use of either (A.5) or (A.6). It will not, however, yield the same results as the AOSIM R statistic in (A.), which is based on ranks. The form of the Mantel test-statistic iven in (A.9) was the one we used in our simulations. To draw further parallels, the Mantel test also has a clear and close kinship with the resemblance-based permutation test statistic described by Good (98) and Smith et al. (990), namely d / d B W. This would also be monotonic under permutation with any of (A.5), (A.6) or (A.9), and thus would yield identical p values to the Mantel test for these one-way model simulations. (Althouh oriinally described in terms of averae similarity, s, rather than dissimilarity, d, enerally one can easily write a simple inverse function d = s, so the result still holds). Other important parallels can be drawn between the methods we have included in our simulations and the multi-response permutation procedure (MRPP, Mielke et al. 98, Mielke and Berry 00). The eneral formulation of the MRPP statistic is iven by C (A.0) where C 0 is a roup weiht, C, and ( ) ( ) (A.) n ( n ) / i ji

9 9 is the averae of pairwise distance function values within each roup ( =,..., ), where ( ) is an indicator such that if sample units i and j are both within roup. The test () statistic ets smaller with increased clumpin of observations within roups, so the p-value for ( ) the MRPP test is calculated as the proportion of values of under permutation that are less than or equal to the observed value of. If we let d and assin the weihts C to be proportional to the roup sample sizes, i.e., C n /, then we have the direct result that, where /[ ( n )]. Thus, the MRPP test is equivalent to the Mantel test coded z c,0) ( W c W in this way for either balanced or unbalanced desins. ote, however, that under permutation the test statistic z ( c W,0) will only be monotonic on the Mantel test statistics of z (,0), z (, ) or z c W, c ) (as iven in equations (A.5), (A.6), and (A.9) above, respectively) when there are equal ( B numbers of replicate sample units per roup. Thus, the MRPP test usin (and with C n / ) will yield equivalent permutation p-values to these more eneral implementations of the Mantel test only for balanced one-way desins. Mielke and Berry (00, p. ) have also shown, for the one-way case, that MRPP, when based on squared Euclidean distances for a sinle variable, yields p values equivalent to the univariate F statistic under permutation. It is therefore easy to show here the relationship between MRPP and PERMAOVA more enerally for one-way models. First, it is important that the distances be squared, i.e., let. Then, let the weihts be C ( n ) /( ) d, and the relationship between the PERMAOVA statistic of (A.9) and the MRPP statistic of (A.0) is F pseudo SST v. (A.) v

10 0 As the values of SS T, v and v are all constant under permutation, based on squared dissimilarities with this choice of weihts will yield a p value for MRPP that is equivalent to PERMAOVA. This relationship holds for either balanced or unbalanced one-way desins. In this study, the resemblance-based permutation tests (PERMAOVA, AOSIM and Mantel) were compared with one another and with the classical MAOVA test statistic described by Pillai (955). Given that the SSCP matrix for the within-roup variation is W ( ) Yc I H Y c and the SSCP matrix for the between-roup variation is B Y HY c c, then ( ) Pillai s trace is defined as V s tr[ B( W B) ]. To obtain a p-value, the followin F- approximation (Pillai 955) was used: F Pillai ( s) (t s ) V (A.3) ( s) (q s )( s V ) with s ( q s ) and s ( t s ) derees of freedom, where, s min( v, p), q v p ) ( and t ( v p ). ote that we must have v p. Also note that for Euclidean distances only, we can write the PERMAOVA pseudo-f as: F pseudo tr[ B]/ v tr[ W]/ v (A.4) which hihlihts how it differs from Pillai s trace. Pseudo-F is a ratio of two traces, each of these bein a pure sum of individual sums of squares, thus inorin all off-diaonal cross-products and hence correlation structure. For Pillai s trace, in contrast, the off-diaonal cross-product terms will play a role throuh the calculation of an inverse followed by the matrix multiplication, both of which occur prior to takin the trace.

11 LITERATURE CITED Anderson, M. J. 00a. A new method for non-parametric multivariate analysis of variance. Austral Ecoloy 6:3 46. Anderson, M. J. 00b. Permutation tests for univariate or multivariate analysis of variance and reression. Canadian Journal of Fisheries and Aquatic Sciences 58: Clarke, K. R onparametric multivariate analyses of chanes in community structure. Australian Journal of Ecoloy 8:7 43. Edinton, E. S Randomization tests, 3rd edition. Marcel Dekker, ew York, USA. Good, I. J. 98. An index of separateness of clusters and a permutation test for its sinificance. Journal of Statistical Computation and Simulation 5:6 75. Gower, J. C Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika 53: Hope, A. C. A A simplified Monte Carlo sinificance test procedure. Journal of the Royal Statistical Society, Series B 30: Leendre, P., and M. J. Anderson Distance-based redundancy analysis: testin multispecies responses in multifactorial ecoloical experiments. Ecoloical Monoraphs 69: 4. Leendre, P., and L. Leendre umerical ecoloy, Second Enlish edition. Elsevier, Amsterdam, The etherlands. Manly, B. F. J Randomization, bootstrap and Monte Carlo methods in bioloy, 3rd edition. Chapman and Hall, London, United Kindom. Mantel, The detection of disease clusterin and a eneralized reression approach. Cancer Research 7:09 0.

12 Mantel,., and R. S. Valand A technique of nonparametric multivariate analysis. Biometrics 6: McArdle, B. H. 99. Detectin and displayin impacts of bioloical monitorin: spatial problems and partial solutions. Paes in Proceedins of Invited Papers, XVth International Biometrics Conference, IBC, Budapest, Hunary. McArdle, B. H., and M. J. Anderson. 00. Fittin multivariate models to community data: a comment on distance-based redundancy analysis. Ecoloy 8: Mielke, P. W., K. J. Berry, P. J. Brockwell, and J. S. Williams. 98. A class of nonparametric tests based on multiresponse permutation procedures. Biometrika 68: Mielke, P. W., and K. J. Berry. 00. Permutation methods: a distance function approach. Spriner-Verla, ew York, USA. Pillai, K. C. S Some new test criteria in multivariate analysis. Annals of Mathematical Statistics 6:7. Pillar, V. D. P., and L. Orlóci On randomization testin in veetation science: multifactor comparisons of relevé roups. Journal of Veetation Science 7: Plackett, R. L A historical note on the method of least squares. Biometrika 36: Smith, E. P., K. W. Pontasch, and J. Cairns Community similarity and the analysis of multispecies environmental data: a unified statistical approach. Water Research 4: Snedecor, G. W Calculation and interpretation of analysis of variance and covariance. Colleiate Press, Ames, Iowa, USA.