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Distance Based Permutation Tests 637, Figure 1 Mean correlation matrices among 38 brain regions of interest for the three age groups indicated The first 24. and last 14 regions belong to two proposed anticorrelated systems known respectively as the task positive and task negative. networks The differentiation of these networks can be seen to progress from childhood to adulthood This figure appears in. color in the electronic version of this article, a grid of locations in the subject s brain Traditional fMRI vations are divided into some sort of a priori groups that ac. studies seek to identify brain regions whose BOLD signal in cording to the null hypothesis are indistinguishable from each. dicates a response to some stimulus A growing body of work other This hypothesis is tested by permutation methods Let. has instead examined the brain in its resting state By scan denote a permutation of the numbers 1 n i e a one. ning subjects while they attend to no stimulus in particular to one function from 1 n to itself and let be a group. one can investigate which brain regions activity levels tend of permutations such that under the null hypothesis d i j. to vary in tandem and how the brain is organized into func has the same distribution as dij 1 i j n for each. tional networks A key finding was the identification by Fox The null hypothesis can thus be tested by computing a test. et al 2005 of two widely distributed brain networks playing statistic t that depends on the dij s and referring t to the null. opposite roles a task positive network showing higher acti distribution of t the corresponding statistic calculated from. vation during overt task performance and a task negative the d i j s for In practice this permutation distri. or default mode network more active during rest bution is usually approximated by Monte Carlo simulation. Some recent work e g Fair et al 2008 Kelly et al 2009 In some settings it may be preferable to permute residuals. has examined how functional networks of this kind mature rather than raw distances see Legendre and Anderson 1999. from childhood to adulthood Consider Figure 1 which dis Anderson and ter Braak 2003 but other than the test of. plays resting state correlation matrices among 38 regions of an interaction term in Section 7 2 permutation of residuals. interest ROIs from Toro Fox and Paus 2008 averaged lies beyond the scope of this article. respectively over 13 child subjects 13 adolescents and 26. adults Due to the high dimension of the correlation matri. ces we employed the shrinkage methodology of Scha fer and Suppose the n observations are divided among g a priori. Strimmer 2005 to estimate them These plots suggest a pro groups G1 Gg of size n1 ng More generally one may. gressive differentiation from childhood to adulthood between include a g 1 th excess group consisting of unclassified. the first 24 ROIs which belong to the task positive network observations but we will assume this group to be empty The. and the last 14 belonging to the default mode network But MRPP statistic is given by. how can we formally test this subjective impression g. One class of approaches is referred to in the neuroimag Ck i j 1. ing literature as mass univariate essentially one tests for nk nk 1. k 1 i j i j Gk, among group differences separately at each of the 38 38 1 2. 703 distinct connections between ROIs using say a one way where ij denotes a measure of dissimilarity between the ith. ANOVA F test and then use some multiple comparisons and jth observations and C 1 Cg are weights summing to. method to infer which of these tests yield significant results 1 In Mielke and Berry s work ij typically equals dvij for some. For instance Fair et al 2008 employ the false discovery v 0 most often dij or d2ij where dij denotes a metric such as. rate Section 7 2 below describes a mass univariate permuta Euclidean distance In general these authors advocate taking. tion test procedure A possible limitation of such methods is ij to be a metric rather than a squared metric based on. that they can detect only particular connections displaying robustness considerations and a notion of congruence between. unusually large differences among groups We were interested the data and analysis spaces Mielke 1986 Recall that a. in a test sensitive to differences in the overall pattern of con distance or dissimilarity function is a metric if in addition. nections which might not be reflected in such large differ to the assumptions given at the beginning of Section 3 all. ences for any particular connection We therefore turned to distinct pairs of points have positive distance and the triangle. the distance based procedures considered in this article inequality holds The choice of weights. 3 Overview of the Two Approaches Ck 2, The methods discussed here take as their starting point an. n n symmetric matrix D dij 1 i j n representing non is efficient in the sense that it minimizes the asymptotic order. negative distances among n observations It is assumed that of the permutation distribution s variance Mielke and Berry. dii 0 for each i and that dij 0 for some i j The n obser 2007 The alternative weights. 638 Biometrics June 2010, nk nk 1 See Web Appendix A for further details F can be presented.
in an analysis of distance table analogous to an ANOVA ta. nm nm 1 ble as in McArdle and Anderson 2001 or Anderson 2001. m 1 When m1 0 i e when testing the full model against the. g intercept only model 5 reduces to the original expression of. reduce 1 to g k 1,n k n k 1 i j i j Gk,McArdle and Anderson 2001. is simply the mean of the within group differences cf Mantel. and Valand 1970 However this choice is not efficient in the tr H GH m 1. above sense tr I H G I H n m, A point of nomenclature the terms distance and dis. The pseudo F statistic does not in general have an F. similarity are ordinarily treated as synonyms in multivariate. distribution under the null hypothesis rather like the MRPP. analysis but here to avoid possible confusion between dij. and MRBP statistics it has an unknown null distribution so. and ij we will refer to dij as a distance and to ij as a. its significance is assessed with a permutation test. dissimilarity, The motivation for the pseudo F statistic 6 and hence for. Consider next a randomized block design in which the n. 5 can be summarized as follows Suppose first that D rep. observations are arranged in g groups and b blocks with one. resents the Euclidean distances among univariate outcomes. observation in each group block pair For this setting Mielke. y 1 yn Then F reduces to the usual F statistic for test. and Iyer 1982 propose a modification of MRPP the mul. ing the model regressing these values on a design matrix X. tivariate randomized block permutation procedure MRBP. versus the null model as can be inferred from an elementary. identity relating the sum of squared residuals to the sum of. b squared distances e g Legendre and Legendre 1998 equa. g i j 4 tions 8 5 and 8 6 More generally suppose that D rep. i 1 j i j b i resents the Euclidean distances among y 1 y n Rp for. where j b i means observations j i are in the same block some p that may be larger than n Even if D was not origi. Although the MRPP and MRBP statistics look somewhat dif nally formed from the distances among such a set of points it. ferent they are motivated by the same underlying idea both can be shown that such points do exist i e D has the Eu. statistics represent average within group differences that un clidean property if and only if G is positive definite Mardia. der the alternative hypothesis should lie in the left tail of the Kent and Bibby 1979 indeed the principal coordinates of. permutation distribution Mielke and Berry 2007 extend the Gower 1966 provide such a set of points In this case if we. MRPP methodology to more general linear models but these performed separate regressions of each component of the y i s. are not considered here with design matrix X and if Ak Bk were the numerator and. denominator of the F statistic,for the kth component then. 3 2 Pseudo F Tests p n,we would have F k 1 Ak k 1 Bk Legendre and An.
Let A 12 d2i j 1 i j n and let G I 11T n A I derson 1999 Finally observe that F can be calculated even. 11T n where 1 is a vector of n 1 s G is the centered if G is not positive definite D is not Euclidean In short the. matrix used in Gower s 1966 development of principal co pseudo F statistic is a generalization of the classical F statis. ordinate analysis Consider three partial design matrices tic that can be calculated directly from the distance matrix. X 0 1 X 1 and X 2 such that for k 0 1 2 X k is an D whether or not D is Euclidean This last property is es. n mk matrix of rank mk n Without loss of generality pecially helpful in the field of ecology where analyses often. for distinct k l 0 1 2 X Tk X l 0m k m l Otherwise we employ non Euclidean distance measures such that defined by. can use the Gram Schmidt process to obtain modified design Bray and Curtis 1957. matrices for which this orthogonality does hold We can then. consider a nested sequence of design matrices representing a. null intercept only model an intermediate model and a full xk yk. model d x y,1 1 X 1 X 1 X1 X 2 xk yk, By definition m0 1 m1 may be zero in which case X 1 is. null whereas m2 is assumed positive The dimension of X where x y are p dimensional vectors of nonnegative numbers. is n m where m 1 m1 m2 For k 0 1 2 let The quantity. H k X k X Tk X k 1 X Tk be the hat matrix associated with. X k i e the matrix of projection onto the column space of tr H k GH k. X k thus H 0 11T n Similarly let tr G, H hi j 1 i j n X X T X 1X T H 0 H 1 H 2 can be viewed as the proportion of variation explained by. X k Legendre and Anderson 1999 McArdle and Anderson. If m1 0 then take H 1 0 The pseudo F statistic is then 2001 This expression like the pseudo F statistic can be mo. given by tivated as a generalization of the corresponding expression for. tr H 2 GH 2 m2 ordinary ANOVA by appealing to a principal coordinates ar. F 5 gument Gower and Krzanowski 1999,tr I H G I H n m. Distance Based Permutation Tests 639, Expression 5 is sufficiently general to encompass the two i if hij 0 for all i j 1 n or. simple designs for which we present equivalence results be ii if G is positive semidefinite i e D is Euclidean. low the one way design see Section 5 and the randomized n n. complete block design Section 6 Modifications would be re The test statistic tr HA 12 i 1 j 1 hi j d2i j of. quired for more complex designs involving nested or random Theorem 1 is a weighted sum of the squared distances. effects or interactions PERMANOVA software Anderson Gor where the weights depend on the design Thus as the next. ley and Clarke 2008 an add on to the PRIMER statistical two sections will show this test statistic serves as a bridge. package for ecologists Clarke and Gorley 2006 implements from the pseudo F statistic to the MRPP and MRBP statis. pseudo F tests for ANOVA and other regression models with tics which are particular weighted sums of dissimilarities. general multiway balanced or unbalanced designs for two simple designs More generally the equivalence of. In order to derive the results below regarding permutation the pseudo F statistic and tr HA under the conditions of. tests it will be helpful to define explicit notation for permuta Theorem 1 provides some insight into what sort of models. tion of distances If we denote by ei the n dimensional vector the pseudo F test treats as good models i e models in. with 1 in the ith position and 0 elsewhere then applying per favor of which we would reject the null model namely. mutation to the distance matrix D means replacing the those for which tr HA is low or equivalently for which. latter with hij tends to be large when dij is small In the context of a. linear model with univariate outcomes y 1 yn hij can. D d i j 1 i j n E DE T 9 be interpreted as yy ji or the effect of yj on y i Thus the. above heuristic criterion for a good model stipulates that. where E e 1 e n T We can similarly define A and, observations that are closer to the ith observation should.
G by replacing D with D in the above from which we. have greater impact on the ith fitted value This is precisely. obtain A E AE T and using the fact that I 11T n, the idea that underlies local regression methodology e g. commutes with E,Fan and Gijbels 1996,G E GE T 10,5 Application to One Way Design. Suppose we wish to test for differences among groups. G1 Gg of sizes n1 ng respectively and that the n, 4 Reduction of the Pseudo F to a Simpler n1 ng rows and columns of H are arranged by these. Test Statistic groups Then, In this article we refer to two permutation tests as equiva. lent if whenever both are conducted using the same set of 1n 1 1Tn 1 n1 0 0. permuted distance matrices 9 the two tests yield the same 0 1n 2 1Tn 2 n2 0. p value The following result provides conditions under which H. F is equivalent in this sense to a simpler test statistic This. equivalence may be of some interest in its own right and will. 0 0 1n g 1Tn g ng, be used in Sections 5 and 6 to connect pseudo F tests with.
MRPP and MRBP Proofs of this result and of those in the and thus. sequel are given in Web Appendix B,tr HA d2i j d2i j. Theorem 1 If 2nk nk,k 1 i j Gk k 1 i j i j Gk, there exists a constant K dij On the other hand if we take dissimilarity ij d2ij and. 2n weights 2 then,such that tr H 1 A K for all 11 n g. tr HA 0 for all 12 Since condition i of Proposition 1 applies 13 leads to the. following result, then a permutation test based on F with rejection region in. the right tail of the permutation distribution is equivalent to a Proposition 2 A pseudo F test of the group effect with. test based on tr HA with rejection region in the left tail distance dij is equivalent to an MRPP test with dissimilarity. For the m1 0 case we can identify two more trans ij d2ij and weights Ck nnk g 1. parent conditions one pertaining to the design one to the In Section 7 1 we illustrate how the MRPP with dissim. distance function either of which implies the conclusion of ilarity ij d2ij e g ij dij and or weights other than. Theorem 1 those in Proposition 2 can yield different conclusions than the. pseudo F test, Proposition 1 If m1 0 then the equivalence of Theo We remark that Proposition 2 can be proved without ref.
rem 1 holds erence to Proposition 1 by combining 13 with Lemma 1 of. 640 Biometrics June 2010, Web Appendix B to express as a strictly decreasing function 7 1 Ecological Data. of F For the one way design Proposition 2 asserts that the pseudo. n F test with distance dij and the MRPP with dissimilarity. d2i j ij are equivalent when ij d2ij and the MRPP employs. i 1 j 1 weights 2 We examine here the effect of removing either. n n g g 1 F of these conditions using the Dutch dune meadow vegetation. data set from Jongman ter Braak and van Tongeren 1995. This equality directly generalizes the relationship between included in the vegan package The data consist of cover class. and the ordinary F statistic in the setting of one way ANOVA values for 30 species along with several environmental vari. Mielke and Berry 2007 Section 2 9 ables such as moisture and land use for each of 20 sites We. Given a distance dij such as the Bray Curtis distance 7 assessed whether the Bray Curtis distances 7 among the. Proposition 1 reveals that the choice between an MRPP with meadows betray a significant effect of the amount of manure. ij dij and appropriate weights and a pseudo F test is not applied This is actually an ordinal variable with levels from. really a choice between the two methods since, the former is 0 to 4 but the tests treat it as categorical The nonmetric. the same as a pseudo F with distance di j while the latter multidimensional scaling NMDS Kruskal 1964 plot in. is tantamount to an MRPP with squared dissimilarity d2ij Figure 2 indicates a marked difference between the meadows. Rather the question, reduces to which distance to use for the with no manure applied and those at levels 1 4. pseudo F dij or di j or equivalently which dissimilarity for Our analyses were performed in R version 2 7 0 R Devel. the MRPP d2ij or dij The above cited congruence concept opment Core Team 2008 with code based on the vegan. Mielke 1986 would seem to favor the second of either pair version 1 15 0 functions mrpp and adonis but modified to. of choices use the same set of 9999 permutations for the two types of. 6 Application to Randomized Block Design tests We confirmed that the MRPP with weights 2 and. squared Bray Curtis measure produced exactly the same sig. Recall the notation b of Section 3 1 and similarly write. nificance level as the pseudo F test p 0 0141 Changing. i g j if observations i j belong to the same group For the. the power to which 7 was raised had little effect for in. randomized block design with X 1 and X 2 representing block. stance the MRPP with raw Bray Curtis measure yielded p. and group effects respectively the generic elements of H 1. 0 0147 However changing the weights to 3 raised the p. H 2 and H are, value to 0 0763 A simple explanation for this loss of signifi. 1 I i b j 1 cance is that 3 places a higher proportion of the weight on. g bg dissimilarities within large groups than does 2 Thus the. rather high dissimilarities among observations in the largest. I i g j 1 group level 0 render not as small relative to the permuta. hi j and 15 tion distribution, Changing the MRPP weights has not had a dramatic effect.
in most examples we have studied But in this case weights. I i b j I i g j 1, hi j 16 2 produced results seemingly more in line with the NMDS. g b bg plot than did weights 3 As noted in Section 3 1 Mielke. respectively These formulas enable us to prove the following and Berry 2007 favor weights 2 over 3 for asymptotic. result efficiency reasons To this argument one might add that by. Proposition 2 the former choice brings MRPP into line with. Proposition 3 If the squared distance function d2ij is a the pseudo F approach. metric then the pseudo F test is equivalent to the MRBP test 7 2 Functional Connectivity Data. based on squared distances i e with ij d2ij, We now return to the functional connectivity data introduced. If the squared distance is a metric then so is the raw dis in Section 2 For comparison with the distance based tests of. tance but not conversely Web Appendix C presents an exam this article we applied a mass univariate permutation proce. ple of a metric distance function whose square is not a metric dure similar to that used by Church et al 2009 for data. and for which the two tests are not equivalent Proposition 3 of the same type for each of 499 permutations of the age. could perhaps be formulated more naturally by expressing group labels we calculated F statistics for the age effect at. the distance in terms of the dissimilarity rather than vice each of the 703 connections and obtained the permutation. versa The result could then be restated as an MRBP test distribution of the maximum of the 703 F statistics to which. for which the dissimilarity ij is a metric equivalent to the the real data F statistics were referred Only one connection. pseudo F test with distance di j i j was found significant even at the 0 4 level that between the. subgenual anterior cingulate cortex and posterior cingulate. 7 Real Data Illustrations cortex for which we obtained p 0 038 The observed mean. In this section we first revisit a previously studied ecologi correlations between these two regions were 0 05 for children. cal data set in light of our equivalence results We then re 0 25 for adolescents and 0 40 for adults These two regions. turn to our motivating application in neuroscience and show are the core nodes of the default mode network and this find. how distance based permutation tests offer a novel inferential ing suggests that the strengthening of their connection may. approach be an important component of brain maturation in line with. Distance Based Permutation Tests 641,Second coordinate. 0 5 0 0 0 5 1 0,First coordinate, Figure 2 Nonmetric multidimensional scaling plot of the 20 Dutch meadows indicating the level of manure applied on a. scale from 0 to 4, the findings of Kelly et al 2009 Still this test procedure but contributed the additional information that age group ex.
has several limitations The finding of a single significantly plains about 4 9 of variation in the sense of 8 Moreover. different connection offers no confirmation of the broader dif the latter method can be extended to include other predic. ferentiation of task positive and task negative networks from tors Adding gender to the model resulted in the analysis of. childhood to adulthood as suggested by Figure 1 Indeed distance table shown in Table 1 Gender appears not to have. had we not ordered the rows and columns in that figure as much effect whereas as we would expect the p value for age. task positive followed by task negative based on a priori con group agrees very closely with the MRPP result Subsequent. siderations the mass univariate finding would not have led tests comparing each pair of age groups found a clear differ. us to suspect such a broad pattern of differential connectivity ence between children and adults p 0 0041 but no signif. among the age groups Furthermore due to the need for sep icant differences between the adolescents and the other two. arate tests at each connection the relatively modest number groups supporting the intuition that adolescent functional. of permutations required over 11 minutes on a MacBook Pro connectivity patterns lie between those of children and adults. with a 2 16 GHz Intel Core Duo processor applications to not Although tests of interactions lie beyond the scope of our. much larger data sets could be excessively time consuming main development they can be performed within either the. In view of these limitations the distance based permuta MRPP or the pseudo F framework Mielke and Berry 2007. tion tests implemented in the vegan package offer a useful Legendre and Anderson 1999 We carried out a pseudo F test. complement to the mass univariate analysis We applied an as described in Web Appendix A and found no significant. MRPP to the three age groups after applying the Fisher age group by gender interaction p 0 215 based on 9999. 1921 z transformation to the correlations as is commonly permutations. done for functional connectivity data The dissimilarity used. was the square of the Frobenius metric e g Bickel and 8 Discussion. Levina 2008 between pairs of correlation matrices this norm Mielke and Berry 2007 describe applications of MRPP in a. can be defined for two matrices A and B as the Euclidean wide array of disciplines from archeology to meteorology to. distance between vec A and vec B It took just 3 8 sec psychometrics Pseudo F tests have a shorter history with ap. onds to run 9999 permutations using the same machine as plications first in ecology then in genetics and more recently. above resulting in a p value of 0 0163 see Figure 3 Without in computer graphics C ad k et al 2008 Thus specialists. squaring the Frobenius norm we obtained p 0 0212 An in different fields may be conducting distance based permu. equivalent permutational MANOVA was slower 9 7 seconds tation tests under these two paradigms with at best limited. 642 Biometrics June 2010,53 0 53 5 54 0 54 5,Test statistic. Figure 3 Permutation distribution of average within group dissimilarity among the fMRI correlation matrices with a. vertical line indicating the observed value This figure appears in color in the electronic version of this article. Table 1 Anderson M J 2005 PERMANOVA A FORTRAN computer pro. Analysis of distance table for the functional connectivity data gram for permutational multivariate analysis of variance Depart. ment of Statistics University of Auckland New Zealand. df SS MS F R2 Pr F Anderson M J and ter Braak C J F 2003 Permutation tests for. multi factorial analysis of variance Journal of Statistical Compu. Age group 2 68 3076 34 1538 1 2735 0 0494 0 0164,tation and Simulation 73 85 113. Gender 1 27 8779 27 8779 1 0395 0 0202 0 3309,Anderson M J Gorley R N and Clarke K R 2008 PER. Residuals 48 1287 3076 26 8189 0 9305, MANOVA for PRIMER Guide to Software and Statistical. Total 51 1383 4932 1,Methods Plymouth U K PRIMER E Ltd.
Bickel P J and Levina E 2008 Covariance regularization by thresh. awareness of the close relationship between them Such a situ olding Annals of Statistics 36 2577 2604. ation contributes to a lack of understanding across disciplines Bray J R and Curtis J T 1957 An ordination of the upland forest. communities of southern Wisconsin Ecological Monographs 27. It is our hope that the equivalence results presented here will. help to reduce this mutual incomprehension In addition we C ad k M Wimmer M Neumann L and Artusi A 2008 Evalu. hope that our neuroimaging application may inspire a few ation of HDR tone mapping methods using essential perceptual. researchers to consider distance based permutation tests as attributes Computers and Graphics 32 330 349. a tool for meeting the challenges posed with increasing fre Church J A Fair D A Dosenbach N U F Cohen A L Miezin F. quency by high dimensional data analyses M Petersen S E and Schlaggar B L 2009 Control networks. in paediatric Tourette syndrome show immature and anomalous. 9 Supplementary Materials patterns of functional connectivity Brain 132 225 238. Web Appendix A referenced in Sections 3 2 and 7 2 Web Clarke K R and Gorley R N 2006 Primer v6 User Man. Appendix B referenced in Sections 4 and 5 and Web ual Tutorial Plymouth U K PRIMER E Ltd. Appendix C referenced in Section 6 are available un Fair D A Cohen A L Dosenbach N U F Church J A Miezin F. M Barch D M Raichle M E Petersen S E and Schlaggar. der the Paper Information link at the Biometrics website. B L 2008 The maturing architecture of the brain s default. http www biometrics tibs org, network Proceedings of the National Academy of Sciences 105. Acknowledgements Fan J and Gijbels I 1996 Local Polynomial Modelling and Its Ap. plications London Chapman and Hall, The authors thank the co editor Prof Geert Molenberghs Fisher R A 1921 On the probable error of a coefficient of corre. and the associate editor and referees for valuable and insight lation deduced from a small sample Metron 1 3 32. ful comments as well as Prof Nik Schork Ondrej Libiger Fox M D Snyder A Z Vincent J L Corbetta M Van Essen. Prof Jari Oksanen and Dr Clare Kelly for their invaluable D C and Raichle M E 2005 The human brain is intrinsi. advice and assistance cally organized into dynamic anticorrelated functional networks. Proceedings of the National Academy of Sciences 102 9673. References, Good I J 1982 An index of separateness of clusters and a permu. Anderson M J 2001 A new method for non parametric multivariate tation test for its significance Journal of Statistical Computation. analysis of variance Austral Ecology 26 32 46 and Simulation 15 81 84. Distance Based Permutation Tests 643, Gower J C 1966 Some distance properties of latent root and vector Mielke P W and Iyer H K 1982 Permutation techniques for an. methods used in multivariate analysis Biometrika 53 325 338 alyzing multi response data from randomized block experiments. Gower J C and Krzanowski W J 1999 Analysis of distance for Communications in Statistics Theory and Methods 11 1427. structured multivariate data and extensions to multivariate anal 1437. ysis of variance Applied Statistics 48 505 519 Mielke P W Berry K J and Johnson E S 1976 Multi response. Jongman R H G ter Braak C J F and van Tongeren O F R permutation procedures for a priori classifications Communica. eds 1995 Data Analysis in Community and Landscape Ecology tions in Statistics Theory and Methods 5 1409 1424. new edition Cambridge U K Cambridge University Press Nievergelt C M Libiger O and Schork N J 2007 Generalized. Kelly A M C Di Martino A Uddin L Q Shehzad Z Gee D G analysis of molecular variance PLoS Genetics 3 4 e51. Reiss P T Margulies D S Castellanos F X and Milham Oksanen J Kindt R Legendre P O Hara B Simpson G L Soly. M P 2009 Development of anterior cingulate functional con mos P Stevens M H H and Wagner H 2008 vegan Com. nectivity from late childhood to early adulthood Cerebral Cortex munity ecology package R package version 1 15 0 http cran. 19 640 657 r project org http vegan r forge r project org. Kruskal J B 1964 Nonmetric multidimensional scaling A numerical R Development Core Team 2008 R A language and environ. method Psychometrika 29 115 129 ment for statistical computing R Foundation for Statistical. Krzanowski W J 2002 Multifactorial analysis of distance in studies Computing Vienna Austria ISBN 3 900051 07 0 http www. of ecological community structure Journal of Agricultural Bio R project org. logical and Environmental Statistics 7 222 232 Scha fer J and Strimmer K 2005 A shrinkage approach to large. Krzanowski W J 2006 Sensitivity in metric scaling and analysis of scale covariance matrix estimation and implications for functional. distance Biometrics 62 239 244 genomics Statistical Applications in Genetics and Molecular Bi. Legendre P and Anderson M J 1999 Distance based redundancy ology 4 1 article 32. analysis Testing multispecies responses in multifactorial ecologi Smith E P Pontasch K W and Cairns J 1990 Community sim. cal experiments Ecological Monographs 69 1 24 ilarity and the analysis of multispecies environmental data A. Legendre P and Legendre L 1998 Numerical Ecology 2nd English unified statistical approach Water Research 24 507 514. edition Amsterdam Elsevier Toro R Fox P T and Paus T 2008 Functional coactivation map. Mantel N and Valand R S 1970 A technique of nonparametric of the human brain Cerebral Cortex 18 2553 2559. multivariate analysis Biometrics 26 547 558 Wessel J and Schork N J 2006 Generalized genomic distance. Mardia K V Kent J T and Bibby J M 1979 Multivariate Anal based regression methodology for multilocus association analysis. ysis London Academic Press American Journal of Human Genetics 79 792 806. McArdle B H and Anderson M J 2001 Fitting multivariate models Zapala M A and Schork N J 2006 Multivariate regression analysis. to community data A comment on distance based redundancy of distance matrices for testing associations between gene expres. analysis Ecology 82 290 297 sion patterns and related variables Proceedings of the National. Mielke P W 1986 Non metric statistical analyses Some metric al Academy of Sciences 103 19430 19435. ternatives Journal of Statistical Planning and Inference 13 377. Mielke P W and Berry K J 2007 Permutation Methods A Dis Received November 2008 Revised February 2009. tance Function Approach 2nd edition New York Springer Accepted March 2009.

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