BI! ae
`
`OF I ee
`
`RESEARCH
`DESIGN
`
`VOLUME1
`
`
`
`
` FDITED BY
`
`NEIL J. SALKIND
`University ofKansas
`
`= ae
`®SAGE| ‘teference
`Leis Acpelee
`(Loon
`tte
`Genome & Co. v. Univ. of Chicago
`rae
`PGR2019-00002
`UNIV. CHICAGO EX. 2068
`
`

`

`Copyright © 2010 by SAGE Publications,Inc.
`
`All rights reserved, No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical,
`including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the
`
`publisher.
`
`For information:
`
`SAGE Publications, Inc.
`
`6 2455 Teller Road
`
`Thousand Oaks, California 91320
`E-mail: order@sagepub.com
`
`SAGE Publications Lrd..
`1 Oliver’s Yard
`S55 City Road
`London ECLY ISP
`United Kingdom
`
`SAGE Publications India Pye. Led.
`B 1/1 Mohan Cooperative Industrial Area
`Mathura Road, New Delhi 110 044
`India
`
`SAGE Publications Asia-Pacific Pre. Ltd.
`33 Pekin Street #02-01
`Far East Square
`Singapore 048763
`
`Printed in the United States of America.
`
`Library of Congress Cataloging-in-Publication Data
`
`Enevelopedia of research design/edited by Neil |. Salkind.
`v.cm.
`
`Includes bibliographical references and index.
`ISBN 978-1-4129-6127-1 (cloth)
`1. Social sciences—Statistical methods—Encyclopedias. 2, Social sciences—Research—Methodology—Eneyclopedias.
`I. Salkind, Neil J.
`
`HA29,E525 2010
`001.403—de22
`
`2010001779
`
`This book is printed on acid-free paper.
`
`It
`
`12
`
`
`
`
`
`
`
`8. 7 6 § 4 FP13 4 0 9 2
`
`
`
`
`
`
`
`
`
`Publisher:
`Acquisitions Editor:
`Editorial Assistant:
`Developmental Editor:
`Reference Systems Coordinators:
`Production Editor:
`Copy Editors:
`Typesetter:
`Proofreaders:
`Indexer:
`Cover Designer:
`Marketing Manager:
`
`Rolf A, Janke
`Jim Brace-Thompson
`Michele Thompson
`Carole Maurer
`Lencia M, Gutierrez, Laura Notton
`Kate Schroeder
`Bonnie Freeman, Liann Lech, Sheree Van Vreede
`C&MDigitals (P) Led.
`Kristin Bergstad, Kevin Gleason, Sally Jaskold, Sandy Zilka
`Virgil Diodato
`Glenn Vogel
`Amberlyn McKay
`
`

`

`Contents
`
`
`
`Volume 1
`
`List of Entries
`
`vit
`
`Reader’s Guide xiii
`
`About the Editors xix
`
`Contributors xxi
`
`Introduction xxix
`
`Entries
`
`|
`
`57
`
`111
`
`321
`
`E
`
`F
`
`G
`
`399
`
`471
`
`519
`
`Volume 2
`
`List of Entries vi
`
`Entries
`
`S61
`589
`
`654
`
`663
`
`681
`
`M
`N
`
`O
`
`P
`
`745
`869
`
`949
`
`98S
`
`Volume 3
`
`List of Entries
`
`vil
`
`Entries
`
`1149
`
`1183
`
`[295
`
`1489
`
`1583
`
`Vv
`
`1589
`
`W 1611
`
`Y
`
`Zz
`
`1645
`
`16453
`
`Index
`
`1675
`
`PeSasSas>
`
`envezyp
`
`

`

`
`
` This material may be protected by Copyright law (Title 17 U.S. Code)
`
`
`
`164 Coefficient of Concordance
`
`of measurement uses, and consequently it should
`be viewed within a much larger system of reliabil-
`ity
`analysis, generalizability theory. Moreover,
`alpha focused attention on reliability coefficients
`when that attention should instead be cast on
`measurement error and the standard error of
`measurement.
`For Cronbach, the extension of alpha (and clas-
`sical test theory) came when Fisherian notions of
`experimental design and analysis of variance were
`put together with the idea that some “treatment”
`conditions could be considered random samples
`from a large universe, as alpha assumes aboutitem
`sampling. Measurement data,
`then, could be col-
`lected in complex designs with multiple variables
`(e.g., items, occasions, and rater effects) and ana-
`lyzed with random-effects analysis of variance
`models. The goal was not so much to estimate
`a reliability coefficient as to estimate the compo-
`nents of variance that arose from multiple vari-
`ables and their interactions in order to account for
`observed score variance. This approach of parti-
`tioning effects into their variance components pro-
`vides information as to the magnitude of each of
`the multiple sources of error and a standard error
`of measurement, as well as an “alpha-like” reliabil-
`ity coefficient for complex measurement designs.
`Moreover,
`the variance-component approach
`can provide the value of “alpha” expected by
`increasing or decreasing the numberof items (or
`raters or occasions) like those in the test. In addi-
`tion, the proportion of observed score variance
`attributable to variance in item difficulty (or, for
`example,
`rater stringency) may also be com-
`puted, which is especially important to contem-
`porary testing programs that seek to determine
`whether examinees have achieved an absolute,
`rather than relative,
`level of proficiency. Once
`these possibilities were envisioned, coefficient
`alpha morphed into generalizability theory, with
`sophisticated analyses
`involving crossed and
`nested designs with random and fixed variables
`(facets) producing variance
`components
`for
`multiple measurement facets such as raters and
`testing occasions so as to provide a complex
`standard error of measurement.
`By all accounts, coefficient alpha—Cronbach’s
`alpha—has been and will continue to be the most
`popular method
`for
`estimating
`behavioral
`measurement
`reliability. As of 2004,
`the 1951
`
`coefficient alpha article had been cited in more
`than 5,000 publications,
`
`JeffreyT. Steedle and RichardJ, Shavelson
`
`See also Classical Test Theory; Generalizability Theory;
`Internal Consistency Reliability; KR-20; Reliability;
`Split-Half Reliability
`
`Further Readings
`
`Brennan, R. L. (2001). Generalizability theory. New
`York: Springer-Verlag.
`Cronbach, L. J., & Shavelson, R. J. (2004), My current
`thoughts on coefficient alpha and successor
`procedures. Educational & Psychological
`Measurement, 64(3), 391-418.
`Haertel, E. H. (2006). Reliability. In R. L. Brennan (Ed.),
`Educational nteasurentent (pp. 65-110). Westport,
`CT: Praeger.
`Shavelson, R. J. (2004). Editor’s preface to Lee
`J. Cronbach’s “MyCurrent Thoughts on Coetficient
`Alpha and Successor Procedures.” Educational &
`Psychological Measurement, 64(3), 389-390,
`Shavelson, R. J., & Webb, N. M. (1991). Generalizability
`theory: A primer. Newbury Park, CA: Sage.
`
`
`
`COEFFICIENT OF CONCORDANCE
`
`Proposed by Maurice G, Kendall and Bernard
`Babington Smith, Kendall’s coefficient of concor-
`dance (W) is a measure of the agreement among
`several
`(#7) quantitative or semiquantitative vari-
`ables that are assessing a set of 1 objects of inter-
`est. In the social sciences,
`the variables are often
`people, called judges, assessing different subjects
`or situations. In community ecology, they may be
`species whose abundances are used to assess habi-
`tat quality at study sites.
`In taxonomy, they may
`be characteristics measured over different species,
`biological populations, or individuals.
`There is a close relationship between Milton
`Friedman’s two-way analysis of variance without
`replication by ranks and Kendall’s coefficient of
`concordance. They address hypotheses concerning
`the same data table, and they use the same y? sta-
`tistic for testing. They differ only in the formula-
`tion of their respective null hypothesis. Consider
`Table 1, which containsillustrative data. In Fried-
`man’s test, the null hypothesis is that there is no
`
`

`

`
`
` Coefficient of Concordance 165
`
`Table |
`
`Site 4
`Site 9
`Site 14
`Site 22
`Site 31
`Site 34
`Site 45
`Site 53
`Site 61
`Site 69
`
`Illustrative Example: Ranked Relative Abundances of Four Soil Mite Species (Variables) at |0 Sites (Objects)
`Sum of Ranks
`Ranks (column-wise)
` Species 13 Species 14 Spectes 18 Species 23 R,
`
`
`
`
`5
`6
`3
`5
`19.0
`1)
`4
`8
`2
`24.0
`7
`§
`3
`4
`24.0
`8
`10
`9
`2
`29.0
`6
`5
`7
`6
`24.0
`9
`7
`10
`7
`33.0
`3
`3
`2
`8
`16.0
`1.5
`2
`4
`9
`16.5
`1.5
`|
`|
`2
`5.5
`4
`9
`6
`10
`29.0
`
`Source: Legendre, P. (2005) Species associations: The Kendall coefficient of concordance revisited. Journal of Agricultural,
`Biological, & Environmental Statistics, 10, 230. Reprinted with permission from the Journal of Agricultural, Biological, &
`Environmental Statistics. Copyright 2005 by the AmericanStatistical Association. All rights reserved.
`
`Nofes: The ranks are computed columnwise with ties. Right-hand column: sumof the ranks for eachsite.
`
`real difference among the » objects (sites, rows of
`Table 1) because they pertain to the samestatisti-
`cal population, Under the null hypothesis,
`they
`should have received random ranks along the vari-
`ous variables, so that their sums of ranks should
`be approximately equal. Kendall’s test focuses on
`the m variables.
`If the null hypothesis of Fried-
`man’s test
`is true,
`this means that
`the variables
`have produced rankings of the objects that are
`independent of one another. This
`is
`the null
`hypothesis of Kendall’s test.
`
`Computing Kendall’s W
`
`There are two ways of computing Kendall’s Wsta-
`tistic (first and second forms of Equations 1 and
`2); they lead to the sameresult, § or S’ is computed
`first
`from the row-marginal
`sums of
`ranks R,
`received by the objects:
`nN
`
`n
`
`Or
`
`W=
`
`128’ —3n(n + 1)
`ye (98 —n) — mT
`
`(2)
`
`the number of objects and m is the
`where # is
`number of variables. T is a correction factor for
`tied ranks:
`
`&
`T= So (t-te),
`k=
`
`(3)
`
`in which f, is the number of tied ranks in each (k)
`of g groups ofties. The sum is computed overall
`groups of ties found inall #7 variables of the data
`table. T = 0 whenthere are no tied values.
`Kendall’s W is an estimate of the variance of the
`row sums of ranks R; divided by the maximum
`possible value the variance can take;
`this occurs
`whenall variables are in total agreement. Hence
`O0<W<1,
`|
`representing perfect concordance. To
`derive the formulas for W (Equation 2), one has to
`know that whenall variables are in perfect agree-
`ment, the sum ofall sums of ranks in the data table
`(right-hand column of Table 1)
`is #z(# + 1)/2 and
`that the sum of squares of the sumsofall ranks is
`penn + 1)(27 + 1)/6 (withoutties).
`There is a close relationship berween Charles
`Spearman’s correlation coefficient rs and Kendall’s
`Wstatistic: W can be directly calculated from the
`
`S = 5° (R,—R)or S' = 5° R? = SSR,
`
`te
`
`T
`
`f=
`
`il
`
`(1)
`
`where § is a sum-of-squares statistic over the row
`sums of ranks R,, and R is the mean of the R,;
`values. Following that, Kendall’s W statistic can be
`obtained fromeither of the following formulas:
`
`128
`WV =—
`me (1 — #2) — mT
`
`

`

`
`
`166 Coefficient of Concordance
`
`mean (7s) of the pairwise Spearman correlations rs
`using the following relationship:
`
`the index, because different
`included in
`be
`groups of species may be associated to different
`environmental conditions.
`
`y aea (4)
`
`WE
`
`where mis the numberof variables (judges) among
`which Spearmancorrelations are computed. Equa-
`tion 4 is strictly true for untied observations only;
`for tied observations, ties are handled in a bivariate
`way in each Spearman rs coefficient whereas in
`Kendall’s W the correction for ties is Computed in
`a single equation (Equation 3)
`for all variables.
`For two variables (judges) only, W is simply a lin-
`ear transformation of rs; W=(re+1)/2.
`In that
`case, a permutation test of W for twovariables is
`the exact equivalent of a permutation test of ry for
`the same variables.
`The relationship described by Equation 4 clearly
`limits the domain of application of the coefficient of
`concordance to variables that are all meant to esti-
`mate the same general property of the objects: vari-
`ables
`are considered concordant only if
`their
`Spearman correlations are positive. Two variables
`that give perfectly opposite ranks to a set of objects
`have a Spearmancorrelation of — 1, hence W=0
`for these two variables (Equation 4);
`this is the
`lower bound of the coefficient of concordance. For
`twovariables only, rs5=0 gives W=0.5. So coeffi-
`cient Wapplies well to rankings given by a panel of
`judges called in to assess overall performance in
`sports or quality of wines or foodin restaurants, to
`rankings obtained from criteria used in quality tests
`of appliances or services by consumer organizations,
`and soforth. It does not apply, however, to variables
`used in multivariate analysis in which negative as
`well as positive relationships are informative. Jerrold
`H. Zar, for example, uses wing length, tail length,
`and bill length of birds to illustrate the use of the
`coefficient of concordance. These data are appropri-
`ate for W because they are all indirect measures of
`a commonproperty, the size of the birds.
`In ecological applications, one can use the
`abundances of various species as indicators of
`the good or bad environmental quality of the
`studysites. If a group of species is used to pro-
`duce a global index of the overall quality (good
`or bad) of the environment at the studysites,
`only the species that are significantly associated
`and positively correlated to one another should
`
`Testing the Significance of W
`
`Friedman’s chi-squarestatistic is obtained from W
`by the formula
`
`7
`x = m(n— 1)W.
`
`(5)
`
`This quantity is asymptotically distributed like
`chi-square with v = (m7 — 1) degrees of freedom; it
`can be used to test Wfor significance, According to
`Kendall and Babington Smith, this approachis satis-
`factory only for moderately large values of m7 and 7,
`Sidney Siegel and N. John Castellan Jr. recom-
`mend the use of a table of critical values for W
`when #1 <7 and #7 < 20; otherwise, they recommend
`testing the chi-square statistic (Equation 5) using the
`chi-square distribution. Their table of critical values
`of W for small # and 1 is derived from a table of
`critical values of S assembled by Friedman using the
`z test of Kendall and Babington Smith and repro-
`duced in Kendall’s classic monograph, Rank Corre-
`lation Methods. Using numerical simulations, Pierre
`Legendre compared results of
`the classical chi-
`square test of the chi-square statistic (Equation 5) to
`the permutation test that Siegel and Castellan also
`recommend for small samples (small 77). The simula-
`tion results showed that the classical chi-square test
`was too conservative for any sample size (7) when
`the number of variables #7 was smaller than 20; the
`test had rejection rates well below the significance
`level, so it remained valid. The classical chi-square
`test had a correct
`level of Type | error (rejecting
`a null hypothesis that is true) for 20 variables and
`more. The permutation test had a correct rate of
`Type I error forall values of #7 and n. The powerof
`the permutation test was higher than that of the
`classical chi-square test because of the differences in
`rates of Type I error between the twotests. The dif-
`ferences in power disappeared asymptotically as the
`number of variables increased.
`An alternative approach is to compute the fol-
`lowing F statistic:
`
`F = (m—1)W/(1-— W),
`
`(6)
`
`which is asymptotically distributed like F with
`my = n—1—(2/m) and v2 = vy(m— 1) degrees
`
`

`

`Smith
`and Babington
`freedom. Kendall
`of
`described this approach using a Fisher ¢ transfor-
`mation of the F statistic. z=0.5 log.(F). They
`recommendedit for testing W for moderate values
`of m and ». Numerical simulations show, however,
`that
`this F statistic has correct
`levels of Type I
`error for any value of 2 and m.
`In permutation tests of Kendall’s W, the objects
`are the permutable units under the null hypothesis
`(the objects are sites in Table 1). For the global test
`of significance, the rank values in all variables are
`permuted at random, independently from variable
`to variable because the null hypothesis is the inde-
`pendence of the rankings produced byall vari-
`ables. The alternative hypothesis is
`that at
`least
`one of the variables is concordant with one, or
`with some, of the other variables. Actually,
`for
`permutation
`testing,
`the
`four
`statistics
`SSR
`(Equation 1), W (Equation 2), x? (Equation 5),
`and F (Equation 6) are monotonic to one another
`since # and m, as well as T, are constant within
`a given permutation test; thus they are equivalent
`statistics for testing, producing the same permuta-
`tional probabilities. The test is one-tailed because
`it
`recognizes only positive associations between
`vectors of ranks. This maybeseen if one considers
`two vectors with exactly opposite rankings: They
`produce a Spearmanstatistic of — 1, hence a value
`of zero for W (Equation 4).
`Manyof the problems subjected to Kendall's
`concordance analysis involve fewer than 20 vari-
`ables. The chi-square test should be avoided in
`these cases. The F test (Equation 6), as well as the
`permutation test, can safely be used withall values
`of nz and n.
`
`Contributions of Individual Variables
`to Kendall’s Concordance
`
`The overall permutation test of W suggests
`a way of testing a posteriori the significance of
`the contributions of individual variables to the
`overall concordance to determine which of the
`individual variables are concordant with one or
`several other variables in the group. There is
`interest in several fields in identifying discordant
`variables or judges. This includes all fields that
`use panels of judges to assess the overall quality
`of the objects or subjects under study (sports,
`
`
`
`Coefficient of Concordance 167
`
`law, consumerprotection, etc.). In other types of
`studies, scientists are interested in identifying
`variables that agree in their estimation of a com-
`mon property of the objects. This is the case in
`environmental
`studies in which scientists are
`interested in identifying groups of concordant
`species that are indicators of some property of
`the environment and can be combined into indi-
`ces of its quality,
`in particular in situations of
`pollution or contamination.
`The contribution of individual variables to
`the W statistic can be assessed by a permutation
`test proposed by Legendre. The null hypothesis
`is the monotonic independence of the variable
`subjected to the test, with respect toall the other
`variables in the group under study. The alterna-
`tive hypothesis is that this variable is concordant
`with other variables in the set under study, hav-
`ing similar rankings of values (one-tailed test).
`The statistic W can be used directly in a poste-
`riori tests. Contrary to the global test, only the
`variable under test is permuted here. If that vari-
`able has values that are monotonically indepen-
`dent of the other variables, permuting its values
`at randomshould havelittle influence on the W
`statistic. If, on the contrary, it is concordant with
`one or several other variables, permuting its
`values at random should break the concordance
`and induce a noticeable decrease on W.
`Two specific partial concordance statistics can
`also be used in a posteriori tests. The first one is the
`mean, 7, of the pairwise Spearman correlations
`between variable ; under test and all the other vari-
`ables. The secondstatistic, W;, is obtained by apply-
`ing, Equation 4 to 7, instead of 7, with 1 the number
`of variables in the group. These twostatistics are
`shown in Table 2 for the example data; 7, and W,
`are monotonic to each other because mis constant
`in a given permutation test. Within a given a poster-
`ior test, W is also monotonic to W; because only
`the values related to variable j are permuted when
`testing variable ;. These three statistics are thus
`equivalent for a posteriori permutationtests, produc-
`ing the same permutational probabilities. Like 7, W,
`can take negative values; this is not the case of W.
`There are advantages to performing a single
`a posteriori
`test for variable /
`instead of (72—1)
`tests
`of
`the Spearman correlation coefficients
`between variable j and all the other variables: The
`tests of the (#:—1) correlation coefficients would
`
`
`
`

`

`168
`
`Coefficient of Concordance
`
`Table 2
`
`Results of (a) the Overall and (b) the A Posteriori Tests of Concordance Amongthe Four Species of Table
`Iz (c) Overall and (d) A Posteriori Tests of Concordance Among Three Species
`
`(a) Overall test of Wstatistic, four species. Hy : The four species are not concordant with one another.
`Permutational p value = .0448*
`Kendall’s W =
`0.44160
`F distribution p value = .0440*
`F statistic =
`2.37252
`1S.89771
`Fricdman’s chi-square =
`Chi-square distribution p value = .0690
`(b) A posteriori rests, four species. Hy : This species is not concordant with theother three.
`
` r, W; p value Corrected p
`
`
`Decision at a = 5%
`0.32657
`0.49493
`0766
`-1532
`Do not reject Ho
`Species 13
`0.39655
`0.54741
`0240
`.0720
`Do not reject Hy
`Species 14
`0.45704
`0.59278
`O05 |
`.0204*
`Reject Hy
`Species 15
`
`
`-0.16813 0.12391 -7070 .7070Species 23 Do notreject Hy
`
`
`
`
`(c) Overall test of W statistic, three species. Hy : The three species are not concordant with one another.
`Kendall’s W =
`0.78273
`Permutational p value = .0005*
`Fdistribution p value = .0003*
`7.20497
`F statistic =
`Chi-square distribution p value = .0121*
`Friedman’s chi-square =
`21.13360
`
`(d) A posteriori tests, three species. Hy : This species is not concordant with the other two.
`
` Decision at a = 5% r W,
`p value
`Corrected p
`0040
`.0120*
`Reject Hy
`0.69909
`0.79939
`Species 13
`0290
`0290*
`Reject Ho
`0.59176
`0.72784
`Species 14
`0050 Reject Hy 0120"
`
`
`Species 15
`0.73158
`O.82105
`Source: (a) and(b); Adapted from Legendre, P. (2005). Species associations: The Kendall coefficient of concordancerevisited. Journal of
`Agricultural, Biological, and EnvironmentalStatistics, 10, 233. Reprinted with permission from the Journal of Agriciltural, Biological
`and Exwironmental Statistics. Copyright 2004 by the American Statistical Association. All rights reserved.
`
`Notes: F; = meanof the Spearman correlations with the other species; W, = partial concordance per species; p value= permutational
`probability(9,999 random permutations); corrected p= Holm-corrected p value. * = Reject Hy ata = 05.
`
`have to be corrected for multiple testing, and they
`could provide discordant information; a single test
`of the contribution of variable j to the Wstatistic
`has greater power and provides a single, clearer
`answer. In order to preserve a correct Or approxi-
`mately correct experimentwise error rate, the proba-
`bilities of the a posteriori
`tests computed forall
`species in a group should be adjusted for multiple
`testing.
`A posteriori tests are useful for identifying the
`variables that are not concordant with the others,
`as in the examples, but they donottell us whether
`there are one or several groups of congruent vari-
`ables among those for which the null hypothesis of
`independenceis rejected. This information can be
`obtained by computing Spearman correlations
`among the variables and clustering them into
`groups of variables that are significantly and posi-
`tively correlated,
`
`The example data are analyzed in Table 2. The
`overall permutational test of the Wstatistic is sig-
`nificant at a=5%, but marginally (Table 2a). The
`cause appears when examining the a_ posteriori
`tests in Table 2b: Species 23 has a negative mean
`correlation with the three other species in the
`group (7, = —.168). This indicates that Species 23
`does not belong in that group. Were we analyzing
`a large group of variables, we could look at
`the
`next partition in an agglomerative clustering den-
`drogram, or the next K-means partition, and pro-
`ceed to the overall and a posteriori tests for the
`members of these new groups. In the present illus-
`trative example, Species 23 clearly differs from the
`other three species. We can nowtest Species 13,
`14, and 15 as a group. Table 2c shows that this
`group hasa highly significant concordance, and all
`individual species contribute significantly to the
`overall concordance of their group (Table 2d).
`
`

`

`In Table 2a and 2c, the Ftest results are concor-
`dant with the permutation test results, but due to
`small #2 and , the chi-squaretest lacks power.
`
`Discussion
`
`The Kendall coefficient of concordance can be
`used to assess the degree to which a groupof vari-
`ables provides a common ranking for a set of
`objects. It should be used only to obtain a state-
`ment about variables that are all meant to measure
`the same general property of the objects. It should
`not be used to analyze sets of variables in which
`the negative and positive correlations have equal
`importance for
`interpretation. When the null
`hypothesis is rejected, one cannot conclude that all
`variables are concordant with one another, as
`shown in Table 2 (a) and (b); only that at least one
`variable is concordant with one or some of the
`others.
`The partial concordance coefficients and a pos-
`teriori
`tests of significance are essential comple-
`ments of the overall test of concordance. In several
`fields,
`there is
`interest
`in identifying discordant
`variables;
`this is the case in all
`fhelds
`that use
`panels of judges to assess the overall quality of the
`objects under study (e.g., sports,
`law, consumer
`protection). In other applications, oneis interested
`in using the sum of ranks, or the sumof values,
`provided by several variables or judges, to create
`an overall indicator of the response of the objects
`under study. It is advisable to look for one or sev-
`eral groups of variables that
`rank the objects
`broadly in the same way, using clustering, and
`then carry out a posteriori tests on the putative
`members of each group. Only then can their values
`or ranks be pooled into an overall index.
`
`Pierre Legendre
`
`See also Friedman Test; Holm’s Sequential Bonferroni
`Procedure; Spearman Rank Order Correlation
`
`Further Readings
`
`Friedman, M. (1937). The use of ranks to avoid the
`assumption of normality implicit in the analysis of
`variance. Journal of the American Statistical
`Association, 32, 675-701.
`
`
`
`Coefficient of Variation 169
`
`Friedman, M.(1940). A comparison of alternative tests
`of significance for the problem of mz rankings. Annals
`of Mathematical Statistics, 11, 86-92.
`Kendall, M. G. (1948). Rank correlation methods (st
`ed.). London: Charles Griffith.
`Kendall, M. G., & Babingron Smith, B. (1939). The
`problem of mrankings. Annals of Mathematical
`Statistics, 10, 275-287.
`Legendre, P. (2005). Species associations: The Kendall
`coefficient of concordance revisited. Journal of
`Agricultural, Biological, & Environmental Statistics,
`10, 226-245,
`Zar, J. H. (1999). Biostatistical analysis (4th ed.). Upper
`Saddle River, NJ]: Prentice Hall.
`
`
`
`COEFFICIENT OF VARIATION
`
`The coefficient of variation measures the vari-
`ability of a series of numbers independent of the
`unit of measurement used for these numbers, In
`order to do so, the coefficient of variation elimi-
`nates the unit of measurement of the standard
`deviation of a series of numbers by dividing the
`standard deviation by the mean of these num-
`bers. The coefficient of variation can be used to
`compare distributions obtained with different
`units, such as the variability of the weights of
`newborns (measured in grams) with the size of
`adults (measured in centimeters). The coefficient
`of variation is meaningful only for measurements
`with a real zero (1.e., “ratio scales”) because the
`mean is meaningful (i.e., unique) only for these
`scales. So, for example, it would be meaningless
`to compute the coefficient of variation of the
`temperature measured in degrees Fahrenheit,
`because changing the measurement
`to degrees
`Celsius will not change the temperature bur will
`change the value of the coefficient of variation
`(because the value of zero for Celsius is 32 for
`Fahrenheit, and therefore the mean of the tem-
`perature will change from one scale to the
`other).
`In addition,
`the values of the measure-
`ment used to compute the coefficient of variation
`are assumed to be always positive or null. The
`coefficient of variation is primarily a descriptive
`statistic, bur it
`is amenable to statistical
`infer-
`ences such as null hypothesis testing or confi-
`dence intervals. Standard procedures are often
`very dependent on the normality assumption,
`
`
`
`