:NCYCLOPED|A
`
`RESEARCH
`DESIGN
`
`VOLUME 1
`
`
`
` —LJ|||—LJ
`“W
`
`
`NEIL]. SALKIND
`Un 1‘\--’ersity ofKemsas
`
`; “
`®SAGE reference
`Genome & Co. v. Univ. of Chicago
`PG R201 9-00002
`
`UNIV. CHICAGO EX. 2068
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`Library of ("rumors (Ii-ital:igr'ng-in-I’irlilimrion Dara
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`Encyclopedia of misc-arch dcsigni’cditcd by Neil I. balkind.
`v. cm.
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`Includes bibliographical references and index.
`ISBN 9734 -*I I 29-6 l 27-l lultathl
`1. Social SCIENCES—~Stal'i5thEll methods—FIiuvclopcdiaii. 2. Souial SCIEDCCS—hRCSCflI‘CI'I—NIt‘ll’IUdUlUgy-——I:.l1\.‘)'Cl(1Pt'di35.
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`Contents
`
`
`
`Volume 1
`
`List of Entries mi
`
`Reader‘s Guide xiii
`
`About the Editors xix
`
`Contributors xxi
`
`Introduction xxi'x
`
`Entrics
`
`I
`
`57
`
`I I I
`
`.32!
`
`E
`
`F
`
`G
`
`399
`
`4.71
`
`SW
`
`Volume 2
`
`List of Entries
`
`vii
`
`Entries
`
`561
`589
`
`655
`663
`
`681'
`
`M
`N
`
`0
`P
`
`745
`869
`
`949
`985
`
`Volume 3
`
`List of Entries
`
`vii
`
`Entries
`
`4' I49
`I I83
`
`HE’S
`
`I489
`
`I583
`
`1589
`V
`W MI I
`
`Y
`
`2
`
`164).
`
`169?
`
`Index
`
`1675
`
`A
`
`B
`
`C
`
`D
`
`H
`l
`
`_|
`K
`
`l
`
`Q
`R
`
`S
`
`T
`
`Ll
`
`

`

`
`This material may be protected by Copyright law (Title 17 US. Code)
`
`
`
`
`
`
`164 Coe fficient 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 extensiou of alpha (and clas-
`sical test theory) came when Fisherian notions of
`experimental design and analysis of vat'ial‘iCe were
`put together with the idea that some “treatment"
`conditions could be considered random samples
`from a large universe, as alpha assumes about item
`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—ctnnponent approach
`can provide the value of “alpha" expected by
`increasing or decreasing the number of 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 he 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
`alphafihas been and will continue to be the mosr
`popular method
`for
`estimating
`behavioral
`measurement
`reliability. As of 2004,
`the 1951
`
`coefficient alpha article had been cited in more
`than 5,000 publications.
`
`jeffrey T. Steedle and Richard]. Sbaeelson
`
`See also Classical Test Theory; Generalizability Theory;
`Interim] Consistency Reliability; KR-ZO; Reliability;
`
`Split-Half Reliability
`
`Further Readings
`
`Brennan, R. L. {2001]. (ierierulizahiliry theory. New
`York: Springer-Verlag.
`(ironbach, I.. ‘I., t’c Shavelson, R. ]. (2004}. My current
`thoughts on coefficient alpha and successor
`procedures. Educational 6" Psychological
`Measurement, 64(3), 391—413.
`Haertel, E. H. {Billie}. Reliability. In R. I.. Brennan {ELL},
`Edm'ttrional measurement (pp. 65—] Id}. Westport,
`CT: I’raeger.
`Shovelson, R. J. {2004'}. Editor's preface to Lee
`_I. Cronbach‘s “My Current Thoughts on Coefficienl
`Alpha and Successor Procedures." Educational c'r
`Psychological A’Ietrsurement. (54(3), 389—390.
`Shavelson, R. _]., 5; Webb, N. M, {1991], Genertrltzdhiltty
`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 (ml quantitative or scmiquantitative varis
`ables that are assessing a set of n 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 x3 sta-
`tistic for testing. They differ only in the formula~
`tion of their respecrive null hypothesis. Consider
`Table l, which contains illustrative data. In Fried‘
`man‘s test, the null hypothesis is that there is no
`
`

`

`
`
` Coefficient of Concordance 165
`
`Table I
`
`Site 4
`Site 9
`Site 14
`Site 23
`Site 31
`Site 34
`Site. 43
`
`Site 53
`Site 61
`Site 69
`
`ll|u5trative Example: Ranked Relative Abundances of Four Soil Mite Species (Variables) at 10 Sites (Objects)
`Sun: of Ranks
`Rtmks (coiunm-ttuse)
`
`Species 1-?
`Species 1.5
`Species 2.3
`R,
`6
`3
`5
`19.1}
`4
`8
`J.
`24.1}
`8
`5
`4
`24.0
`It)
`9
`2
`29.0
`5
`7
`6
`24.0
`3’
`10
`7"
`33.0
`3
`2
`8
`16.0
`
`Species 1 i
`S
`111
`7
`8
`ii
`9
`3
`
`1.5
`1.5
`4
`
`2
`l
`9
`
`4
`l
`o
`
`9
`2
`10
`
`16.5
`5.5
`29.11
`
`.‘irmrce: Legendre. P. 12005} Species associations: The Kendall coefficient of concordance revisited. journal! of.—1gricnimml,
`Biological. (3‘ Environmental Statistics. 11?. 231}. Reprinted with permission from thcjonrnal rif'Agrit'nlmnn'. Biological. ci-
`lim'i'rwmicntm' Statistics. Copyright 21105 by the American Statistical Association. All rights reserved.
`
`Notes: The ranks are computed colnmnwisc with ties. Right-hand column: sum of live ranks for each site.
`
`real difference among the H obiects (sites, rows Of
`Table 11 because they pertain to the same Statisti—
`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 tCSt focuses on
`the or variables. 1f
`the null hypothesis of Fried-
`man’s test
`is true,
`this means that
`the variables
`have produced rankings of the obiects 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 W sta-
`tistic {firsr and second forms of Equations 1 and
`2); they lead to the same result. S or S’ is computed
`first
`from the row-marginal
`sums of
`ranks R,
`received by the objects:
`I!
`
`5 : Elli—iii]: ms
`t
`l
`
`;:R-= 55R {1)
`
`where S is a sum-oi-squarcs statistic over the row
`sums of ranks Ry, and F is the mean of the R,
`values. Following that. Kendall‘s W statistic can be
`obtained From either of the following formulas:
`
`l 23
`W = —-—.—--——-—-
`mliir‘ # it} — mi"
`
`01'
`
`W = 125’1f3iriln(ri 4- Hi
`”Pin" — n1 — mT
`
`{2)
`
`the number of objects and m is the
`where n is
`number of variables. T is a correction factor For
`tied ranks:
`
`.k'
`
`T = Zn; —rn.
`11-..1
`
`(3i
`
`in which It. is the number of tied ranks in each (it)
`of g groups of ties. The sum is computed over all
`groups of ties fOund in all or variables of the data
`table. T = 0 when there 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
`when all variables are in toral agreement. Hence
`0 5 W5 1,
`1
`representing perfect concordance. To
`derive the formulas for W {Equation 2), one has to
`know that when all variables are in perfect agree—
`ment, the sum of all sums of ranks in the data table
`(right—hand column of Table l) is mntn + 11/2 and
`that the sum of squares of the sums of all ranks is
`mini” + ”1fo i 1116 (without ties).
`There is a close relationship betWeen Charles
`Spearman's correlation coefficient r3- and Kendall‘s
`W statistic: W can be directly calculated from the
`
`

`

`
`
`166 Coefficient of Concordance
`
`mean (in of the pairwise Spearman correlations r5
`using the following relationship:
`
`the index, because different
`included in
`be
`groups of species may be associated to different
`environmental conditions.
`
`W:(i-n-ljr_q+l,
`I?!
`
`(4}
`
`where m is the number of variables {judges} among
`which Spearman correlations 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 r5 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 13,-: WTlTs'd' llll.
`In that
`case, a permutation test of W for two variables is
`the exact equivalent of a permutation test of 11,- 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 Spearman correlation of — l, hence erl
`for these two variables {Equation 4};
`this is the
`lower bound of the coefficient of concordance. For
`
`two variables only, art-=0 gives W:ll.5. So coeffi—
`cient “7 applies well to rankings given by a panel of
`judges called in to assess overall performance in
`sports or quality of wines or food in restaurants, to
`rankings obtained from criteria used in quality tests
`of appliances or services by consumer organizations,
`and so forth. 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 lior W because they are all indirect measures of
`:1 common property, the size of the birds.
`ln ecological applications, one can use the
`abundances of various species as indicators of
`the good or bad environmental quality of the
`study sites. It a group of species is used to pro-
`duce a global index of the overall quality (good
`or bad} of the environment at the study sites.
`only the species that are significantly associated
`and positively correlated to one another should
`
`Testing the Significance of W
`
`Friedman's chi-square statistic is obtained from W
`by the formula
`
`‘1
`
`x : Nil” — MW.
`
`{5}
`
`This quantity is asymptotically distributed like
`chi-square with 1' = in — 1} degrees of freedom; it
`can be used to test W for significance. According to
`Kendall and Babingtori Smith, this approach is satis-
`factory only for moderately large values of m and n.
`Sidney Siegel and N. john Casrellan ‘lr. recom-
`mend the use of a table of critical values for W
`
`when it 5 7 and m 5 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 a and m is derived from a table of
`
`critical values of S assembled by Friedman using the
`3 test of Kendall and Babington Smith and repro—
`duced in Kendall‘s classic monograph, Rank Corre-
`lation Merbt'ids. 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 Siege] and Castellan also
`recommend for small samples (small a}. The simula»
`tion results showed that the classical chi-square test
`was too conservative for any sample size [in when
`the number of variables or 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 1 error {reiecting
`a null hypothesis that is true) for .20 variables and
`more. The permutation test had a correct rate of
`Type I error for all values of m and H. The power of
`the permutation test was higher than that of the
`classical chi—square tesr because of the differences in
`rates of Type I error between the two tests. The dif-
`lerences in power disappeared asymptotically as the
`number of variables increased.
`
`An alternative approach is to Compute the fol—
`lowing } statistic:
`
`F .— (m — lZJW/(l — W},
`
`(6)
`
`which is asymptotically distributed like i- with
`I'] = u —
`1— (2/312) and v3 _— 11mm — 1} degrees
`
`

`

`Smith
`and Babington
`freedom. Kendall
`of
`described this approach using :1 Fisher 3 transfor—
`mation of the F statistic, 320.3 logiF}. They
`recommended it for testing W for moderate values
`of m and ii. Numerical simulations show, however,
`that
`this F statistic has correct
`levels of Type I
`error for any value of it and m.
`In permutation tests of Kendall‘s W, the objects
`are the permutablc units under the null hypothesis
`{the objects are sites in Table I]. For the global tesr
`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 by all 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}, xi {Equation 5],
`and F {Equation 6] are monotonic to one another
`since a and or, as well as T, are constant within
`
`a given permutation test; thus they are equivalent
`statisrics for testing, producing the same permuta—
`tional probabilities. The test is one—tailed because
`it
`recognizes only positive associations between
`vectors of ranks. This may be seen if one considers
`two vectors with exacrly opposite rankings: They
`produce a Spearman statistic of — I, hence a value
`of zero for W [Equation 4}.
`Many of 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 lest {Equation 6], as well as the
`permutation test, can safely be used with all values
`ot a: and iv.
`
`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. consumer protection, 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 ease 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-
`
`in particular in situations of
`ces of its quality,
`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 to all 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 random should have little 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.
`f,. of the pairwise Speatman correlations
`between variable i under test and all the other vari—
`ables. The second statisric, W“ is obtained by apply—
`ing Equation 4 to r, instead of F, with m the number
`of variables in the group. These two statistics are
`shown in Table 2 for the example data; :7, and W,
`are monotonic to each other because at is constant
`
`in a given permutation test. Within a given a poster-
`iot‘i test, W is also monotonic to W, because only
`the values related to variable i are. permuted when
`testing variable i. These three statistics are thus
`equivalent for a posteriori permutation tests, produc—
`ing the same permutational probabilities. Like Ff, W,
`can take negative values; this is not the case of W.
`There are advantages to performing a single
`a posteriori rest for variable 1
`instead of im— 1}
`tests
`of
`the Spearman correlation coefficients
`between variable ,i and all the other variables: The
`tests of the [iii— 1} correlation coefficients would
`
`
`
`

`

`168
`
`Coefficient of Concordance
`
`Table 2
`
`Results of (a) the Overall and (b) the A Posteriori Tests of Concordance Among the Four Species of Table
`l: Sc) Overall and Ed) A Posteriori Tests of Concordance Among Three Species
`
`la} Overall test of W statistic, four species. H“ ; The four species are not concordant with one another.
`Kendall's \V —
`0.44160
`
`F statistic =
`
`2.3?252
`
`l’erinutational p value —— .0448?
`i- distribution ,0 value = .0440”
`
`
`Friedman‘s chi-square : .0690#15.89}?! (Ihi—squa re distribution ,9 value —-
`
`
`
`{hi A posteriori tests. four species. HU : This species is not concordant with the other three.
`
`i‘,
`WI-
`.0 value
`Corrected p
`
`Decision at ct = 5%
`
`Do not reiect Hi:
`.1332
`.0?66
`0.49493
`0.32657
`Species 13
`Do nor reiect Hi1
`.0720
`.0240
`0.54?“
`0.39655
`Species 14
`Reiect Ho
`.0204“
`.0051
`0.59278
`0.45704
`Species 15
`
`
`
`
`
`0.12391 .7070 .70704.116313Species 23 Do not reject H“
`
`{c} Overall test of W Statistic. three species. H1, : The three species are not ct'incordant with one another.
`
`Kendall's W =
`0.78273
`Permutation-a] p value -— .0005 1’
`F distribution )9 value - .0003“
`
`F statistic =
`
`7.20497
`
`21.13360
`Friedman‘s chi—square —-
`ChI—square distribution p value 2 .0121‘“
`
`(dl A posteriori tests, three species. HI; : This species is not concordant with the other two.
`
`r,
`W,
`p value
`Corrected ,0
`Decision at o = 3%
`
`ReieCt Ho
`.0120”
`.0040
`0.?9939
`0.69909
`Species I3
`Reject Ho
`.l_l:.".‘30"r
`.0290
`0.72784
`0.59 l 76
`Species 14
`
`
`0.73153Species 15 Reject H” 0.81l05 .0050 .0120”
`
`
`
`Source: la} and lb]: Adapted from legendre, P. {1005 1. Species associations: The Kendall coefficient of concordance revisited. joimiril of
`Agricultural. Biological. and i-Liii'iromnental Statistics,
`if}. 233. Reprinted with pennission from the Journal inf—Agricultural. Biological
`and Environmental Statistics. Copyright 2005 by the American Statistical Association. All rights resen'ed.
`
`partial concordance per species; in value
`Notes: r, : mean of the Swarm-an col‘relat‘ii'it‘ls with the other species; ll”.
`prolmbilio' {9,999 random permutations}; corrected p .— HoIm-corttcted p value. *
`= Reicct H“ at u : .05.
`
`pern'nitational
`
`have to be corrected for multiple testing, and they
`could provide discordant information; a single test
`of the contribution of variable i to the W statistic
`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 for all
`species in a group should be adjusted for multiple
`tesring.
`A posterioti tests are usuful for identifying the
`variables that are not concordant with the others,
`
`as in the examples, but they do not tell us whether
`there are one or several groups of congruent vari-
`ables among those for which the null hypothesis of
`independence is rejected. This information can be
`obtained b_v 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 “3” statistic is sig—
`nificant at
`or = 5%, but marginally {Table 2a}. The
`cause appears when examining the a posteriori
`tests in Table 3h: Species 23 has a negative mean
`correlation with the three other species in the
`group (r, = — .163}. This indicates that Species 23
`Clt}ES 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-rneans 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 now test Species 131
`14, and 15 as a group. Table 2c shows that this
`group has a highly significant concordance1 and all
`individual species contribute significantly to the
`overall concordance. of their group {Table 1d].
`
`

`

`In Table la and 3c. the F test results are concor-
`
`dant with the permutation test results, but due to
`small ”I and n. the chi-square test lacks pchr.
`
`Discussion
`
`The Kendall coefficient of concordance can he
`
`used to assess the degree to which a group of 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
`
`in identifying discordant
`interest
`there is
`fields,
`variables;
`this is the case in all
`fields that use
`panels of judges to assess the overall quality of the
`objects under study (e.g., sports,
`law, consumer
`protection). In other applications. one is interested
`in using the sum of ranks, or the sum of values,
`provided by several variables or judges, to create
`an overall indicator of the response of the obiects
`under study. It is advisable to look for one or sev—
`eral groups of variables that
`rank the obiects
`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 (iorrelatirm
`
`Further Readings
`
`Friedman, M- 1193?}. The use of ranks to avoid the
`
`assumption of normality implicit in the analysis of
`variance. journal of the American Statistical
`."tSSi'Jt’JrJfli"il'I. 3.2. 6?.‘3—70 I .
`
`
`
`Coefficient of Variation 169
`
`Friedman. M. {1940}. A comparison of alternative tests
`of significance For the problem of in rankings. Annals
`of Mulbematiml Statistics. ll. lib—92.
`Kendall, M. G. [1948). Rank correlation methods i lst
`mi]. London; Charles ('irii‘fith.
`Kendall, M. (1. 5c Babington Smith. 3. {1939}. The
`problem of m rankings. Annals of Mathematical
`Statistics.
`it]. 275—287.
`
`Legendre. l’. {200% Species associations: The Kendall
`coefficient of concordance revisited. journal of
`Agricultural. Biological. cit rim-arrinmenrai Statistics.
`Hi. 236—145.
`
`Zar, ‘I. H. {[999}. Biosmtisriml analysis {4th cd.J. Upper
`Saddle River, N]: Prentice Hall.
`
`
`
`COEFFICIENT or 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 tie. “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 but 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, but
`it
`is amenable to statisrical
`infer—
`ences such as null hypothesis testing, or confi—
`dence intervals. Standard procedures are often
`very dependent on the normality assumption,
`
`
`
`