Table Of ContentKenneth J. Berry · Janis E. Johnston
Paul W. Mielke, Jr.
The
Measurement
of Association
A Permutation Statistical Approach
The Measurement of Association
Kenneth J. Berry • Janis E. Johnston (cid:129)
Paul W. Mielke, Jr.
The Measurement
of Association
A Permutation Statistical Approach
123
KennethJ.Berry JanisE.Johnston
DepartmentofSociology Alexandria
ColoradoStateUniversity Virginia,USA
FortCollins
Colorado,USA
PaulW.Mielke,Jr.
DepartmentofStatistics
ColoradoStateUniversity
FortCollins
Colorado,USA
ISBN978-3-319-98925-9 ISBN978-3-319-98926-6 (eBook)
https://doi.org/10.1007/978-3-319-98926-6
LibraryofCongressControlNumber:2018954500
MathematicsSubjectClassification(2010):62gxx,62-07,62-03,62axx
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E.Berry, andLauraB. Berry;Lindsay
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Malia,Ollie,Cami,and Brian;andRoberta
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Spear,andLynn (Mielke) Basila.
Preface
TheMeasurementofAssociation:APermutationStatisticalApproachutilizesexact
andMonte Carlo resamplingpermutationstatistical proceduresto generateproba-
bilityvaluesforavarietyofmeasuresofassociation.Associationisbroadlydefined
to include measures of correlation for two interval-levelvariables; association for
twonominal-level;twoordinal-level,ortwointerval-levelvariables,andagreement
for two nominal-level or two ordinal-level variables. Measures of association
have historically been constructed for three levels of measurement, i.e., nominal,
ordinal, and interval. Additionally, measures of association for mixtures of the
threelevelsofmeasurementhavebeenconsidered,i.e.,nominal–ordinal,nominal–
interval, and ordinal–interval.The bookis structuredaccordingto the three levels
ofmeasurement.
S.S.Stevenspromotedthetypologyofscalescontainingfourlevelsofmeasure-
ment:nominal,ordinal,interval,andratio,butitshouldbenotedthatanumberof
writers have taken exception to the organization of statistical tests and measures
by levels of measurement, arguing that there is no relationship between levels of
measurementandstatisticaltechniquesused,whileothershavesuggesteddifferent
typologies. Stevens also recognized that a too rigid adoption of his suggested
typology could be counterproductive. In this book, the interval and ratio scales
areconsideredtogetherassimply“interval”andthenominal,ordinal,andinterval
typology is utilized strictly as a pragmatic organizational framework. The 10
chaptersofthebookprovide:
Chapter1:Anintroductionto,andthecriterianecessaryfor,creatingvalidmeasures
ofassociation.
Chapter 2: A description and comparison of two models of statistical inference:
thepopulationmodelandthepermutationmodel.Permutationmethods,which
are used almost exclusively in this book, are further detailed and illustrated,
includingexact,moment-approximation,andMonteCarloresamplingpermu-
tationmethods.
Chapter 3: Presentation, discussion, and examples of measures of association
for two nominal-level variables that are based on Pearson’s chi-squared test
statistic.
vii
viii Preface
Chapter 4: Presentation, discussion, and examples of measures of association for
twonominal-levelvariablesthatarebasedoncriteriaotherthanPearson’schi-
squaredteststatistic.
Chapter 5: Presentation, discussion, and examples of measures of association for
two ordinal-level variables that are based on pairwise comparisons between
rankscores.
Chapter 6: Presentation, discussion, and examples of measures of association for
two ordinal-level variables that are based on criteria other than pairwise
comparisonsbetweenrankscores.
Chapter 7: Presentation, discussion, and examples of measures of association for
twointerval-levelvariables.
Chapter 8: Presentation, discussion, and examples of measures of association for
two variables at different levels of measurement:nominal–ordinal,nominal–
interval,andordinal–interval.
Chapter9:Presentation,discussion,andexamplesoffourfoldcontingencytablesas
aspecialapplicationofmeasuresofassociation.
Chapter 10: Presentation, discussion, and examples of measures of association
appliedtosymmetricalfourfoldcontingencytables.
The Measurementof Associationadoptsa permutationapproachforgenerating
exact and resampling probability values for various measures of association.
Permutationstatisticalmeasurespossessseveraladvantagesoverclassicalstatistical
methods in that they are optimal for small samples, can be utilized to analyze
nonrandom samples, are completely data dependent, are free of distributional
assumptions,andyieldexactprobabilityvalues.Today,permutationstatisticaltests
areconsideredbymanytobeagoldstandardagainstwhichconventionalstatistical
tests should be evaluated and validated. An obvious drawback to permutation
statisticalmethodsistheamountofcomputationrequired.Whileittooktheadvent
ofhigh-speedcomputingtomakepermutationmethodsfeasibleformanyproblems,
todaypowerfulcomputationalalgorithmsandmoderncomputersmakepermutation
analysespracticalformanyresearchapplications.
A comparison of two models of statistical inference begins the book: the con-
ventionalpopulationmodel and the permutation statistical model. The population
model assumes random sampling from one or more specified populations. Under
thepopulationmodel,thelevelofstatisticalsignificancethatresultsfromapplying
a statistical test to the results of an experiment or survey corresponds to the
frequency with which the null hypothesis would be rejected in repeated random
samplingsfromaspecifiedpopulation.Becauserepeatedsamplingofthespecified
population is impractical, it is assumed that the sampling distribution of test
statisticsgeneratedunderrepeatedrandomsamplingconformstoanapproximating
theoreticaldistribution,suchasthenormaldistribution.Thesizeofastatisticaltest
istheprobabilityunderthenullhypothesisthatrepeatedoutcomesbasedonrandom
samplesofthesamesizeareequaltoormoreextremethantheobservedoutcome.
In contrast, the permutation model does not assume, nor require, random
sampling from a specified population. For the exact permutation model, a test
Preface ix
statistic is computed for the observed data. The observations are then permuted
over all possible arrangementsof the observed data, and the selected test statistic
iscomputedforeachofthepossiblearrangements.Theproportionofarrangements
with test statistic values equal to or more extreme than the observed test statistic
yields the exact probability of the observed test statistic value. When the number
of possible arrangements of the observed data is very large, exact permutation
methodsareimpracticalandMonteCarloresamplingpermutationmethodsbecome
necessary.Resamplingmethodsgeneratearandomsampleofallpossiblearrange-
mentsoftheobserveddata,andtheresamplingprobabilityvalueistheproportion
ofarrangementswithteststatisticvaluesequaltoormoreextremethanthevalueof
theobservedteststatistic.
Asdescribed,videsupra,thisbookprovidespermutationstatisticalmethodsfor
differentmeasuresofassociationfornominal-,ordinal-,andinterval-levelvariables
andisorganizedinto10chapters.
Chapter 1 defines association in general terms and examines four dimensions
of association: symmetry and asymmetry; one- and two-way association; models
of association including maximum-corrected,chance-corrected, and proportional-
reduction-in-error measures; and measures of correlation, association, and agree-
ment. Chapter 1 concludes with sections on choosing criteria for creating useful
measures of association, assessing the strength of association, and selecting an
appropriatemeasureofassociation.
Chapter2comparesandcontraststwomodelsofstatisticalinference:thepopu-
lationmodelandthepermutationmodel.Underthepermutationmodel,threetypes
of permutationtests are described:exact,Monte Carlo resampling-approximation,
and moment-approximationstatistical tests. Permutationand parametricstatistical
testsarecomparedandcontrastedintermsofsamplesize,datadependency,andthe
assumptionsofrandomsamplingandnormality.
Chapter3introducespermutationstatisticalmethodsformeasuresofassociation
designedfortwonominal-level(categorical)variables.IncludedinChap.3arethe
usualchi-squared-basedmeasures,Pearson’sφ2,Tschuprov’sT2,Cramér’sV2,and
Pearson’scontingencycoefficient,C.Thediscussionofthefourchi-squared-based
measuresofassociationisfollowedbyananalysisofpermutation-basedgoodness-
of-fit tests. Chapter 3 concludes with an examination of the relationship between
chi-squaredandPearson’sproduct-momentcorrelationcoefficient.
Chapter4introducespermutationstatisticalmethodsformeasuresofassociation
designed for two nominal-level variables that are based on criteria other than
Pearson’schi-squaredteststatistic.IncludedinChap.4arediscussionsofGoodman
and Kruskal’s two asymmetric measures of nominal-level association, λ and τ,
McNemar’sQandCochran’sQtestsforchange,Cohen’sunweightedκ measureof
inter-raterchance-correctedagreement,the Mantel–Haenszeltest ofindependence
forcombined2×2contingencytables,andFisher’sexactprobabilitytestappliedto
avarietyofr×ccontingencytables.
Chapter5introducespermutationstatisticalmethodsformeasuresofassociation
designed for ordinal-levelvariables based on pairwise comparisons between rank
scores.IncludedinChap.5areKendall’sτ andτ measures,Stuart’sτ measure,
a b c
x Preface
GoodmanandKruskal’sγ measure,Somers’dyx anddxy measures,Kim’sdy·x and
dx·y measures, Wilson’s e measure, Whitfield’s S measure of ordinal association
betweenanordinal-levelvariableandabinaryvariable,andCureton’srank-biserial
correlationcoefficient.
Chapter6introducespermutationstatisticalmethodsformeasuresofassociation
designedfortwoordinal-levelvariablesthatarebasedoncriteriaotherthanpairwise
comparisons between rank scores. Included in Chap.6 are Spearman’s rank-
ordercorrelationcoefficient,Spearman’sfootrulemeasureofinter-rateragreement,
Kendall’s coefficientof concordance,Kendall’s u measure of agreement,Cohen’s
weighted kappa measure of agreement with both linear and quadratic weighting,
andBross’sriditanalysis.
Chapter7introducespermutationstatisticalmethodsformeasuresofassociation
designed for interval-level variables. Included in Chap.7 are simple and multiple
ordinaryleast squares (OLS) and least absolute deviation(LAD) regression using
permutation statistical methodology. Fisher’s r to z transform is described and
xy
evaluatedas to its utility in transformingskewed distributionsfor both hypothesis
testingandconfidenceintervals.Point-biserialandbiserialcorrelationaredescribed
andtestedwithexactandMonteCarloresamplingpermutationmethods.Chapter7
concludeswithadiscussionoftheintraclasscorrelation.
Chapter8introducespermutationstatisticalmethodsformeasuresofassociation
designed for mixed variables: nominal–ordinal, nominal–interval, and ordinal–
interval. Included in Chap.8 are Freeman’s θ, Agresti’s δˆ, Piccarreta’s τˆ, and
Berry and Mielke’s (cid:2) for the measurement of nominal–ordinalassociation. Also,
Whitfield’s S measure and Cureton’s rank-biserial measure for a dichotomous
nominal-level variable and an ordinal-level variable are described. For nominal–
intervalassociation:Pearson’sη2, Kelley’s(cid:10)2, andHays’ωˆ2 are presented.Chap-
ter 8 concludes with a discussion of permutation statistical methods for Jaspen’s
multiserialcorrelationcoefficientforanordinal-levelvariableandaninterval-level
variable.
Chapter9introducespermutationstatisticalmethodsformeasuresofassociation
usually reserved for 2×2 contingency tables. Included in Chap.9 are discussions
of Yule’s Q and Yule’s Y measures of nominal-level association, Pearson’s φ2
measure,simplepercentagedifferences,GoodmanandKruskal’st andt measures,
a b
Somers’d andd measures,theMantel–Haenszeltest,Fisher’sexactprobability
yx xy
test,tetrachoriccorrelation,andtheoddsratio.
Chapter 10 continues the discussion of 2×2 contingency tables initiated in
Chap.9 with consideration of symmetrical 2×2 contingency tables. Included in
Chap.10are permutationstatistical methodsappliedto Pearson’sφ2, Tschuprov’s
T2, Cramér’s V2, Pearson’s product-moment correlation coefficient, Leik and
Gove’s dc measure, Goodman and Kruskal’s t and t asymmetric measures,
N a b
Kendall’sτ andStuart’sτ measures,Somers’d andd asymmetricmeasures,
b c yx xy
simple percentage differences, Yule’s Y measure of nominal association, and
Cohen’s unweighted and weighted κ measures of inter-rater chance-corrected
agreement.
Preface xi
Acknowledgments. The authors wish to thank the editors and staff at Springer-
Verlag.VeryspecialthankstoDr.EvaHiripi,StatisticsEditor,Springer,whoguided
theprojectfrombeginningtoend.WearegratefultoRobertaMielkewhoreadthe
entiremanuscript.Finally,wewish to thankSteveandLindaJones,proprietorsof
the Rainbow Restaurant, 212 West Laurel Street, FortCollins, Colorado,for their
gracious hospitality. Like our previous books, much of this book was written at
Table22intheirrestaurantadjacenttotheColoradoStateUniversitycampus.
FortCollins,CO,USA KennethJ.Berry
Alexandria,VA,USA JanisE.Johnston
FortCollins,CO,USA PaulW.Mielke,Jr.
August2017
