Analysis of Clustered Binary Data

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nalysis of Clustered Binary DataValerie Durkalski aaDepartment of Biostatistics, Bioinformatics and Epidemiology, Medical University of South Carolina,Charleston, South Carolina, U.S.A.

Online Publication Date: 13 April 2005

To cite this SectionDurkalski, Valerie(2005)'Analysis of Clustered Binary Data',Encyclopedia of Biopharmaceutical Statistics,1:1,1 6

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Analysis of Clustered Binary Data

Valerie DurkalskiMedical University of South Carolina, Charleston, South Carolina, U.S.A.

INTRODUCTION

Clinical trial designs often incorporate binary outcomes(i.e., proportion of responders; success probabilities) orchange continuous outcomes into binary outcomes forinterpretation purposes. When a subject contributesmore than one binary response (paired or unpaired), itresults in clustered data in which the subject is thecluster and each response within a cluster is a unit. Inthis scenario, responsesare considered dependent withina cluster, and independent between clusters.

One can imagine several distinct contexts in whichclustered binary data might arise. Examples includeophthalmology studies, in which the unit is each eye,but more generally the number of units may varyacross clusters. This more general formulation thenincludes dental studies, in which the unit of analysisis each tooth, oncology studies, in which the units areinfected nodes or lesions within a patient, and commu-nity intervention studies, in which the unit is the indi-vidual within the community. When dealing with twotreatments, it would be possible for some clusters,and every unit in those clusters, to be assigned to onetreatment group, and for the other clusters, and every

unit in these other clusters, to be assigned to the othertreatment group. For example, to examine the successof two teaching programs on the graduation rate inone school, one may randomize classrooms (cluster)to the treatment and have students be the unit of ana-lysis. Fear of contamination would preclude one fromrandomizing students within the same classroom todifferent teaching conditions. This is also the contextfor repeated measures studies, in which the cluster isthe patient and the units are the sampling times. Sucha design allows for the separation of effects due tobetween-subject variability from effects due to within-subject variability.[1] It is also possible that each cluster

receives both treatments, but each unit within eachcluster receives one treatment or the other, such as insome ophthalmology studies, where one eye is givenone treatment and the other is given the comparisontreatment. A third context would be in which each unitreceives both treatments, such as in matched-pairdesigns, where clusters (i.e., subjects) act as their owncontrol and receive both procedures=interventions, orclusters are matched with each receiving one of theprocedures=interventions.

At the unit level, the correlation between unitswithin a cluster violates the independence assumptionof many of the statistical methods for analyzing binaryoutcomes, such as Pearsons chi-square, McNemarstest, and logistic regression. This violation maydecrease standard error estimates and the p-value asso-ciated with the test statistic, thereby yielding mislead-ing statistical conclusions. The challenge that ariseswhen analyzing clustered binary data, therefore, ishandling the correlation among units within a cluster.While cluster summary approaches such as collapsing

data within a cluster or focusing the design on clus-ter-level analyses are valid in some cases, they do notuse all available data, and so more informative analysistechniques would be expected to yield more preciseresults. Unit-level analyses are the focus of this entry.

METHODS OF ANALYSIS

Methods for the analysis of clustered binary data(paired and unpaired) are well developed.[2,3] Mostresearch directly addresses important theoretical andapplied issues regarding the effect of clustering and

the consequential effect on the variance and on the sta-tistical conclusions. Adjustments for correlated binaryresponses within a cluster have been incorporated intostatistical tests using a correlated binomial model,[4]

Fishers permutation test,[5] binomial ratio estimators,[6,7]

pooled estimators,[8] method of moments estimators,[9]

and intracluster correlation (ICC).[10] Anotherapproach that is widely applied is marginal regressionmodeling using the general estimating equation(GEE) approach of Zeger and Liang.[11] Althoughthese comparison methods have been developed forclustered binary data, not all tests are consistent interms of performance. Depending on the cluster sizes

and the correlation structure within clusters, the testsperformances may vary in terms of size and power.This entry focuses on the ICC, pooled estimators,method of moments estimators, and GEEapproaches.

To review different analysis methods, three sub-scripts are defined in the case of two proportions, asfollows. Let the response variable Yijkbe dichotomous(1 success, 0 failure), where i ( 1, 2) is thetreatment or intervention, j( 1, 2, . . . , nk) is the unit

Encyclopedia of Biopharmaceutical Statistics DOI: 10.1081/E-EBS-120029859Copyright # 2005 by Taylor & Francis. All rights reserved. 1


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within the cluster, and k( 1, 2, . . . , K) is the cluster.Let ppik n

1ik

Pnkj1Yijkbe the event rate (i.e., propor-

tion of successes) in group ifor cluster k; Kis the totalnumber of clusters in the study population, and

N PK

k1nk is the total number of units across allthe clusters for both treatments. The nk units in the

kth cluster are assumed to be fixed for all Kclusters.With this framework, the data from each unit withineach cluster can be summarized as a standard 2 2contingency table, as both treatment and outcomevary, and are binary variables (assuming that thereare only two treatments). The frequencies of theresponses per group may be summed over allKclusters.This data display is presented in Table 1 for inde-pendent bivariate random variables and in Table 2for dependent bivariate variables (matched-pair data).

Hypothesis Testing

When dealing with bivariate random variables, one for

each of two treatment conditions, it is often of interestto test the hypothesis of equality of the probabilities ofa positive response across the treatment conditions.Doing so is fairly straightforward when subjects areallocated to treatment groups by random allocation,with either no restrictions at all on the randomizationor with the only restriction being that the group totalsare fixed. In either case (conditioning on the randomgroup totals in the first case), the design-based analysisis Fishers exact test.[12] When clustering is present,

however, this would not be an appropriate analysis,because it would ignore the clustering.

To avoid underestimating the variance of theparameter in the presence of clustering, Donnerand colleagues explore a model using the intraclustercorrelation coefficient (ICC), which adjusts the chi-square test for correlation within clustered data.[13,14]

The ICC, originally applied to interobserver agreementanalyses as an alternative to Cohens kappa, an index

of agreement between two or more raters, is estimatedusing the mean square errors of analysis of variance(ANOVA) for mixed models, where the treatment isconstant and the cluster is random. The estimatedICC represents the proportion of variation due tobetween-cluster differences. Donner presents a detailed

explanation of how a consistent estimate of the ICC is

calculated under the assumption of a constant ICCacross clusters.[5] The ICC adjustment is applied tothe Pearson chi-square test,[13] chi-square test for lineartrend,[14] MantelHaenszel test,[15] and McNemartest.[16] The test statistic is adjusted by dividing it byan inflation factor, Ci 1 m 1rr, where mis the average number of units per cluster for treatmentgroup i and ris the estimated ICC for clustering,

rr BMS WMS

BMS S0 1WMS

In this equation, BMS is the mean squared errorbetween subjects, WMS is the mean squared errorwithin subjects, and S0 is the adjusted mean clustersize. The ICC approach for adjustment of a chi-squaretest is an extension of a standard chi-square test,because when r 0, the adjusted test reduces to thestandard test with one degree of freedom. Therefore,

if there is only one unit per cluster, the standard testto compare overall event rates can be adopted,

w2 X2

i1

nipi pp2

Cipp1 pp

where ppi PK

k1pik and pp is the overall event rate.In addition, the test can be extended to the comparisonof more than two treatment groups.[14] Although theICC appears to be quite adaptable to adjusting avariety of chi-square tests, this method has limitations

regarding the required size of the study population andthe test assumptions. Donald and Donner[13]

suggest that the number of units per cluster shouldbe greater than 10 and the ratio of the number of unitsto the number of observations per unit should begreater than two when applying the adjusted chi-square to Pearsons chi-square test for homogeneityof proportions. They also note that the methodsassumption, that the correlation between responses

within a cluster is equal, may not be appropriate whenthe probability of a correct response varies betweenunits within a cluster or when the correlation is depen-dent upon the size of the cluster. However, Junget al.[17] illustrate through simulation studies that theadjustment method performs well even if the assump-

tion of a common intracluster correlation is not met,

Table 1 Contingency table for clustered binary data

Response

Success Failure

Group 1PK

k1

Pnkj1y1jk

PKk1

Pnkj1 n1 y1jk

PKk1

Pnkj1n1jk

Group 2PK

k1

Pnkj1y2jk

PKk1

Pnkj1 n2 y2jk

PKk1

Pnkj1n2jkPK

k1

Pnkj1 y1jk y2jk

PKk1

Pnkj1 n1jk n2jk

2 Analysis of Clustered Binary Data


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data. Moreover, this method has been extended toaccount for non-inferiority study designs.[18]

General Estimating Equations

Although adjusting the chi-square test for the equalityof probabilities is popular, more complex analyses thatincorporate covariate effects may also be of interest.These analyses require sophisticated models thataccount for the correlation within clusters. A popularmethod due to the availability of statistical softwareis the application of generalized linear models.[19]

Based on this model, marginal regression modelingusing GEEs was developed as a semiparametricapproach to fitting logistic regression models forbinary clustered data.[11] The logistic regression modelusing the GEE estimating function obtains similarresults as the adjusted chi-square tests previously dis-cussed in this entry when covariates are not present.

This model is defined as

logit PrYijk 1 b0 b1xijk

wherexijkidentifies theith treatment variable from thejth unit of the kth cluster (in the case of two treat-ments) and can be run using PROC GENMOD inSAS.[20] The p-value is generated from a comparisonof the log odds ratio to the estimated variance. Becauseclustering affects the standard errors of the parameterestimates rather than the parameter estimates them-selves, the parameter estimates can be obtained byrunning a regression analysis on each response. The

logit link expresses the linear relationship between aclusters responses and the corresponding covariates.The correlation within a cluster is accounted forin the variancecovariance matrix. To estimate thevariance of the response, a specified workingcorrelation matrix that defines the association amongunits within a cluster substitutes for the true, oftenunknown, correlation matrix. Although the workingcorrelation matrix may be mis-specified, which couldpossibly result in a decrease of efficiency, the GEEmethod can produce consistent estimates.[11] Further-more, the decreased efficiency may be minimal if thenumber of clusters is large.[21] Prentice[22] extends

Zeger and Liangs model to incorporate the modelingof correlations within a cluster in order to improveupon the efficiency of the GEE estimates of the regres-sion parameters. Although the GEE method is morecomplex than the straightforward comparison of eventrates, the attractiveness of this approach is that it easilyincorporates adjustments for covariates when needed.Methods for binary regression using clustered datadeserve an entry of their own and are not discussedfurther.

EXAMPLE

To illustrate the clinical application of commonly usedmethods for analyzing clustered binary data and theimportance of accounting for the clustered nature ofthe data, Donner and Klar[23] consider data collectedfrom schools randomly allocated to one of two inter-ventions. The outcome of interest is the proportionof children who use smokeless tobacco after two yearsof follow up. The performance of various methodsfor testing the equality of clustered binary data isobserved, including the test statistics and p-values ofthe unadjusted Pearson chi-square, the ICC inflationfactor, the design effect inflation factor, and the GEEapproach. The test statistics andp-values illustrate thatstatistical conclusions can be false when clustering isignored (using the standard chi-square test), while allmethods that adjust for clustering give relatively thesame conclusions with some variability in the actualtest values and significance levels.

For matched-pair data, a data set containingclustered matched-pair data that appeared inObuchowskis paper[8] is assessed with the unadjustedMcNemar test, the ICC adjustment, pooled estimators,and the method of moments approach. A trial in diag-nostic methods for hyperparathyroidism is designed tocompare the sensitivity and specificity of two tech-niques, positron emission tomography (PET) and asingle photon emission CT (SPECT) scan. The dataconsist of 21 patients whose glands were examinedfor the presence of hyperparathyroidism (Table 3).Of the 21 patients, a total of 72 glands were evaluatedby both diagnostic tools. The specificity of the two

scanners is considered to be of interest and, therefore,only the glands that are confirmed negative by surgery(considered the gold standard) are evaluated. Of the 72glands among the 21 patients, a total of 51 glands wereconfirmed negative. The estimated intracluster correla-tion is equal to 0.46. Obuchowskis, Donners, andDurkalskis test results are chi-square values of 2.86(p 0.091), 3.66 (p 0.056), and 2.32 (p 0.128),respectively. The unadjusted McNemar test statistic is4.5 (p 0.034).

All the tests that account for clustering fail to rejectthe null hypothesis of no difference between the twoprocedures (at the 0.05 level), whereas the McNemar

test (which does not account for the clustering) con-cludes that a statistically significant difference doesexist (at the 0.05 level) in the specificity of PET andSPECT. This is not surprising, because as seen inDonner and Klars comparisons, accounting for clus-tering means assigning less weight to multiple observa-tions from the same cluster. Without doing this, thesemultiple observations would have more influence thanthey perhaps should, and this could easily lead to pseudo-power, or a false rejection of a true null hypothesis.



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CONCLUSIONS

Clustered binary data are prevalent in medicalresearch. Ophthalmology studies, dental research,oncology studies, and community research programsoften involve multiple layers, which violate the basicassumption of independence for common statisticalmethods. Methods for analyzing clustered binary dataare available and continue to be developed andenhanced. All methods account for the within-cluster

correlation; yet, it is the approach to doing so thatmakes them different. The appropriate strategy isdependent on the research question being explored

and the data structure in terms of cluster size andcluster covariates.

Due to the availability of current statistical soft-ware, the majority of methods discussed in this entryare simple to implement in practice. Therefore, thecollapsing of the data to a single-level model should

be avoided. Ignoring the cluster effect or collapsingof data has been shown throughout the literatureto create biased estimates, which can lead to falsestatistical conclusions (as was the case in our

examples). If a simple test of the equality of event ratesis being performed and a relatively small differencein mean cluster size between comparison groups ispresent, then the ICC approach is convenient toimplement and can be extended to a number of datascenarios, including stratified or matched-pair data.

For more complex study designs, such as those thatincorporate covariates or multiple cluster levels (per-haps classrooms within school districts or designs that

have a combination of clustered and longitudinaldata), hierarchical modeling approaches are availableand continue to be explored.[1,3] It is worthwhile tomention that during the planning stages of a study thatinvolves clustered binary data, the statistician needs toconsider inflating the sample size to offset the loss ofinformation that occurs due to the clustering. Leeand Dubin offer considerations for computing theseinflation factors in practice.[24]

ACKNOWLEDGMENT

The author would like to thank Dr. Vance Bergerfor his constructive comments, which have added tothis entry.

REFERENCES

1. Neuhaus, J. Assessing change with longitudinaland clustered binary data. Annu. Rev. Public

Health 2001, 22, 115128.2. Ashby, M.; Neuhaus, J.M.; Hauck, W.W.;

Bacchetti, P.; Heilbron, D.C.; Jewell, N.P.; Segal,M.R.; Fusaro, R.E. An annotated bibliography of

method for analyzing correlated categorical data.Stat. Med.1992, 11, 6799.

3. Pendergast, J.F.; Gange, S.J.; Newton, M.A.;

Lindstrom, M.J.; Palta, M.; Fisher, M.R. Asurvey of methods for analyzing clustered binaryresponse data. Int. Stat. Rev.1996,64 (1), 89118.

4. Hujoel, P.P.; Moulton, L.H.; Loesche, W.J.Estimation of sensitivity and specificity of site-specific diagnostic tests. J. Periodont. Res. 1990,

25, 193196.5. Donner, A. Statistical methodology for paired

cluster designs. Am. J. Epidemiol. 1987, 126 (5),

972979.6. Rao, J.N.K.; Scott, A.J. A simple method for the

analysis of clustered binary data. Biometrics1992,48, 577585.

7. Lee, E.W.; Dubin, N. Estimation and sample sizeconsiderations for clustered binary responses.Stat. Med.1994, 13, 12411252.

Table 3 Hyperthyroid data

k nk yik yi0k ak bk ck dk

1 3 0 2 0 0 2 1

2 3 2 3 2 0 1 0

3 3 3 3 3 0 0 0

4 1 1 1 1 0 0 0

5 3 2 3 2 0 1 0

6 4 4 4 4 0 0 0

7 3 3 3 3 0 0 0

8 2 2 2 2 0 0 0

9 2 2 1 1 1 0 0

10 1 1 1 1 0 0 0

11 3 2 2 2 0 0 1

12 2 2 2 2 0 0 0

13 3 3 3 3 0 0 0

14 2 2 2 2 0 0 0

15 2 0 2 0 0 2 0

16 3 2 2 2 0 0 117 3 2 2 2 0 0 1

18 3 2 3 2 0 1 0

19 2 2 2 2 0 0 0

20 1 1 1 1 0 0 0

21 2 2 2 2 0 0 0

K 21, N PK

k1nk 51, yik Pnk

j1yijk 40, yi0k Pnkj1yi0jk 46, a

PKk1

Pnkj1ajk 39, b

PKk1

Pnkj1 bjk 1,

c PK

k1

Pnkj1cjk 7, d

PKk1

Pnkj1djk 4.

(From Ref.[8].)

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8. Obuchowski, N.A. On the comparison of corre-lated proportions for clustered data. Stat. Med.1998, 17, 14951507.

9. Durkalski, V.; Palesch, Y.; Lipsitz, S.; Rust, P.The analysis of clustered matched-pair data. Stat.Med.2003, 22, 24172428.

10. Donner, A. The analysis of intraclass correlationin multiple samples. Ann. Hum. Genet. 1985, 49,7582.

11. Zeger, S.L.; Liang, K. Longitudinal data analysisfor discrete and continuous outcomes. Biometrics1986, 42, 121130.

12. Berger, V.W. Pros and cons of permutation testsin clinical trials. Stat. Med. 2000, 19, 13191328.

13. Donner, A.; Donald, A. The statistical analysis ofmultiple binary measurements. J. Clin. Epidemiol.1988, 41(9), 899905.

14. Donner, A.; Banting, D. Adjustment of frequentlyused chi-square procedures for the effect of site-

to-site dependencies in the analysis of dental data.J. Dent. Res. 1989, 68(9), 13501354.15. Donald, A.; Donner, A. Adjustments to the

MantelHaenszel chi-square statistic and oddsratio variance estimator when the data arefclustered. Stat. Med. 1987, 6, 491499.

16. Donner, A.; Eliasziw, M. Application of matchedpair procedures to site-specific data in periodontal

research. J. Clin. Periodontol. 1991, 18,755759.

17. Jung, S.; Ahn, C.; Donner, A. Evaluation of anadjusted chi-square statistic as applied to observa-tional studies involving clustered binary data.Stat. Med. 2001, 20, 21492161.

18. Durkalski, V.; Palesch, Y.; Lipsitz, S.; Rust, P.The analysis of clustered matched-pair data undera non-inferiority study design. Stat. Med. 2003,22, 279290.

19. Nelder, J.A.; Wedderburn, R.W.M. Generalizedlinear models. J. R. Stat. Soc. A 1972, 135,370384.

20. SAS Statistical Software V8 or V9, Cary, NorthCarolina.

21. Ahn, C. Statistical methods for the estimation ofsensitivity and specificity of site-specific diagnos-tic tests. J. Periodont. Res. 1997, 32, 351354.

22. Prentice, R.L. Correlated binary regression withcovariates specific to each binary observation.

Biometrics1988, 44, 10331048.23. Donner, A.; Klar, N. Analysis of binary out-comes. In Cluster Randomization Trials; Arnold,Hodder Headline Group: London, 2000; 8695(Chapter 6).

24. Lee, E.W.; Dubin, N. Estimation and sample sizeconsiderations for clustered binary responses.Stat. Med. 1994, 13, 12411252.


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