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REGRESSION

MODELS FOR DISCRETE

Although all the estimatorsperformwell in terms of bias when the data are MCAR, in practice,when there are missing responses, the distinctionbetween responses MCAR and MAR can usually not be made with much certainty.With incompleteresponses,the GEE approach performsremarkablywell when the responses are MAR and the design does not include covariates. For both the GEE and likelitime-varying hood-basedapproaches,thereis a substantialreduction in the bias of the timeeffectswhen a close approximation to cov(Yi) is obtained. Thus, when interestis focussed primarilyon the time effectsand there are missingresponses,the "workingindependence"estimators cannot be recommended.For group effects,this distinctionis not quite so clear,and it seems to depend or time-varying on whethergroupis a time-stationary covariate. Finally,althoughin many instances the asymptoticbiases of the GEE and likelihood-basedapproaches are comparable, there may be substantial in termsof efficiency. differences In conclusion,the importanceof accuratelymodelling the correlationamong the repeatedresponsesin a longitudinalstudywill depend on a numberof factors: the design ofthe study,the parametersofinterestand whetheror not there are missing data. Fortunately, for many practical situations,it appears that nearly and unbiased estimates of the regressionpaefficient rametersforthe marginalexpectationcan be obtained even whenthe true association betweenthe responses is only crudelyapproximated. ACKNOWLEDGMENT This research was supported by Grants GM29745 and MH17119 fromthe National Institutesof Health. REFERENCES designwitht2experimental L. N. (1968).A two-period units.Biometrics24 61-73. ofthejointdistribution BAHADUR, R. R. (1961).A representation of responsesto n dichotomousitems. In Studies in Item Analysisand Prediction.StanfordMathematicalStudies in theSocial Sciences VI (H. Solomon,ed.) 158-168. Stanford Univ.Press. and Exponential BARNDORFF-NIELSEN, 0. E. (1978).Information Familiesin StatisticalTheory.Wiley,Chichester. BISHOP, Y. M. M., FIENBERG, S. E. and HOLLAND, P. W. (1975). DiscreteMultivariateAnalysis:Theoryand Practice.MIT Press. in estimation CHAMBERLAIN, G. (1987). Asymptoticefficiency 34 305J.Econometrics momentrestriction. withconditional 324. binarydata. J. Cox, D. R. (1972).The analysisof multivariate Roy. Statist.Soc. Ser. C 21 113-120. and Cox, D. R. and REID, N. (1987). Parameterorthogonality (withdiscussion).J. Roy. conditionalinference approximate Statist.Soc. Ser. C 49 1-39. Cox, D. R. and REID, N. (1989). On the stabilityof maximumlikelihoodestimatorsof orthogonalparameters.Canad. J. Statist.17 229-233. BAALAM,

LONGITUDINAL

RESPONSES

299

W. E. and STEPHAN, F. F. (1940). On a least squares tablewhentheexpected adjustmentofa sampledfrequency marginaltotalsare known.Ann.Math. Statist.11 427-444. DEMPSTER, A. P., LAIRD, N. M. and RUBIN, D. B. (1977).Maxidata via theEM fromincomplete estimation mumlikelihood algorithm(withdiscussion).J. Roy. Statist.Soc. Ser. B 39 1-38. FITZMAURICE, G. M. and LAIRD, N. M. (1993).A likelihood-based binaryresponses.Biomemethodforanalysinglongitudinal trika80 141-151. HUBER, P. J. (1967). The behaviorof maximumlikelihoodestimates undernonstandardconditions.Proc. FifthBerkeley Symp.Math. Statist.Probab. 1 221-233. Univ. California Press,Berkeley. JONES, B. and KENWARD, M. (1989). Design and Analysis of Cross-OverTrials.Chapmanand Hall, London. KALBFLEISCH, J. D. and PRENTICE, R. L. (1980). The Statistical AnalysisofFailureTimeData. Wiley,New York. KORN, E. L. and WHITTEMORE, A. S. (1979).Methodsforanalyzing panel studies of acute healtheffectsof air pollution. Biometrics35 795-802. studies.StatisLAIRD, N. M. (1988).Missing data in longitudinal ticsin Medicine7 305-315. methods for LAIRD, N. M. (1991). Topics in likelihood-based data analysis.Statist.Sinica 1 33-50. longitudinal data analysis LIANG, K. Y. and ZEGER, S. L. (1986).Longitudinal usinggeneralizedlinearmodels.Biometrika73 13-22. LIANG, K. Y., ZEGER, S. L. and QAQISH, B. (1992).Multivariate regressionanalysesforcategoricaldata (withdiscussion).J. Roy. Statist.Soc. Ser. B 54 3-40. LIPsITZ, S. R., LAIRD, N. M. and HARRINGTON, D. P. (1991). binarydata: equationsforcorrelated estimating Generalized Usingtheoddsratioas a measureofassociation.Biometrika 78 153-160. LITTLE, R. J. A. and RUBIN, D. B. (1987). StatisticalAnalysis withMissingData. Wiley,New York. MCCULLAGH, P. and NELDER, J. A. (1989). GeneralizedLinear Models,2nd ed. Chapmanand Hall, New York. bounds.Journal efficiency NEWEY, W. K. (1990).Semiparametric ofAppliedEconometrics5 99-135. DEMING,

PRENTICE, R. L. (1988). Correlated binary regressionwith covari-

ates specificto each binary observation.Biometrics 44 1033-

1048.

R. L., and ZHAO, L. P. (1991).Estimatingequations disforparametersin meanand covariancesofmultivariate creteand continuousresponses.Biometrics47 825-839. ROTNITZKY, A. and WYPIJ,D. (1992). A note on the bias of PRENTICE,

estimators with missing data. Amer. Statist. To appear.

ROYALL,

intervalsusing R. M. (1986).Model robustconfidence

maximum likelihood estimators. Internat. Statist. Rev. 54

221-226. J.H. (1985).Linearmodelsfortheanalysisoflongitudinal

WARE,

studies. Amer. Statist. 39 95-101.

WHITE,

ofmisspecified estimation H. (1982).Maximumlikelihood

ZEGER,

data analysis S. L. and LIANG, K. Y. (1986).Longitudinal

models. Econometrica 50 1-26. ZEGER, S. L. (1988). The analysis of discrete longitudinal data: commentary.Statistics in Medicine 7 161-168.

for discrete and continuous outcomes. Biometrics 42 121-

130. binaryregresL. P. and PRENTICE, R. L. (1990).Correlated

ZHAO,

sion using a quadratic exponential model. Biometrika 77

642-648. ZHAO, L. P., PRENTICE, R. L. and SELF, S. G. (1992).Multivariate mean parameter estimation by using a partly exponential

model.J. Roy. Statist.Soc. Ser. B 54 805-811.