Statistical Science

Marginalized multilevel models and likelihood inference (with comments and a rejoinder by the authors)

Patrick J. Heagerty and Scott L. Zeger

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Hierarchical or ‘‘multilevel’’ regression models typically parameterize the mean response conditional on unobserved latent variables or ‘‘random’’ effects and then make simple assumptions regarding their distribution. The interpretation of a regression parameter in such a model is the change in possibly transformed mean response per unit change in a particular predictor having controlled for all conditioning variables including the random effects. An often overlooked limitation of the conditional formulation for nonlinear models is that the interpretation of regression coefficients and their estimates can be highly sensitive to difficult-to-verify assumptions about the distribution of random effects, particularly the dependence of the latent variable distribution on covariates. In this article, we present an alternative parameterization for the multilevel model in which the marginal mean, rather than the conditional mean given random effects, is regressed on covariates. The impact of random effects model violations on the marginal and more traditional conditional parameters is compared through calculation of asymptotic relative biases. A simple two-level example from a study of teratogenicity is presented where the binomial overdispersion depends on the binary treatment assignment and greatly influences likelihood-based estimates of the treatment effect in the conditional model. A second example considers a three-level structure where attitudes toward abortion over time are correlated with person and district level covariates. We observe that regression parameters in conditionally specified models are more sensitive to random effects assumptions than their counterparts in the marginal formulation.

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Statist. Sci., Volume 15, Number 1 (2000), 1-26.

First available in Project Euclid: 24 December 2001

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Generalized linear model latent variable logistic regression random effects model


Heagerty, Patrick J.; Zeger, Scott L. Marginalized multilevel models and likelihood inference (with comments and a rejoinder by the authors). Statist. Sci. 15 (2000), no. 1, 1--26. doi:10.1214/ss/1009212671.

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  • ABRAMOWITZ, K. M. and STEGUN, I. A. 1972. Handbook of Mathematical Functions. Dover, New York. Z.
  • AERTS, M. and CLAESKENS, G. 1997. Local polynomial estimation in multiparameter likelihood models. J. Amer. Statist. Assoc. 92 1536 1545. Z.
  • AZZALINI, A. 1994. Logistic regression for autocorrelated data with application to repeated measures. Biometrika 81 767 775. Z.
  • BISHOP, Y., FEINBERG, S. and HOLLAND, P. 1975. Discrete Multivariate Analysis. MIT Press.Z.
  • BOOTH, J. G. and HOBERT, J. P. 1999. Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm. J. Roy. Statist. Soc. Ser. B 61 265 285.
  • BRESLOW, N. and CLAYTON, D. G. 1993. Approximate inference in generalized linear mixed models. J. Amer. Statist. Assoc. 88 9 25. Z.
  • CAREY, V. C., ZEGER, S. L. and DIGGLE, P. J. 1993. Modelling multivariate binary data with alternating logistic regressions. Biometrika 80 517 526. Z.
  • CONAWAY, M. 1989. Analysis of repeated categorical measurements with conditional likelihood methods. J. Amer. Statist. Assoc. 84 53 62. Z.
  • COX, D. R. 1990. Role of models in statistical analysis. Statist. Sci. 5 169 174. Z.
  • DALE, J. R. 1986. Global cross-ratio models for bivariate discrete ordered responses. Biometrics 42 909 917. Z.
  • DEMING, W. E. and STEPHAN, F. F. 1940. On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. Ann. Math. Statist. 11 427 444. Z.
  • DIGGLE, P. J. 1988. An approach to the analysis of repeated measures. Biometrics 44 959 971. Z.
  • DIGGLE, P. J., LIANG, K.-Y. and ZEGER, S. L. 1994. Analysis of Longitudinal Data. Oxford Univ. Press. Z.
  • EFRON, B. and MORRIS, C. 1973. Stein's estimation rule and its competitors an empirical Bayes approach. J. Amer. Statist. Assoc. 68 117 130. Z.
  • FITZMAURICE, G. M. and LAIRD, N. M. 1993. A likelihood-based method for analysing longitudinal binary responses. Biometrika 80 141 151. Z.
  • GIBBONS, R. D. and HEDEKER, D. 1997. Random effects probit and logistic regression models for three-level data. Biometrics 53 1527 1537. Z.
  • GILKS, W., RICHARDSON, S. and SPIEGELHALTER, D. 1996. Markov Chain Monte Carlo in Practice. Chapman and Hall, London. Z.
  • GLONEK, G. F. V. and MCCULLAGH, P. 1995. Multivariate logistic models. J. Roy. Statist. Soc. Ser. B 57 533 546. Z.
  • GODAMBE, V. P. 1960. An optimum property of regular maximum likelihood estimation. Ann. Math. Statist. 31 1208 1212.Z.
  • GOLDSTEIN, H. 1991. Nonlinear multilevel models with an application to discrete response data. Biometrika 78 45 51. Z.
  • GOLDSTEIN, H. 1995a. Multilevel Statistical Models. Arnold, London. Z.
  • GOURIEROUX, C., MONFORT, A. and TROGNON, A 1984. Pseudomaximum likelihood methods: theory. Econometrica 52 681 700. Z.
  • GRAUBARD, B. I. and KORN, E. L. 1994. Regression analysis with clustered data. Statistics in Medicine 13 509 522. Z.
  • HEAGERTY, P. J. 1999. Marginally specified logistic-normal models for longitudinal binary data. Biometrics 55 688 698. Z.
  • HEAGERTY, P. J. and ZEGER, S. L. 1996. Marginal regression models for clustered ordinal measurements. J. Amer. Statist. Assoc. 91 1024 1036. Z.
  • HEDEKER, D. and GIBBONS, R. 1994. A random-effects ordinal regression model for multilevel analysis. Biometrics 50 933 944. Z.
  • HOLLAND, P. W. 1986. Statistics and causal inference. J. Amer. Statist. Assoc. 81 945 970.Z.
  • LANG, J. B. and AGRESTI, A. 1994. Simultaneously modeling joint and marginal distributions of multivariate categorical responses. J. Amer. Statist. Assoc. 89 625 632. Z.
  • LIANG, K.-Y. and HANFELT, J. 1994. On the use of the quasilikelihood method in teratological experiments. Biometrics 50 872 880. Z.
  • LIANG, K.-Y. and ZEGER, S. L. 1986. Longitudinal data analysis using generalized linear models. Biometrika 73 13 22. Z.
  • LIPSITZ, S., LAIRD, N. and HARRINGTON, D. 1991. Generalized estimating equations for correlated binary data: using odds ratios as a measure of association. Biometrika 78 153 160. Z.
  • MACDONALD, I. L. and ZUCCHINI, W. 1997. Hidden Markov and Other Models for Discrete-Valued Time Series. Chapman and Hall, London. Z.
  • MCCULLAGH, P. and NELDER, J. 1989. Generalized Linear Models. Chapman and Hall, London. Z.
  • MCCULLOCH, C. E. 1997. Maximum likelihood algorithms for generalized linear mixed models. J. Amer. Statist. Assoc. 92, 162 170. Z.
  • MCGRATH, K. and WATERTON, J. 1986. British social attitudes, 1983 1986, panel survey. Technical report, London Social and Community Planning Research. Z.
  • MOLENBERGHS, G. and LESAFFRE, E. 1994. Marginal modeling of correlated ordinal data using a multivariate Plackett distribution. J. Amer. Statist. Assoc. 89 633 644. Z.
  • MONAHAN, J. F. and STEFANSKI, L. A. 1992. Normal scale Z. mixture approximations to F* x and computation of the logistic-normal integral. In Handbook of the Logistic DistriZ. bution N. Balakrishnan, ed. 529 540. Dekker, New York. Z.
  • NEUHAUS, J. M., HAUCK, W. W. and KALBFLEISCH, J. D. 1992. The effects of mixture distribution misspecification when fitting mixed-effects logistic models. Biometrika 79 755 762. Z.
  • NEUHAUS, J. M. and JEWELL, N. 1993. A geometric approach to assess bias due to omitted covariates in generalized linear models. Biometrika 80 807 815. Z.
  • NEUHAUS, J. M. and KALBFLEISCH, J. D. 1998. Betweenand within-cluster covariate effects in the analysis of clustered data. Biometrics 54 638 645. Z.
  • NEUHAUS, J. M., KALBFLEISCH, J. D. and HAUCK, W. W. 1991. A comparison of cluster-specific and population-averaged approaches for analyzing correlated binary data. Internat. Statist. Rev. 59 25 35.
  • PENDERGAST, J. F., GANGE, S. J., NEWTON, M. A., LINDSTROM, M. Z. J., PALTA, M. and FISHER, M. R. 1996. A survey of methods for analyzing clustered binary response data. Internat. Statist. Rev. 64 89 118. Z.
  • PLACKETT, R. L. 1965. A class of bivariate distributions. J. Amer. Statist. Assoc. 60 516 522. Z.
  • ROTNITZKY, A. and WYPIJ, D. 1994. A note on the bias of estimators with missing data. Biometrics 50 1163 1170. Z.
  • SHEN, W. and LOUIS, T. A. 1998. Triple-goal estimates in two-stage hierarchical models. J. Roy. Statist. Soc. Ser. B 60 455 471. Z.
  • STIRATELLI, R., LAIRD, N. and WARE, J. 1984. Random effects models for serial observations with binary responses. Biometrics 40 961 970.Z.
  • WEDDERBURN, R. W. M. 1974. Quasilikelihood functions, generalized linear models and the Gauss Newton method. Biometrika 61 439 447. Z.
  • WEIL, C. S. 1970. Selection of the valid number of sampling units and a consideration of their combination in toxicological studies involving reproduction, teratogenesis or carcinogenesis. Food and Cosmetics Toxicology 8 177 182. Z.
  • WHITE, H. 1982. Maximum likelihood estimation of misspecified models. Econometrica 50 1 25.
  • GALBRAITH, J. I. 1990. Multilevel analysis of attitudes to abortion. The Statistician 40 225 234. Z.
  • WOLFINGER, R. and O'CONNELL, M. 1993. Generalized linear mixed models: a pseudolikelihood approach. J. Statist. Comput. Simulation 48 233 243. Z.
  • ZEGER, S. L. and KARIM, M. R. 1991. Generalized linear models with random effects: a Gibbs sampling approach. J. Amer. Statist. Assoc. 86 79 86. Z.
  • ZEGER, S. L., LIANG, K.-Y. and ALBERT, P. A. 1988. Models for longitudinal data: a generalized estimating equation approach. Biometrics 44 1049 1060.
  • M, that is, the marginal linear regression model.
  • KEYSER, P. 1998. Twelve weeks of continuous onychomycosis caused by dermatophytes: a double blind comparative trial of terbafine 250 mg day versus itraconazole 200 mg day. J. Amer. Acad. Dermatology 38 5, 3, S57 S63. Z.
  • HEDEKER, D., and GIBBONS, R. D. 1996. MIXOR: a computer program for mixed-effects ordinal regression analysis. Computer Methods and Programs in Biomedicine 49 157 176. Z.
  • LESAFFRE, E., VERBEKE, G. and KENWARD, M. 2000. On the two-stage interpretation of the linear mixed model. UnpubZ. lished manuscript in preparation. Z.
  • LINDSAY, B. 1995. Mixture Models: Theory, Geometry and Applications IMS, Hayward, CA. Z.
  • NEUHAUS, J. M. and SEGAL, M. R. 1997. An assessment of approximate maximum likelihood estimators in generalized linear mixed models. In Modelling Longitudinal and SpaZ tially Correlated Data T. Gregoire, D. R. Brillinger, P. J. Diggle, E. Russek-Cohen, W. Warren and R. D. Wolfinger,. eds. 11 22. Springer, New York. Z.
  • BOCK, R. D. 1989. Multilevel Analysis of Educational Data. Academic Press, New York.Z.
  • BRYK, A. and RAUDENBUSH, S. 1992. Hierarchical Linear Models for Social and Behavioral Research: Applications and Data Analysis Methods. Sage, Newbury Park, CA. Z.
  • FULLER, W. A. 1987. Measurement Error Models. Wiley, New York. Z.
  • GOLDSTEIN, H. 1995b. Multilevel Statistical Models, 2nd ed. Wiley, New York. Z.
  • LAIRD, N. M. and WARE, J. H. 1982. Random-effects models for longitudinal data. Biometrika 65 581 590. Z.
  • LILLARD, L. A. and FARMER, M. M. 1998. Functional limitations, disability and perceived health of the oldest old: an examination of health status in AHEAD. Paper presented to the Survey Research Center, Univ. Michigan. Z.
  • LINDLEY, D. and SMITH, A. 1972. Bayes estimates for the linear model. J. Roy. Statist. Soc. Ser. B 34 1 41. Z.
  • LITTLE, R. and RUBIN, D. 1987. Statistical Analysis with Missing Data. Wiley, New York. Z.
  • LONGFORD, N. 1993. Random Coefficient Models. Clarendon, Oxford.Z.
  • MORRIS, C. 1983. Parametric empirical Bayes inference: theory and applications. J. Amer. Statist. Assoc. 78 47 65. Z.
  • MORRIS, C. and NORMAND, S. 1992. Hierarchical models for combining information and for meta-analysis. In Bayesian Z Statistics 4 J. O. Berger, J. M. Bernardo, A. P. Dawid, D. V.. Lindley and A. F. M. Smith, eds. 321 344. Oxford Univ. Press. Z.
  • RAUDENBUSH, S. 1988. Educational applications of hierarchical Z. linear models: a review. J. Educational Statist. 13 20 85 116. Z.
  • RAUDENBUSH, S. and BRYK, A. 1985. Empirical Bayes metaanalysis. J. Educational Statist. 10 75 98. Z.
  • RAUDENBUSH, S. W. and SAMPSON, R. 1999a. Ecometrics: toward a science of assessing ecological settings, with application to the systematic social observation of neighborhoods. Sociological Methodology 29 1 41. Z.
  • RAUDENBUSH, S. W. and SAMPSON, R. 1999b. Assessing direct and indirect associations in multilevel designs with latent variables. Sociological Methods and Research 28 123 153. Z.
  • SELTZER, M. 1993. Sensitivity analysis for fixed effects in the hierarchical model: a Gibbs sampling approach. J. Educ. Statist. 18 207 235. Z.
  • TSUTAKAWA, R. 1988. Mixed model for studying geographic variability in mortality rates. J. Amer. Statist. Assoc. 83 37 42. Z.
  • PINHEIRO, J. C. and BATES, D. M. 1995. Approximations to the log-likelihood function in the non-linear mixed-effects model. J. Comput. Graph. Statist. 4 12 35. Z.
  • GREENLAND, S., ROBINS, J. M. and PEARL, J. 1999. Confounding and collapsibility in causal inference. Statist. Sci. 14 29 46.