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Equivalence of conditional and marginal regression models for clustered and longitudinal data

Departments of Biostatistics and Epidemiology, Harvard School of Public Health, Boston, MA, USA, stjor{at}channing.harvard.edu

Donna Spiegelman

Departments of Biostatistics and Epidemiology, Harvard School of Public Health, Boston, MA, USA

Certain statistical models specify a conditional mean function, given a random effect and covariates of interest. On the other hand, one may instead model a marginal mean only in terms of the covariates. We discuss some common situations where conditional and marginal means coincide. In a Gaussian linear mixed effects model we have equivalent interpretations of the conditional and marginal regression parameter estimates. Similar results exist for more general link functions. In this paper we give a short overview of some models, where conditional and marginal results are equivalent and we illustrate this with some examples. When the conditional mean is additive in a random effect on the log scale, it is seen that the marginal mean equals the conditional mean plus a constant, such that slope parameters have the same interpretation in both formulations. No further distributional assumptions are needed in either of these cases. With a logit link and a double exponential random effect, a closed form marginal link function is derived from the conditional model. When a logit or probit link is used with a normal random effect, the marginal mean parameters become attenuated by a factor which depends on parameters of the distribution of the covariates. In a conditional Weibull proportional hazards model with a positive stable frailty, the marginal hazards are again Weibull but with slope parameters attenuated towards zero.

Statistical Methods in Medical Research, Vol. 13, No. 4, 309-323 (2004)
DOI: 10.1191/0962280204sm368ra


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