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

Abstract

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.