Using Hierarchical Likelihood for Missing Data Problems

Most statistical solutions to the problem of statistical inferencewith missing data involve integration or expectation. This canbe done in many ways: directly or indirectly, analytically ornumerically, deterministically or stochastically. Missing-dataproblems can be formulated in terms of latent random variables,so that hierarchical likelihood methods of Lee & Nelder(1996) can be applied to missing-value problems to provide onesolution to the problem of integration of the likelihood. Theresulting methods effectively use a Laplace approximation tothe marginal likelihood with an additional adjustment to themeasures of precision to accommodate the estimation of the fixedeffects parameters. We first consider missing at random caseswhere problems are simpler to handle because the integrationdoes not need to involve the missing-value mechanism and thenconsider missing not at random cases. We also study tobit regressionand refit the missing not at random selection model to the antidepressanttrial data analyzed in Diggle & Kenward (1994).