Summary: Most statistical solutions to the problem of statistical inference with missing data involve integration or expectation. This can be done in many ways: directly or indirectly, analytically or numerically, deterministically or stochastically. Missing-data problems can be formulated in terms of latent random variables, so that the hierarchical likelihood methods of Y. Lee and J. A. Nelder [J. R. Stat. Soc., Ser. B 58, No. 4, 619–678 (1996; Zbl 0880.62076)] can be applied to missing-value problems to provide one solution to the problem of integration of the likelihood. The resulting methods effectively use a Laplace approximation to the marginal likelihood with an additional adjustment to the measures of precision to accommodate the estimation of the fixed effects parameters.
We first consider missing at random cases where problems are simpler to handle because the integration does not need to involve the missing-value mechanism and then consider missing not at random cases. We also study tobit regression and refit the missing not at random selection model to the antidepressant trial data analyzed by P. Diggle amd M. G. Kenward [ibid., Ser. C 43, No. 1, 49–93 (1994; Zbl 0825.62010)].