The proposed research concerns theoretical and computational issues arising in the context of generalized linear and mixed effects models. A new multivariate model is suggested as a construct for analysis of covariance which provides a unified framework for adjusting treatment means in balanced and unbalanced design settings. Second, a new approximation for the number of contingency tables and tables of zeros and ones, meeting certain linear constraints, is proposed. This is relevant, for example, in determining the feasibility of exact conditional analysis of log-linear models. The approximation arises from a novel formulation of the problem in terms of a generalized linear model for geometric responses. The approximation is much more generally applicable, and appears to be far more accurate, than exiting competitors. Third, a practical fitting algorithm for a broad class of mixed effects models with analytically intractable likelihood functions is proposed. The approach involves an implementation of the Monte Carlo EM algorithm that uses a randomized spherical-radial integration rule at the E-step. Use of this integration rule reduces the required Monte Carlo sample size by two orders of magnitude in test cases.
Statistical models are ubiquitous in almost all areas of modern research, including such diverse fields as agriculture, economics, medicine, and sociology. Advances in computing power enable statisticians to consider models and do calculations that were not feasible even a few years ago. This research targets three problems related to widely-used statistical models. The first concerns a technique for adjusting treatment means in designed experiments to account for observed covariates related to the response of interest. This is a classical problem with its roots in agricultural field trials. The technique has a long history dating back to the mid-20th century. It is somewhat surprising then that there is still disagreement on the correct way to make the adjustments, even in simple balanced experiments. An explanation is that the mathematical tools and computing power necessary for a complete solution were not available when the method was first developed. The second problem relates to the feasibility of exact statistical tests when data is sparse, and the standard approximations break down. Exact methods are used, for example, in medical studies testing for factors associated with various diseases. Finally, a new fitting algorithm is proposed for an important class of statistical models. Test cases suggest that the methods will significantly extend the range of models for which the computations are practically feasible.
