Optimal experimental designs for multilevel logistic models

When designing experiments in multilevel populations the following questions arise: what is the optimal lever of randomization, and what is the optimal allocation of units? In this paper these questions wilt be dealt with for populations with two levels of nesting and binary outcomes. The multilevel logistic model, which is used to describe the relationship between treatment condition and outcome, is linearized. The variance of the regression coefficient associated with treatment condition in the linearized model is used to find the optimal level of randomization and the optimal allocation of units. An analytical expression for this Variance can only be obtained for the first-order marginal quasi-likelihood linearization method, which is known to be biased. A simulation study shows that penalized quasi-likelihood linearization and numerical integration of the likelihood lead to conclusions about the optimal design that are similar to those from the analytical derivations for first-order marginal quasi-likelihood.