Ordinal regression models are suitable methods for the analysis of ordinal data which are categorized based on a latent unobserved continuous variable. Due to the contribution of clusters or subjects in variety of applications in such models, the usage of random parameters has been taken into consideration and as a consequence the mixed effects ordinal regression models are constructed. Indeed, they can be defined in the frame of generalized linear mixed models with the logit and the probit link functions connecting the response variable expectation to the linear predictor.
In the current project we aim to achieve D-optimal design for the contemplated model. For this reason, it is required to maximize the D-optimality criterion with respect to the design measure. This criterion is a function of the Fisher information matrix which can be obtained in an explicit form in the ordinal model with fixed parameters while this does not occur when random parameters are added. The major reason arises from unclosed integral(s) over the normally distributed random effects in the marginal likelihood function on which the Fisher information matrix depends. In this situation, one suggestion is to build up the quasi Fisher information matrix instead, which relies merely on the first and the second moment of the model equations. However, for such models the closed form expression is not still found due to unclosed integrals with respect to random effects. As a consequence, the analytical approximations of the integrals are proposed which can be reliable in specific design regions. Finally, the D-optimal designs found by the new approach are compared with D-optimal designs obtained through numerical computations in special cases of the mixed effects ordinal regression model. Moreover, these designs are compared with D-optimal designs when the random effects are assumed to be zero in order to detect the impact of the random effects on the ultimate designs.