Approximation Methods for Mixed Models with Probit Link Functions

TL;DR

This paper compares approximation methods for mixed models with probit links, analyzing when to use Gaussian weighted integrals versus Gaussian cumulative density functions, based on simulations and data examples.

Contribution

It provides guidance on choosing between approximation methods for complex probit mixed models, a topic not thoroughly investigated before.

Findings

Neither approximation form is universally preferable.

Guidelines are provided for selecting the appropriate approximation method.

The study evaluates multiple methods for complex mixed effects models.

Abstract

We study approximation methods for a large class of mixed models with a probit link function that includes mixed versions of the binomial model, the multinomial model, and generalized survival models. The class of models is special because the marginal likelihood can be expressed as Gaussian weighted integrals or as multivariate Gaussian cumulative density functions. The latter approach is unique to the probit link function models and has been proposed for parameter estimation in complex, mixed effects models. However, it has not been investigated in which scenarios either form is preferable. Our simulations and data example show that neither form is preferable in general and give guidance on when to approximate the cumulative density functions and when to approximate the Gaussian weighted integrals and, in the case of the latter, which general purpose method to use among a large list of methods.