  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.
