Regression Models with Correlated Binary Response Variables: A Comparison of Different Methods in Finite Samples

The present paper deals with the comparison of the performance of
different estimation methods for regression models with correlated
binary responses. Throughout, we consider probit models where an
underlying latent continous random variable crosses a threshold.
The error variables in the unobservable latent model are assumed to
be normally distributed. The estimation procedures considered are
(1) marginal maximum likelihood estimation using Gauss-Hermite
quadrature, (2) generalized estimation equations (GEE) techniques
with an extension to estimate tetrachoric correlations in a second
step, and, (3) the MECOSA approach proposed by Schepers, Arminger
and Küsters (1991) using hierarchical mean and covariance structure
models. We present the results of a simulation study designed to
evaluate the small sample properties of the different estimators
and to make some comparisons with respect to technical aspects of
the estimation procedures and to bias and mean squared error of the
estimators. The results show that the calculation of the ML
estimator requires the most computing time, followed by the MECOSA
estimator. For small and moderate sample sizes the calculation of
the MECOSA estimator is problematic because of problems of
convergence as well as a tendency of underestimating the variances.
In large samples with moderate or high correlations of the errors
in the latent model, the MECOSA estimators are not as efficient as
ML or GEE estimators. The higher the `true' value of an
equicorrelation structure in the latent model and the larger the
sample sizes are, the more is the efficiency gain of the ML
estimator compared to the GEE and MECOSA estimators. Using the GEE
approach, the ML estimates of tetrachoric correlations calculated
in a second step are biased to a smaller extent than using the
MECOSA approach.