Improved estimation procedures for multilevel models with binary response:a case-study

During recent years, analysts have been relying on approximate methods of i nference to estimate multilevel models for binary or count data. In an earl ier study of random-intercept models for binary outcomes we used simulated data to demonstrate that one such approximation, known as marginal quasi-li kelihood, leads to a substantial attenuation bias in the estimates of both fixed and random effects whenever the random effects are non-trivial. In th is paper, we fit three-level random-intercept models to actual data for two binary outcomes, to assess whether refined approximation procedures, namel y penalized quasi-likelihood and second-order improvements to marginal and penalized quasi-likelihood, also underestimate the underlying parameters. T he extent of the bias is assessed by two standards of comparison: exact max imum likelihood estimates, based on a Gauss-Hermite numerical quadrature pr ocedure, and a set of Bayesian estimates, obtained from Gibbs sampling with diffuse priors. We also examine the effectiveness of a parametric bootstra p procedure for reducing the bias. The results indicate that second-order p enalized quasi-likelihood estimates provide a considerable improvement over the other approximations, but all the methods of approximate inference res ult in a substantial underestimation of the fixed and random effects when t he random effects are sizable. We also find that the parametric bootstrap m ethod can eliminate the bias but is computationally very intensive.