Second-stage least squares versus penalized quasi-likelihood for fitting hierarchical models in epidemiologic analyses.

Hierarchical regression analysis holds much promise for epidemiologic analysis, but has as yet seen limited application because of lack of easily used software and the relatively lengthy run times of preferred fitting methods (such as true maximum likelihood and Bayesian approaches). This paper compares three relatively simple choices for estimation of the regression coefficients: maximum-likelihood first stage combined with a weighted-least-squares second stage (MLLS); joint iteratively reweighted least squares fitting of first and second stage (JILS); and empirically penalized quasi-likelihood (EPQL). These choices can be combined with various methods for estimating the second-stage variance; the two contrasted here are based on first and second-stage residuals. JILS and EPQL yielded indistinguishable results, and had small sample performance superior to MLLS. In larger samples there was little practical difference among the methods. Use of first-stage residuals to estimate the prior variance required considerably more computation than use of second-stage residuals, but produced no discernible improvement in regression coefficient estimates. All three methods performed well for estimation of first-stage parameters but were less satisfactory for estimation of second-stage parameters.