Generalized linear mixed models (GLMM) have been used in many areas of research to analyze longitudinal and clustered data with non-normal responses. In addition to the fixed effects parameters found in the generalized linear model (GLM), variance components associated with unobservable random effects are estimated in the GLMM. Moreover, it is well understood that order restricted inference methods that properly incorporate additional information by way of a restricted parameter space are more efficient than procedures that ignore this information. In this thesis, a distance statistic
based on the Wald statistic is suggested for order restricted tests on the random components in the mixed model. The null distributions of the distance and the likelihood ratio test statistics are shown to be asymptotically equivalent and that of a chi-bar-square. An analysis conducted on data extracted from the 2011 National Youth Tobacco Survey will serve as an illustration of the proposed testing procedure.