An orthodox blup approach to generalized linear mixed models

We introduce a new class of generalized linear mixed models assuming Tweedie exponential dispersion model distributions for both the response and the random effects. This class of models accommodates a wide range of discrete, continuous and mixed data. By letting the random effects enter as weights as well as means in the conditional distributions, the variance matrix may be expressed as a sum of variance components. We consider an orthodox BLUP approach to parameter estimation and random effects prediction for this new class of models based on a predictor of the random effects that is truly best linear and unbiased, in contrast to the conventional BLUP which is the conditional mode. We obtain an optimal estimating equation based on the orthodox BLUP, which is solved by a modified Newton algorithm. This approach facilitates analysis of residuals and allows justification of asymptotic results under realistic conditions through standard estimating equation theory. An important feature of this approach is that the principal results depend only on the first and second moment assumptions of unobserved random effects. The common fitting algorithm based on orthodox BLUP enables us to study this new class of models as a single class, rather than as a collection of unrelated different models. This approach is illustrated with the analyses of seed germination data, epilepsy data and cake baking data. By means of asymptotic justifications, simulations and worked examples, we conclude that the orthodox BLUP approach is of practical value for analysis of clustered non-normal data.