The link function, that relates the linear predictor to the expected value, just as in the generalized linear models, is usually assumed to be known and fixed. In some cases it might be useful to improve the modeling flexibility, allowing the link to be a member of a class indexed by one or more unknown parameters, that can be estimated. Some families of link functions have been introduced, which for certain choices of the parameters reduce to some of the well known link functions, as logit, probit and complementary log-log. While the performance of these families has been investigated for the generalized linear models, not much work has been done for hierarchical generalized linear models. Here we consider, in particular, the global performance of the model within a family of link functions, possible changes in the set of statistically significant parameters for fixed and random effects, as well as problems related to comparison of clusters. All considerations will be based on a re-analysis of a data-set related to the study of burnout syndrome among teachers at various levels of instruction.