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This paper proposes a novel generalized partial linear mixed models (GPLMMs) with the random effects distribution distributed as a multivariate skew-normal distribution for discrete longitudinal data over time. Under the Bayesian framework, we extended the Bayesian version of adaptive lasso (least absolute shrinkage and selection operator) for linear regression to the Bayesian adaptive lasso with the least squares approximation (LSA) method for generalized partial linear mixed models. A hybrid algorithm combining the Gibbs sampler and the Metropolis-Hastings algorithm in conjunction with the Bayesian adaptive lasso with LSA is used to simultaneously obtain the Bayesian estimates of unknown parameters, smoothing function and random effects, as well as select important covariates in GPLMMs. Several simulation studies and a real example are presented to illustrate the proposed methodologies. [ABSTRACT FROM AUTHOR]

Semiparametric reproductive dispersion mixed model (SPRDMM) is a natural extension of the reproductive dispersion model and the semiparametric mixed model. In this paper, we relax the normality assumption of random effects in SPRDMM and use a truncated and centred Dirichlet process prior to specify random effects, and present the Bayesian P-spline to approximate the smoothing unknown function. A hybrid algorithm combining the block Gibbs sampler and the Metropolis–Hastings algorithm is implemented to sample observations from the posterior distribution. Also, we develop Bayesian case deletion influence measure for SPRDMM based on theφ-divergence and present those computationally feasible formulas. Several simulation studies and a real example are presented to illustrate the proposed methodologies. [ABSTRACT FROM AUTHOR]