Generalized additive mixed models are proposed for overdispersed and correlated data, which arise frequently in studies involving clustered, hierarchical and spatial designs. This class of models allows¯exible functional dependence of an outcome variable on covariates by using nonparametric regression, while accounting for correlation between observations by using random effects. We estimate nonparametric functions by using smoothing splines and jointly estimate smoothing parameters and variance components by using marginal quasi-likelihood. Because numerical integration is often required by maximizing the objective functions, double penalized quasilikelihood is proposed to make approximate inference. Frequentist and Bayesian inferences are compared. A key feature of the method proposed is that it allows us to make systematic inference on all model components within a uni®ed parametric mixed model framework and can be easily implemented by ®tting a working generalized linear mixed model by using existing statistical software. A bias correction procedure is also proposed to improve the performance of double penalized quasi-likelihood for sparse data. We illustrate the method with an application to infectious disease data and we evaluate its performance through simulation.

Normality of random effects is a routine assumption for the linear mixed model, but it may be unrealistic, obscuring important features of among-individual variation. We relax this assumption by approximating the random effects density by the seminonparameteric (SNP) representation of Gallant and Nychka (1987, Econometrics 55, 363-390), which includes normality as a special case and provides flexibility in capturing a broad range of nonnormal behavior, controlled by a user-chosen tuning parameter. An advantage is that the marginal likelihood may be expressed in closed form, so inference may be carried out using standard optimization techniques. We demonstrate that standard information criteria may be used to choose the tuning parameter and detect departures from normality, and we illustrate the approach via simulation and using longitudinal data from the Framingham study.

We consider inference for a semiparametric stochastic mixed model for longitudinal data. This model uses parametric fixed effects to represent the covariate effects and an arbitrary smooth function to model the time effect and accounts for the within-subject correlation using random effects and a stationary or nonstationary stochastic process. We derive maximum penalized likelihood estimators of the regression coefficients and the nonparametric function. The resulting estimator of the nonparametric function is a smoothing spline. We propose and compare frequentist inference and Bayesian inference on these model components. We use restricted maximum likelihood to estimate the smoothing parameter and the variance components simultaneously. We show that estimation of all model components of interest can proceed by fitting a modified linear mixed model. We illustrate the proposed method by analyzing a hormone dataset and evaluate its performance through simulations.

AMH, an endocrine marker that reflects the transition of resting primordial follicles to growing follicles, declined to a time point 5 yr prior to the FMP; this may represent a critical biological juncture in the menopause transition. Low and nondetectable levels inhibin B levels also were observed 4-5 yr prior to the FMP but were less predictive of time to FMP or age at FMP.

The effect of lactation on weight retention was investigated longitudinally, with data collected at 0.5, 2, 4, 6, 12, and 18 mo after parturition in 110 women aged 20-40 y who had been nulliparous or primiparous. At each evaluation women were categorized as fully breast-feeding, partly breast-feeding, or bottle-feeding including infants weaned to a bottle (bottle feeding/weaned). Postpartum weight retention was calculated by subtracting weight before pregnancy from weight at each evaluation. Lactation practices were found to be significantly associated (P < 0.05) with postpartum weight retention by longitudinal regression analysis. Women who bottle-fed their infants retained more weight over time than women who breast-fed their infants. Significantly slower rates of weight loss were observed when women ceased breast-feeding or switched from fully to partly breast-feeding. Weight retention over time was greater in women who were older, unmarried, or had greater weight gain during pregnancy (P < 0.05). A pattern of weight gain rather than weight loss was observed in unmarried women. Our findings suggest that lactation influences the pattern of postpartum weight retention; however, the effect of lactation on weight retention was sufficiently limited to warrant minimal emphasis on lactation as a means of minimizing postpartum weight retention.