Semiparametric estimation with profile algorithm for longitudinal binary data

This article considers analyzing longitudinal binary data semiparametrically and proposing GEE-Smoothing spline in the estimation of parametric and nonparametric components. The method is an extension of the parametric generalized estimating equation to semiparametric. The nonparametric component is estimated by smoothing spline approach, i.e., natural cubic spline. We use profile algorithm in the estimation of both parametric and nonparametric components. Properties of the estimators are evaluated by simulation

Semiparametric GEE analysis in partially linear single-index models for longitudinal data

In this article, we study a partially linear single-index model for longitudinal data under a general framework which includes both the sparse and dense longitudinal data cases. A semiparametric estimation method based on a combination of the local linear smoothing and generalized estimation equations (GEE) is introduced to estimate the two parameter vectors as well as the unknown link function. Under some mild conditions, we derive the asymptotic properties of the proposed parametric and nonparametric estimators in different scenarios, from which we find that the convergence rates and asymptotic variances of the proposed estimators for sparse longitudinal data would be substantially different from those for dense longitudinal data. We also discuss the estimation of the covariance (or weight) matrices involved in the semiparametric GEE method. Furthermore, we provide some numerical studies including Monte Carlo simulation and an empirical application to illustrate our methodology and theory.Comment: Published at http://dx.doi.org/10.1214/15-AOS1320 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Gee-Smoothing Spline for Semiparametric Estimation of Longitudinal Categorical Data

In this thesis we propose estimation methods of semiparametric marginal models for longitudinal (correlated) categorical data, where the systematic component of the model consists of parametric and nonparametric forms. We develop GEE-Smoothing spline as a method to analyze semiparametric model for longitudinal data. The proposed methods are an extension of parametric generalized estimating equation (GEE) to semiparametric GEE by introducing smoothing spline into parametric GEE. We derive estimation method of GEE-Smoothing spline in the case of longitudinal binary, ordinal, and nominal data. Derivation of the estimating equation of GEE-Smoothing spline for these three types ofcategorical data is the same. However their estimating equations have different forms of the covariance and correlation matrices. In the estimation of the association (correlation) parameter for binary data, we use moment method of Liang & Zeger’s and method of Prentice’s. For ordinal and nominal data, we use different models of the covariance matrices than of binary data. These models need smaller number of the association parameter to be estimated which is different from the existing models of parametric GEE for ordinal data. We also derive and propose the methods to estimate the association parameter for these two types of data. The properties of the estimate for both parametric and nonparametric components of GEE-Smoothing spline are evaluated using simulation studies. We obtained that the estimates of parametric component for binary and ordinal data are unbiased. Whilst for nominal data, the estimates of parametric componentare almost unbiased. Meanwhile the estimates of the nonparametric component for all types of data are biased, with the bias decreases when the samplesize increases. The estimators of both parametric and nonparametric components are also consistent, and the consistency is not affected by the correct or incorrect working correlation used in model. This consistency property holds for correlated and independent data. The efficiency of the estimates of using independent or correlated working correlation in the estimation depends on the type of covariate, such as time varying, subject specific, or mean-balanced covariates. The estimates of both parametric and nonparametric components also follow the central limit theorem (CLT), for both independent and correlated data, and using correct or incorrect working correlation. Both components estimate have normal distribution

Monte Carlo modified profile likelihood in models for clustered data

The main focus of the analysts who deal with clustered data is usually not on the clustering variables, and hence the group-specific parameters are treated as nuisance. If a fixed effects formulation is preferred and the total number of clusters is large relative to the single-group sizes, classical frequentist techniques relying on the profile likelihood are often misleading. The use of alternative tools, such as modifications to the profile likelihood or integrated likelihoods, for making accurate inference on a parameter of interest can be complicated by the presence of nonstandard modelling and/or sampling assumptions. We show here how to employ Monte Carlo simulation in order to approximate the modified profile likelihood in some of these unconventional frameworks. The proposed solution is widely applicable and is shown to retain the usual properties of the modified profile likelihood. The approach is examined in two instances particularly relevant in applications, i.e. missing-data models and survival models with unspecified censoring distribution. The effectiveness of the proposed solution is validated via simulation studies and two clinical trial applications

A semiparametric extension of the stochastic block model for longitudinal networks

To model recurrent interaction events in continuous time, an extension of the stochastic block model is proposed where every individual belongs to a latent group and interactions between two individuals follow a conditional inhomogeneous Poisson process with intensity driven by the individuals' latent groups. The model is shown to be identifiable and its estimation is based on a semiparametric variational expectation-maximization algorithm. Two versions of the method are developed, using either a nonparametric histogram approach (with an adaptive choice of the partition size) or kernel intensity estimators. The number of latent groups can be selected by an integrated classification likelihood criterion. Finally, we demonstrate the performance of our procedure on synthetic experiments, analyse two datasets to illustrate the utility of our approach and comment on competing methods

A semiparametric regression model for paired longitudinal outcomes with application in childhood blood pressure development

This research examines the simultaneous influences of height and weight on longitudinally measured systolic and diastolic blood pressure in children. Previous studies have shown that both height and weight are positively associated with blood pressure. In children, however, the concurrent increases of height and weight have made it all but impossible to discern the effect of height from that of weight. To better understand these influences, we propose to examine the joint effect of height and weight on blood pressure. Bivariate thin plate spline surfaces are used to accommodate the potentially nonlinear effects as well as the interaction between height and weight. Moreover, we consider a joint model for paired blood pressure measures, that is, systolic and diastolic blood pressure, to account for the underlying correlation between the two measures within the same individual. The bivariate spline surfaces are allowed to vary across different groups of interest. We have developed related model fitting and inference procedures. The proposed method is used to analyze data from a real clinical investigation.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS567 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Semiparametric inference in mixture models with predictive recursion marginal likelihood

Predictive recursion is an accurate and computationally efficient algorithm for nonparametric estimation of mixing densities in mixture models. In semiparametric mixture models, however, the algorithm fails to account for any uncertainty in the additional unknown structural parameter. As an alternative to existing profile likelihood methods, we treat predictive recursion as a filter approximation to fitting a fully Bayes model, whereby an approximate marginal likelihood of the structural parameter emerges and can be used for inference. We call this the predictive recursion marginal likelihood. Convergence properties of predictive recursion under model mis-specification also lead to an attractive construction of this new procedure. We show pointwise convergence of a normalized version of this marginal likelihood function. Simulations compare the performance of this new marginal likelihood approach that of existing profile likelihood methods as well as Dirichlet process mixtures in density estimation. Mixed-effects models and an empirical Bayes multiple testing application in time series analysis are also considered