Efficient Estimation in Marginal Partially Linear Models for Longitudinal/Clustered Data Using Splines

Abstract.  We consider marginal semiparametric partially linear models for longitudinal/clustered data and propose an estimation procedure based on a spline approximation of the non-parametric part of the model and an extension of the parametric marginal generalized estimating equations (GEE). Our estimates of both parametric part and non-parametric part of the model have properties parallel to those of parametric GEE, that is, the estimates are efficient if the covariance structure is correctly specified and they are still consistent and asymptotically normal even if the covariance structure is misspecified. By showing that our estimate achieves the semiparametric information bound, we actually establish the efficiency of estimating the parametric part of the model in a stronger sense than what is typically considered for GEE. The semiparametric efficiency of our estimate is obtained by assuming only conditional moment restrictions instead of the strict multivariate Gaussian error assumption.     

Partial Linear Models for Longitudinal Data Based on Quadratic Inference Functions

In this paper, we consider improved estimating equations for semiparametric partial linear models (PLM) for longitudinal data, or clustered data in general. We approximate the non‐parametric function in the PLM by a regression spline, and utilize quadratic inference functions (QIF) in the estimating equations to achieve a more efficient estimation of the parametric part in the model, even when the correlation structure is misspecified. Moreover, we construct a test which is an analogue to the likelihood ratio inference function for inferring the parametric component in the model. The proposed methods perform well in simulation studies and real data analysis conducted in this paper.     

Marginal Regression of Multivariate Event Times Based on Linear Transformation Models

Multivariate event time data are common in medical studies and have received much attention recently. In such data, each study subject may potentially experience several types of events or recurrences of the same type of event, or event times may be clustered. Marginal distributions are specified for the multivariate event times in multiple events and clustered events data, and for the gap times in recurrent events data, using the semiparametric linear transformation models while leaving the dependence structures for related events unspecified. We propose several estimating equations for simultaneous estimation of the regression parameters and the transformation function. It is shown that the resulting regression estimators are asymptotically normal, with variance–covariance matrix that has a closed form and can be consistently estimated by the usual plug-in method. Simulation studies show that the proposed approach is appropriate for practical use. An application to the well-known bladder cancer tumor recurrences data is also given to illustrate the methodology.     

Multivariate Latent Growth Models for Mixed Data with Covariate Effects

The paper presents an extension of a new class of multivariate latent growth models (Bianconcini and Cagnone, 2012) to allow for covariate effects on manifest, latent variables and random effects. The new class of models combines: (i) multivariate latent curves that describe the temporal behavior of the responses, and (ii) a factor model that specifies the relationship between manifest and latent variables. Based on the Generalized Linear and Latent Variable Model framework (Bartholomew and Knott, 1999), the response variables are assumed to follow different distributions of the exponential family, with item-specific linear predictors depending on both latent variables and measurement errors. A full maximum likelihood method is used to estimate all the model parameters simultaneously. Data coming from the Data WareHouse of the University of Bologna are used to illustrate the methodology.     

Rank Regression Analysis of Multivariate Failure Time Data Based on Marginal Linear Models

Abstract.  Multivariate failure time data arises when each study subject can potentially ex-perience several types of failures or recurrences of a certain phenomenon, or when failure times are sampled in clusters. We formulate the marginal distributions of such multivariate data with semiparametric accelerated failure time models (i.e. linear regression models for log-transformed failure times with arbitrary error distributions) while leaving the dependence structures for related failure times completely unspecified. We develop rank-based monotone estimating functions for the regression parameters of these marginal models based on right-censored observations. The estimating equations can be easily solved via linear programming. The resultant estimators are consistent and asymptotically normal. The limiting covariance matrices can be readily estimated by a novel resampling approach, which does not involve non-parametric density estimation or evaluation of numerical derivatives. The proposed estimators represent consistent roots to the potentially non-monotone estimating equations based on weighted log-rank statistics. Simulation studies show that the new inference procedures perform well in small samples. Illustrations with real medical data are provided.     

Longitudinal Data Estimating Equations for Dispersion Models

Liang & Zeger's generalized estimating equation approach for analysis of longitudinal data is extended to marginal distributions of dispersion model type. This includes for example the von Mises and simplex distributions, suitable for angles and proportions, respectively. Both modelling of position and joint modelling of position and dispersion is considered, and the method is applied to a set of bird orientation data.     

Simple Fitting Algorithms for Incomplete Categorical Data

A popular approach to estimation based on incomplete data is the EM algorithm. For categorical data, this paper presents a simple expression of the observed data log-likelihood and its derivatives in terms of the complete data for a broad class of models and missing data patterns. We show that using the observed data likelihood directly is easy and has some advantages. One can gain considerable computational speed over the EM algorithm and a straightforward variance estimator is obtained for the parameter estimates. The general formulation treats a wide range of missing data problems in a uniform way. Two examples are worked out in full.     

Smooth-Threshold GEE Variable Selection in High-Dimensional Partially Linear Models with Longitudinal Data

We consider the problem of variable selection in high-dimensional partially linear models with longitudinal data. A variable selection procedure is proposed based on the smooth-threshold generalized estimating equation (SGEE). The proposed procedure automatically eliminates inactive predictors by setting the corresponding parameters to be zero, and simultaneously estimates the nonzero regression coefficients by solving the SGEE. We establish the asymptotic properties in a high-dimensional framework where the number of covariates p_n increases as the number of clusters n increases. Extensive Monte Carlo simulation studies are conducted to examine the finite sample performance of the proposed variable selection procedure.     

Binary Dynamic Logit for Correlated Ordinal: estimation,application and simulation

We evaluate the estimation performance of the Binary Dynamic Logit model for correlated ordinal variables (BDLCO model), and compare it to GEE and Ordinal Logistic Regression performance in terms of bias and Mean Absolute Percentage Error (MAPE) via Monte Carlo simulation. Our results indicate that when the proportional-odds assumption does not hold, the proposed BDLCO method is superior to existing models in estimating correlated ordinal data. Moreover, this method is flexible in terms of modeling dependence and allows unequal slopes for each category, and can be used to estimate an apple bloom data set where the proportional-odds assumption is violated. We also provide a function in R to implement BDLCO.     

GEEs for repeated categorical responses based on generalized residuals

Clustered or correlated samples of categorical response data arise frequently in many fields of application. The method of generalized estimating equations (GEEs) introduced in Liang and Zeger [Longitudinal data analysis using generalized linear models, Biometrika 73 (1986), pp. 13–22] is often used to analyse this type of data. GEEs give consistent estimates of the regression parameters and their variance based upon the Pearson residuals. Park et al. [Alternative GEE estimation procedures for discrete longitudinal data, Comput. Stat. Data Anal. 28 (1998), pp. 243–256] considered a modification of the GEE approach using the Anscombe residual and the deviance residual. In this work, we propose to extend this idea to a family of generalized residuals. A wide simulation study is conducted for binary and Poisson correlated outcomes and also two numerical illustrations are presented.     

On the application of extended quasi‐likelihood to the clustered data case

The author describes the relationship between the extended generalized estimating equations (EGEEs) of Hall & Severini (1998) and various similar methods. He proposes a true extended quasi‐likelihood approach for the clustered data case and explores restricted maximum likelihood‐like versions of the EGEE and extended quasi‐likelihood estimating equations. He also presents simulation results comparing the various estimators in terms of mean squared error of estimation based on three moderate sample size, discrete data situations.     

Weighted Estimator for the Linear Transformation Models with Multivariate Failure Time Data

In this article, a simple and efficient weighted method is proposed to improve the estimation efficiency for the linear transformation models with multivariate failure time data. Asymptotic properties of the estimators with a closed-form variance-covariance matrix are established. In addition, a goodness-of-fit test is developed to evaluate the adequacy of the model. The performance of proposed method and the comparison on the efficiency between the proposed method and the working independence method (Lu, 2005) are conducted in finite-sample situation by simulation studies. Finally a real data set from the Busselton Population Health Surveys is illustrated to validate the proposed methodology. The related proofs of the theorems are given in the Appendix.     

A new approach to modeling positive random variables with repeated measures

In many situations, it is common to have more than one observation per experimental unit, thus generating the experiments with repeated measures. In the modeling of such experiments, it is necessary to consider and model the intra-unit dependency structure. In the literature, there are several proposals to model positive continuous data with repeated measures. In this paper, we propose one more with the generalization of the beta prime regression model. We consider the possibility of dependence between observations of the same unit. Residuals and diagnostic tools also are discussed. To evaluate the finite-sample performance of the estimators, using different correlation matrices and distributions, we conducted a Monte Carlo simulation study. The methodology proposed is illustrated with an analysis of a real data set. Finally, we create an R package for easy access to publicly available the methodology described in this paper.     

An ECM Estimation Approach for Analyzing Multivariate Skew-Normal Data with Dropout

In this article, an ECM algorithm is developed to obtain the maximum likelihood estimates of parameters where multivariate skew-normal distribution is used for analyzing longitudinal skewed normal regression data with dropout. A simulation study is performed to investigate the performance of the presented algorithm. Also, the methodology is illustrated through two applications and the results of proposed methodology are compared with ECM under multivariate normal assumption using AIC and BIC criteria. Standard errors of parameter estimates are obtained by asymptotic observed information matrix.     

Dependence Estimation for Marginal Models of Multivariate Survival Data

We have previously(Segal and Neuhaus, 1993) devised methods for obtaining marginal regression coefficients and associated variance estimates for multivariate survival data, using a synthesis of the Poisson regression formulation for univariate censored survival analysis and generalized estimating equations (GEE's). The method is parametric in that a baseline survival distribution is specified. Analogous semiparametric models, with unspecified baseline survival, have also been developed (Wei, Lin and Weissfeld, 1989; Lin, 1994).Common to both these approaches is the provision of robust variances for the regression parameters. However, none of this work has addressed the more difficult area of dependence estimation. While GEE approaches ostensibly provide such estimates, we show that there are problems adopting these with multivariate survival data. Further, we demonstrate that these problems can affect estimation of the regression coefficients themselves. An alternate, ad hoc approach to dependence estimation, based on design effects, is proposed and evaluated via simulation and illustrative examples. This revised version was published online in July 2006 with corrections to the Cover Date.     

Model Selection Criterion Based on the Multivariate Quasi‐Likelihood for Generalized Estimating Equations

The generalized estimating equations (GEE) approach has attracted considerable interest for the analysis of correlated response data. This paper considers the model selection criterion based on the multivariate quasi‐likelihood (MQL) in the GEE framework. The GEE approach is closely related to the MQL. We derive a necessary and sufficient condition for the uniqueness of the risk function based on the MQL by using properties of differential geometry. Furthermore, we establish a formal derivation of model selection criterion as an asymptotically unbiased estimator of the prediction risk under this condition, and we explicitly take into account the effect of estimating the correlation matrix used in the GEE procedure.     

Hypothesis Testing of Hazard Ratio Parameters in Marginal Models for Multivariate Failure Time Data

Marginal hazard models for multivariate failure time data have been studied extensively in recent literature. However, standard hypothesis test statistics based on the likelihood method are not exactly appropriate for this kind of model. In this paper, extensions of the three commonly used likelihood hypothesis test statistics are discussed. Generalized Wald, generalized score and generalized likelihood ratio tests for hazard ratio parameters in a marginal hazard model for multivariate failure time data are proposed and their asymptotic distributions examined. The finite sample properties of these statistics are studied through simulations. The proposed method is applied to data from Busselton Population Health Surveys.     

Bayesian Proportional Odds Models for Analyzing Current Status Data: Univariate,Clustered, and Multivariate

Current status data commonly arise in many fields such as epidemiological studies and cross-sectional tumorigenicity studies. In this article, we propose a semiparametric Bayesian approach for analyzing current status data with the proportional odds model. The use of monotone splines for the baseline odds function and a novel data augmentation with Poisson latent variables enable simple updating all of the parameters in the posterior computation. The proposed approach shows good performance and is compared with the approach in Wang and Dunson (2010 Wang , L. , Dunson , D. B. ( 2010 ). Semiparametric Bayes proportional odds models for current status data with under-reporting . Biometrics. Online early, DOI: 10.1111/j.1541-0420.2010.01532.x [Web of Science ®] , [Google Scholar]) in a simulation study. We also generalize the proposed approach to analyze clustered and multivariate current status data under the frailty proportional odds models.     

Comparisons of Test Statistics Arising from Marginal Analyses of Multivariate Survival Data

We investigate the properties of several statistical tests for comparing treatment groups with respect to multivariate survival data, based on the marginal analysis approach introduced by Wei, Lin and Weissfeld [Regression Analysis of multivariate incomplete failure time data by modelling marginal distributians, JASA vol. 84 pp. 1065–1073]. We consider two types of directional tests, based on a constrained maximization and on linear combinations of the unconstrained maximizer of the working likelihood function, and the omnibus test arising from the same working likelihood. The directional tests are members of a larger class of tests, from which an asymptotically optimal test can be found. We compare the asymptotic powers of the tests under general contiguous alternatives for a variety of settings, and also consider the choice of the number of survival times to include in the multivariate outcome. We illustrate the results with simulations and with the results from a clinical trial examining recurring opportunistic infections in persons with HIV.     

Multivariate Dispersion Models Generated From Gaussian Copula

In this paper a class of multivariate dispersion models generated from the multivariate Gaussian copula is presented. Being a multivariate extension of Jørgensen's (1987a) dispersion models, this class of multivariate models is parametrized by marginal position, dispersion and dependence parameters, producing a large variety of multivariate discrete and continuous models including the multivariate normal as a special case. Properties of the multivariate distributions are investigated, some of which are similar to those of the multivariate normal distribution, which makes these models potentially useful for the analysis of correlated non-normal data in a way analogous to that of multivariate normal data. As an example, we illustrate an application of the models to the regression analysis of longitudinal data, and establish an asymptotic relationship between the likelihood equation and the generalized estimating equation of Liang & Zeger (1986).