Nonparametric regression with cross‐classified responses

In this article, we develop regression models with cross‐classified responses. Conditional independence structures can be explored/exploited through the selective inclusion/exclusion of terms in a certain functional ANOVA decomposition, and the estimation is done nonparametrically via the penalized likelihood method. A cohort of computational and data analytical tools are presented, which include cross‐validation for smoothing parameter selection, Kullback–Leibler projection for model selection, and Bayesian confidence intervals for odds ratios. Random effects are introduced to model possible correlations such as those found in longitudinal and clustered data. Empirical performances of the methods are explored in simulation studies of limited scales, and a real data example is presented using some eyetracking data from linguistic studies. The techniques are implemented in a suite of R functions, whose usage is briefly described in the appendix. The Canadian Journal of Statistics 39: 591–609; 2011. © 2011 Statistical Society of Canada     

On a Unified Generalized Quasi–likelihood Approach for Familial–Longitudinal Non‐Stationary Count Data

Abstract. In this paper, conditional on random family effects, we consider an auto‐regression model for repeated count data and their corresponding time‐dependent covariates, collected from the members of a large number of independent families. The count responses, in such a set up, unconditionally exhibit a non‐stationary familial–longitudinal correlation structure. We then take this two‐way correlation structure into account, and develop a generalized quasilikelihood (GQL) approach for the estimation of the regression effects and the familial correlation index parameter, whereas the longitudinal correlation parameter is estimated by using the well‐known method of moments. The performance of the proposed estimation approach is examined through a simulation study. Some model mis‐specification effects are also studied. The estimation methodology is illustrated by analysing real life healthcare utilization count data collected from 36 families of size four over a period of 4 years.     

Progressive multi-state models for informatively incomplete longitudinal data

Progressive multi-state models provide a convenient framework for characterizing chronic disease processes where the states represent the degree of damage resulting from the disease. Incomplete data often arise in studies of such processes, and standard methods of analysis can lead to biased parameter estimates when observation of data is response-dependent. This paper describes a joint analysis useful for fitting progressive multi-state models to data arising in longitudinal studies in such settings. Likelihood based methods are described and parameters are shown to be identifiable. An EM algorithm is described for parameter estimation, and variance estimation is carried out using the Louis’ method. Simulation studies demonstrate that the proposed method works well in practice under a variety of settings. An application to data from a smoking prevention study illustrates the utility of the method.     

Inference in Semi‐Parametric Dynamic Models for Repeated Count Data

This paper deals with a longitudinal semi‐parametric regression model in a generalised linear model setup for repeated count data collected from a large number of independent individuals. To accommodate the longitudinal correlations, we consider a dynamic model for repeated counts which has decaying auto‐correlations as the time lag increases between the repeated responses. The semi‐parametric regression function involved in the model contains a specified regression function in some suitable time‐dependent covariates and a non‐parametric function in some other time‐dependent covariates. As far as the inference is concerned, because the non‐parametric function is of secondary interest, we estimate this function consistently using the independence assumption‐based well‐known quasi‐likelihood approach. Next, the proposed longitudinal correlation structure and the estimate of the non‐parametric function are used to develop a semi‐parametric generalised quasi‐likelihood approach for consistent and efficient estimation of the regression effects in the parametric regression function. The finite sample performance of the proposed estimation approach is examined through an intensive simulation study based on both large and small samples. Both balanced and unbalanced cluster sizes are incorporated in the simulation study. The asymptotic performances of the estimators are given. The estimation methodology is illustrated by reanalysing the well‐known health care utilisation data consisting of counts of yearly visits to a physician by 180 individuals for four years and several important primary and secondary covariates.     

Analyzing multivariate longitudinal binary data: A generalized estimating equations approach

The authors consider regression analysis for binary data collected repeatedly over time on members of numerous small clusters of individuals sharing a common random effect that induces dependence among them. They propose a mixed model that can accommodate both these structural and longitudinal dependencies. They estimate the parameters of the model consistently and efficiently using generalized estimating equations. They show through simulations that their approach yields significant gains in mean squared error when estimating the random effects variance and the longitudinal correlations, while providing estimates of the fixed effects that are just as precise as under a generalized penalized quasi‐likelihood approach. Their method is illustrated using smoking prevention data.     

A robust pairwise likelihood method for incomplete longitudinal binary data arising in clusters

Clustered longitudinal data feature cross‐sectional associations within clusters, serial dependence within subjects, and associations between responses at different time points from different subjects within the same cluster. Generalized estimating equations are often used for inference with data of this sort since they do not require full specification of the response model. When data are incomplete, however, they require data to be missing completely at random unless inverse probability weights are introduced based on a model for the missing data process. The authors propose a robust approach for incomplete clustered longitudinal data using composite likelihood. Specifically, pairwise likelihood methods are described for conducting robust estimation with minimal model assumptions made. The authors also show that the resulting estimates remain valid for a wide variety of missing data problems including missing at random mechanisms and so in such cases there is no need to model the missing data process. In addition to describing the asymptotic properties of the resulting estimators, it is shown that the method performs well empirically through simulation studies for complete and incomplete data. Pairwise likelihood estimators are also compared with estimators obtained from inverse probability weighted alternating logistic regression. An application to data from the Waterloo Smoking Prevention Project is provided for illustration. The Canadian Journal of Statistics 39: 34–51; 2011 © 2010 Statistical Society of Canada     

Likelihood analysis of joint marginal and conditional models for longitudinal categorical data

The authors develop a Markov model for the analysis of longitudinal categorical data which facilitates modelling both marginal and conditional structures. A likelihood formulation is employed for inference, so the resulting estimators enjoy the optimal properties such as efficiency and consistency, and remain consistent when data are missing at random. Simulation studies demonstrate that the proposed method performs well under a variety of situations. Application to data from a smoking prevention study illustrates the utility of the model and interpretation of covariate effects. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Comparing crossing hazard rate functions by joint modelling of survival and longitudinal data

Abstract

It is one of the important issues in survival analysis to compare two hazard rate functions to evaluate treatment effect. It is quite common that the two hazard rate functions cross each other at one or more unknown time points, representing temporal changes of the treatment effect. In certain applications, besides survival data, we also have related longitudinal data available regarding some time-dependent covariates. In such cases, a joint model that accommodates both types of data can allow us to infer the association between the survival and longitudinal data and to assess the treatment effect better. In this paper, we propose a modelling approach for comparing two crossing hazard rate functions by joint modelling survival and longitudinal data. Maximum likelihood estimation is used in estimating the parameters of the proposed joint model using the EM algorithm. Asymptotic properties of the maximum likelihood estimators are studied. To illustrate the virtues of the proposed method, we compare the performance of the proposed method with several existing methods in a simulation study. Our proposed method is also demonstrated using a real dataset obtained from an HIV clinical trial.     

A Two-Latent-Class Model for Smoking Cessation Data with Informative Dropouts

Non ignorable missing data is a common problem in longitudinal studies. Latent class models are attractive for simplifying the modeling of missing data when the data are subject to either a monotone or intermittent missing data pattern. In our study, we propose a new two-latent-class model for categorical data with informative dropouts, dividing the observed data into two latent classes; one class in which the outcomes are deterministic and a second one in which the outcomes can be modeled using logistic regression. In the model, the latent classes connect the longitudinal responses and the missingness process under the assumption of conditional independence. Parameters are estimated by the method of maximum likelihood estimation based on the above assumptions and the tetrachoric correlation between responses within the same subject. We compare the proposed method with the shared parameter model and the weighted GEE model using the areas under the ROC curves in the simulations and the application to the smoking cessation data set. The simulation results indicate that the proposed two-latent-class model performs well under different missing procedures. The application results show that our proposed method is better than the shared parameter model and the weighted GEE model.     

Semi‐parametric small‐area estimation by combining time‐series and cross‐sectional data methods

In survey sampling, policymaking regarding the allocation of resources to subgroups (called small areas) or the determination of subgroups with specific properties in a population should be based on reliable estimates. Information, however, is often collected at a different scale than that of these subgroups; hence, the estimation can only be obtained on finer scale data. Parametric mixed models are commonly used in small‐area estimation. The relationship between predictors and response, however, may not be linear in some real situations. Recently, small‐area estimation using a generalised linear mixed model (GLMM) with a penalised spline (P‐spline) regression model, for the fixed part of the model, has been proposed to analyse cross‐sectional responses, both normal and non‐normal. However, there are many situations in which the responses in small areas are serially dependent over time. Such a situation is exemplified by a data set on the annual number of visits to physicians by patients seeking treatment for asthma, in different areas of Manitoba, Canada. In cases where covariates that can possibly predict physician visits by asthma patients (e.g. age and genetic and environmental factors) may not have a linear relationship with the response, new models for analysing such data sets are required. In the current work, using both time‐series and cross‐sectional data methods, we propose P‐spline regression models for small‐area estimation under GLMMs. Our proposed model covers both normal and non‐normal responses. In particular, the empirical best predictors of small‐area parameters and their corresponding prediction intervals are studied with the maximum likelihood estimation approach being used to estimate the model parameters. The performance of the proposed approach is evaluated using some simulations and also by analysing two real data sets (precipitation and asthma).     

Analysis of ordinal outcomes with longitudinal covariates subject to missingness

We propose a mixture model for data with an ordinal outcome and a longitudinal covariate that is subject to missingness. Data from a tailored telephone delivered, smoking cessation intervention for construction laborers are used to illustrate the method, which considers as an outcome a categorical measure of smoking cessation, and evaluates the effectiveness of the motivational telephone interviews on this outcome. We propose two model structures for the longitudinal covariate, for the case when the missing data are missing at random, and when the missing data mechanism is non-ignorable. A generalized EM algorithm is used to obtain maximum likelihood estimates.     

Semiparametric analysis of recurrent events: artificial censoring,truncation, pairwise estimation and inference

The analysis of recurrent failure time data from longitudinal studies can be complicated by the presence of dependent censoring. There has been a substantive literature that has developed based on an artificial censoring device. We explore in this article the connection between this class of methods with truncated data structures. In addition, a new procedure is developed for estimation and inference in a joint model for recurrent events and dependent censoring. Estimation proceeds using a mixed U-statistic based estimating function approach. New resampling-based methods for variance estimation and model checking are also described. The methods are illustrated by application to data from an HIV clinical trial as with a limited simulation study.     

Hybrid pairwise‐likelihood estimation methods for incomplete longitudinal binary data

In longitudinal data, missing observations occur commonly with incomplete responses and covariates. Missing data can have a ‘missing not at random’ mechanism, a non‐monotone missing pattern, and moreover response and covariates can be missing not simultaneously. To avoid complexities in both modelling and computation, a two‐stage estimation method and a pairwise‐likelihood method are proposed. The two‐stage estimation method enjoys simplicities in computation, but incurs more severe efficiency loss. On the other hand, the pairwise approach leads to estimators with better efficiency, but can be cumbersome in computation. In this paper, we develop a compromise method using a hybrid pairwise‐likelihood framework. Our proposed approach has better efficiency than the two‐stage method, but its computational cost is still reasonable compared to the pairwise approach. The performance of the methods is evaluated empirically by means of simulation studies. Our methods are used to analyse longitudinal data obtained from the National Population Health Study.     

Joint modeling of longitudinal data with a dependent terminal event

ABSTRACT

Longitudinal data often arise in longitudinal follow-up studies, and there may exist a dependent terminal event such as death that stops the follow-up. In this article, we propose a new joint modeling for the analysis of longitudinal data with informative observation times via a dependent terminal event and two latent variables. Estimating equations are developed for parameter estimation, and asymptotic properties of the resulting estimators are established. In addition, a generalization of the joint model with time-varying coefficients for the longitudinal response variable is considered, and goodness-of-fit methods for assessing the adequacy of the model are also provided. The proposed method works well in our simulation studies, and is applied to a data set from a bladder cancer study.     

A Case Study for Modelling Cancer Incidence Using Bayesian Spatio‐Temporal Models

Researchers familiar with spatial models are aware of the challenge of choosing the level of spatial aggregation. Few studies have been published on the investigation of temporal aggregation and its impact on inferences regarding disease outcome in space–time analyses. We perform a case study for modelling individual disease outcomes using several Bayesian hierarchical spatio‐temporal models, while taking into account the possible impact of spatial and temporal aggregation. Using longitudinal breast cancer data from South East Queensland, Australia, we consider both parametric and non‐parametric formulations for temporal effects at various levels of aggregation. Two temporal smoothness priors are considered separately; each is modelled with fixed effects for the covariates and an intrinsic conditional autoregressive prior for the spatial random effects. Our case study reveals that different model formulations produce considerably different model performances. For this particular dataset, a classical parametric formulation that assumes a linear time trend produces the best fit among the five models considered. Different aggregation levels of temporal random effects were found to have little impact on model goodness‐of‐fit and estimation of fixed effects.     

Α Markov model for longitudinal studies with incomplete dichotomous outcomes

Missing outcome data constitute a serious threat to the validity and precision of inferences from randomized controlled trials. In this paper, we propose the use of a multistate Markov model for the analysis of incomplete individual patient data for a dichotomous outcome reported over a period of time. The model accounts for patients dropping out of the study and also for patients relapsing. The time of each observation is accounted for, and the model allows the estimation of time‐dependent relative treatment effects. We apply our methods to data from a study comparing the effectiveness of 2 pharmacological treatments for schizophrenia. The model jointly estimates the relative efficacy and the dropout rate and also allows for a wide range of clinically interesting inferences to be made. Assumptions about the missingness mechanism and the unobserved outcomes of patients dropping out can be incorporated into the analysis. The presented method constitutes a viable candidate for analyzing longitudinal, incomplete binary data.     

The impact of period effects on dose level contrasts in alternating cross‐over designs for first‐time‐in‐human studies

For first‐time‐in‐human studies with small molecules alternating cross‐over designs are often employed and at study end are analyzed using linear models. We discuss the impact of including a period effect in the model on the precision with which dose level contrasts can be estimated and quantify the bias of least squares estimators if a period effect is inherent in the data that is not accounted for in the model. We also propose two alternative designs that allow a more precise estimation of dose level contrasts compared with the standard design when period effects are included in the model. Copyright © 2010 John Wiley & Sons, Ltd.     

Comparing Joint GQL Estimation and GMM Adaptive Estimation in COM-Poisson Longitudinal Regression Model

It is of scientific interest to study the application of COM-Poisson model to the case of longitudinal response data, the analysis of which is quite challenging due to the fact that longitudinal responses of a subject are correlated and the correlation pattern is usually unknown. In this article, we extend the COM-Poisson GLM to the generalized linear longitudinal model. We also develop a joint generalized quasi-likelihood estimating equation approach based on a stationary autocorrelation structure for the repeated count data. We further compare the performance of this estimation method with that of Generalized Method of Moments through a simulation study.     

Expected estimating equations to accommodate covariate measurement error

Estimating equations which are not necessarily likelihood-based score equations are becoming increasingly popular for estimating regression model parameters. This paper is concerned with estimation based on general estimating equations when true covariate data are missing for all the study subjects, but surrogate or mismeasured covariates are available instead. The method is motivated by the covariate measurement error problem in marginal or partly conditional regression of longitudinal data. We propose to base estimation on the expectation of the complete data estimating equation conditioned on available data. The regression parameters and other nuisance parameters are estimated simultaneously by solving the resulting estimating equations. The expected estimating equation (EEE) estimator is equal to the maximum likelihood estimator if the complete data scores are likelihood scores and conditioning is with respect to all the available data. A pseudo-EEE estimator, which requires less computation, is also investigated. Asymptotic distribution theory is derived. Small sample simulations are conducted when the error process is an order 1 autoregressive model. Regression calibration is extended to this setting and compared with the EEE approach. We demonstrate the methods on data from a longitudinal study of the relationship between childhood growth and adult obesity.     

Small Area Estimation via Multivariate Fay–Herriot Models with Latent Spatial Dependence

The Fay–Herriot model is a standard model for direct survey estimators in which the true quantity of interest, the superpopulation mean, is latent and its estimation is improved through the use of auxiliary covariates. In the context of small area estimation, these estimates can be further improved by borrowing strength across spatial regions or by considering multiple outcomes simultaneously. We provide here two formulations to perform small area estimation with Fay–Herriot models that include both multivariate outcomes and latent spatial dependence. We consider two model formulations. In one of these formulations the outcome‐by‐space dependence structure is separable. The other accounts for the cross dependence through the use of a generalized multivariate conditional autoregressive (GMCAR) structure. The GMCAR model is shown, in a state‐level example, to produce smaller mean square prediction errors, relative to equivalent census variables, than the separable model and the state‐of‐the‐art multivariate model with unstructured dependence between outcomes and no spatial dependence. In addition, both the GMCAR and the separable models give smaller mean squared prediction error than the state‐of‐the‐art model when conducting small area estimation on county level data from the American Community Survey.