Empirical‐likelihood‐based criteria for model selection on marginal analysis of longitudinal data with dropout missingness

Longitudinal data are common in clinical trials and observational studies, where missing outcomes due to dropouts are always encountered. Under such context with the assumption of missing at random, the weighted generalized estimating equation (WGEE) approach is widely adopted for marginal analysis. Model selection on marginal mean regression is a crucial aspect of data analysis, and identifying an appropriate correlation structure for model fitting may also be of interest and importance. However, the existing information criteria for model selection in WGEE have limitations, such as separate criteria for the selection of marginal mean and correlation structures, unsatisfactory selection performance in small‐sample setups, and so forth. In particular, there are few studies to develop joint information criteria for selection of both marginal mean and correlation structures. In this work, by embedding empirical likelihood into the WGEE framework, we propose two innovative information criteria named a joint empirical Akaike information criterion and a joint empirical Bayesian information criterion, which can simultaneously select the variables for marginal mean regression and also correlation structure. Through extensive simulation studies, these empirical‐likelihood‐based criteria exhibit robustness, flexibility, and outperformance compared to the other criteria including the weighted quasi‐likelihood under the independence model criterion, the missing longitudinal information criterion, and the joint longitudinal information criterion. In addition, we provide a theoretical justification of our proposed criteria, and present two real data examples in practice for further illustration.     

Joint model selection of marginal mean regression and correlation structure for longitudinal data with missing outcome and covariates

This work develops a joint model selection criterion for simultaneously selecting the marginal mean regression and the correlation/covariance structure in longitudinal data analysis where both the outcome and the covariate variables may be subject to general intermittent patterns of missingness under the missing at random mechanism. The new proposal, termed “joint longitudinal information criterion” (JLIC), is based on the expected quadratic error for assessing model adequacy, and the second‐order weighted generalized estimating equation (WGEE) estimation for mean and covariance models. Simulation results reveal that JLIC outperforms existing methods performing model selection for the mean regression and the correlation structure in a two stage and hence separate manner. We apply the proposal to a longitudinal study to identify factors associated with life satisfaction in the elderly of Taiwan.     

Modified Gaussian estimation for correlated binary data

In this paper, we develop a Gaussian estimation (GE) procedure to estimate the parameters of a regression model for correlated (longitudinal) binary response data using a working correlation matrix. A two‐step iterative procedure is proposed for estimating the regression parameters by the GE method and the correlation parameters by the method of moments. Consistency properties of the estimators are discussed. A simulation study was conducted to compare 11 estimators of the regression parameters, namely, four versions of the GE, five versions of the generalized estimating equations (GEEs), and two versions of the weighted GEE. Simulations show that (i) the Gaussian estimates have the smallest mean square error and best coverage probability if the working correlation structure is correctly specified and (ii) when the working correlation structure is correctly specified, the GE and the GEE with exchangeable correlation structure perform best as opposed to when the correlation structure is misspecified.     

Model Selection for Generalized Estimating Equations Accommodating Dropout Missingness

Summary The generalized estimating equation (GEE) has been a popular tool for marginal regression analysis with longitudinal data, and its extension, the weighted GEE approach, can further accommodate data that are missing at random (MAR). Model selection methodologies for GEE, however, have not been systematically developed to allow for missing data. We propose the missing longitudinal information criterion (MLIC) for selection of the mean model, and the MLIC for correlation (MLICC) for selection of the correlation structure in GEE when the outcome data are subject to dropout/monotone missingness and are MAR. Our simulation results reveal that the MLIC and MLICC are effective for variable selection in the mean model and selecting the correlation structure, respectively. We also demonstrate the remarkable drawbacks of naively treating incomplete data as if they were complete and applying the existing GEE model selection method. The utility of proposed method is further illustrated by two real applications involving missing longitudinal outcome data.     

Model selection of generalized estimating equations with multiply imputed longitudinal data

Longitudinal data often encounter missingness with monotone and/or intermittent missing patterns. Multiple imputation (MI) has been popularly employed for analysis of missing longitudinal data. In particular, the MI‐GEE method has been proposed for inference of generalized estimating equations (GEE) when missing data are imputed via MI. However, little is known about how to perform model selection with multiply imputed longitudinal data. In this work, we extend the existing GEE model selection criteria, including the “quasi‐likelihood under the independence model criterion” (QIC) and the “missing longitudinal information criterion” (MLIC), to accommodate multiple imputed datasets for selection of the MI‐GEE mean model. According to real data analyses from a schizophrenia study and an AIDS study, as well as simulations under nonmonotone missingness with moderate proportion of missing observations, we conclude that: (i) more than a few imputed datasets are required for stable and reliable model selection in MI‐GEE analysis; (ii) the MI‐based GEE model selection methods with a suitable number of imputations generally perform well, while the naive application of existing model selection methods by simply ignoring missing observations may lead to very poor performance; (iii) the model selection criteria based on improper (frequentist) multiple imputation generally performs better than their analogies based on proper (Bayesian) multiple imputation.     

Directly parameterized regression conditioning on being alive: analysis of longitudinal data truncated by deaths

For observational longitudinal studies of geriatric populations, outcomes such as disability or cognitive functioning are often censored by death. Statistical analysis of such data may explicitly condition on either vital status or survival time when summarizing the longitudinal response. For example a pattern-mixture model characterizes the mean response at time t conditional on death at time S = s (for s > t), and thus uses future status as a predictor for the time t response. As an alternative, we define regression conditioning on being alive as a regression model that conditions on survival status, rather than a specific survival time. Such models may be referred to as partly conditional since the mean at time t is specified conditional on being alive (S > t), rather than using finer stratification (S = s for s > t). We show that naive use of standard likelihood-based longitudinal methods and generalized estimating equations with non-independence weights may lead to biased estimation of the partly conditional mean model. We develop a taxonomy for accommodation of both dropout and death, and describe estimation for binary longitudinal data that applies selection weights to estimating equations with independence working correlation. Simulation studies and an analysis of monthly disability status illustrate potential bias in regression methods that do not explicitly condition on survival.     

Double‐Robust Semiparametric Estimator for Differences in Restricted Mean Lifetimes in Observational Studies

Summary Restricted mean lifetime is often of direct interest in epidemiologic studies involving censored survival times. Differences in this quantity can be used as a basis for comparing several groups. For example, transplant surgeons, nephrologists, and of course patients are interested in comparing posttransplant lifetimes among various types of kidney transplants to assist in clinical decision making. As the factor of interest is not randomized, covariate adjustment is needed to account for imbalances in confounding factors. In this report, we use semiparametric theory to develop an estimator for differences in restricted mean lifetimes although accounting for confounding factors. The proposed method involves building working models for the time‐to‐event and coarsening mechanism (i.e., group assignment and censoring). We show that the proposed estimator possesses the double robust property; i.e., when either the time‐to‐event or coarsening process is modeled correctly, the estimator is consistent and asymptotically normal. Simulation studies are conducted to assess its finite‐sample performance and the method is applied to national kidney transplant data.     

Selection criterion of work matrix as a function of limiting estimates of the covariance matrix of correlated data in GEE

The modeling of generalized estimating equations used in the analysis of longitudinal data whether in continuous or discrete variables, necessarily requires the prior specification of a correlation matrix in its iterative process in order to obtain the estimates of the regression parameters. Such a matrix is called working correlation matrix and its incorrect specification produces less efficient estimates for the model parameters. Due to this fact, this study aims to propose a selection criterion of working correlation matrix based on the covariance matrix estimates of correlated responses resulting from the limiting values of the association parameter estimates. For validation of the criterion, we used simulation studies considering normal and binary correlated responses. Compared to some criteria in the literature, it was concluded that the proposed criterion resulted in a better performance when the correlation structure for exchangeable working correlation matrix was considered as true structure in the simulated samples and for large samples, the proposed criterion showed similar behavior to the other criteria, resulting in higher success rates.     

A modified pseudolikelihood approach for analysis of longitudinal data

We consider the analysis of longitudinal data when the covariance function is modeled by additional parameters to the mean parameters. In general, inconsistent estimators of the covariance (variance/correlation) parameters will be produced when the "working" correlation matrix is misspecified, which may result in great loss of efficiency of the mean parameter estimators (albeit the consistency is preserved). We consider using different "working" correlation models for the variance and the mean parameters. In particular, we find that an independence working model should be used for estimating the variance parameters to ensure their consistency in case the correlation structure is misspecified. The designated "working" correlation matrices should be used for estimating the mean and the correlation parameters to attain high efficiency for estimating the mean parameters. Simulation studies indicate that the proposed algorithm performs very well. We also applied different estimation procedures to a data set from a clinical trial for illustration.     

Penalized generalized estimating equations for high-dimensional longitudinal data analysis

We consider the penalized generalized estimating equations (GEEs) for analyzing longitudinal data with high-dimensional covariates, which often arise in microarray experiments and large-scale health studies. Existing high-dimensional regression procedures often assume independent data and rely on the likelihood function. Construction of a feasible joint likelihood function for high-dimensional longitudinal data is challenging, particularly for correlated discrete outcome data. The penalized GEE procedure only requires specifying the first two marginal moments and a working correlation structure. We establish the asymptotic theory in a high-dimensional framework where the number of covariates p(n) increases as the number of clusters n increases, and p(n) can reach the same order as n. One important feature of the new procedure is that the consistency of model selection holds even if the working correlation structure is misspecified. We evaluate the performance of the proposed method using Monte Carlo simulations and demonstrate its application using a yeast cell-cycle gene expression data set.     

Pitfalls of hypothesis tests and model selection on bootstrap samples: Causes and consequences in biometrical applications

The bootstrap method has become a widely used tool applied in diverse areas where results based on asymptotic theory are scarce. It can be applied, for example, for assessing the variance of a statistic, a quantile of interest or for significance testing by resampling from the null hypothesis. Recently, some approaches have been proposed in the biometrical field where hypothesis testing or model selection is performed on a bootstrap sample as if it were the original sample. P‐values computed from bootstrap samples have been used, for example, in the statistics and bioinformatics literature for ranking genes with respect to their differential expression, for estimating the variability of p‐values and for model stability investigations. Procedures which make use of bootstrapped information criteria are often applied in model stability investigations and model averaging approaches as well as when estimating the error of model selection procedures which involve tuning parameters. From the literature, however, there is evidence that p‐values and model selection criteria evaluated on bootstrap data sets do not represent what would be obtained on the original data or new data drawn from the overall population. We explain the reasons for this and, through the use of a real data set and simulations, we assess the practical impact on procedures relevant to biometrical applications in cases where it has not yet been studied. Moreover, we investigate the behavior of subsampling (i.e., drawing from a data set without replacement) as a potential alternative solution to the bootstrap for these procedures.     

Regularized estimation in the accelerated failure time model with high-dimensional covariates

We consider two regularization approaches, the LASSO and the threshold-gradient-directed regularization, for estimation and variable selection in the accelerated failure time model with multiple covariates based on Stute's weighted least squares method. The Stute estimator uses Kaplan-Meier weights to account for censoring in the least squares criterion. The weighted least squares objective function makes the adaptation of this approach to multiple covariate settings computationally feasible. We use V-fold cross-validation and a modified Akaike's Information Criterion for tuning parameter selection, and a bootstrap approach for variance estimation. The proposed method is evaluated using simulations and demonstrated on a real data example.     

Quadratic inference functions for varying-coefficient models with longitudinal data

Nonparametric smoothing methods are used to model longitudinal data, but the challenge remains to incorporate correlation into nonparametric estimation procedures. In this article, we propose an efficient estimation procedure for varying-coefficient models for longitudinal data. The proposed procedure can easily take into account correlation within subjects and deal directly with both continuous and discrete response longitudinal data under the framework of generalized linear models. The proposed approach yields a more efficient estimator than the generalized estimation equation approach when the working correlation is misspecified. For varying-coefficient models, it is often of interest to test whether coefficient functions are time varying or time invariant. We propose a unified and efficient nonparametric hypothesis testing procedure, and further demonstrate that the resulting test statistics have an asymptotic chi-squared distribution. In addition, the goodness-of-fit test is applied to test whether the model assumption is satisfied. The corresponding test is also useful for choosing basis functions and the number of knots for regression spline models in conjunction with the model selection criterion. We evaluate the finite sample performance of the proposed procedures with Monte Carlo simulation studies. The proposed methodology is illustrated by the analysis of an acquired immune deficiency syndrome (AIDS) data set.     

Extension of the Haseman-Elston regression model to longitudinal data

We propose an extension to longitudinal data of the Haseman and Elston regression method for linkage analysis. The proposed model is a mixed model having several random effects. As response variable, we investigate the sibship sample mean corrected cross-product (smHE) and the BLUP-mean corrected cross product (pmHE), comparing them with the original squared difference (oHE), the overall mean corrected cross-product (rHE), and the weighted average of the squared difference and the squared mean-corrected sum (wHE). The proposed model allows for the correlation structure of longitudinal data. Also, the model can test for gene x time interaction to discover genetic variation over time. The model was applied in an analysis of the Genetic Analysis Workshop 13 (GAW13) simulated dataset for a quantitative trait simulating systolic blood pressure. Independence models did not preserve the test sizes, while the mixed models with both family and sibpair random effects tended to preserve size well.     

Doubly robust multiple imputation using kernel‐based techniques

We consider the problem of estimating the marginal mean of an incompletely observed variable and develop a multiple imputation approach. Using fully observed predictors, we first establish two working models: one predicts the missing outcome variable, and the other predicts the probability of missingness. The predictive scores from the two models are used to measure the similarity between the incomplete and observed cases. Based on the predictive scores, we construct a set of kernel weights for the observed cases, with higher weights indicating more similarity. Missing data are imputed by sampling from the observed cases with probability proportional to their kernel weights. The proposed approach can produce reasonable estimates for the marginal mean and has a double robustness property, provided that one of the two working models is correctly specified. It also shows some robustness against misspecification of both models. We demonstrate these patterns in a simulation study. In a real‐data example, we analyze the total helicopter response time from injury in the Arizona emergency medical service data.     

A weighted partial likelihood approach for zero‐truncated models

Zero‐truncated data arises in various disciplines where counts are observed but the zero count category cannot be observed during sampling. Maximum likelihood estimation can be used to model these data; however, due to its nonstandard form it cannot be easily implemented using well‐known software packages, and additional programming is often required. Motivated by the Rao–Blackwell theorem, we develop a weighted partial likelihood approach to estimate model parameters for zero‐truncated binomial and Poisson data. The resulting estimating function is equivalent to a weighted score function for standard count data models, and allows for applying readily available software. We evaluate the efficiency for this new approach and show that it performs almost as well as maximum likelihood estimation. The weighted partial likelihood approach is then extended to regression modelling and variable selection. We examine the performance of the proposed methods through simulation and present two case studies using real data.     

Restricted fence method for covariate selection in longitudinal data analysis

Fence method (Jiang and others 2008. Fence methods for mixed model selection. Annals of Statistics 36, 1669-1692) is a recently proposed strategy for model selection. It was motivated by the limitation of the traditional information criteria in selecting parsimonious models in some nonconventional situations, such as mixed model selection. Jiang and others (2009. A simplified adaptive fence procedure, Statistics & Probability Letters 79, 625-629) simplified the adaptive fence method of Jiang and others (2008) to make it more suitable and convenient to use in a wide variety of problems. Still, the current modification encounters computational difficulties when applied to high-dimensional and complex problems. To address this concern, we proposed a restricted fence procedure that combines the idea of the fence with that of the restricted maximum likelihood. Furthermore, we propose to use the wild bootstrap for choosing adaptively the tuning parameter used in the restricted fence. We focus on problems of longitudinal studies and demonstrate the performance of the new procedure and its comparison with other procedures of variable selection, including the information criteria and shrinkage methods, in simulation studies. The method is further illustrated by an example of real-data analysis.     

Quantile regression models with multivariate failure time data

As an alternative to the mean regression model, the quantile regression model has been studied extensively with independent failure time data. However, due to natural or artificial clustering, it is common to encounter multivariate failure time data in biomedical research where the intracluster correlation needs to be accounted for appropriately. For right-censored correlated survival data, we investigate the quantile regression model and adapt an estimating equation approach for parameter estimation under the working independence assumption, as well as a weighted version for enhancing the efficiency. We show that the parameter estimates are consistent and asymptotically follow normal distributions. The variance estimation using asymptotic approximation involves nonparametric functional density estimation. We employ the bootstrap and perturbation resampling methods for the estimation of the variance-covariance matrix. We examine the proposed method for finite sample sizes through simulation studies, and illustrate it with data from a clinical trial on otitis media.     

Generalized additive selection models for the analysis of studies with potentially nonignorable missing outcome data

Rotnitzky, Robins, and Scharfstein (1998, Journal of the American Statistical Association 93, 1321-1339) developed a methodology for conducting sensitivity analysis of studies in which longitudinal outcome data are subject to potentially nonignorable missingness. In their approach, they specify a class of fully parametric selection models, indexed by a non- or weakly identified selection bias function that indicates the degree to which missingness depends on potentially unobservable outcomes. Estimation of the parameters of interest proceeds by varying the selection bias function over a range considered plausible by subject-matter experts. In this article, we focus on cross-sectional, univariate outcome data and extend their approach to a class of semiparametric selection models, using generalized additive restrictions. We propose a backfitting algorithm to estimate the parameters of the generalized additive selection model. For estimation of the mean outcome, we propose three types of estimating functions: simple inverse weighted, doubly robust, and orthogonal. We present the results of a data analysis and a simulation study.     

Mixture models, robustness, and the weighted likelihood methodology

Problems associated with the analysis of data from a mixture of distributions include the presence of outliers in the sample, the fact that a component may not be well represented in the data, and the problem of biases that occur when the model is slightly misspecified. We study the performance of weighted likelihood in this context. The method produces estimates with low bias and mean squared error, and it is useful in that it unearths data substructures in the form of multiple roots. This in turn indicates multiple potential mixture model fits due to the presence of more components than originally specified in the model. To compute the weighted likelihood estimates, we use as starting values the method of moment estimates computed on bootstrap subsamples drawn from the data. We address a number of important practical issues involving bootstrap sample size selection, the role of starting values, and the behavior of the roots. The algorithm used to compute the weighted likelihood estimates is competitive with EM, and it is similar to EM when the components are not well separated. Moreover, we propose a new statistical stopping rule for the termination of the algorithm. An example and a small simulation study illustrate the above points.