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.     

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 in the weighted generalized estimating equations for longitudinal data with dropout

We propose criteria for variable selection in the mean model and for the selection of a working correlation structure in longitudinal data with dropout missingness using weighted generalized estimating equations. The proposed criteria are based on a weighted quasi‐likelihood function and a penalty term. Our simulation results show that the proposed criteria frequently select the correct model in candidate mean models. The proposed criteria also have good performance in selecting the working correlation structure for binary and normal outcomes. We illustrate our approaches using two empirical examples. In the first example, we use data from a randomized double‐blind study to test the cancer‐preventing effects of beta carotene. In the second example, we use longitudinal CD4 count data from a randomized double‐blind study.     

On the Impact of Parametric Assumptions and Robust Alternatives for Longitudinal Data Analysis

Models for longitudinal data are employed in a wide range of behavioral, biomedical, psychosocial, and health‐care‐related research. One popular model for continuous response is the linear mixed‐effects model (LMM). Although simulations by recent studies show that LMM provides reliable estimates under departures from the normality assumption for complete data, the invariable occurrence of missing data in practical studies renders such robustness results less useful when applied to real study data. In this paper, we show by simulated studies that in the presence of missing data estimates of the fixed effect of LMM are biased under departures from normality. We discuss two robust alternatives, the weighted generalized estimating equations (WGEE) and the augmented WGEE (AWGEE), and compare their performances with LMM using real as well as simulated data. Our simulation results show that both WGEE and AWGEE provide valid inference for skewed non‐normal data when missing data follows the missing at random, the most popular missing data mechanism for real study data.     

Model selection for G‐estimation of dynamic treatment regimes

Dynamic treatment regimes (DTRs) aim to formalize personalized medicine by tailoring treatment decisions to individual patient characteristics. G‐estimation for DTR identification targets the parameters of a structural nested mean model, known as the blip function, from which the optimal DTR is derived. Despite its potential, G‐estimation has not seen widespread use in the literature, owing in part to its often complex presentation and implementation, but also due to the necessity for correct specification of the blip. Using a quadratic approximation approach inspired by iteratively reweighted least squares, we derive a quasi‐likelihood function for G‐estimation within the DTR framework, and show how it can be used to form an information criterion for blip model selection. We outline the theoretical properties of this model selection criterion and demonstrate its application in a variety of simulation studies as well as in data from the Sequenced Treatment Alternatives to Relieve Depression study.     

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.     

Variable selection with incomplete covariate data

SUMMARY: Application of classical model selection methods such as Akaike's information criterion (AIC) becomes problematic when observations are missing. In this article we propose some variations on the AIC, which are applicable to missing covariate problems. The method is directly based on the expectation maximization (EM) algorithm and is readily available for EM-based estimation methods, without much additional computational efforts. The missing data AIC criteria are formally derived and shown to work in a simulation study and by application to data on diabetic retinopathy.     

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.     

Bayesian analysis for generalized linear models with nonignorably missing covariates

We propose Bayesian methods for estimating parameters in generalized linear models (GLMs) with nonignorably missing covariate data. We show that when improper uniform priors are used for the regression coefficients, phi, of the multinomial selection model for the missing data mechanism, the resulting joint posterior will always be improper if (i) all missing covariates are discrete and an intercept is included in the selection model for the missing data mechanism, or (ii) at least one of the covariates is continuous and unbounded. This impropriety will result regardless of whether proper or improper priors are specified for the regression parameters, beta, of the GLM or the parameters, alpha, of the covariate distribution. To overcome this problem, we propose a novel class of proper priors for the regression coefficients, phi, in the selection model for the missing data mechanism. These priors are robust and computationally attractive in the sense that inferences about beta are not sensitive to the choice of the hyperparameters of the prior for phi and they facilitate a Gibbs sampling scheme that leads to accelerated convergence. In addition, we extend the model assessment criterion of Chen, Dey, and Ibrahim (2004a, Biometrika 91, 45-63), called the weighted L measure, to GLMs and missing data problems as well as extend the deviance information criterion (DIC) of Spiegelhalter et al. (2002, Journal of the Royal Statistical Society B 64, 583-639) for assessing whether the missing data mechanism is ignorable or nonignorable. A novel Markov chain Monte Carlo sampling algorithm is also developed for carrying out posterior computation. Several simulations are given to investigate the performance of the proposed Bayesian criteria as well as the sensitivity of the prior specification. Real datasets from a melanoma cancer clinical trial and a liver cancer study are presented to further illustrate the proposed methods.     

A Positive Stable Frailty Model for Clustered Failure Time Data with Covariate‐Dependent Frailty

Summary In this article, we propose a positive stable shared frailty Cox model for clustered failure time data where the frailty distribution varies with cluster‐level covariates. The proposed model accounts for covariate‐dependent intracluster correlation and permits both conditional and marginal inferences. We obtain marginal inference directly from a marginal model, then use a stratified Cox‐type pseudo‐partial likelihood approach to estimate the regression coefficient for the frailty parameter. The proposed estimators are consistent and asymptotically normal and a consistent estimator of the covariance matrix is provided. Simulation studies show that the proposed estimation procedure is appropriate for practical use with a realistic number of clusters. Finally, we present an application of the proposed method to kidney transplantation data from the Scientific Registry of Transplant Recipients.     

Effects of genotyping errors,missing values and segregation distortion in molecular marker data on the construction of linkage maps

A simulation study was performed to investigate the effects of missing values, typing errors and distorted segregation ratios in molecular marker data on the construction of genetic linkage maps, and to compare the performance of three locus-ordering criteria (weighted least squares, maximum likelihood and minimum sum of adjacent recombination fractions criteria) in the presence of such effects. The study was based upon three linkage groups of 10 loci at 2, 6, and 10 cM spacings simulated from a doubled-haploid population of size 150. Criteria performance were assessed using the number of replicates with correctly estimated orders, the mean rank correlation between the estimated and the true order and the mean total map length. Bootstrap samples from replicates in the maximum likelihood analysis produced a measure of confidence in the estimated locus order. The effects of missing values and/or typing errors in the data are to reduce the proportion of correctly ordered maps, and this problem worsens as the distances between loci decreases. The maximum likelihood criterion is most successful at ordering loci correctly, but gives estimated map lengths, which are substantially inflated when typing errors are present. The presence of missing values in the data produces shorter map lengths for more widely spaced markers, especially under the weighted least-squares criterion. Overall, the presence of segregation distortion has little effect on this population.     

Variable selection – A review and recommendations for the practicing statistician

Statistical models support medical research by facilitating individualized outcome prognostication conditional on independent variables or by estimating effects of risk factors adjusted for covariates. Theory of statistical models is well‐established if the set of independent variables to consider is fixed and small. Hence, we can assume that effect estimates are unbiased and the usual methods for confidence interval estimation are valid. In routine work, however, it is not known a priori which covariates should be included in a model, and often we are confronted with the number of candidate variables in the range 10–30. This number is often too large to be considered in a statistical model. We provide an overview of various available variable selection methods that are based on significance or information criteria, penalized likelihood, the change‐in‐estimate criterion, background knowledge, or combinations thereof. These methods were usually developed in the context of a linear regression model and then transferred to more generalized linear models or models for censored survival data. Variable selection, in particular if used in explanatory modeling where effect estimates are of central interest, can compromise stability of a final model, unbiasedness of regression coefficients, and validity of p‐values or confidence intervals. Therefore, we give pragmatic recommendations for the practicing statistician on application of variable selection methods in general (low‐dimensional) modeling problems and on performing stability investigations and inference. We also propose some quantities based on resampling the entire variable selection process to be routinely reported by software packages offering automated variable selection algorithms.     

Latent class regression: Inference and estimation with two‐stage multiple imputation

Latent class regression (LCR) is a popular method for analyzing multiple categorical outcomes. While nonresponse to the manifest items is a common complication, inferences of LCR can be evaluated using maximum likelihood, multiple imputation, and two‐stage multiple imputation. Under similar missing data assumptions, the estimates and variances from all three procedures are quite close. However, multiple imputation and two‐stage multiple imputation can provide additional information: estimates for the rates of missing information. The methodology is illustrated using an example from a study on racial and ethnic disparities in breast cancer severity.     

The weighted generalized estimating equations approach for the evaluation of medical diagnostic test at subunit level

Sensitivity and specificity are common measures used to evaluate the performance of a diagnostic test. A diagnostic test is often administrated at a subunit level, e.g. at the level of vessel, ear or eye of a patient so that the treatment can be targeted at the specific subunit. Therefore, it is essential to evaluate the diagnostic test at the subunit level. Often patients with more negative subunit test results are less likely to receive the gold standard tests than patients with more positive subunit test results. To account for this type of missing data and correlation between subunit test results, we proposed a weighted generalized estimating equations (WGEE) approach to evaluate subunit sensitivities and specificities. A simulation study was conducted to evaluate the performance of the WGEE estimators and the weighted least squares (WLS) estimators (Barnhart and Kosinski, 2003) under a missing at random assumption. The results suggested that WGEE estimator is consistent under various scenarios of percentage of missing data and sample size, while the WLS approach could yield biased estimators due to a misspecified missing data mechanism. We illustrate the methodology with a cardiology example.     

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.     

Estimating Incidence Rates from Population‐Based Case‐Control Studies in the Presence of Nonrespondents

In population‐based case‐control studies, it is of great public‐health importance to estimate the disease incidence rates associated with different levels of risk factors. This estimation is complicated by the fact that in such studies the selection probabilities for the cases and controls are unequal. A further complication arises when the subjects who are selected into the study do not participate (i.e. become nonrespondents) and nonrespondents differ systematically from respondents. In this paper, we show how to account for unequal selection probabilities as well as differential nonresponses in the incidence estimation. We use two logistic models, one relating the disease incidence rate to the risk factors, and one modelling the predictors that affect the nonresponse probability. After estimating the regression parameters in the nonresponse model, we estimate the regression parameters in the disease incidence model by a weighted estimating function that weights a respondent's contribution to the likelihood score function by the inverse of the product of his/her selection probability and his/her model‐predicted response probability. The resulting estimators of the regression parameters and the corresponding estimators of the incidence rates are shown to be consistent and asymptotically normal with easily estimated variances. Simulation results demonstrate that the asymptotic approximations are adequate for practical use and that failure to adjust for nonresponses could result in severe biases. An illustration with data from a cardiovascular study that motivated this work is presented.     

Information‐theoretic model selection affects home‐range estimation and habitat preference inference: a case study of male Reeves’s Pheasants Syrmaticus reevesii

Reeves’s Pheasant Syrmaticus reevesii is a vulnerable forest bird inhabiting broadleaved habitats dominated by oaks Quercus spp. in central China. Identifying home‐ranges and habitat associations is important for understanding the biology of this species and developing effective management and conservation plans. We used information‐theoretic criteria to evaluate the relative performance of four parametric (exponential power, one‐mode bivariate normal, two‐mode bivariate normal and two‐mode bivariate circle) and two non‐parametric models (adaptive and fixed kernel) for estimating home‐ranges and habitat associations of Reeves’s Pheasants. For parametric models, Akaike’s information criterion (AIC_c) and the likelihood cross‐validation criterion (CVC) were relatively consistent in ranking the bivariate exponential power model the least acceptable, whereas the two‐mode bivariate models performed better. The CVC suggested that kernel models, particularly the adaptive kernel, performed best among all six models evaluated. The average core area and 95% contour area based on the model with greatest support were 6.1 and 54.9 ha, respectively, and were larger than those estimated from other models. The discrepancy in estimates between models with highest and the lowest support decreased as the contour size increased; however, home‐range shapes differed between models. Minimum convex polygons that removed 5% of extreme data points (MCP95) were roughly half the size of home‐ranges based on kernel models. Estimates of home‐range and model evaluation were not affected by sample size (> 50 observations for each bird). Inference about habitat preference based on composition analysis and home‐range overlap varied between models. That with strongest support suggested that Reeves’s Pheasants selected mature fir and mixed forest, avoided farmland, and had mean among‐individual home‐range overlaps of 20%. We recommend non‐parametric methods, particularly the adaptive kernel method, for estimating home‐ranges and core areas for species with complex multi‐polar habitat preferences in heterogeneous environments with large habitat patches. However, we caution against the traditional convenience of using a single model to estimate home‐ranges and recommend exploration of multiple models for describing and understanding the ecological processes underlying space use and habitat associations.     

Marginalized zero‐inflated Poisson models with missing covariates

Unlike zero‐inflated Poisson regression, marginalized zero‐inflated Poisson (MZIP) models for counts with excess zeros provide estimates with direct interpretations for the overall effects of covariates on the marginal mean. In the presence of missing covariates, MZIP and many other count data models are ordinarily fitted using complete case analysis methods due to lack of appropriate statistical methods and software. This article presents an estimation method for MZIP models with missing covariates. The method, which is applicable to other missing data problems, is illustrated and compared with complete case analysis by using simulations and dental data on the caries preventive effects of a school‐based fluoride mouthrinse program.     

Variable selection for logistic regression using a prediction-focused information criterion

In biostatistical practice, it is common to use information criteria as a guide for model selection. We propose new versions of the focused information criterion (FIC) for variable selection in logistic regression. The FIC gives, depending on the quantity to be estimated, possibly different sets of selected variables. The standard version of the FIC measures the mean squared error of the estimator of the quantity of interest in the selected model. In this article, we propose more general versions of the FIC, allowing other risk measures such as the one based on L(p) error. When prediction of an event is important, as is often the case in medical applications, we construct an FIC using the error rate as a natural risk measure. The advantages of using an information criterion which depends on both the quantity of interest and the selected risk measure are illustrated by means of a simulation study and application to a study on diabetic retinopathy.     

Focused information criterion on predictive models in personalized medicine

Instead of assessing the overall fit of candidate models like the traditional model selection criteria, the focused information criterion focuses attention directly on the parameter of the primary interest and aims to select the model with the minimum estimated mean squared error for the estimate of the focused parameter. In this article we apply the focused information criterion for personalized medicine. By using individual‐level information from clinical observations, demographics, and genetics, we obtain the personalized predictive models to make the prognosis and diagnosis individually. The consideration of the heterogeneity among the individuals helps reduce the prediction uncertainty and improve the prediction accuracy. Two real data examples from biomedical research are studied as illustrations.