Strategies to fit pattern-mixture models

Whereas most models for incomplete longitudinal data are formulated within the selection model framework, pattern-mixture models have gained considerable interest in recent years (Little, 1993, 1994). In this paper, we outline several strategies to fit pattern-mixture models, including the so-called identifying restrictions strategy. Multiple imputation is used to apply this strategy to realistic settings, such as quality-of-life data from a longitudinal study on metastatic breast cancer patients.     

A flexible B-spline model for multiple longitudinal biomarkers and survival

Often when jointly modeling longitudinal and survival data, we are interested in a multivariate longitudinal measure that may not fit well by linear models. To overcome this problem, we propose a joint longitudinal and survival model that has a nonparametric model for the longitudinal markers. We use cubic B-splines to specify the longitudinal model and a proportional hazards model to link the longitudinal measures to the hazard. To fit the model, we use a Markov chain Monte Carlo algorithm. We select the number of knots for the cubic B-spline model using the Conditional Predictive Ordinate (CPO) and the Deviance Information Criterion (DIC). The method and model selection approach are validated in a simulation. We apply this method to examine the link between viral load, CD4 count, and time to event in data from an AIDS clinical trial. The cubic B-spline model provides a good fit to the longitudinal data that could not be obtained with simple parametric models.     

Neural networks for clustered and longitudinal data using mixed effects models

Although most statistical methods for the analysis of longitudinal data have focused on retrospective models of association, new advances in mobile health data have presented opportunities for predicting future health status by leveraging an individual's behavioral history alongside data from similar patients. Methods that incorporate both individual-level and sample-level effects are critical to using these data to its full predictive capacity. Neural networks are powerful tools for prediction, but many assume input observations are independent even when they are clustered or correlated in some way, such as in longitudinal data. Generalized linear mixed models (GLMM) provide a flexible framework for modeling longitudinal data but have poor predictive power particularly when the data are highly nonlinear. We propose a generalized neural network mixed model that replaces the linear fixed effect in a GLMM with the output of a feed-forward neural network. The model simultaneously accounts for the correlation structure and complex nonlinear relationship between input variables and outcomes, and it utilizes the predictive power of neural networks. We apply this approach to predict depression and anxiety levels of schizophrenic patients using longitudinal data collected from passive smartphone sensor data.     

A semiparametric likelihood approach to joint modeling of longitudinal and time-to-event data

Joint models for a time-to-event (e.g., survival) and a longitudinal response have generated considerable recent interest. The longitudinal data are assumed to follow a mixed effects model, and a proportional hazards model depending on the longitudinal random effects and other covariates is assumed for the survival endpoint. Interest may focus on inference on the longitudinal data process, which is informatively censored, or on the hazard relationship. Several methods for fitting such models have been proposed, most requiring a parametric distributional assumption (normality) on the random effects. A natural concern is sensitivity to violation of this assumption; moreover, a restrictive distributional assumption may obscure key features in the data. We investigate these issues through our proposal of a likelihood-based approach that requires only the assumption that the random effects have a smooth density. Implementation via the EM algorithm is described, and performance and the benefits for uncovering noteworthy features are illustrated by application to data from an HIV clinical trial and by simulation.     

Marginalized transition shared random effects models for longitudinal binary data with nonignorable dropout

In longitudinal studies investigators frequently have to assess and address potential biases introduced by missing data. New methods are proposed for modeling longitudinal categorical data with nonignorable dropout using marginalized transition models and shared random effects models. Random effects are introduced for both serial dependence of outcomes and nonignorable missingness. Fisher‐scoring and Quasi–Newton algorithms are developed for parameter estimation. Methods are illustrated with a real dataset.     

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.     

Competing regression models for longitudinal data

The choice of an appropriate family of linear models for the analysis of longitudinal data is often a matter of concern for practitioners. To attenuate such difficulties, we discuss some issues that emerge when analyzing this type of data via a practical example involving pretest–posttest longitudinal data. In particular, we consider log‐normal linear mixed models (LNLMM), generalized linear mixed models (GLMM), and models based on generalized estimating equations (GEE). We show how some special features of the data, like a nonconstant coefficient of variation, may be handled in the three approaches and evaluate their performance with respect to the magnitude of standard errors of interpretable and comparable parameters. We also show how different diagnostic tools may be employed to identify outliers and comment on available software. We conclude by noting that the results are similar, but that GEE‐based models may be preferable when the goal is to compare the marginal expected responses.     

Median regression models for longitudinal data with dropouts

Summary .  Recently, median regression models have received increasing attention. When continuous responses follow a distribution that is quite different from a normal distribution, usual mean regression models may fail to produce efficient estimators whereas median regression models may perform satisfactorily. In this article, we discuss using median regression models to deal with longitudinal data with dropouts. Weighted estimating equations are proposed to estimate the median regression parameters for incomplete longitudinal data, where the weights are determined by modeling the dropout process. Consistency and the asymptotic distribution of the resultant estimators are established. The proposed method is used to analyze a longitudinal data set arising from a controlled trial of HIV disease ( Volberding et al., 1990 , The New England Journal of Medicine 322, 941–949). Simulation studies are conducted to assess the performance of the proposed method under various situations. An extension to estimation of the association parameters is outlined.     

Multiple-Imputation-Based Residuals and Diagnostic Plots for Joint Models of Longitudinal and Survival Outcomes

Summary .  The majority of the statistical literature for the joint modeling of longitudinal and time-to-event data has focused on the development of models that aim at capturing specific aspects of the motivating case studies. However, little attention has been given to the development of diagnostic and model-assessment tools. The main difficulty in using standard model diagnostics in joint models is the nonrandom dropout in the longitudinal outcome caused by the occurrence of events. In particular, the reference distribution of statistics, such as the residuals, in missing data settings is not directly available and complex calculations are required to derive it. In this article, we propose a multiple-imputation-based approach for creating multiple versions of the completed data set under the assumed joint model. Residuals and diagnostic plots for the complete data model can then be calculated based on these imputed data sets. Our proposals are exemplified using two real data sets.     

Expected estimating equations for missing data, measurement error, and misclassification, with application to longitudinal nonignorable missing data

Summary .   Missing data, measurement error, and misclassification are three important problems in many research fields, such as epidemiological studies. It is well known that missing data and measurement error in covariates may lead to biased estimation. Misclassification may be considered as a special type of measurement error, for categorical data. Nevertheless, we treat misclassification as a different problem from measurement error because statistical models for them are different. Indeed, in the literature, methods for these three problems were generally proposed separately given that statistical modeling for them are very different. The problem is more challenging in a longitudinal study with nonignorable missing data. In this article, we consider estimation in generalized linear models under these three incomplete data models. We propose a general approach based on expected estimating equations (EEEs) to solve these three incomplete data problems in a unified fashion. This EEE approach can be easily implemented and its asymptotic covariance can be obtained by sandwich estimation. Intensive simulation studies are performed under various incomplete data settings. The proposed method is applied to a longitudinal study of oral bone density in relation to body bone density.     

Dynamic prediction of time to a clinical event with sparse and irregularly measured longitudinal biomarkers

In clinical research and practice, landmark models are commonly used to predict the risk of an adverse future event, using patients' longitudinal biomarker data as predictors. However, these data are often observable only at intermittent visits, making their measurement times irregularly spaced and unsynchronized across different subjects. This poses challenges to conducting dynamic prediction at any post-baseline time. A simple solution is the last-value-carry-forward method, but this may result in bias for the risk model estimation and prediction. Another option is to jointly model the longitudinal and survival processes with a shared random effects model. However, when dealing with multiple biomarkers, this approach often results in high-dimensional integrals without a closed-form solution, and thus the computational burden limits its software development and practical use. In this article, we propose to process the longitudinal data by functional principal component analysis techniques, and then use the processed information as predictors in a class of flexible linear transformation models to predict the distribution of residual time-to-event occurrence. The measurement schemes for multiple biomarkers are allowed to be different within subject and across subjects. Dynamic prediction can be performed in a real-time fashion. The advantages of our proposed method are demonstrated by simulation studies. We apply our approach to the African American Study of Kidney Disease and Hypertension, predicting patients' risk of kidney failure or death by using four important longitudinal biomarkers for renal functions.     

Joint Models for Time-to-Event Data and Longitudinal Biomarkers of High Dimension

Statistics in Biosciences - Joint models for longitudinal biomarkers and time-to-event data are widely used in longitudinal studies. Many joint modeling approaches have been proposed to handle...     

A two-part mixed-effects pattern-mixture model to handle zero-inflation and incompleteness in a longitudinal setting

Two-part regression models are frequently used to analyze longitudinal count data with excess zeros, where the same set of subjects is repeatedly observed over time. In this context, several sources of heterogeneity may arise at individual level that affect the observed process. Further, longitudinal studies often suffer from missing values: individuals dropout of the study before its completion, and thus present incomplete data records. In this paper, we propose a finite mixture of hurdle models to face the heterogeneity problem, which is handled by introducing random effects with a discrete distribution; a pattern-mixture approach is specified to deal with non-ignorable missing values. This approach helps us to consider overdispersed counts, while allowing for association between the two parts of the model, and for non-ignorable dropouts. The effectiveness of the proposal is tested through a simulation study. Finally, an application to real data on skin cancer is provided.     

Estimation in Semiparametric Transition Measurement Error Models for Longitudinal Data

Summary .  We consider semiparametric transition measurement error models for longitudinal data, where one of the covariates is measured with error in transition models, and no distributional assumption is made for the underlying unobserved covariate. An estimating equation approach based on the pseudo conditional score method is proposed. We show the resulting estimators of the regression coefficients are consistent and asymptotically normal. We also discuss the issue of efficiency loss. Simulation studies are conducted to examine the finite-sample performance of our estimators. The longitudinal AIDS Costs and Services Utilization Survey data are analyzed for illustration.     

Mixed effects models with censored data with application to HIV RNA levels

Mixed effects models are often used for estimating fixed effects and variance components in longitudinal studies of continuous data. When the outcome being modelled is a laboratory measurement, however, it may be subject to lower and upper detection limits (i.e., censoring). In this paper, the usual EM estimation procedure for mixed effects models is modified to account for left and/or right censoring.     

Dynamic conditionally linear mixed models for longitudinal data

We develop a new class of models, dynamic conditionally linear mixed models, for longitudinal data by decomposing the within-subject covariance matrix using a special Cholesky decomposition. Here 'dynamic' means using past responses as covariates and 'conditional linearity' means that parameters entering the model linearly may be random, but nonlinear parameters are nonrandom. This setup offers several advantages and is surprisingly similar to models obtained from the first-order linearization method applied to nonlinear mixed models. First, it allows for flexible and computationally tractable models that include a wide array of covariance structures; these structures may depend on covariates and hence may differ across subjects. This class of models includes, e.g., all standard linear mixed models, antedependence models, and Vonesh-Carter models. Second, it guarantees the fitted marginal covariance matrix of the data is positive definite. We develop methods for Bayesian inference and motivate the usefulness of these models using a series of longitudinal depression studies for which the features of these new models are well suited.     

Temporal Patterns of Variable Relationships in Person-Oriented Research: Prediction and Auto-Association Models of Configural Frequency Analysis

Longitudinal Configural Frequency Analysis (CFA) seeks to identify, at the manifest variable level, those temporal patterns that are observed more frequently (CFA types) or less frequently (CFA antitypes) than expected with reference to a base model. This article discusses, compares, and extends two base models of interest in longitudinal data analysis. The first of these, Prediction CFA (P-CFA), is a base model that can be used in the configural analysis of both cross-sectional and longitudinal data. This model takes the associations among predictors and among criteria into account. The second base model, Auto-Association CFA (A-CFA), was specifically designed for longitudinal data. This model takes the auto-associations among repeatedly observed variables into account. Both models are extended to accommodate covariates, for example, stratification variables. Application examples are given using data from a longitudinal study of domestic violence. It is illustrated that CFA is able to yield results that are not redundant with results from log-linear modeling or multinomial regression. It is concluded that CFA is particularly useful in the context of person-oriented research.     

The effect of miss-specified baseline characteristics on inference for longitudinal trends in linear mixed models

The main advantage of longitudinal studies is that they can distinguish changes over time within individuals (longitudinal effects) from differences among subjects at the start of the study (baseline characteristics, cross-sectional effects). Often, especially in observational studies, longitudinal trends are studied after correction for many potentially important baseline differences between subjects. We show that, in the context of linear mixed models, inference for longitudinal trends is in general biased if a wrong model for the baseline characteristics is used. However, we will argue that this bias is small in most practical situations and completely vanishes in the special case of a growth curve model for complete balanced data. In the latter case, inference for longitudinal trends is completely independent of additional baseline covariates that might have been omitted from the model.     

A marginalized conditional linear model for longitudinal binary data when informative dropout occurs in continuous time

Within the pattern-mixture modeling framework for informative dropout, conditional linear models (CLMs) are a useful approach to deal with dropout that can occur at any point in continuous time (not just at observation times). However, in contrast with selection models, inferences about marginal covariate effects in CLMs are not readily available if nonidentity links are used in the mean structures. In this article, we propose a CLM for long series of longitudinal binary data with marginal covariate effects directly specified. The association between the binary responses and the dropout time is taken into account by modeling the conditional mean of the binary response as well as the dependence between the binary responses given the dropout time. Specifically, parameters in both the conditional mean and dependence models are assumed to be linear or quadratic functions of the dropout time; and the continuous dropout time distribution is left completely unspecified. Inference is fully Bayesian. We illustrate the proposed model using data from a longitudinal study of depression in HIV-infected women, where the strategy of sensitivity analysis based on the extrapolation method is also demonstrated.     

Marginalized models for moderate to long series of longitudinal binary response data

Marginalized models (Heagerty, 1999, Biometrics 55, 688-698) permit likelihood-based inference when interest lies in marginal regression models for longitudinal binary response data. Two such models are the marginalized transition and marginalized latent variable models. The former captures within-subject serial dependence among repeated measurements with transition model terms while the latter assumes exchangeable or nondiminishing response dependence using random intercepts. In this article, we extend the class of marginalized models by proposing a single unifying model that describes both serial and long-range dependence. This model will be particularly useful in longitudinal analyses with a moderate to large number of repeated measurements per subject, where both serial and exchangeable forms of response correlation can be identified. We describe maximum likelihood and Bayesian approaches toward parameter estimation and inference, and we study the large sample operating characteristics under two types of dependence model misspecification. Data from the Madras Longitudinal Schizophrenia Study (Thara et al., 1994, Acta Psychiatrica Scandinavica 90, 329-336) are analyzed.