Joint regression analysis for discrete longitudinal data

Summary We introduce an approximation to the Gaussian copula likelihood of Song, Li, and Yuan (2009, Biometrics 65, 60–68) used to estimate regression parameters from correlated discrete or mixed bivariate or trivariate outcomes. Our approximation allows estimation of parameters from response vectors of length much larger than three, and is asymptotically equivalent to the Gaussian copula likelihood. We estimate regression parameters from the toenail infection data of De Backer et al. (1996, British Journal of Dermatology 134, 16–17), which consist of binary response vectors of length seven or less from 294 subjects. Although maximizing the Gaussian copula likelihood yields estimators that are asymptotically more efficient than generalized estimating equation (GEE) estimators, our simulation study illustrates that for finite samples, GEE estimators can actually be as much as 20% more efficient.     

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.     

Gaussian process functional regression modeling for batch data

A Gaussian process functional regression model is proposed for the analysis of batch data. Covariance structure and mean structure are considered simultaneously, with the covariance structure modeled by a Gaussian process regression model and the mean structure modeled by a functional regression model. The model allows the inclusion of covariates in both the covariance structure and the mean structure. It models the nonlinear relationship between a functional output variable and a set of functional and nonfunctional covariates. Several applications and simulation studies are reported and show that the method provides very good results for curve fitting and prediction.     

Structured additive regression for categorical space-time data: a mixed model approach

Motivated by a space-time study on forest health with damage state of trees as the response, we propose a general class of structured additive regression models for categorical responses, allowing for a flexible semiparametric predictor. Nonlinear effects of continuous covariates, time trends, and interactions between continuous covariates are modeled by penalized splines. Spatial effects can be estimated based on Markov random fields, Gaussian random fields, or two-dimensional penalized splines. We present our approach from a Bayesian perspective, with inference based on a categorical linear mixed model representation. The resulting empirical Bayes method is closely related to penalized likelihood estimation in a frequentist setting. Variance components, corresponding to inverse smoothing parameters, are estimated using (approximate) restricted maximum likelihood. In simulation studies we investigate the performance of different choices for the spatial effect, compare the empirical Bayes approach to competing methodology, and study the bias of mixed model estimates. As an application we analyze data from the forest health survey.     

Behavioral testing of antidepressant compounds: an analysis of crossover design for correlated binary data

The differential reinforcement of low-rate 72 seconds schedule (DRL-72) is a standard behavioral test procedure for screening potential antidepressant compounds. The protocol for the DRL-72 experiment, proposed by Evenden et al. (1993), consists of using a crossover design for the experiment and one-way ANOVA for the statistical analysis. In this paper we discuss the choice of several crossover designs for the DRL-72 experiment and propose to estimate the treatment effects using either generalized linear mixed models (GLMM) or generalized estimating equation (GEE) models for clustered binary data.     

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.     

Competing risks analysis of correlated failure time data

Summary .   We develop methods for competing risks analysis when individual event times are correlated within clusters. Clustering arises naturally in clinical genetic studies and other settings. We develop a nonparametric estimator of cumulative incidence, and obtain robust pointwise standard errors that account for within-cluster correlation. We modify the two-sample Gray and Pepe–Mori tests for correlated competing risks data, and propose a simple two-sample test of the difference in cumulative incidence at a landmark time. In simulation studies, our estimators are asymptotically unbiased, and the modified test statistics control the type I error. The power of the respective two-sample tests is differentially sensitive to the degree of correlation; the optimal test depends on the alternative hypothesis of interest and the within-cluster correlation. For purposes of illustration, we apply our methods to a family-based prospective cohort study of hereditary breast/ovarian cancer families. For women with BRCA1 mutations, we estimate the cumulative incidence of breast cancer in the presence of competing mortality from ovarian cancer, accounting for significant within-family correlation.     

The analysis of longitudinal data using mixed model L-splines

Summary .   L-splines are a large family of smoothing splines defined in terms of a linear differential operator. This article develops L-splines within the context of linear mixed models and uses the resulting mixed model L-spline to analyze longitudinal data from a grassland experiment. In the spirit of time-series analysis, a periodic mixed model L-spline is developed, which partitions data into a smooth periodic component plus smooth long-term trend.     

Flexible copula model for integrating correlated multi-omics data from single-cell experiments

With recent advances in technologies to profile multi-omics data at the single-cell level, integrative multi-omics data analysis has been increasingly popular. It is increasingly common that information such as methylation changes, chromatin accessibility, and gene expression are jointly collected in a single-cell experiment. In biomedical studies, it is often of interest to study the associations between various data types and to examine how these associations might change according to other factors such as cell types and gene regulatory components. However, since each data type usually has a distinct marginal distribution, joint analysis of these changes of associations using multi-omics data is statistically challenging. In this paper, we propose a flexible copula-based framework to model covariate-dependent correlation structures independent of their marginals. In addition, the proposed approach could jointly combine a wide variety of univariate marginal distributions, either discrete or continuous, including the class of zero-inflated distributions. The performance of the proposed framework is demonstrated through a series of simulation studies. Finally, it is applied to a set of experimental data to investigate the dynamic relationship between single-cell RNA sequencing, chromatin accessibility, and DNA methylation at different germ layers during mouse gastrulation.     

Effects of variance-function misspecification in analysis of longitudinal data

The approach of generalized estimating equations (GEE) is based on the framework of generalized linear models but allows for specification of a working matrix for modeling within-subject correlations. The variance is often assumed to be a known function of the mean. This article investigates the impacts of misspecifying the variance function on estimators of the mean parameters for quantitative responses. Our numerical studies indicate that (1) correct specification of the variance function can improve the estimation efficiency even if the correlation structure is misspecified; (2) misspecification of the variance function impacts much more on estimators for within-cluster covariates than for cluster-level covariates; and (3) if the variance function is misspecified, correct choice of the correlation structure may not necessarily improve estimation efficiency. We illustrate impacts of different variance functions using a real data set from cow growth.     

Marginal hazards regression for retrospective studies within cohort with possibly correlated failure time data

Summary .  A retrospective dental study was conducted to evaluate the degree to which pulpal involvement affects tooth survival. Due to the clustering of teeth, the survival times within each subject could be correlated and thus the conventional method for the case–control studies cannot be directly applied. In this article, we propose a marginal model approach for this type of correlated case–control within cohort data. Weighted estimating equations are proposed for the estimation of the regression parameters. Different types of weights are also considered for improving the efficiency. Asymptotic properties of the proposed estimators are investigated and their finite sample properties are assessed via simulations studies. The proposed method is applied to the aforementioned dental study.     

Marginal mark regression analysis of recurrent marked point process data

Summary .  Longitudinal studies typically collect information on the timing of key clinical events and on specific characteristics that describe those events. Random variables that measure qualitative or quantitative aspects associated with the occurrence of an event are known as marks. Recurrent marked point process data consist of possibly recurrent events, with the mark (and possibly exposure) measured if and only if an event occurs. Analysis choices depend on which aspect of the data is of primary scientific interest. First, factors that influence the occurrence or timing of the event may be characterized using recurrent event analysis methods. Second, if there is more than one event per subject, then the association between exposure and the mark may be quantified using repeated measures regression methods. We detail assumptions required of any time-dependent exposure process and the event time process to ensure that linear or generalized linear mixed models and generalized estimating equations provide valid estimates. We provide theoretical and empirical evidence that if these conditions are not satisfied, then an independence estimating equation should be used for consistent estimation of association. We conclude with the recommendation that analysts carefully explore both the exposure and event time processes prior to implementing a repeated measures analysis of recurrent marked point process data.     

Bayesian inference for stationary points in Gaussian process regression models for event-related potentials analysis

Stationary points embedded in the derivatives are often critical for a model to be interpretable and may be considered as key features of interest in many applications. We propose a semiparametric Bayesian model to efficiently infer the locations of stationary points of a nonparametric function, which also produces an estimate of the function. We use Gaussian processes as a flexible prior for the underlying function and impose derivative constraints to control the function's shape via conditioning. We develop an inferential strategy that intentionally restricts estimation to the case of at least one stationary point, bypassing possible mis-specifications in the number of stationary points and avoiding the varying dimension problem that often brings in computational complexity. We illustrate the proposed methods using simulations and then apply the method to the estimation of event-related potentials derived from electroencephalography (EEG) signals. We show how the proposed method automatically identifies characteristic components and their latencies at the individual level, which avoids the excessive averaging across subjects that is routinely done in the field to obtain smooth curves. By applying this approach to EEG data collected from younger and older adults during a speech perception task, we are able to demonstrate how the time course of speech perception processes changes with age.     

A mixed-effects regression model for longitudinal multivariate ordinal data

A mixed-effects item response theory model that allows for three-level multivariate ordinal outcomes and accommodates multiple random subject effects is proposed for analysis of multivariate ordinal outcomes in longitudinal studies. This model allows for the estimation of different item factor loadings (item discrimination parameters) for the multiple outcomes. The covariates in the model do not have to follow the proportional odds assumption and can be at any level. Assuming either a probit or logistic response function, maximum marginal likelihood estimation is proposed utilizing multidimensional Gauss-Hermite quadrature for integration of the random effects. An iterative Fisher scoring solution, which provides standard errors for all model parameters, is used. An analysis of a longitudinal substance use data set, where four items of substance use behavior (cigarette use, alcohol use, marijuana use, and getting drunk or high) are repeatedly measured over time, is used to illustrate application of the proposed model.     

Marginal regression analysis of a multivariate binary response

We propose the use of the mean parameter for regression analysisof a multivariate binary response. We model the associationusing dependence ratios defined in terms of the mean parameter,the components of which are the joint success probabilitiesof all orders. This permits flexible modelling of higher-orderassociations, using maximum likelihood estimation. We reanalysetwo data sets, one with variable cluster size and the othera longitudinal data set with constant cluster size.     

Bayesian hierarchical spatially correlated functional data analysis with application to colon carcinogenesis

Summary .   In this article, we present new methods to analyze data from an experiment using rodent models to investigate the role of p27, an important cell-cycle mediator, in early colon carcinogenesis. The responses modeled here are essentially functions nested within a two-stage hierarchy. Standard functional data analysis literature focuses on a single stage of hierarchy and conditionally independent functions with near white noise. However, in our experiment, there is substantial biological motivation for the existence of spatial correlation among the functions, which arise from the locations of biological structures called colonic crypts: this possible functional correlation is a phenomenon we term crypt signaling . Thus, as a point of general methodology, we require an analysis that allows for functions to be correlated at the deepest level of the hierarchy. Our approach is fully Bayesian and uses Markov chain Monte Carlo methods for inference and estimation. Analysis of this data set gives new insights into the structure of p27 expression in early colon carcinogenesis and suggests the existence of significant crypt signaling. Our methodology uses regression splines, and because of the hierarchical nature of the data, dimension reduction of the covariance matrix of the spline coefficients is important: we suggest simple methods for overcoming this problem.     

Nonparametric regression using local kernel estimating equations for correlated failure time data

We study nonparametric regression for correlated failure timedata. Kernel estimating equations are used to estimate nonparametriccovariate effects. Independent and weighted-kernel estimatingequations are studied. The derivative of the nonparametric functionis first estimated and the nonparametric function is then estimatedby integrating the derivative estimator. We show that the nonparametrickernel estimator is consistent for any arbitrary working correlationmatrix and that its asymptotic variance is minimized by assumingworking independence. We evaluate the performance of the proposedkernel estimator using simulation studies, and apply the proposedmethod to the western Kenya parasitaemia data.