Multivariate probit linear mixed models for multivariate longitudinal binary data.

When analyzing multivariate longitudinal binary data, we estimate the effects on the responses of the covariates while accounting for three types of complex correlations present in the data. These include the correlations within separate responses over time, cross-correlations between different responses at different times, and correlations between different responses at each time point. The number of parameters thus increases quadratically with the dimension of the correlation matrix, making parameter estimation difficult; the estimated correlation matrix must also meet the positive definiteness constraint. The correlation matrix may additionally be heteroscedastic; however, the matrix structure is commonly considered to be homoscedastic and constrained, such as exchangeable or autoregressive with order one. These assumptions are overly strong, resulting in skewed estimates of the covariate effects on the responses. Hence, we propose probit linear mixed models for multivariate longitudinal binary data, where the correlation matrix is estimated using hypersphere decomposition instead of the strong assumptions noted above. Simulations and real examples are used to demonstrate the proposed methods. An open source R package, BayesMGLM, is made available on GitHub at https://github.com/kuojunglee/BayesMGLM/ with full documentation to produce the results.

Determination of correlations in multivariate longitudinal data with modified Cholesky and hypersphere decomposition using Bayesian variable selection approach.

In this article, we present a Bayesian framework for multivariate longitudinal data analysis with a focus on selection of important elements in the generalized autoregressive matrix. An efficient Gibbs sampling algorithm was developed for the proposed model and its implementation in a comprehensive R package called MLModelSelection is available on the comprehensive R archive network. The performance of the proposed approach was studied via a comprehensive simulation study. The effectiveness of the methodology was illustrated using a nonalcoholic fatty liver disease dataset to study correlations in multiple responses over time to explain the joint variability of lung functions and body mass index. Supplementary materials for this article, including a standardized description of the materials needed to reproduce the work, are available as an online supplement.

Estimation of covariance matrix of multivariate longitudinal data using modified Choleksky and hypersphere decompositions.

Linear models are typically used to analyze multivariate longitudinal data. With these models, estimating the covariance matrix is not easy because the covariance matrix should account for complex correlated structures: the correlation between responses at each time point, the correlation within separate responses over time, and the cross-correlation between different responses at different times. In addition, the estimated covariance matrix should satisfy the positive definiteness condition, and it may be heteroscedastic. However, in practice, the structure of the covariance matrix is assumed to be homoscedastic and highly parsimonious, such as exchangeable or autoregressive with order one. These assumptions are too strong and result in inefficient estimates of the effects of covariates. Several studies have been conducted to solve these restrictions using modified Cholesky decomposition (MCD) and linear covariance models. However, modeling the correlation between responses at each time point is not easy because there is no natural ordering of the responses. In this paper, we use MCD and hypersphere decomposition to model the complex correlation structures for multivariate longitudinal data. We observe that the estimated covariance matrix using the decompositions is positive-definite and can be heteroscedastic and that it is also interpretable. The proposed methods are illustrated using data from a nonalcoholic fatty liver disease study.

ARMA Cholesky Factor Models for the Covariance Matrix of Linear Models.

In longitudinal studies, serial dependence of repeated outcomes must be taken into account to make correct inferences on covariate effects. As such, care must be taken in modeling the covariance matrix. However, estimation of the covariance matrix is challenging because there are many parameters in the matrix and the estimated covariance matrix should be positive definite. To overcomes these limitations, two Cholesky decomposition approaches have been proposed: modified Cholesky decomposition for autoregressive (AR) structure and moving average Cholesky decomposition for moving average (MA) structure, respectively. However, the correlations of repeated outcomes are often not captured parsimoniously using either approach separately. In this paper, we propose a class of flexible, nonstationary, heteroscedastic models that exploits the structure allowed by combining the AR and MA modeling of the covariance matrix that we denote as ARMACD. We analyze a recent lung cancer study to illustrate the power of our proposed methods.

Flexible marginalized models for bivariate longitudinal ordinal data.

Random effects models are commonly used to analyze longitudinal categorical data. Marginalized random effects models are a class of models that permit direct estimation of marginal mean parameters and characterize serial correlation for longitudinal categorical data via random effects (Heagerty, 1999). Marginally specified logistic-normal models for longitudinal binary data. Biometrics 55, 688-698; Lee and Daniels, 2008. Marginalized models for longitudinal ordinal data with application to quality of life studies. Statistics in Medicine 27, 4359-4380). In this paper, we propose a Kronecker product (KP) covariance structure to capture the correlation between processes at a given time and the correlation within a process over time (serial correlation) for bivariate longitudinal ordinal data. For the latter, we consider a more general class of models than standard (first-order) autoregressive correlation models, by re-parameterizing the correlation matrix using partial autocorrelations (Daniels and Pourahmadi, 2009). Modeling covariance matrices via partial autocorrelations. Journal of Multivariate Analysis 100, 2352-2363). We assess the reasonableness of the KP structure with a score test. A maximum marginal likelihood estimation method is proposed utilizing a quasi-Newton algorithm with quasi-Monte Carlo integration of the random effects. We examine the effects of demographic factors on metabolic syndrome and C-reactive protein using the proposed models.

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.

Causal inference for bivariate longitudinal quality of life data in presence of death by using global odds ratios.

In longitudinal clinical trials, if a subject drops out due to death, certain responses, such as those measuring quality of life (QoL), will not be defined after the time of death. Thus, standard missing data analyses, e.g., under ignorable dropout, are problematic because these approaches implicitly 'impute' values of the response after death. In this paper we define a new survivor average causal effect for a bivariate response in a longitudinal quality of life study that had a high dropout rate with the dropout often due to death (or tumor progression). We show how principal stratification, with a few sensitivity parameters, can be used to draw causal inferences about the joint distribution of these two ordinal quality of life measures.

Model-free predictor tests in survival regression through sufficient dimension reduction.

In this article, we test the effects of predictors in survival regression through two well-known sufficient dimension reduction methods. Since the usual sufficient dimension reduction methods do not require pre-specified models, the predictor effect tests can be considered model-free. All of the test statistics have χ (2) distributions. Numerical studies of the proposed predictor effect tests in various simulations and real data application are presented.

Normalization of dye bias in microarray data using the mixture of splines model.

This paper discusses characteristics of dye biases in microarray data that the conventional normalization methods do not handle, and proposes a new normalization method involving a mixture of splines model. We also develop a test for between-group comparisons of each gene that is designed to be used with our proposed method.

CAUSAL EFFECTS OF TREATMENTS FOR INFORMATIVE MISSING DATA DUE TO PROGRESSION/DEATH.

In longitudinal clinical trials, when outcome variables at later time points are only defined for patients who survive to those times, the evaluation of the causal effect of treatment is complicated. In this paper, we describe an approach that can be used to obtain the causal effect of three treatment arms with ordinal outcomes in the presence of death using a principal stratification approach. We introduce a set of flexible assumptions to identify the causal effect and implement a sensitivity analysis for non-identifiable assumptions which we parameterize parsimoniously. Methods are illustrated on quality of life data from a recent colorectal cancer clinical trial.

Marginalized random effects models for multivariate longitudinal binary data.

Generalized linear models with random effects are often used to explain the serial dependence of longitudinal categorical data. Marginalized random effects models (MREMs) permit likelihood-based estimations of marginal mean parameters and also explain the serial dependence of longitudinal data. In this paper, we extend the MREM to accommodate multivariate longitudinal binary data using a new covariance matrix with a Kronecker decomposition, which easily explains both the serial dependence and time-specific response correlation. A maximum marginal likelihood estimation is proposed utilizing a quasi-Newton algorithm with quasi-Monte Carlo integration of the random effects. Our approach is applied to analyze metabolic syndrome data from the Korean Genomic Epidemiology Study for Korean adults.

Marginalized models for longitudinal ordinal data with application to quality of life studies.

Random effects are often used in generalized linear models to explain the serial dependence for longitudinal categorical data. Marginalized random effects models (MREMs) for the analysis of longitudinal binary data have been proposed to permit likelihood-based estimation of marginal regression parameters. In this paper, we propose a model to extend the MREM to accommodate longitudinal ordinal data. Maximum marginal likelihood estimation is proposed utilizing quasi-Newton algorithms with Monte Carlo integration of the random effects. Our approach is applied to analyze the quality of life data from a recent colorectal cancer clinical trial. Dropout occurs at a high rate and is often due to tumor progression or death. To deal with events due to progression/death, we used a mixture model for the joint distribution of longitudinal measures and progression/death times and use principal stratification to draw causal inferences about survivors.

A class of markov models for longitudinal ordinal data.

Generalized linear models with serial dependence are often used for short longitudinal series. Heagerty (2002, Biometrics58, 342-351) has proposed marginalized transition models for the analysis of longitudinal binary data. In this article, we extend this work to accommodate longitudinal ordinal data. Fisher-scoring algorithms are developed for estimation. Methods are illustrated on quality-of-life data from a recent colorectal cancer clinical trial.