Bayesian analysis of covariance matrices and dynamic models for longitudinal data

Bayesian dynamic probit models for the analysis of longitudinal data

Analysis of binary longitudinal data with time-varying effects

There are various analysis methods for CT perfusion (CTP). Although the advantages of Bayesian estimation algorithms have been newly suggested, comparisons with other analysis methods on clinical data are still limited. In this study, we compared the Bayesian estimation method with the singular value decomposition (SVD) method in the evaluation of patients with acute cerebral infarction and examined its usefulness. CTP data from 13 patients with acute stroke were analyzed using the SVD and Bayesian estimation methods implemented in Vitrea. Evaluation of visual clarity of the ischemic area and quantitative values of the healthy side-affected side ratio using the mean values of the left and right region of interest (ROI) on the images were compared using the SVD and Bayesian estimation methods. In visual evaluation, there were significant differences in CBV in four cases, and in CBF, MTT, and TTP in many cases. The healthy side-affected side ratio of the SVD and Bayesian estimation methods were as follows: CBF 1.19, 1.84; CBV 1.09, 1.02; MTT 1.12, 1.79; and TTP 1.48, 1.19. For CBF and MTT, the Bayesian estimation method had a larger ratio of the healthy side to the affected side, and for TTP, the SVD method had a larger ratio of the test side to the affected side. We suggest that the Bayesian estimation method is more useful than the SVD method for assessing CBF and MTT in CTP analysis of patients with acute stroke.

The main difficulty in parametric analysis of longitudinal data lies in specifying covariance structure. Several covariance structures, which usually reflect one series of measurements collected over time, have been presented in the literature. However there is a lack of literature on covariance structures designed for repeated measures specified by more than one repeated factor. In this paper a new, general method of modelling covariance structure based on the Kronecker product of underlying factor specific covariance profiles is presented. The method has an attractive interpretation in terms of independent factor specific contribution to overall within subject covariance structure and can be easily adapted to standard software.

Longitudinal designs for biomarker traits are very appealing for a number of reasons. First, the collection of repeated biomarker measurements over time allows investigators to characterize both the overall and individual trajectories in these measurements. This helps investigators to understand overall effects as well as how individuals vary around these average effects. Distinguishing individual heterogeneity from average trend is impossible without longitudinal data. This distinction is important for understanding individual between-subject in its own right, as well as in developing predictors of disease outcome from longitudinal marker data. There are many statistical issues that are unique to longitudinal data analysis ranging from correctly specifying realistic models which characterize average changes over time and individual variation, accounting for observations that are taken at unequally spaced observation times which may differ between individuals, to accounting for missing data due to occasionally missed measurements or loss to follow-up. When the reason for missing data is not related to the underlying observation process, most longitudinal data analysis methods can be routinely applied. However, complex modelling is necessary when the reason for missing an observation or dropout is directly related to the underlying outcome process [1–2]. Statistical techniques that appropriately analyze continuous and discrete longitudinal data have been discussed by many authors. Diggle, et al. [3] provides a good summary of these techniques for the practitioner, while Fitzmaurice et al. [4] discusses recent advances from a more technical perspective. The three papers in this section present innovative approaches for solving important methodological problems in biomarker studies. Most standard longitudinal data analysis techniques assume that the measurement process is independent of the outcome process. However, when the outcome is related to the underlying observation process, failure to properly account for the observation process can result in bias [5]. Schildcrout et al.[6] proposed an approach for regression modelling for longitudinal binary data where the outcome is measured with probabilities determined by an auxiliary variable that is related, but not identical to the binary outcome. Modelling multivariate longitudinal biomarker data is an increasingly common issue with the advent of an increasing number of important biomarker techniques (e.g., multiple cytokines in a multiplex assay or multiple reproductive hormones that describe complex biological phenomenon). Although various authors have proposed techniques for analyzing multivariate longitudinal data [7–8 and others], little work has focused on estimating multivariate linear mixed models under known constraints in the relationship between the multivariate outcomes. Roy et al [9] proposes such an approach for analyzing multivariate reproductive hormones over the menstrual cycle, where established relationships have solid biological justification. Finally, the collection of longitudinal biomarker data as disease predictors is becoming an increasingly important design in diagnostic medicine. Pfeiffer et al. [10] proposes flexible non-parametric methodology for combining biomarkers that exploits the longitudinal structure in formulating a score that is used for predicting a binary outcome.

Robust growth mixture models with non-ignorable missingness: Models, estimation, selection, and application

Summary. In long-term follow-up studies, irregular longitudinal data are observed when individuals are assessed repeatedly over time, but at uncommon and irregularly spaced time points. Modeling the covariance structure for this type of data is challenging as it requires specification of a covariance function that is positive definite. Moreover, in certain settings, careful modeling of the covariance structure for irregular longitudinal data can be crucial in order to ensure no bias arises in the mean structure. Two common settings where this occurs are studies with ‘outcome-dependent follow-up’ and studies with ‘ignorable dropout’. ‘outcome-dependent follow-up’ occurs when individuals with a history of poor health outcomes had more follow-up measurements and the intervals between the repeated measurements were shorter. When the follow-up time process only depends on previous outcomes, likelihood-based methods can still provide consistent estimates of the regression parameters given that both the mean and covariance structures of the irregular longitudinal data are correctly specified, while no model for the follow-up time process is required. For ‘ignorable missing data’, the missing data mechanism does not need to be specified, but valid likelihood-based inference requires correct specification of the covariance structure. In both cases, flexible modeling approaches for the covariance structure are essential. In this paper, we develop a flexible approach to modeling the covariance structure for irregular continuous longitudinal data using the partial autocorrelation function (PACF) and the variance function. In particular, we propose semiparametric non-stationary PACF models which do not suffer from complex positive definiteness restrictions like the autocorrelation function. We describe a Bayesian approach, discuss computational issues, and apply the proposed methods to CD4 count data from a pediatric AIDS clinical trial.

Growth curve mixture models with unknown covariance structures

Experiments involving repeated observations of multivariate outcomes are common in biomedical and public health researches which lead to multivariate longitudinal data. These kinds of data have a unique property in the sense that they allow the researcher to study the joint evolution of the multiple outcomes over the time. Recently, there has been a considerable amount of interest on using Bayesian modelling of longitudinal data, data which commonly suffer from incomplete observations. Those Bayesian models for longitudinal data that rely on the ignorability assumption of the dropout mechanism might give misleading inferences. Hence, there is a need to further study the impact of departures from the ignorability assumption on the Bayesian estimates of the model parameters. Current methodology for Bayesian sensitivity analysis mostly involves single response variable in both cross-sectional and longitudinal studies. In this paper, we propose a multivariate extension of the Bayesian index of sensitivity to non-ignorability for the general case of multivariate longitudinal studies with the possibility of having mixed correlated outcomes and a vector of multiple non-ignorability parameters in the missing mechanism. To simultaneously model the mixed responses over the time, we use a random effect latent variable approach. We illustrate the method conducting some simulation studies and analyzing a real data set from a longitudinal study for the comparison of two oral treatments for toenail dermatophyte onychomycosis.

The smoothness of subdiagonals of the Cholesky factor of large covariance matrices is closely related to the degree of nonstationarity of autoregressive models for time series data. Heuristically, one expects for nearly stationary covariance matrix entries in each subdiagonal of the Cholesky factor of its inverse to be approximately the same in the sense that the sum of the absolute values of successive differences is small. Statistically, such smoothness is achieved by regularizing each subdiagonal using fused-type lasso penalties. We rely on the standard Cholesky factor as the new parameter within a regularized normal likelihood setup which guarantees: (a) joint convexity of the likelihood function, (b) strict convexity of the likelihood function restricted to each subdiagonal even when n < p, and (c) positive-definiteness of the estimated covariance matrix. A block coordinate descent algorithm, where each block is a subdiagonal, is proposed, and its convergence is established under mild conditions. Lack of decoupling of the penalized likelihood function into a sum of functions involving individual subdiagonals gives rise to some computational challenges and advantages relative to two recent algorithms for sparse estimation of the Cholesky factor, which decouple row-wise. Simulation results and real data analysis show the scope and good performance of the proposed methodology. Software for our method is freely available in R language. Supplementary materials for this article are available online.

The linear mixed-effects model (LMM) is widely used in the analysis of clustered or longitudinal data. This paper aims to address analytic challenges arising from estimation and selection in the application of the LMM to high-dimensional longitudinal data. We develop a doubly regularized approach in the LMM to simultaneously select fixed and random effects. On the theoretical front, we establish large sample properties for the proposed method under the high-dimensional setting, allowing both numbers of fixed effects and random effects to be much larger than the sample size. We present new regularity conditions for the diverging rates, under which the proposed method achieves both estimation and selection consistency. In addition, we propose a new algorithm that solves the related optimization problem effectively so that its computational cost is comparable with that of the Newton-Raphson algorithm for maximum likelihood estimator in the LMM. Through simulation studies we assess performances of the proposed regularized LMM in both aspects of variable selection and estimation. We also illustrate the proposed method by two data analysis examples.

Model selection is an essential issue in longitudinal data analysis since many different models have been proposed to fit the covariance structure. The likelihood criterion is commonly used and allows to compare the fit of alternative models. Its value does not reflect, however, the potential improvement that can still be reached in fitting the data unless a reference model with the actual covariance structure is available. The score test approach does not require the knowledge of a reference model, and the score statistic has a meaningful interpretation in itself as a goodness-of-fit measure. The aim of this paper was to show how the score statistic may be separated into the genetic and environmental parts, which is difficult with the likelihood criterion, and how it can be used to check parametric assumptions made on variance and correlation parameters. Selection of models for genetic analysis was applied to a dairy cattle example for milk production.

Estimating large covariance and precision (inverse covariance) matrices has become increasingly important in high dimensional statistics because of its wide applications. The estimation problem is challenging not only theoretically due to the constraint of its positive definiteness, but also computationally because of the curse of dimensionality. Many types of estimators have been proposed such as thresholding under the sparsity assumption of the target matrix, banding and tapering the sample covariance matrix. However, these estimators are not always guaranteed to be positive-definite, especially, for finite samples, and the sparsity assumption is rather restrictive. We propose a novel two-stage adaptive method based on the Cholesky decomposition of a general covariance matrix. By banding the precision matrix in the first stage and adapting the estimates to the second stage estimation, we develop a computationally efficient and statistically accurate method for estimating high dimensional precision matrices. We demonstrate the finite-sample performance of the proposed method by simulations from autoregressive, moving average, and long-range dependent processes. We illustrate its wide applicability by analyzing financial data such S&amp;P 500 index and IBM stock returns, and electric power consumption of individual households. The theoretical properties of the proposed method are also investigated within a large class of covariance matrices.

This paper is motivated from a neurophysiological study of muscle fatigue, in which biomedical researchers are interested in understanding the time-dependent relationships of handgrip force and electromyography measures. A varying coefficient model is appealing here to investigate the dynamic pattern in the longitudinal data. The response variable in the study is continuous but bounded on the standard unit interval (0, 1) over time, while the longitudinal covariates are contaminated with measurement errors. We propose a generalization of varying coefficient models for the longitudinal proportional data with errors-in-covariates. We describe two estimation methods with penalized splines, which are formalized under a Bayesian inferential perspective. The first method is an adaptation of the popular regression calibration approach. The second method is based on a joint likelihood under the hierarchical Bayesian model. A simulation study is conducted to evaluate the efficacy of the proposed methods under different scenarios. The analysis of the neurophysiological data is presented to demonstrate the use of the methods.

We provide sufficient conditions for estimating from longitudinal data the causal effect of a time-dependent exposure or treatment on the marginal probability of response for a dichotomous outcome. We then show how one can estimate this effect under these conditions using the g-computation algorithm of Robins. We also derive the conditions under which some current approaches to the analysis of longitudinal data, such as the generalized estimating equations (GEE) approach of Zeger and Liang, the feedback model techniques of Liang and Zeger, and within-subject conditional methods, can provide valid tests and estimates of causal effects. We use our methods to estimate the causal effect of maternal stress on the marginal probability of a child's illness from the Mothers' Stress and Children's Morbidity data and compare our results with those previously obtained by Zeger and Liang using a GEE approach.