Bivariate random effect model using skew-normal distribution with application to HIV-RNA

Multivariate linear mixed models with censored and nonignorable missing outcomes, with application to AIDS studies

The analysis of multivariate longitudinal data could encounter some complications due to censorship induced by detection limits of the assay and nonresponse occurring when participants missed scheduled visits intermittently or discontinued participation. This paper establishes a generalization of the multivariate linear mixed model that can accommodate censored responses and nonignorable missing outcomes simultaneously. To account for the nonignorable missingness, the selection approach which decomposes the joint distribution as a marginal distribution for the primary outcome variables and a model describing the missing process conditional on the hypothetical complete data is used. A computationally feasible Monte Carlo expectation conditional maximization algorithm is developed for parameter estimation with the maximum likelihood (ML) method. Furthermore, a general information-based approach is presented to assess the variability of ML estimators. The techniques for the prediction of censored responses and imputation of missing outcomes are also discussed. The methodology is motivated and exemplified by a real dataset concerning HIV-AIDS clinical trials. A simulation study is conducted to examine the performance of the proposed method compared with other traditional approaches.

A Comparison of Bayesian Spatial Models for HIV Mapping in South Africa

Despite making significant progress in tackling its HIV epidemic, South Africa, with 7.7 million people living with HIV, still has the biggest HIV epidemic in the world. The Government, in collaboration with developmental partners and agencies, has been strengthening its responses to the HIV epidemic to better target the delivery of HIV care, treatment strategies and prevention services. Population-based household HIV surveys have, over time, contributed to the country’s efforts in monitoring and understanding the magnitude and heterogeneity of the HIV epidemic. Local-level monitoring of progress made against HIV and AIDS is increasingly needed for decision making. Previous studies have provided evidence of substantial subnational variation in the HIV epidemic. Using HIV prevalence data from the 2016 South African Demographic and Health Survey, we compare three spatial smoothing models, namely, the intrinsically conditionally autoregressive normal, Laplace and skew-t (ICAR-normal, ICAR-Laplace and ICAR-skew-t) in the estimation of the HIV prevalence across 52 districts in South Africa. The parameters of the resulting models are estimated using Bayesian approaches. The skewness parameter for the ICAR-skew-t model was not statistically significant, suggesting the absence of skewness in the HIV prevalence data. Based on the deviance information criterion (DIC) model selection, the ICAR-normal and ICAR-Laplace had DIC values of 291.3 and 315, respectively, which were lower than that of the ICAR-skewed t (348.1). However, based on the model adequacy criterion using the conditional predictive ordinates (CPO), the ICAR-skew-t distribution had the lowest CPO value. Thus, the ICAR-skew-t was the best spatial smoothing model for the estimation of HIV prevalence in our study.

A curvilinear bivariate random changepoint model to assess temporal order of markers

In biomedical research, various longitudinal markers measuring different quantities are often collected over time. For example, repeated measures of psychometric scores are very informative about the degradation process toward dementia. These trajectories are generally nonlinear with an acceleration of the decline a few years before the diagnosis and a large heterogeneity between psychometric tests depending on the underlying cognitive function to be evaluated and the metrological properties of the test. Comparing the times of acceleration of the decline before diagnosis between cognitive tests is useful to better understand the natural history of the disease. Our objective is to propose a bivariate random changepoint model that allows for the comparison of the mean time of change between two markers. A frequentist approach is proposed that gives validated statistical tests to assess the temporal order of the changepoints. Using a spline transformation function, the model is designed to handle non-Gaussian data, that are common for cognitive scores which frequently exhibit a strong ceiling effect. The procedure is assessed through a simulation study and applied to a French cohort of elderly to identify the order of the decline of several cognitive scores. The whole methodology has been implemented in a R package freely available.

Cumulative viral load as a predictor of CD4+ T-cell response to antiretroviral therapy using Bayesian statistical models

INTRODUCTION

There are Challenges in statistically modelling immune responses to longitudinal HIV viral load exposure as a function of covariates. We define Bayesian Markov Chain Monte Carlo mixed effects models to incorporate priors and examine the effect of different distributional assumptions. We prospectively fit these models to an as-yet-unpublished data from the Tshwane District Hospital HIV treatment clinic in South Africa, to determine if cumulative log viral load, an indicator of long-term viral exposure, is a valid predictor of immune response.

METHODS

Models are defined, to express 'slope', i.e. mean annual increase in CD4 counts, and 'asymptote', i.e. the odds of having a CD4 count ≥500 cells/μL during antiretroviral treatment, as a function of covariates and random-effects. We compare the effect of using informative versus non-informative prior distributions on model parameters. Models with cubic splines or Skew-normal distributions are also compared using the conditional Deviance Information Criterion.

RESULTS

The data of 750 patients are analyzed. Overall, models adjusting for cumulative log viral load provide a significantly better fit than those that do not. An increase in cumulative log viral load is associated with a decrease in CD4 count slope (19.6 cells/μL (95% credible interval: 28.26, 10.93)) and a reduction in the odds of achieving a CD4 counts ≥500 cells/μL (0.42 (95% CI: 0.236, 0.730)) during 5 years of therapy. Using informative priors improves the cumulative log viral load estimate, and a skew-normal distribution for the random-intercept and measurement error results is a better fit compared to using classical Gaussian distributions.

DISCUSSION

We demonstrate in an unpublished South African cohort that cumulative log viral load is a strong and significant predictor of both CD4 count slope and asymptote. We argue that Bayesian methods should be used more frequently for such data, given their flexibility to incorporate prior information and non-Gaussian distributions.

A Bayesian Framework for Estimating the Concordance Correlation Coefficient Using Skew-elliptical Distributions

The concordance correlation coefficient (CCC) is a widely used scaled index in the study of agreement. In this article, we propose estimating the CCC by a unified Bayesian framework that can (1) accommodate symmetric or asymmetric and light- or heavy-tailed data; (2) select model from several candidates; and (3) address other issues frequently encountered in practice such as confounding covariates and missing data. The performance of the proposal was studied and demonstrated using simulated as well as real-life biomarker data from a clinical study of an insomnia drug. The implementation of the proposal is accessible through a package in the Comprehensive R Archive Network.

Analysis of Censored Longitudinal Data with Skewness and a Terminal Event

In HIV/AIDS study, the measurements viral load are often highly skewed and left-censored because of a lower detection limit. Furthermore, a terminal event (e.g., death) stops the follow-up process. The time to terminal event may be dependent on the viral load measurements. In this article, we present a joint analysis framework to model the censored longitudinal data with skewness and a terminal event process. The estimation is carried out by adaptive Gaussian quadrature techniques in SAS procedure NLMIXED. The proposed model is evaluated by a simulation study and is applied to the motivating Multicenter AIDS Cohort Study (MACS).

Testing multiple hypotheses with skewed alternatives

In many practical cases of multiple hypothesis problems, it can be expected that the alternatives are not symmetrically distributed. If it is known a priori that the distributions of the alternatives are skewed, we show that this information yields high power procedures as compared to the procedures based on symmetric alternatives when testing multiple hypotheses. We propose a Bayesian decision theoretic rule for multiple directional hypothesis testing, when the alternatives are distributed as skewed, under a constraint on a mixed directional false discovery rate. We compare the proposed rule with a frequentist's rule of Benjamini and Yekutieli (2005) using simulations. We apply our method to a well-studied HIV dataset.

Log-gamma linear-mixed effects models for multiple outcomes with application to a longitudinal glaucoma study

Glaucoma is a progressive disease due to damage in the optic nerve with associated functional losses. Although the relationship between structural and functional progression in glaucoma is well established, there is disagreement on how this association evolves over time. In addressing this issue, we propose a new class of non-Gaussian linear-mixed models to estimate the correlations among subject-specific effects in multivariate longitudinal studies with a skewed distribution of random effects, to be used in a study of glaucoma. This class provides an efficient estimation of subject-specific effects by modeling the skewed random effects through the log-gamma distribution. It also provides more reliable estimates of the correlations between the random effects. To validate the log-gamma assumption against the usual normality assumption of the random effects, we propose a lack-of-fit test using the profile likelihood function of the shape parameter. We apply this method to data from a prospective observation study, the Diagnostic Innovations in Glaucoma Study, to present a statistically significant association between structural and functional change rates that leads to a better understanding of the progression of glaucoma over time.

Intravitreal bevacizumab injection alone or combined with triamcinolone versus macular photocoagulation in bilateral diabetic macular edema; application of bivariate generalized linear mixed model with asymmetric random effects in a subgroup of a clinical trial

PURPOSE

To compare the efficacy of intravitreal bevacizumab (IVB) injection alone or with intravitreal triamcinolone acetonide (IVB/IVT) versus macular photocoagulation (MPC) in bilateral diabetic macular edema (DME).

METHODS

In this study we revisited data from a subset of subjects previously enrolled in a randomized clinical trial. The original study included 150 eyes randomized to three treatment arms: 1.25 mg IVB alone, combined injection of 1.25 mg IVB and 2 mg IVT, and focal or modified grid MPC. To eliminate the possible effects of systemic confounders, we selected fellow eyes of bilaterally treated subjects who had undergone different treatments; eventually 30 eyes of 15 patients were re-evaluated at baseline, 6, 12, 18, and 24 months. Using mixed model analysis, we compared the treatment protocols regarding visual acuity (VA) and central macular thickness (CMT).

RESULTS

Improvement in VA in the IVB group was significantly greater compared to MPC at months 6 and 12 (P = 0.037 and P = 0.035, respectively) but this difference did not persist thereafter up to 24 months. Other levels of VA were comparable at different follow-up intervals (all P > 0.05). The only significant difference in CMT was observed in favor of the IVB group as compared to IVB/IVT group at 24 months (P = 0.048).

CONCLUSION

Overall VA was superior in IVB group as compared to MPC up to 12 months. Although the IVB group showed superiority regarding CMT reduction over 24 months as compared to IVB/IVT group, it was comparable to the MPC group through the same period of follow up.

Bayesian bivariate linear mixed-effects models with skew-normal/independent distributions, with application to AIDS clinical studies

Bivariate correlated (clustered) data often encountered in epidemiological and clinical research are routinely analyzed under a linear mixed-effected (LME) model with normality assumptions for the random-effects and within-subject errors. However, those analyses might not provide robust inference when the normality assumptions are questionable if the data set particularly exhibits skewness and heavy tails. In this article, we develop a Bayesian approach to bivariate linear mixed-effects (BLME) models replacing the Gaussian assumptions for the random terms with skew-normal/independent (SNI) distributions. The SNI distribution is an attractive class of asymmetric heavy-tailed parametric structure which includes the skew-normal, skew-t, skew-slash, and skew-contaminated normal distributions as special cases. We assume that the random-effects and the within-subject (random) errors, respectively, follow multivariate SNI and normal/independent (NI) distributions, which provide an appealing robust alternative to the symmetric normal distribution in a BLME model framework. The method is exemplified through an application to an AIDS clinical data set to compare potential models with different distribution specifications, and clinically important findings are reported.

Comparison of models for analyzing two-group, cross-sectional data with a Gaussian outcome subject to a detection limit

A potential difficulty in the analysis of biomarker data occurs when data are subject to a detection limit. This detection limit is often defined as the point at which the true values cannot be measured reliably. Multiple, regression-type models designed to analyze such data exist. Studies have compared the bias among such models, but few have compared their statistical power. This simulation study provides a comparison of approaches for analyzing two-group, cross-sectional data with a Gaussian-distributed outcome by exploring statistical power and effect size confidence interval coverage of four models able to be implemented in standard software. We found using a Tobit model fit by maximum likelihood provides the best power and coverage. An example using human immunodeficiency virus type 1 ribonucleic acid data is used to illustrate the inferential differences in these models.

SNP_NLMM: A SAS Macro to Implement a Flexible Random Effects Density for Generalized Linear and Nonlinear Mixed Models

Generalized linear and nonlinear mixed models (GMMMs and NLMMs) are commonly used to represent non-Gaussian or nonlinear longitudinal or clustered data. A common assumption is that the random effects are Gaussian. However, this assumption may be unrealistic in some applications, and misspecification of the random effects density may lead to maximum likelihood parameter estimators that are inconsistent, biased, and inefficient. Because testing if the random effects are Gaussian is difficult, previous research has recommended using a flexible random effects density. However, computational limitations have precluded widespread use of flexible random effects densities for GLMMs and NLMMs. We develop a SAS macro, SNP_NLMM, that overcomes the computational challenges to fit GLMMs and NLMMs where the random effects are assumed to follow a smooth density that can be represented by the seminonparametric formulation proposed by Gallant and Nychka (1987). The macro is flexible enough to allow for any density of the response conditional on the random effects and any nonlinear mean trajectory. We demonstrate the SNP_NLMM macro on a GLMM of the disease progression of toenail infection and on a NLMM of intravenous drug concentration over time.

Multivariate skew-normal at linear mixed models for multi-outcome longitudinal data

More than one series of longitudinal data frequently encountered in biomedical, psychological and clinical research are routinely analyzed under a multivariate linear mixed model framework with underlying multivariate normality assumptions for the random effects and within-subject errors. However, such normality assumption might not offer robust inference if the data, even after being transformed, particularly exhibit skewness. In this paper, we propose a multivariate skew-normal linear mixed model constructed by assuming a multivariate skew-normal distribution for the random effects and a multivariate normal distribution for the random errors. A damped exponential correlation structure is adopted to address the within-subject autocorrelation possibly existing among irregularly observed measures. We present an efficient alternating expectation-conditional maximization (AECM) algorithm for maximum likelihood estimation of parameters. The techniques for estimation of random effects and prediction of future outcomes are discussed. Our proposed model is motivated by, and used for, the analysis of AIDS clinical trials in which we investigate the ‘association-of-the-evolutions’ and the ‘evolution-of-the-association’ of HIV-1 RNA copies and CD4+T cell counts during antiviral therapies.

Multivariate t linear mixed models for irregularly observed multiple repeated measures with missing outcomes

Missing outcomes or irregularly timed multivariate longitudinal data frequently occur in clinical trials or biomedical studies. The multivariate t linear mixed model (MtLMM) has been shown to be a robust approach to modeling multioutcome continuous repeated measures in the presence of outliers or heavy-tailed noises. This paper presents a framework for fitting the MtLMM with an arbitrary missing data pattern embodied within multiple outcome variables recorded at irregular occasions. To address the serial correlation among the within-subject errors, a damped exponential correlation structure is considered in the model. Under the missing at random mechanism, an efficient alternating expectation-conditional maximization (AECM) algorithm is used to carry out estimation of parameters and imputation of missing values. The techniques for the estimation of random effects and the prediction of future responses are also investigated. Applications to an HIV-AIDS study and a pregnancy study involving analysis of multivariate longitudinal data with missing outcomes as well as a simulation study have highlighted the superiority of MtLMMs on the provision of more adequate estimation, imputation and prediction performances.

Skew-normal/independent linear mixed models for censored responses with applications to HIV viral loads

Often in biomedical studies, the routine use of linear mixed-effects models (based on Gaussian assumptions) can be questionable when the longitudinal responses are skewed in nature. Skew-normal/elliptical models are widely used in those situations. Often, those skewed responses might also be subjected to some upper and lower quantification limits (QLs; viz., longitudinal viral-load measures in HIV studies), beyond which they are not measurable. In this paper, we develop a Bayesian analysis of censored linear mixed models replacing the Gaussian assumptions with skew-normal/independent (SNI) distributions. The SNI is an attractive class of asymmetric heavy-tailed distributions that includes the skew-normal, skew-t, skew-slash, and skew-contaminated normal distributions as special cases. The proposed model provides flexibility in capturing the effects of skewness and heavy tail for responses that are either left- or right-censored. For our analysis, we adopt a Bayesian framework and develop a Markov chain Monte Carlo algorithm to carry out the posterior analyses. The marginal likelihood is tractable, and utilized to compute not only some Bayesian model selection measures but also case-deletion influence diagnostics based on the Kullback-Leibler divergence. The newly developed procedures are illustrated with a simulation study as well as an HIV case study involving analysis of longitudinal viral loads.

Combining structural and functional measurements to improve detection of glaucoma progression using Bayesian hierarchical models

PURPOSE

To present and evaluate a new methodology for combining longitudinal information from structural and functional tests to improve detection of glaucoma progression and estimation of rates of change.

METHODS

This observational cohort study included 434 eyes of 257 participants observed for an average of 4.2 ± 1.1 years and recruited from the Diagnostic Innovations in Glaucoma Study (DIGS). The subjects were examined annually with standard automated perimetry, optic disc stereophotographs, and scanning laser polarimetry with enhanced corneal compensation. Rates of change over time were measured using the visual field index (VFI) and average retinal nerve fiber layer thickness (TSNIT average). A bayesian hierarchical model was built to integrate information from the longitudinal measures and classify individual eyes as progressing or not. Estimates of sensitivity and specificity of the bayesian method were compared with those obtained by the conventional approach of ordinary least-squares (OLS) regression.

RESULTS

The bayesian method identified a significantly higher proportion of the 405 glaucomatous and suspect eyes as having progressed when compared with the OLS method (22.7% vs. 12.8%; P < 0.001), while having the same specificity of 100% in 29 healthy eyes. In addition, the bayesian method identified a significantly higher proportion of eyes with progression by optic disc stereophotographs compared with the OLS method (74% vs. 37%; P = 0.001).

CONCLUSIONS

A bayesian hierarchical modeling approach for combining functional and structural tests performed significantly better than the OLS method for detection of glaucoma progression. (ClinicalTrials.gov number, NCT00221897.).

A DYNAMIC BAYESIAN NONLINEAR MIXED-EFFECTS MODEL OF HIV RESPONSE INCORPORATING MEDICATION ADHERENCE, DRUG RESISTANCE AND COVARIATES()

HIV dynamic studies have contributed significantly to the understanding of HIV pathogenesis and antiviral treatment strategies for AIDS patients. Establishing the relationship of virologic responses with clinical factors and covariates during long-term antiretroviral (ARV) therapy is important to the development of effective treatments. Medication adherence is an important predictor of the effectiveness of ARV treatment, but an appropriate determinant of adherence rate based on medication event monitoring system (MEMS) data is critical to predict virologic outcomes. The primary objective of this paper is to investigate the effects of a number of summary determinants of MEMS adherence rates on virologic response measured repeatedly over time in HIV-infected patients. We developed a mechanism-based differential equation model with consideration of drug adherence, interacted by virus susceptibility to drug and baseline characteristics, to characterize the long-term virologic responses after initiation of therapy. This model fully integrates viral load, MEMS adherence, drug resistance and baseline covariates into the data analysis. In this study we employed the proposed model and associated Bayesian nonlinear mixed-effects modeling approach to assess how to efficiently use the MEMS adherence data for prediction of virologic response, and to evaluate the predicting power of each summary metric of the MEMS adherence rates. In particular, we intend to address the questions: (i) how to summarize the MEMS adherence data for efficient prediction of virologic response after accounting for potential confounding factors such as drug resistance and covariates, and (ii) how to evaluate treatment effect of baseline characteristics interacted with adherence and other clinical factors. The approach is applied to an AIDS clinical trial involving 31 patients who had available data as required for the proposed model. Results demonstrate that the appropriate determinants of MEMS adherence rates are important in order to more efficiently predict virologic response, and investigations of adherence to ARV treatment would benefit from measuring not only adherence rate but also its summary metric assessment. Our study also shows that the mechanism-based dynamic model is powerful and effective to establish a relationship of virologic responses with medication adherence, virus resistance to drug and baseline covariates.

Skew-normal Bayesian nonlinear mixed-effects models with application to AIDS studies

Studies of HIV dynamics in AIDS research are very important in understanding the pathogenesis of HIV-1 infection and also in assessing the effectiveness of antiviral therapies. Nonlinear mixed-effects (NLME) models have been used for modeling between-subject and within-subject variations in viral load measurements. Mostly, normality of both within-subject random error and random-effects is a routine assumption for NLME models, but it may be unrealistic, obscuring important features of between-subject and within-subject variations, particularly, if the data exhibit skewness. In this paper, we develop a Bayesian approach to NLME models and relax the normality assumption by considering both model random errors and random-effects to have a multivariate skew-normal distribution. The proposed model provides flexibility in capturing a broad range of non-normal behavior and includes normality as a special case. We use a real data set from an AIDS study to illustrate the proposed approach by comparing various candidate models. We find that the model with skew-normality provides better fit to the observed data and the corresponding estimates of parameters are significantly different from those based on the model with normality when skewness is present in the data. These findings suggest that it is very important to assume a model with skew-normal distribution in order to achieve robust and reliable results, in particular, when the data exhibit skewness.