Autoregressive and cross‐lagged model for bivariate non‐commensurate outcomes

Autoregressive and cross‐lagged models have been widely used to understand the relationship between bivariate commensurate outcomes in social and behavioral sciences, but not much work has been carried out in modeling bivariate non‐commensurate (e.g., mixed binary and continuous) outcomes simultaneously. We develop a likelihood‐based methodology combining ordinary autoregressive and cross‐lagged models with a shared subject‐specific random effect in the mixed‐model framework to model two correlated longitudinal non‐commensurate outcomes. The estimates of the cross‐lagged and the autoregressive effects from our model are shown to be consistent with smaller mean‐squared error than the estimates from the univariate generalized linear models. Inclusion of the subject‐specific random effects in the proposed model accounts for between‐subject variability arising from the omitted and/or unobservable, but possibly explanatory, subject‐level predictors. Our model is not restricted to the case with equal number of events per subject, and it can be extended to different types of bivariate outcomes. We apply our model to an ecological momentary assessment study with complex dependence and sampling data structures. Specifically, we study the dependence between the condom use and sexual satisfaction based on the data reported in a longitudinal study of sexually transmitted infections. We find negative cross‐lagged effect between these two outcomes and positive autoregressive effect within each outcome. Copyright © 2017 John Wiley & Sons, Ltd.     

A linear mixed model approach to compare the evolution of multiple biological rhythms

The assessment and comparison of multiple biological rhythms represent an important challenge in chronobiology. They allow the investigation of whether a well‐defined time‐qualified relationship between biorhythms of the same frequency range is maintained in the presence of functional alterations, which may lead to chronodisruption or internal desynchronization. We propose a multivariate linear mixed model approach where functions of several biorhythms are jointly modeled in a multivariate longitudinal fashion, handling both the correlation between biorhythms of multiple outcomes and the correlation between measurements collected over time within the same biological entity. Furthermore, between‐subject heterogeneity is also taken into account with the inclusion of random effects. Pairwise comparisons between biorhythms are performed by means of proper contrasts. As an example, we define contrasts which allow us testing whether or not two biorhythms are identical or opposing, providing additional support in clinical practice. Moreover, we illustrate the proposed method using both simulated and biological real data, concerning the comparison of three specific lymphocytes profiles which modulate the function of immune system between healthy subjects and non‐small lung cancer patients. Finally, the corresponding SAS syntax is provided. Copyright © 2012 John Wiley & Sons, Ltd.     

Bayesian multivariate meta‐analysis with multiple outcomes

There has been a recent growth in developments of multivariate meta‐analysis. We extend the methodology of Bayesian multivariate meta‐analysis to the situation when there are more than two outcomes of interest, which is underexplored in the current literature. Our objective is to meta‐analyse summary data from multiple outcomes simultaneously, accounting for potential dependencies among the data. One common issue is that studies do not all report all of the outcomes of interests, and we take an approach relying on marginal modelling of only the reported data. We employ a separation prior for the between‐study variance–covariance matrix, which offers an improvement on the conventional inverse‐Wishart prior, showing robustness in estimation and flexibility in incorporating prior information. Particular challenges arise when the number of outcomes is large relative to the number of studies because the number of parameters in the variance–covariance matrix can become substantial and there can be very little information with which to estimate between‐study correlation coefficients. We explore assumptions that reduce the number of parameters in this matrix, including assumptions of homogenous variances, homogenous correlations for certain outcomes and positive correlation coefficients. We illustrate the methods with an example data set from the Cochrane Database of Systematic Reviews. Copyright © 2013 John Wiley & Sons, Ltd.     

Kinetics of maternal antibodies against rubella and varicella in infants

Kinetics of maternal rubella and varicella antibodies in 213 mother-infant pairs are described in a longitudinal study in Belgium.Blood samples are taken at 7 time points (week 36 of pregnancy, birth (cord), 1, 3, 6, 9, and 12 months), and analyzed for anti-rubella IgG and anti-varicella IgG by enzyme linked immunosorbent assay (ELISA). A generalized exponential model is used to analyse maternal antibody decay in infants.Model based, the mean duration of passive immunity is 2.1 months for rubella and 2.4 months for varicella.Infants are susceptible at young age for rubella, a disease with high vaccination coverage, as well as for varicella, an endemic disease in Western Europe.     

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. Copyright © 2010 John Wiley & Sons, Ltd.     

Multivariate t nonlinear mixed‐effects models for multi‐outcome longitudinal data with missing values

The multivariate nonlinear mixed‐effects model (MNLMM) has emerged as an effective tool for modeling multi‐outcome longitudinal data following nonlinear growth patterns. In the framework of MNLMM, the random effects and within‐subject errors are assumed to be normally distributed for mathematical tractability and computational simplicity. However, a serious departure from normality may cause lack of robustness and subsequently make invalid inference. This paper presents a robust extension of the MNLMM by considering a joint multivariate t distribution for the random effects and within‐subject errors, called the multivariate t nonlinear mixed‐effects model. Moreover, a damped exponential correlation structure is employed to capture the extra serial correlation among irregularly observed multiple repeated measures. An efficient expectation conditional maximization algorithm coupled with the first‐order Taylor approximation is developed for maximizing the complete pseudo‐data likelihood function. The techniques for the estimation of random effects, imputation of missing responses and identification of potential outliers are also investigated. The methodology is motivated by a real data example on 161 pregnant women coming from a study in a private fertilization obstetrics clinic in Santiago, Chile and used to analyze these data. Copyright © 2014 John Wiley & Sons, Ltd.     

Pseudo maximum likelihood approach for the analysis of multivariate left‐censored longitudinal data

The linear mixed effects model based on a full likelihood is one of the few methods available to model longitudinal data subject to left censoring. However, a full likelihood approach is complicated algebraically because of the large dimension of the numeric computations, and maximum likelihood estimation can be computationally prohibitive when the data are heavily censored. Moreover, for mixed models, the complexity of the computation increases as the dimension of the random effects in the model increases. We propose a method based on pseudo likelihood that simplifies the computational complexities, allows a wide class of multivariate models, and that can be used for many different data structures including settings where the level of censoring is high. The motivation for this work comes from the need for a joint model to assess the joint effect of pro‐inflammatory and anti‐inflammatory biomarker data on 30‐day mortality status while simultaneously accounting for longitudinal left censoring and correlation between markers in the analysis of Genetic and Inflammatory Markers for Sepsis study conducted at the University of Pittsburgh. Two markers, interleukin‐6 and interleukin‐10, which naturally are correlated because of a shared similar biological pathways and are left‐censored because of the limited sensitivity of the assays, are considered to determine if higher levels of these markers is associated with an increased risk of death after accounting for the left censoring and their assumed correlation. Copyright © 2016 John Wiley & Sons, Ltd.     

A tutorial on Bayesian bivariate meta‐analysis of mixed binary‐continuous outcomes with missing treatment effects

Bivariate random‐effects meta‐analysis (BVMA) is a method of data synthesis that accounts for treatment effects measured on two outcomes. BVMA gives more precise estimates of the population mean and predicted values than two univariate random‐effects meta‐analyses (UVMAs). BVMA also addresses bias from incomplete reporting of outcomes. A few tutorials have covered technical details of BVMA of categorical or continuous outcomes. Limited guidance is available on how to analyze datasets that include trials with mixed continuous‐binary outcomes where treatment effects on one outcome or the other are not reported. Given the advantages of Bayesian BVMA for handling missing outcomes, we present a tutorial for Bayesian BVMA of incompletely reported treatment effects on mixed bivariate outcomes. This step‐by‐step approach can serve as a model for our intended audience, the methodologist familiar with Bayesian meta‐analysis, looking for practical advice on fitting bivariate models. To facilitate application of the proposed methods, we include our WinBUGS code. As an example, we use aggregate‐level data from published trials to demonstrate the estimation of the effects of vitamin K and bisphosphonates on two correlated bone outcomes, fracture, and bone mineral density. We present datasets where reporting of the pairs of treatment effects on both outcomes was ‘partially’ complete (i.e., pairs completely reported in some trials), and we outline steps for modeling the incompletely reported data. To assess what is gained from the additional work required by BVMA, we compare the resulting estimates to those from separate UVMAs. We discuss methodological findings and make four recommendations. Copyright © 2015 John Wiley & Sons, Ltd.     

Latent time‐varying factors in longitudinal analysis: a linear mixed hidden Markov model for heart rates

Longitudinal data are often segmented by unobserved time‐varying factors, which introduce latent heterogeneity at the observation level, in addition to heterogeneity across subjects. We account for this latent structure by a linear mixed hidden Markov model. It integrates subject‐specific random effects and Markovian sequences of time‐varying effects in the linear predictor. We propose an expectation?‐maximization algorithm for maximum likelihood estimation, based on data augmentation. It reduces to the iterative maximization of the expected value of a complete likelihood function, derived from an augmented dataset with case weights, alternated with weights updating. In a case study of the Survey on Stress Aging and Health in Russia, the model is exploited to estimate the influence of the observed covariates under unobserved time‐varying factors, which affect the cardiovascular activity of each subject during the observation period. Copyright © 2014 John Wiley & Sons, Ltd.     

Estimating within‐study covariances in multivariate meta‐analysis with multiple outcomes

Multivariate meta‐analysis allows the joint synthesis of effect estimates based on multiple outcomes from multiple studies, accounting for the potential correlations among them. However, standard methods for multivariate meta‐analysis for multiple outcomes are restricted to problems where the within‐study correlation is known or where individual participant data are available. This paper proposes an approach to approximating the within‐study covariances based on information about likely correlations between underlying outcomes. We developed methods for both continuous and dichotomous data and for combinations of the two types. An application to a meta‐analysis of treatments for stroke illustrates the use of the approximated covariance in multivariate meta‐analysis with correlated outcomes. Copyright © 2012 John Wiley & Sons, Ltd.     

Correlated bivariate continuous and binary outcomes: Issues and applications

Increasingly multiple outcomes are collected in order to characterize treatment effectiveness or to evaluate the impact of large policy initiatives. Often the multiple outcomes are non‐commensurate, e.g. measured on different scales. The common approach to inference is to model each outcome separately ignoring the potential correlation among the responses. We describe and contrast several full likelihood and quasi‐likelihood multivariate methods for non‐commensurate outcomes. We present a new multivariate model to analyze binary and continuous correlated outcomes using a latent variable. We study the efficiency gains of the multivariate methods relative to the univariate approach. For complete data, all approaches yield consistent parameter estimates. When the mean structure of all outcomes depends on the same set of covariates, efficiency gains by adopting a multivariate approach are negligible. In contrast, when the mean outcomes depend on different covariate sets, large efficiency gains are realized. Three real examples illustrate the different approaches. Copyright © 2009 John Wiley & Sons, Ltd.     

Multiple imputation in the presence of non‐normal data

Multiple imputation (MI) is becoming increasingly popular for handling missing data. Standard approaches for MI assume normality for continuous variables (conditionally on the other variables in the imputation model). However, it is unclear how to impute non‐normally distributed continuous variables. Using simulation and a case study, we compared various transformations applied prior to imputation, including a novel non‐parametric transformation, to imputation on the raw scale and using predictive mean matching (PMM) when imputing non‐normal data. We generated data from a range of non‐normal distributions, and set 50% to missing completely at random or missing at random. We then imputed missing values on the raw scale, following a zero‐skewness log, Box–Cox or non‐parametric transformation and using PMM with both type 1 and 2 matching. We compared inferences regarding the marginal mean of the incomplete variable and the association with a fully observed outcome. We also compared results from these approaches in the analysis of depression and anxiety symptoms in parents of very preterm compared with term‐born infants. The results provide novel empirical evidence that the decision regarding how to impute a non‐normal variable should be based on the nature of the relationship between the variables of interest. If the relationship is linear in the untransformed scale, transformation can introduce bias irrespective of the transformation used. However, if the relationship is non‐linear, it may be important to transform the variable to accurately capture this relationship. A useful alternative is to impute the variable using PMM with type 1 matching. Copyright © 2016 John Wiley & Sons, Ltd.     

Inference for correlated effect sizes using multiple univariate meta‐analyses

Multivariate meta‐analysis, which involves jointly analyzing multiple and correlated outcomes from separate studies, has received a great deal of attention. One reason to prefer the multivariate approach is its ability to account for the dependence between multiple estimates from the same study. However, nearly all the existing methods for analyzing multivariate meta‐analytic data require the knowledge of the within‐study correlations, which are usually unavailable in practice. We propose a simple non‐iterative method that can be used for the analysis of multivariate meta‐analysis datasets, that has no convergence problems, and does not require the use of within‐study correlations. Our approach uses standard univariate methods for the marginal effects but also provides valid joint inference for multiple parameters. The proposed method can directly handle missing outcomes under missing completely at random assumption. Simulation studies show that the proposed method provides unbiased estimates, well‐estimated standard errors, and confidence intervals with good coverage probability. Furthermore, the proposed method is found to maintain high relative efficiency compared with conventional multivariate meta‐analyses where the within‐study correlations are known. We illustrate the proposed method through two real meta‐analyses where functions of the estimated effects are of interest. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Multivariate meta‐analysis for non‐linear and other multi‐parameter associations

In this paper, we formalize the application of multivariate meta‐analysis and meta‐regression to synthesize estimates of multi‐parameter associations obtained from different studies. This modelling approach extends the standard two‐stage analysis used to combine results across different sub‐groups or populations. The most straightforward application is for the meta‐analysis of non‐linear relationships, described for example by regression coefficients of splines or other functions, but the methodology easily generalizes to any setting where complex associations are described by multiple correlated parameters. The modelling framework of multivariate meta‐analysis is implemented in the package mvmeta within the statistical environment R . As an illustrative example, we propose a two‐stage analysis for investigating the non‐linear exposure–response relationship between temperature and non‐accidental mortality using time‐series data from multiple cities. Multivariate meta‐analysis represents a useful analytical tool for studying complex associations through a two‐stage procedure. Copyright © 2012 John Wiley & Sons, Ltd.     

Two‐sample tests and one‐way MANOVA for multivariate biomarker data with nondetects

Testing whether the mean vector of a multivariate set of biomarkers differs between several populations is an increasingly common problem in medical research. Biomarker data is often left censored because some measurements fall below the laboratory's detection limit. We investigate how such censoring affects multivariate two‐sample and one‐way multivariate analysis of variance tests. Type I error rates, power and robustness to increasing censoring are studied, under both normality and non‐normality. Parametric tests are found to perform better than non‐parametric alternatives, indicating that the current recommendations for analysis of censored multivariate data may have to be revised. Copyright © 2016 John Wiley & Sons, Ltd.     

Semiparametric regression models for repeated measures of mortal cohorts with non‐monotone missing outcomes and time‐dependent covariates

We propose a semiparametric marginal modeling approach for longitudinal analysis of cohorts with data missing due to death and non‐response to estimate regression parameters interpreted as conditioned on being alive. Our proposed method accommodates outcomes and time‐dependent covariates that are missing not at random with non‐monotone missingness patterns via inverse‐probability weighting. Missing covariates are replaced by consistent estimates derived from a simultaneously solved inverse‐probability‐weighted estimating equation. Thus, we utilize data points with the observed outcomes and missing covariates beyond the estimated weights while avoiding numerical methods to integrate over missing covariates. The approach is applied to a cohort of elderly female hip fracture patients to estimate the prevalence of walking disability over time as a function of body composition, inflammation, and age. Copyright © 2010 John Wiley & Sons, Ltd.     

An index of local sensitivity to non‐ignorability for multivariate longitudinal mixed data with potential non‐random dropout

Multivariate longitudinal data with mixed continuous and discrete responses with the possibility of non‐ignorable missingness are often common in follow‐up medical studies and their analysis needs to be developed. Standard methods of analysis based on the strong and the unverifiable assumption of missing at random (MAR) mechanism could be highly misleading. A way out of this problem is to start with methods that simultaneously allow modelling non‐ignorable mechanism, which includes somehow troubling computations that are often time consuming, then we can use a sensitivity analysis, in which one estimates models under a range of assumptions about non‐ignorability parameters to study the impact of these parameters on key inferences. A general index of sensitivity to non‐ignorability (ISNI) to measure sensitivity of key inferences in a neighborhood of MAR model without fitting a complicated not MAR (NMAR) model for univariate generalized linear models and for models used for univariate longitudinal normal and non‐Gaussian data with potentially NMAR dropout are well presented in the literature. In this paper we extend ISNI methodology to analyze multivariate longitudinal mixed data subject to non‐ignorable dropout in which the non‐ignorable dropout model could be dependent on the mixed responses. The approach is illustrated by analyzing a longitudinal data set in which the general substantive goal of the study is to better understand the relations between parental assessment of child's antisocial behavior and child's reading recognition skill. Copyright © 2010 John Wiley & Sons, Ltd.     

Linear mixed models for skew‐normal/independent bivariate responses with an application to periodontal disease

Bivariate clustered (correlated) data often encountered in epidemiological and clinical research are routinely analyzed under a linear mixed model (LMM) framework with underlying normality assumptions of the random effects and within‐subject errors. However, such normality assumptions might be questionable if the data set particularly exhibits skewness and heavy tails. Using a Bayesian paradigm, we use the skew‐normal/independent (SNI) distribution as a tool for modeling clustered data with bivariate non‐normal responses in an LMM framework. The SNI distribution is an attractive class of asymmetric thick‐tailed parametric structure which includes the skew‐normal distribution as a special case. We assume that the random effects follow multivariate SNI distributions and the random errors follow SNI distributions which provides substantial robustness over the symmetric normal process in an LMM framework. Specific distributions obtained as special cases, viz. the skew‐t, the skew‐slash and the skew‐contaminated normal distributions are compared, along with the default skew‐normal density. The methodology is illustrated through an application to a real data which records the periodontal health status of an interesting population using periodontal pocket depth (PPD) and clinical attachment level (CAL). Copyright © 2010 John Wiley & Sons, Ltd.     

Multivariate Phenotype Association Analysis by Marker‐Set Kernel Machine Regression

Genetic studies of complex diseases often collect multiple phenotypes relevant to the disorders. As these phenotypes can be correlated and share common genetic mechanisms, jointly analyzing these traits may bring more power to detect genes influencing individual or multiple phenotypes. Given the advancement brought by the multivariate phenotype approaches and the multimarker kernel machine regression, we construct a multivariate regression based on kernel machine to facilitate the joint evaluation of multimarker effects on multiple phenotypes. The kernel machine serves as a powerful dimension‐reduction tool to capture complex effects among markers. The multivariate framework incorporates the potentially correlated multidimensional phenotypic information and accommodates common or different environmental covariates for each trait. We derive the multivariate kernel machine test based on a score‐like statistic, and conduct simulations to evaluate the validity and efficacy of the method. We also study the performance of the commonly adapted strategies for kernel machine analysis on multiple phenotypes, including the multiple univariate kernel machine tests with original phenotypes or with their principal components. Our results suggest that none of these approaches has the uniformly best power, and the optimal test depends on the magnitude of the phenotype correlation and the effect patterns. However, the multivariate test retains to be a reasonable approach when the multiple phenotypes have none or mild correlations, and gives the best power once the correlation becomes stronger or when there exist genes that affect more than one phenotype. We illustrate the utility of the multivariate kernel machine method through the Clinical Antipsychotic Trails of Intervention Effectiveness antibody study.     

Multivariate network meta‐analysis to mitigate the effects of outcome reporting bias

Outcome reporting bias (ORB) is recognized as a threat to the validity of both pairwise and network meta‐analysis (NMA). In recent years, multivariate meta‐analytic methods have been proposed to reduce the impact of ORB in the pairwise setting. These methods have shown that multivariate meta‐analysis can reduce bias and increase efficiency of pooled effect sizes. However, it is unknown whether multivariate NMA (MNMA) can similarly reduce the impact of ORB. Additionally, it is quite challenging to implement MNMA due to the fact that correlation between treatments and outcomes must be modeled; thus, the dimension of the covariance matrix and number of components to estimate grows quickly with the number of treatments and number of outcomes. To determine whether MNMA can reduce the effects of ORB on pooled treatment effect sizes, we present an extensive simulation study of Bayesian MNMA. Via simulation studies, we show that MNMA reduces the bias of pooled effect sizes under a variety of outcome missingness scenarios, including missing at random and missing not at random. Further, MNMA improves the precision of estimates, producing narrower credible intervals. We demonstrate the applicability of the approach via application of MNMA to a multi‐treatment systematic review of randomized controlled trials of anti‐depressants for the treatment of depression in older adults.