Flexible bivariate correlated count data regression

Multivariate count data are common in many disciplines. The variables in such data often exhibit complex positive or negative dependency structures. We propose three Bayesian approaches to modeling bivariate count data by simultaneously considering covariate-dependent means and correlation. A direct approach utilizes a bivariate negative binomial probability mass function developed in Famoye (2010, Journal of Applied Statistics). The second approach fits bivariate count data indirectly using a bivariate Poisson-gamma mixture model. The third approach is a bivariate Gaussian copula model. Based on the results from simulation analyses, the indirect and copula approaches perform better overall than the direct approach in terms of model fitting and identifying covariate-dependent association. The proposed approaches are applied to two RNA-sequencing data sets for studying breast cancer and melanoma (BRCA-US and SKCM-US), respectively, obtained through the International Cancer Genome Consortium.     

A general frailty model to accommodate individual heterogeneity in the acquisition of multiple infections: An application to bivariate current status data

The analysis of multivariate time-to-event (TTE) data can become complicated due to the presence of clustering, leading to dependence between multiple event times. For a long time, (conditional) frailty models and (marginal) copula models have been used to analyze clustered TTE data. In this article, we propose a general frailty model employing a copula function between the frailty terms to construct flexible (bivariate) frailty distributions with the application to current status data. The model has the advantage to impose a less restrictive correlation structure among latent frailty variables as compared to traditional frailty models. Specifically, our model uses a copula function to join the marginal distributions of the frailty vector. In this article, we considered different copula functions, and we relied on marginal gamma distributions due to their mathematical convenience. Based on a simulation study, our novel model outperformed the commonly used additive correlated gamma frailty model, especially in the case of a negative association between the frailties. At the end of the article, the new methodology is illustrated on real-life data applications entailing bivariate serological survey data.     

Modeling the association of bivariate interval-censored data using the copula approach

In a survival study, it may not be possible to record the exact event time but only that the event has occurred between two time points or still has to occur, leading to interval-censored survival times.Recently, Sun et al. (Scand. J. Stat. 2006; 33(4):637-649) suggested to fit a Clayton copula with nonparametric marginal distributions to estimate the association for bivariate interval-censored failure data. We propose here to model the marginal distributions with an accelerated failure time model with a flexible error term as suggested by Komárek et al. (J. Comput. Graph. Stat. 2005; 14(3):726-745) in combination with a one parameter copula. In addition, we allow the association parameter of the copula to depend on covariates.The performance of our method is illustrated by an extensive simulation study and is applied to tooth emergence data of permanent teeth measured on 4468 children from a longitudinal dental study.     

A trivariate continual reassessment method for phase I/II trials of toxicity,efficacy, and surrogate efficacy

Recently, many Bayesian methods have been developed for dose finding when simultaneously modeling both toxicity and efficacy outcomes in a blended phase I/II fashion. A further challenge arises when all the true efficacy data cannot be obtained quickly after the treatment so that surrogate markers are instead used (e.g., in cancer trials). We propose a framework to jointly model the probabilities of toxicity, efficacy, and surrogate efficacy given a particular dose. Our trivariate binary model is specified as a composition of two bivariate binary submodels. In particular, we extend the bivariate continual reassessment method (CRM), as well as utilize a particular Gumbel copula. The resulting trivariate algorithm utilizes all the available data at any given time point and can flexibly stop the trial early for either toxicity or efficacy. Our simulation studies demonstrate that our proposed method can successfully improve dosage targeting efficiency and guard against excess toxicity over a variety of true model settings and degrees of surrogacy. Copyright © 2012 John Wiley & Sons, Ltd.     

Meta‐analysis of diagnostic tests accounting for disease prevalence: a new model using trivariate copulas

In real life and somewhat contrary to biostatistical textbook knowledge, sensitivity and specificity (and not only predictive values) of diagnostic tests can vary with the underlying prevalence of disease. In meta‐analysis of diagnostic studies, accounting for this fact naturally leads to a trivariate expansion of the traditional bivariate logistic regression model with random study effects. In this paper, a new model is proposed using trivariate copulas and beta‐binomial marginal distributions for sensitivity, specificity, and prevalence as an expansion of the bivariate model. Two different copulas are used, the trivariate Gaussian copula and a trivariate vine copula based on the bivariate Plackett copula. This model has a closed‐form likelihood, so standard software (e.g., SAS PROC NLMIXED ) can be used. The results of a simulation study have shown that the copula models perform at least as good but frequently better than the standard model. The methods are illustrated by two examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Copula bivariate probit models: with an application to medical expenditures

The bivariate probit model is frequently used for estimating the effect of an endogenous binary regressor (the ‘treatment’) on a binary health outcome variable. This paper discusses simple modifications that maintain the probit assumption for the marginal distributions while introducing non‐normal dependence using copulas. In an application of the copula bivariate probit model to the effect of insurance status on the absence of ambulatory health care expenditure, a model based on the Frank copula outperforms the standard bivariate probit model. Copyright © 2011 John Wiley & Sons, Ltd.     

Meta‐analysis for diagnostic accuracy studies: a new statistical model using beta‐binomial distributions and bivariate copulas

There are still challenges when meta‐analyzing data from studies on diagnostic accuracy. This is mainly due to the bivariate nature of the response where information on sensitivity and specificity must be summarized while accounting for their correlation within a single trial. In this paper, we propose a new statistical model for the meta‐analysis for diagnostic accuracy studies. This model uses beta‐binomial distributions for the marginal numbers of true positives and true negatives and links these margins by a bivariate copula distribution. The new model comes with all the features of the current standard model, a bivariate logistic regression model with random effects, but has the additional advantages of a closed likelihood function and a larger flexibility for the correlation structure of sensitivity and specificity. In a simulation study, which compares three copula models and two implementations of the standard model, the Plackett and the Gauss copula do rarely perform worse but frequently better than the standard model. We use an example from a meta‐analysis to judge the diagnostic accuracy of telomerase (a urinary tumor marker) for the diagnosis of primary bladder cancer for illustration. Copyright © 2013 John Wiley & Sons, Ltd.     

Advanced methods in meta-analysis: multivariate approach and meta-regression

This tutorial on advanced statistical methods for meta-analysis can be seen as a sequel to the recent Tutorial in Biostatistics on meta-analysis by Normand, which focused on elementary methods. Within the framework of the general linear mixed model using approximate likelihood, we discuss methods to analyse univariate as well as bivariate treatment effects in meta-analyses as well as meta-regression methods. Several extensions of the models are discussed, like exact likelihood, non-normal mixtures and multiple endpoints. We end with a discussion about the use of Bayesian methods in meta-analysis. All methods are illustrated by a meta-analysis concerning the efficacy of BCG vaccine against tuberculosis. All analyses that use approximate likelihood can be carried out by standard software. We demonstrate how the models can be fitted using SAS Proc Mixed.     

Copula selection models for non-Gaussian outcomes that are missing not at random

Missing not at random (MNAR) data pose key challenges for statistical inference because the substantive model of interest is typically not identifiable without imposing further (eg, distributional) assumptions. Selection models have been routinely used for handling MNAR by jointly modeling the outcome and selection variables and typically assuming that these follow a bivariate normal distribution. Recent studies have advocated parametric selection approaches, for example, estimated by multiple imputation and maximum likelihood, that are more robust to departures from the normality assumption compared with those assuming that nonresponse and outcome are jointly normally distributed. However, the proposed methods have been mostly restricted to a specific joint distribution (eg, bivariate t-distribution). This paper discusses a flexible copula-based selection approach (which accommodates a wide range of non-Gaussian outcome distributions and offers great flexibility in the choice of functional form specifications for both the outcome and selection equations) and proposes a flexible imputation procedure that generates plausible imputed values from the copula selection model. A simulation study characterizes the relative performance of the copula model compared with the most commonly used selection models for estimating average treatment effects with MNAR data. We illustrate the methods in the REFLUX study, which evaluates the effect of laparoscopic surgery on long-term quality of life in patients with reflux disease. We provide software code for implementing the proposed copula framework using the R package GJRM .     

Meta‐analysis for the comparison of two diagnostic tests—A new approach based on copulas

Meta‐analysis of diagnostic studies is still field of ongoing biometrical research. Especially, clinical researchers call for methods that allow for a comparison of different diagnostic tests to a common gold standard. Focussing on two diagnostic tests, the main parameters of interest are differences of sensitivities and specificities (with their corresponding confidence intervals) between the two diagnostic tests while accounting for the various associations across the two tests and the single studies. Similar to our previous work using generalized linear mixed models to this task, we propose a model with a quadrivariate response consisting of the two sensitivities and the two specificities of both tests. This new approach uses the ideas of copula modelling, and especially a quadrivariate Gaussian copula and a quadrivariate vine copula, which is built from bivariate Plackett copulas. The different copulas are compared in a simulation study and illustrated by the application of population‐based screening for type 2 diabetes.     

Bivariate poisson-poisson model of zero-inflated absenteeism data

Bimodal distributions of counts with one mode at zero are often seen in medical research. In a health survey parents were asked the number of days their children missed their activities (Y(1)) and the number of days their children spent in bed (Y(2)) due to illness in the past four weeks. Both variables exhibited zero inflation. We consider a bivariate Poisson-Poisson regression model, in which the two variables are regarded as indicators of an unobserved health status variable. Based on this, we further develop a bivariate Poisson-Poisson model that constrains Y(1)>or=Y(2). It is often claimed that there is a critical window of growth and nutrition in foetal life and infancy during which subsequent health status is affected. It is not clear whether the claim is true and whether childhood growth matters more. We analyse the bivariate data in relation to weight-for-age in infancy and weight gain from infancy to age 7 years. The findings do not support the existence of a critical window in infancy. There is some indication that childhood weight gain might affect health status.     

Performance of variable and function selection methods for estimating the nonlinear health effects of correlated chemical mixtures: A simulation study

Statistical methods for identifying harmful chemicals in a correlated mixture often assume linearity in exposure-response relationships. Nonmonotonic relationships are increasingly recognized (eg, for endocrine-disrupting chemicals); however, the impact of nonmonotonicity on exposure selection has not been evaluated. In a simulation study, we assessed the performance of Bayesian kernel machine regression (BKMR), Bayesian additive regression trees (BART), Bayesian structured additive regression with spike-slab priors (BSTARSS), generalized additive models with double penalty (GAMDP) and thin plate shrinkage smoothers (GAMTS), multivariate adaptive regression splines (MARS), and lasso penalized regression. We simulated realistic exposure data based on pregnancy exposure to 17 phthalates and phenols in the US National Health and Nutrition Examination Survey using a multivariate copula. We simulated data sets of size N = 250 and compared methods across 32 scenarios, varying by model size and sparsity, signal-to-noise ratio, correlation structure, and exposure-response relationship shapes. We compared methods in terms of their sensitivity, specificity, and estimation accuracy. In most scenarios, BKMR, BSTARSS, GAMDP, and GAMTS achieved moderate to high sensitivity (0.52-0.98) and specificity (0.21-0.99). BART and MARS achieved high specificity (≥0.90), but low sensitivity in low signal-to-noise ratio scenarios (0.20-0.51). Lasso was highly sensitive (0.71-0.99), except for quadratic relationships (≤0.27). Penalized regression methods that assume linearity, such as lasso, may not be suitable for studies of environmental chemicals hypothesized to have nonmonotonic relationships with outcomes. Instead, BKMR, BSTARSS, GAMDP, and GAMTS are attractive methods for flexibly estimating the shapes of exposure-response relationships and selecting among correlated exposures.     

A BAYESIAN ANALYSIS OF REGRESSION MODELS WITH CONTINUOUS ERRORS WITH APPLICATION TO LONGITUDINAL STUDIES

We employ a regression model with errors that follow a continuous autoregressive process to analyse longitudinal studies. In this way, unequally spaced observations do not present a problem in the analysis. We employ a Bayesian approach, where our inferences are based on a direct resampling process that generates values from the posterior distribution of the parameters of the model. We illustrate these Bayesian inferences with an analysis of a longitudinal study that involves the regression of foetal head circumference on menstrual age. Using these same data, we contrast the Bayesian approach with a maximum likelihood technique. © 1997 by John Wiley & Sons, Ltd.     

A bivariate autoregressive Poisson model and its application to asthma-related emergency room visits

There are no gold standard methods that perform well in every situation when it comes to the analysis of multiple time series of counts. In this paper, we consider a positively correlated bivariate time series of counts and propose a parameter-driven Poisson regression model for its analysis. In our proposed model, we employ a latent autoregressive process, AR(p) to accommodate the temporal correlations in the two series. We compute the familiar maximum likelihood estimators of the model parameters and their standard errors via a Bayesian data cloning approach. We apply the model to the analysis of a bivariate time series arising from asthma-related visits to emergency rooms across the Canadian province of Ontario.     

A GLOBAL ODDS RATIO REGRESSION MODEL FOR BIVARIATE ORDERED CATEGORICAL DATA FROM OPHTHALMOLOGIC STUDIES

In typical clinical trials or epidemiologic studies of a bilateral eye disease, the primary outcome data consist of pairs of ordered categorical responses that tend to be highly correlated. In such studies, interest often centres in associating the outcome data with a grouping variable such as the treatment indicator or the exposure status and other person- and eye-specific covariates. In this paper, we propose a latent variable regression model to analyse such bivariate ordered categorical data. We use as a joint distribution for bivariate latent random variables the cross ratio distribution proposed by Plackett, which results in modelling the dependency between the fellow eyes with the global odds ratio. We illustrate the proposed model with data from the Wisconsin Epidemiologic Study of Diabetic Retinopathy, a study that seeks to identify risk factors among younger-onset diabetics.     

A mixed effect model for bivariate meta‐analysis of diagnostic test accuracy studies using a copula representation of the random effects distribution

Diagnostic test accuracy studies typically report the number of true positives, false positives, true negatives and false negatives. There usually exists a negative association between the number of true positives and true negatives, because studies that adopt less stringent criterion for declaring a test positive invoke higher sensitivities and lower specificities. A generalized linear mixed model (GLMM) is currently recommended to synthesize diagnostic test accuracy studies. We propose a copula mixed model for bivariate meta‐analysis of diagnostic test accuracy studies. Our general model includes the GLMM as a special case and can also operate on the original scale of sensitivity and specificity. Summary receiver operating characteristic curves are deduced for the proposed model through quantile regression techniques and different characterizations of the bivariate random effects distribution. Our general methodology is demonstrated with an extensive simulation study and illustrated by re‐analysing the data of two published meta‐analyses. Our study suggests that there can be an improvement on GLMM in fit to data and makes the argument for moving to copula random effects models. Our modelling framework is implemented in the package CopulaREMADA within the open source statistical environment R . Copyright © 2015 John Wiley & Sons, Ltd.     

The joint effects of maternal prepregnancy body mass index and age on the risk of gastroschisis

Young maternal age has been associated with an increased risk of gastroschisis, while high maternal weight status has been associated with a decreased risk. We were interested in investigating the joint effect of these two risk factors to identify thresholds in risk associated with body mass index (BMI) for a given age. Data from the National Birth Defects Prevention Study included 464 case infants with gastroschisis and 4842 healthy controls. A generalised additive model with a bivariate spline for continuous maternal age and prepregnancy BMI was used to model the probability of gastroschisis.
The bivariate spline in BMI and maternal age was significantly associated with gastroschisis ( P  = 0.0001) after adjustment for study centre, maternal race/ethnicity, education, income and number of persons supported by income, smoking, alcohol use, vitamin use, vasoconstrictor medication use and gestational diabetes. The data indicate that women who are younger and who have lower BMI are at the greatest risk; a woman with a BMI of 17 who gives birth at age 15 has 7 times the odds (adjusted odds ratio = 7.0 [95% CI 4.2, 11.5]) of having an offspring with gastroschisis compared with a woman of age 24 with a BMI of 23. Furthermore, there was an interaction between maternal age and BMI for this risk. The increased risk of low maternal age and prepregnancy BMI associated with gastroschisis appears to suggest an aetiological role related to biological immaturity for this particular birth defect.     

Bayesian spatial modeling of disease risk in relation to multivariate environmental risk fields

The relationship between exposure to environmental chemicals during pregnancy and early childhood development is an important issue that has a spatial risk component. In this context, we have examined mental retardation and developmental delay (MRDD) outcome measures for children in a Medicaid population in South Carolina and sampled measures of soil chemistry (e.g. As, Hg, etc.) on a network of sites that are misaligned to the outcome residential addresses during pregnancy. The true chemical concentration at the residential addresses is not observed directly and must be interpolated from soil samples. In this study, we have developed a Bayesian joint model that interpolates soil chemical fields and estimates the associated MRDD risk simultaneously. Having multiple spatial fields to interpolate, we have considered a low‐rank Kriging method for the interpolation that requires less computation than the Bayesian Kriging. We performed a sensitivity analysis for a bivariate smoothing, changing the number of knots and the smoothing parameter. These analyses show that a low‐rank Kriging method can be used as an alternative to a full‐rank Kriging, reducing the computational burden. However, the number of knots for the low‐rank Kriging model needs to be selected with caution as a bivariate surface estimation can be sensitive to the choice of the number of knots. Copyright © 2009 John Wiley & Sons, Ltd.     

Bayesian neural networks for bivariate binary data: an application to prostate cancer study

Prostate cancer is one of the most common cancers in American men. The cancer could either be locally confined, or it could spread outside the organ. When locally confined, there are several options for treating and curing this disease. Otherwise, surgery is the only option, and in extreme cases of outside spread, it could very easily recur within a short time even after surgery and subsequent radiation therapy. Hence, it is important to know, based on pre-surgery biopsy results how likely the cancer is organ-confined or not.The paper considers a hierarchical Bayesian neural network approach for posterior prediction probabilities of certain features indicative of non-organ confined prostate cancer. In particular, we find such probabilities for margin positivity (MP) and seminal vesicle (SV) positivity jointly. The available training set consists of bivariate binary outcomes indicating the presence or absence of the two. In addition, we have certain covariates such as prostate specific antigen (PSA), gleason score and the indicator for the cancer to be unilateral or bilateral (i.e. spread on one or both sides) in one data set and gene expression microarrays in another data set. We take a hierarchical Bayesian neural network approach to find the posterior prediction probabilities for a test and validation set, and compare these with the actual outcomes for the first data set. In case of the microarray data we use leave one out cross-validation to access the accuracy of our method. We also demonstrate the superiority of our method to the other competing methods through a simulation study. The Bayesian procedure is implemented by an application of the Markov chain Monte Carlo numerical integration technique. For the problem at hand, our Bayesian bivariate neural network procedure is shown to be superior to the classical neural network, Radford Neal's Bayesian neural network as well as bivariate logistic models to predict jointly the MP and SV in a patient in both the data sets as well as in the simulation study.     

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.