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.     

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.     

Score test for age at onset genetic linkage analysis in selected sibling pairs

A new score statistic is derived, which uses information from registries (age‐specific incidences) and family studies (sib–sib marginal correlation) to weight affected sibling pairs according to their age at onset. Age at onset of sibling pairs is modelled by a gamma frailty model. From this model we derive a bivariate survival function, which depends on the marginal survival and on the marginal correlation. The score statistic for linkage is a classical nonparametric linkage (NPL) statistic where the identical by descent sharing is weighted by a particular function of the age at onset data. Since the statistic is based on survival models, it can also be applied to discordant and healthy sibling pairs. Simulation studies show that the proposed method is robust and more powerful than standard NPL methods. As illustration we apply the new score statistic to data from a breast cancer study. Copyright © 2009 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.     

Analysis of paediatric visual acuity using Bayesian copula models with sinh-arcsinh marginal densities

We analyse paediatric ophthalmic data from a large sample of children aged between 3 and 8 years. We use a Bayesian additive conditional bivariate copula regression model with sinh-arcsinh marginal densities with location, scale, and shape parameters that depend smoothly on a covariate. We perform Bayesian inference about the unknown quantities of our model using a specially tailored Markov chain Monte Carlo algorithm. We gain new insights about the processes, which determine transformations in visual acuity with respect to age, including the nature of joint changes in both eyes as modelled with the age-related copula dependence parameter. We analyse posterior predictive distributions to identify children with unusual sight characteristics, distinguishing those who are bivariate, but not univariate outliers. In this way, we provide an innovative tool that enables clinicians to identify children with unusual sight who may otherwise be missed. We compare our simultaneous Bayesian method with a two-step frequentist generalised additive modelling approach.     

The correlated and shared gamma frailty model for bivariate current status data: An illustration for cross‐sectional serological data

Frailty models are often used to study the individual heterogeneity in multivariate survival analysis. Whereas the shared frailty model is widely applied, the correlated frailty model has gained attention because it elevates the restriction of unobserved factors to act similar within clusters. Estimating frailty models is not straightforward due to various types of censoring. In this paper, we study the behavior of the bivariate‐correlated gamma frailty model for type I interval‐censored data, better known as current status data. We show that applying a shared rather than a correlated frailty model to cross‐sectionally collected serological data on hepatitis A and B leads to biased estimates for the baseline hazard and variance parameters. Copyright © 2009 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.     

Mixed binary-continuous copula regression models with application to adverse birth outcomes

Bivariate copula regression allows for the flexible combination of two arbitrary, continuous marginal distributions with regression effects being placed on potentially all parameters of the resulting bivariate joint response distribution. Motivated by the risk factors for adverse birth outcomes, many of which are dichotomous, we consider mixed binary-continuous responses that extend the bivariate continuous framework to the situation where one response variable is discrete (more precisely, binary) whereas the other response remains continuous. Utilizing the latent continuous representation of binary regression models, we implement a penalized likelihood–based approach for the resulting class of copula regression models and employ it in the context of modeling gestational age and the presence/absence of low birth weight. The analysis demonstrates the advantage of the flexible specification of regression impacts including nonlinear effects of continuous covariates and spatial effects. Our results imply that racial and spatial inequalities in the risk factors for infant mortality are even greater than previously suggested.     

Non-parametric estimation and model checking procedures for marginal gap time distributions for recurrent events

For recurrent events there is evidence that misspecification of the frailty distribution can cause severe bias in estimated regression coefficients (Am. J. Epidemiol 1998; 149:404-411; Statist. Med. 2006; 25:1672-1684). In this paper we adapt a procedure originally suggested in (Biometrika 1999; 86:381-393) for parallel data for checking the gamma frailty to recurrent events. To apply the model checking procedure, a consistent non-parametric estimator for the marginal gap time distributions is needed. This is in general not possible due to induced dependent censoring in the recurrent events setting, however, in (Biometrika 1999; 86:59-70) a non-parametric estimator for the joint gap time distributions based on the principle of inverse probability of censoring weights is suggested. Here, we attempt to apply this estimator in the model checking procedure and the performance of the method is investigated with simulations and applied to Danish registry data. The method is further investigated using the usual Kaplan-Meier estimator and a marginalized estimator for the marginal gap time distributions. We conclude that the procedure only works when the recurrent event is common and when the intra-individual association between gap times is weak.     

Multivariate logit copula model with an application to dental data

Applications of copulas for multivariate continuous data abound but there are only a few that treat multivariate binary data. In the present paper, we model multivariate binary data based on copulas using mixtures of max-infinitely divisible copulas, introduced by Joe and Hu (J. Multivar. Anal. 1996; 57(2): 240-265). When applying copulas to binary data the marginal distributions also contribute to the dependence measures. We propose the use of covariate information in the copula parameters to obtain a direct effect of a covariate on dependence. To deal with model uncertainty due to selecting among several candidate models, we use a model averaging technique. We apply the model to data from the Signal-Tandmobiel dental study and, in particular, to four binary responses that refer to caries experience in the mandibular and maxillary left and right molars. We aim to model Kendall's tau associations between them, and examine how covariate information affects these associations. We found that there are systematically larger associations between the two mandibular and the two maxillary molars. Using covariates to model these associations more closely, we found that the systematic fluoride and age of the children affect the associations. Note that such relationships could not have been revealed by methods that focus on the marginal models.     

Impact of model misspecification in shared frailty survival models

Survival models incorporating random effects to account for unmeasured heterogeneity are being increasingly used in biostatistical and applied research. Specifically, unmeasured covariates whose lack of inclusion in the model would lead to biased, inefficient results are commonly modeled by including a subject-specific (or cluster-specific) frailty term that follows a given distribution (eg, gamma or lognormal). Despite that, in the context of parametric frailty models, little is known about the impact of misspecifying the baseline hazard or the frailty distribution or both. Therefore, our aim is to quantify the impact of such misspecification in a wide variety of clinically plausible scenarios via Monte Carlo simulation, using open-source software readily available to applied researchers. We generate clustered survival data assuming various baseline hazard functions, including mixture distributions with turning points, and assess the impact of sample size, variance of the frailty, baseline hazard function, and frailty distribution. Models compared include standard parametric distributions and more flexible spline-based approaches; we also included semiparametric Cox models. The resulting bias can be clinically relevant. In conclusion, we highlight the importance of fitting models that are flexible enough and the importance of assessing model fit. We illustrate our conclusions with two applications using data on diabetic retinopathy and bladder cancer. Our results show the importance of assessing model fit with respect to the baseline hazard function and the distribution of the frailty: misspecifying the former leads to biased relative and absolute risk estimates, whereas misspecifying the latter affects absolute risk estimates and measures of heterogeneity.     

On robustness of marginal regression coefficient estimates and hazard functions in multivariate survival analysis of family data when the frailty distribution is mis-specified

The shared frailty model is an extension of the Cox model to correlated failure times and, essentially, a random effects model for failure time outcomes. In this model, the latent frailty shared by individual members in a cluster acts multiplicatively as a factor on the hazard function and is typically modelled parametrically. One commonly used distribution is gamma, where both shape and scale parameters are set to be the same to allow for unique identification of baseline hazard function. It is popular because it is a conjugate prior, and the posterior distribution possesses the same form as gamma. In addition, the parameter can be interpreted as a time-independent cross-ratio function, a natural extension of odds ratio to failure time outcomes. In this paper, we study the effect of frailty distribution mis-specification on the marginal regression estimates and hazard functions under assumed gamma distribution with an application to family studies. The simulation results show that the biases are generally 10% and lower, even when the true frailty distribution deviates substantially from the assumed gamma distribution. This suggests that the gamma frailty model can be a practical choice in real data analyses if the regression parameters and marginal hazard function are of primary interest and individual cluster members are exchangeable with respect to their dependencies.     

Generalized gamma frailty model

In this article, we present a frailty model using the generalized gamma distribution as the frailty distribution. It is a power generalization of the popular gamma frailty model. It also includes other frailty models such as the lognormal and Weibull frailty models as special cases. The flexibility of this frailty distribution makes it possible to detect a complex frailty distribution structure which may otherwise be missed. Due to the intractable integrals in the likelihood function and its derivatives, we propose to approximate the integrals either by Monte Carlo simulation or by a quadrature method and then determine the maximum likelihood estimates of the parameters in the model. We explore the properties of the proposed frailty model and the computation method through a simulation study. The study shows that the proposed model can potentially reduce errors in the estimation, and that it provides a viable alternative for correlated data. The merits of proposed model are demonstrated in analysing the effects of sublingual nitroglycerin and oral isosorbide dinitrate on angina pectoris of coronary heart disease patients based on the data set in Danahy et al. (sustained hemodynamic and antianginal effect of high dose oral isosorbide dinitrate. Circulation 1977; 55:381-387).     

Copula-based regression models for a bivariate mixed discrete and continuous outcome

This paper is concerned with regression models for correlated mixed discrete and continuous outcomes constructed using copulas. Our approach entails specifying marginal regression models for the outcomes, and combining them via a copula to form a joint model. Specifically, we propose marginal regression models (e.g. generalized linear models) to link the outcomes' marginal means to covariates. To account for associations between outcomes, we adopt the Gaussian copula to indirectly specify their joint distributions. Our approach has two advantages over current methods: one, regression parameters in models for both outcomes are marginally meaningful, and two, the association is 'margin-free', in the sense that it is characterized by the copula alone. By assuming a latent variable framework to describe discrete outcomes, the copula used still uniquely determines the joint distribution. In addition, association measures between outcomes can be interpreted in the usual way. We report results of simulations concerning the bias and efficiency of two likelihood-based estimation methods for the model. Finally, we illustrate the model using data on burn injuries.     

Nonproportional hazards and unobserved heterogeneity in clustered survival data: When can we tell the difference?

Multivariate survival data are frequently encountered in biomedical applications in the form of clustered failures (or recurrent events data). A popular way of analyzing such data is by using shared frailty models, which assume that the proportional hazards assumption holds conditional on an unobserved cluster-specific random effect. Such models are often incorporated in more complicated joint models in survival analysis. If the random effect distribution has finite expectation, then the conditional proportional hazards assumption does not carry over to the marginal models. It has been shown that, for univariate data, this makes it impossible to distinguish between the presence of unobserved heterogeneity (eg, due to missing covariates) and marginal nonproportional hazards. We show that time-dependent covariate effects may falsely appear as evidence in favor of a frailty model also in the case of clustered failures or recurrent events data, when the cluster size or number of recurrent events is small. When true unobserved heterogeneity is present, the presence of nonproportional hazards leads to overestimating the frailty effect. We show that this phenomenon is somewhat mitigated as the cluster size grows. We carry out a simulation study to assess the behavior of test statistics and estimators for frailty models in such contexts. The gamma, inverse Gaussian, and positive stable shared frailty models are contrasted using a novel software implementation for estimating semiparametric shared frailty models. Two main questions are addressed in the contexts of clustered failures and recurrent events: whether covariates with a time-dependent effect may appear as indication of unobserved heterogeneity and whether the additional presence of unobserved heterogeneity can be detected in this case. Finally, the practical implications are illustrated in a real-world data analysis example.     

A bivariate survival model with compound Poisson frailty

A correlated frailty model is suggested for analysis of bivariate time‐to‐event data. The model is an extension of the correlated power variance function (PVF) frailty model (correlated three‐parameter frailty model) (J. Epidemiol. Biostat. 1999; 4:53–60). It is based on a bivariate extension of the compound Poisson frailty model in univariate survival analysis (Ann. Appl. Probab. 1992; 4:951–972). It allows for a non‐susceptible fraction (of zero frailty) in the population, overcoming the common assumption in survival analysis that all individuals are susceptible to the event under study. The model contains the correlated gamma frailty model and the correlated inverse Gaussian frailty model as special cases. A maximum likelihood estimation procedure for the parameters is presented and its properties are studied in a small simulation study. This model is applied to breast cancer incidence data of Swedish twins. The proportion of women susceptible to breast cancer is estimated to be 15 per cent. Copyright © 2009 John Wiley & Sons, Ltd.     

Modeling smoking cessation data with alternating states and a cure fraction using frailty models

We propose a flexible parametric model to describe alternating states recurrent‐event data where there is a possibility of cure with each type of event. We begin by introducing a novel cure model in which a common frailty influences both the cure probability and the hazard function given not cured. We then extend our model to data with recurring events of two alternating types. We assume that each type of event has a gamma frailty, and we link the frailties by a Clayton copula. We illustrate the model with an analysis of data from two smoking cessation trials comparing bupropion and placebo, in which each subject potentially experienced a series of lapse and recovery events. Our analysis suggests that bupropion increases the probability of permanent cure and decreases the hazard of lapse, but does not affect the distribution of time to recovery during a lapse. The data suggest a positive but non‐significant association between the lapse and recovery frailties. A simulation study suggests that the estimates have little bias and that their 95 per cent confidence intervals have nearly nominal coverage in samples of practical size. Copyright © 2010 John Wiley & Sons, Ltd.     

A copula‐based mixed Poisson model for bivariate recurrent events under event‐dependent censoring

In many chronic disease processes subjects are at risk of two or more types of events. We describe a bivariate mixed Poisson model in which a copula function is used to model the association between two gamma distributed random effects. The resulting model is a bivariate negative binomial process in which each type of event arises from a negative binomial process. Methods for parameter estimation are described for parametric and semiparametric models based on an EM algorithm. We also consider the issue of event‐dependent censoring based on one type of event, which arises when one event is sufficiently serious that its occurence may influence the decision of whether to withdraw a patient from a study. The asymptotic biases of estimators of rate and mean functions from naive marginal analyses are discussed, as well as associated treatment effects. Because the joint model is fit based on a likelihood, consistent estimates are obtained. Simulation studies are carried out to evaluate the empirical performance of the proposed estimators with independent and event‐dependent censoring and applications to a trial of breast cancer patients with skeletal metastases and a study of patients with chronic obstructive pulmonary disease illustrate the approach. Copyright © 2010 John Wiley & Sons, Ltd.     

The partly Aalen's model for recurrent event data with a dependent terminal event

Recurrent event data are commonly observed in biomedical longitudinal studies. In many instances, there exists a terminal event, which precludes the occurrence of additional repeated events, and usually there is also a nonignorable correlation between the terminal event and recurrent events. In this article, we propose a partly Aalen's additive model with a multiplicative frailty for the rate function of recurrent event process and assume a Cox frailty model for terminal event time. A shared gamma frailty is used to describe the correlation between the two types of events. Consequently, this joint model can provide the information of temporal influence of absolute covariate effects on the rate of recurrent event process, which is usually helpful in the decision‐making process for physicians. An estimating equation approach is developed to estimate marginal and association parameters in the joint model. The consistency of the proposed estimator is established. Simulation studies demonstrate that the proposed approach is appropriate for practical use. We apply the proposed method to a peritonitis cohort data set for illustration. Copyright © 2015 John Wiley & Sons, Ltd.     

Meta‐analysis of studies with bivariate binary outcomes: a marginal beta‐binomial model approach

When conducting a meta‐analysis of studies with bivariate binary outcomes, challenges arise when the within‐study correlation and between‐study heterogeneity should be taken into account. In this paper, we propose a marginal beta‐binomial model for the meta‐analysis of studies with binary outcomes. This model is based on the composite likelihood approach and has several attractive features compared with the existing models such as bivariate generalized linear mixed model (Chu and Cole, 2006) and Sarmanov beta‐binomial model (Chen et al., 2012). The advantages of the proposed marginal model include modeling the probabilities in the original scale, not requiring any transformation of probabilities or any link function, having closed‐form expression of likelihood function, and no constraints on the correlation parameter. More importantly, because the marginal beta‐binomial model is only based on the marginal distributions, it does not suffer from potential misspecification of the joint distribution of bivariate study‐specific probabilities. Such misspecification is difficult to detect and can lead to biased inference using currents methods. We compare the performance of the marginal beta‐binomial model with the bivariate generalized linear mixed model and the Sarmanov beta‐binomial model by simulation studies. Interestingly, the results show that the marginal beta‐binomial model performs better than the Sarmanov beta‐binomial model, whether or not the true model is Sarmanov beta‐binomial, and the marginal beta‐binomial model is more robust than the bivariate generalized linear mixed model under model misspecifications. Two meta‐analyses of diagnostic accuracy studies and a meta‐analysis of case–control studies are conducted for illustration. Copyright © 2015 John Wiley & Sons, Ltd.