Factor Tree Copula Models for Item Response Data.

Factor copula models for item response data are more interpretable and fit better than (truncated) vine copula models when dependence can be explained through latent variables, but are not robust to violations of conditional independence. To circumvent these issues, truncated vines and factor copula models for item response data are joined to define a combined model, the so-called factor tree copula model, with individual benefits from each of the two approaches. Rather than adding factors and causing computational problems and difficulties in interpretation and identification, a truncated vine structure is assumed on the residuals conditional on one or two latent variables. This structure can be better explained as a conditional dependence given a few interpretable latent variables. On the one hand, the parsimonious feature of factor models remains intact and any residual dependencies are being taken into account on the other. We discuss estimation along with model selection. In particular, we propose model selection algorithms to choose a plausible factor tree copula model to capture the (residual) dependencies among the item responses. Our general methodology is demonstrated with an extensive simulation study and illustrated by analyzing Post-Traumatic Stress Disorder.

Bi-factor and Second-Order Copula Models for Item Response Data.

Bi-factor and second-order models based on copulas are proposed for item response data, where the items are sampled from identified subdomains of some larger domain such that there is a homogeneous dependence within each domain. Our general models include the Gaussian bi-factor and second-order models as special cases and can lead to more probability in the joint upper or lower tail compared with the Gaussian bi-factor and second-order models. Details on maximum likelihood estimation of parameters for the bi-factor and second-order copula models are given, as well as model selection and goodness-of-fit techniques. Our general methodology is demonstrated with an extensive simulation study and illustrated for the Toronto Alexithymia Scale. Our studies suggest that there can be a substantial improvement over the Gaussian bi-factor and second-order models both conceptually, as the items can have interpretations of discretized maxima/minima or mixtures of discretized means in comparison with discretized means, and in fit to data.

Factor copula models for mixed data.

We develop factor copula models to analyse the dependence among mixed continuous and discrete responses. Factor copula models are canonical vine copulas that involve both observed and latent variables, hence they allow tail, asymmetric and nonlinear dependence. They can be explained as conditional independence models with latent variables that do not necessarily have an additive latent structure. We focus on important issues of interest to the social data analyst, such as model selection and goodness of fit. Our general methodology is demonstrated with an extensive simulation study and illustrated by reanalysing three mixed response data sets. Our studies suggest that there can be a substantial improvement over the standard factor model for mixed data and make the argument for moving to factor copula models.

An extended trivariate vine copula mixed model for meta-analysis of diagnostic studies in the presence of non-evaluable outcomes.

A recent paper proposed an extended trivariate generalized linear mixed model (TGLMM) for synthesis of diagnostic test accuracy studies in the presence of non-evaluable index test results. Inspired by the aforementioned model we propose an extended trivariate vine copula mixed model that includes the TGLMM as special case, but can also operate on the original scale of sensitivity, specificity, and disease prevalence. The performance of the proposed vine copula mixed model is examined by extensive simulation studies in comparison with the TGLMM. Simulation studies showed that the TGLMM leads to biased meta-analytic estimates of sensitivity, specificity, and prevalence when the univariate random effects are misspecified. The vine copula mixed model gives nearly unbiased estimates of test accuracy indices and disease prevalence. Our general methodology is illustrated by meta-analysing coronary CT angiography studies.

A multinomial quadrivariate D-vine copula mixed model for meta-analysis of diagnostic studies in the presence of non-evaluable subjects.

Diagnostic test accuracy studies observe the result of a gold standard procedure that defines the presence or absence of a disease and the result of a diagnostic test. They typically report the number of true positives, false positives, true negatives and false negatives. However, diagnostic test outcomes can also be either non-evaluable positives or non-evaluable negatives. We propose a novel model for the meta-analysis of diagnostic studies in the presence of non-evaluable outcomes, which assumes independent multinomial distributions for the true and non-evaluable positives, and, the true and non-evaluable negatives, conditional on the latent sensitivity, specificity, probability of non-evaluable positives and probability of non-evaluable negatives in each study. For the random effects distribution of the latent proportions, we employ a drawable vine copula that can successively model the dependence in the joint tails. Our methodology is demonstrated with an extensive simulation study and applied to data from diagnostic accuracy studies of coronary computed tomography angiography for the detection of coronary artery disease. The comparison of our method with the existing approaches yields findings in the real data application that change the current conclusions.

A D-vine copula mixed model for joint meta-analysis and comparison of diagnostic tests.

For a particular disease, there may be two diagnostic tests developed, where each of the tests is subject to several studies. A quadrivariate generalised linear mixed model (GLMM) has been recently proposed to joint meta-analyse and compare two diagnostic tests. We propose a D-vine copula mixed model for joint meta-analysis and comparison of two diagnostic tests. Our general model includes the quadrivariate GLMM as a special case and can also operate on the original scale of sensitivities and specificities. The method allows the direct calculation of sensitivity and specificity for each test, as well as the parameters of the summary receiver operator characteristic (SROC) curve, along with a comparison between the SROCs of each test. Our methodology is demonstrated with an extensive simulation study and illustrated by meta-analysing two examples where two tests for the diagnosis of a particular disease are compared. Our study suggests that there can be an improvement on GLMM in fit to data since our model can also provide tail dependencies and asymmetries.

Hybrid copula mixed models for combining case-control and cohort studies in meta-analysis of diagnostic tests.

Copula mixed models for trivariate (or bivariate) meta-analysis of diagnostic test accuracy studies accounting (or not) for disease prevalence have been proposed in the biostatistics literature to synthesize information. However, many systematic reviews often include case-control and cohort studies, so one can either focus on the bivariate meta-analysis of the case-control studies or the trivariate meta-analysis of the cohort studies, as only the latter contains information on disease prevalence. In order to remedy this situation of wasting data we propose a hybrid copula mixed model via a combination of the bivariate and trivariate copula mixed model for the data from the case-control studies and cohort studies, respectively. Hence, this hybrid model can account for study design and also due to its generality can deal with dependence in the joint tails. We apply the proposed hybrid copula mixed model to a review of the performance of contemporary diagnostic imaging modalities for detecting metastases in patients with melanoma.

A vine copula mixed effect model for trivariate meta-analysis of diagnostic test accuracy studies accounting for disease prevalence.

A bivariate copula mixed model has been recently proposed to synthesize diagnostic test accuracy studies and it has been shown that it is superior to the standard generalized linear mixed model in this context. Here, we call trivariate vine copulas to extend the bivariate meta-analysis of diagnostic test accuracy studies by accounting for disease prevalence. Our vine copula mixed model includes the trivariate generalized linear mixed model as a special case and can also operate on the original scale of sensitivity, specificity, and disease prevalence. Our general methodology is illustrated by re-analyzing the data of two published meta-analyses. Our study suggests that there can be an improvement on trivariate generalized linear mixed model in fit to data and makes the argument for moving to vine copula random effects models especially because of their richness, including reflection asymmetric tail dependence, and computational feasibility despite their three dimensionality.

Correlation structure and variable selection in generalized estimating equations via composite likelihood information criteria.

The method of generalized estimating equations (GEE) is popular in the biostatistics literature for analyzing longitudinal binary and count data. It assumes a generalized linear model for the outcome variable, and a working correlation among repeated measurements. In this paper, we introduce a viable competitor: the weighted scores method for generalized linear model margins. We weight the univariate score equations using a working discretized multivariate normal model that is a proper multivariate model. Because the weighted scores method is a parametric method based on likelihood, we propose composite likelihood information criteria as an intermediate step for model selection. The same criteria can be used for both correlation structure and variable selection. Simulations studies and the application example show that our method outperforms other existing model selection methods in GEE. From the example, it can be seen that our methods not only improve on GEE in terms of interpretability and efficiency but also can change the inferential conclusions with respect to GEE. Copyright © 2016 John Wiley & Sons, Ltd.

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.

Factor copula models for item response data.

Factor or conditional independence models based on copulas are proposed for multivariate discrete data such as item responses. The factor copula models have interpretations of latent maxima/minima (in comparison with latent means) and can lead to more probability in the joint upper or lower tail compared with factor models based on the discretized multivariate normal distribution (or multidimensional normal ogive model). Details on maximum likelihood estimation of parameters for the factor copula model are given, as well as analysis of the behavior of the log-likelihood. Our general methodology is illustrated with several item response data sets, and it is shown that there is a substantial improvement on existing models both conceptually and in fit to data.

Weighted scores method for regression models with dependent data.

There are copula-based statistical models in the literature for regression with dependent data such as clustered and longitudinal overdispersed counts, for which parameter estimation and inference are straightforward. For situations where the main interest is in the regression and other univariate parameters and not the dependence, we propose a "weighted scores method", which is based on weighting score functions of the univariate margins. The weight matrices are obtained initially fitting a discretized multivariate normal distribution, which admits a wide range of dependence. The general methodology is applied to negative binomial regression models. Asymptotic and small-sample efficiency calculations show that our method is robust and nearly as efficient as maximum likelihood for fully specified copula models. An illustrative example is given to show the use of our weighted scores method to analyze utilization of health care based on family characteristics.

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.

Short-term effects of ambient particles on cardiovascular and respiratory mortality.

BACKGROUND: Particulate air pollution is associated with increased mortality. There is a need for European results from multicountry databases concerning cause-specific mortality to obtain more accurate effect estimates. METHODS: We report the estimated effects of ambient particle concentrations (black smoke and particulate matter less than 10 mum [PM10]) on cardiovascular and respiratory mortality, from 29 European cities, within the Air Pollution and Health: a European Approach (APHEA2) project. We applied a 2-stage hierarchical modeling approach assessing city-specific effects first and then overall effects. City characteristics were considered as potential effect modifiers. RESULTS: An increase in PM10 by 10 microg/m (lag 0 + 1) was associated with increases of 0.76% (95% confidence interval = 0.47 to 1.05%) in cardiovascular deaths and 0.58% (0.21 to 0.95%) in respiratory deaths. The same increase in black smoke was associated with increases of 0.62% (0.35 to 0.90%) and 0.84% (0.11 to 1.57%), respectively. CONCLUSIONS: These effect estimates are appropriate for health impact assessment and standard-setting procedures.