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The composite likelihood (CL) is amongst the computational methods used for the estimation of high-dimensional multivariate normal (MVN) copula models with discrete responses. Its computational advantage, as a surrogate likelihood method, is that is based on the independence likelihood for the univariate regression and non-regression parameters and pairwise likelihood for the correlation parameters, but the efficiency of estimating the univariate regression and non-regression parameters can be low. For a high-dimensional discrete response, we propose weighted versions of the composite likelihood estimating equations and an iterative approach to determine good weight matrices. The general methodology is applied to the MVN copula with univariate ordinal regressions as the marginals. Efficiency calculations show that our method is nearly as efficient as the maximum likelihood for fully specified MVN copula models. Illustrations include simulations and real data applications regarding longitudinal (low-dimensional) and time (high-dimensional) series ordinal response data with covariates and it is shown that there is a substantial gain in efficiency via the weighted CL method.

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 models based on copulas are proposed for item response data, where the items can be split into non-overlapping groups such that there is a homogeneous dependence within each group. 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 latent maxima/minima or mixtures of means in comparison with latent means, and in fit to data.

There is an extensive literature on methods for meta-analysis of diagnostic studies, but it mainly focuses on a single test. However, the better understanding of a particular disease has led to the development of multiple tests. A multinomial generalized linear mixed model (GLMM) is recently proposed for the joint meta-analysis of studies comparing multiple tests. We propose a novel model for the joint meta-analysis of multiple tests, which assumes independent multinomial distributions for the counts of each combination of test results in diseased and non-diseased patients, conditional on the latent vector of probabilities of each combination of test results in diseased and non-diseased patients. For the random effects distribution of the latent proportions, we employ a truncated drawable vine copula that can cover flexible dependence structures. The proposed model includes the multinomial GLMM as a special case, but can also operate on the original scale of the latent proportions. Our methodology is demonstrated with a simulation study and using a meta-analysis of screening for Down syndrome with two tests: shortened humerus and shortened femur. The comparison of our method with the multinomial GLMM yields findings in the real data meta-analysis that change the current conclusions., Comment: arXiv admin note: text overlap with arXiv:2006.09278

As the meta-analysis of more than one diagnostic tests can impact clinical decision making and patient health, there is an increasing body of research in models and methods for meta-analysis of studies comparing multiple diagnostic tests. The application of the existing models to compare the accuracy of three or more tests suffers from the curse of multi-dimensionality, i.e., either the number of model parameters increase rapidly or high dimensional integration is required. To overcome these issues in joint meta-analysis of studies comparing $T >2$ diagnostic tests in a multiple tests design with a gold standard, we propose a model that assumes the true positives and true negatives for each test are conditionally independent and binomially distributed given the $2T$-variate latent vector of sensitivities and specificities. For the random effects distribution, we employ an one-factor copula that provides tail dependence or tail asymmetry. Maximum likelihood estimation of the model is straightforward as the derivation of the likelihood requires bi-dimensional instead of $2T$-dimensional integration. Our methodology is demonstrated with an extensive simulation study and an application example that determines which is the best test for the diagnosis of rheumatoid arthritis.

There exist many bivariate parametric copulas to model bivariate data with different dependence features. We propose a new bivariate parametric copula family that cannot only handle various dependence patterns that appear in the existing parametric bivariate copula families, but also provides a more enriched dependence structure. The proposed copula construction exploits finite mixtures of bivariate normal distributions. The mixing operation, the distinct correlation and mean parameters at each mixture component introduce quite a flexible dependence. The new parametric copula is theoretically investigated, compared with a set of classical bivariate parametric copulas and illustrated on two empirical examples from astrophysics and agriculture where some of the variables have peculiar and asymmetric dependence, respectively.

We develop factor copula models for analysing 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 non-linear dependence. They can be explained as conditional independence models with latent variables that don't necessarily have an additive latent structure. We focus on important issues that would interest 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 re-analysing three mixed response datasets. Our study suggests that there can be a substantial improvement over the standard factor model for mixed data and makes the argument for moving to factor copula models.

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 meta-analysis of diagnostic studies in the presence of non-evaluable outcomes that 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., Comment: arXiv admin note: substantial text overlap with arXiv:1812.03685 and text overlap with arXiv:1506.03920, arXiv:1805.09674, arXiv:1502.07505

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 overestimates the 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., Comment: arXiv admin note: text overlap with arXiv:1506.03920, arXiv:1502.07505

For a particular disease there may be two diagnostic tests developed, where each of the tests is subject to several studies. A quadrivariate generalized 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 2 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., Comment: arXiv admin note: substantial text overlap with arXiv:1506.03920, arXiv:1502.07505

The composite likelihood (CL) is amongst the computational methods used for estimation of the generalized linear mixed model (GLMM) in the context of bivariate meta-analysis of diagnostic test accuracy studies. Its advantage is that the likelihood can be derived conveniently under the assumption of independence between the random effects, but there has not been a clear analysis of the merit or necessity of this method. For synthesis of diagnostic test accuracy studies, a copula mixed model has been proposed in the biostatistics literature. This general model includes the GLMM as a special case and can also allow for flexible dependence modelling, different from assuming simple linear correlation structures, normality and tail independence in the joint tails. A maximum likelihood (ML) method, which is based on evaluating the bi-dimensional integrals of the likelihood with quadrature methods has been proposed, and in fact it eases any computational difficulty that might be caused by the double integral in the likelihood function. Both methods are thoroughly examined with extensive simulations and illustrated with data of a published meta-analysis. It is shown that the ML method has no non-convergence issues or computational difficulties and at the same time allows estimation of the dependence between study-specific sensitivity and specificity and thus prediction via summary receiver operating curves., Comment: arXiv admin note: text overlap with arXiv:1502.07505

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 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., Comment: arXiv admin note: text overlap with arXiv:1506.03920, arXiv:1502.07505

A novel copula-based multivariate panel ordinal model is developed to estimate structural relations among components of well-being. Each ordinal time-series is modelled using a copula-based Markov model to relate the marginal distributions of the response at each time of observation and then, at each observation time, the conditional distributions of each ordinal time-series are joined using a multivariate t copula. Maximum simulated likelihood based on evaluating the multidimensional integrals of the likelihood with randomized quasi Monte Carlo methods is used for the estimation. Asymptotic calculations show that our method is nearly as efficient as maximum likelihood for fully specified multivariate copula models. Our findings highlight the importance of one's relative position in evaluating their well-being with no direct effects of socio-economic characteristics on well-being but strong indirect effects through their impact on components of well-being. Temporal resilience, habit formation and behavioural traits can explain the dependence in the joint tails over time and across well-being components.

We investigate patterns of assortative matching on risk attitude, using self-reported (ordinal) data on risk attitudes for males and females within married couples, from the German Socio-Economic Panel over the period 2004-2012. We apply a novel copula-based bivariate panel ordinal model. Estimation is in two steps: firstly, a copula-based Markov model is used to relate the marginal distribution of the response in different time periods, separately for males and females; secondly, another copula is used to couple the males' and females' conditional (on the past) distributions. We find positive dependence, both in the middle of the distribution, and in the joint tails, and we interpret this as positive assortative matching (PAM). Hence we reject standard assortative matching theories based on risk-sharing assumptions, and favour models based on alternative assumptions such as the ability of agents to control income risk. We also find evidence of "assimilation"; that is, PAM appearing to increase with years of marriage.

Extending generalized estimating equations (GEE) to ordinal response data requires a conversion of the ordinal response to a vector of binary category indicators. That leads to a rather complicated association structure, and the introduction of large matrices when the number of categories and dimension of the cluster are large. To allow a richer specification of working correlation assumptions, we adopt the weighted scores method which is essentially an extension of the GEE approach, since it can also be applied to families that are not in the GLM class. The weighted scores method stems from the lack of a theoretically sound methodology for analyzing multivariate discrete data based only on moments up to second order and it is robust to dependence and nearly as efficient as maximum likelihood. There is no need to convert the ordinal response to binary indicators, thus the weight matrices have smaller dimensions and it is not necessary to guess the correlations of indicator variables for different categories. We focus on important issues that would interest the data analyst, such as choice of the structure of the correlation matrix and of explanatory variables, comparison of results obtained from our methods versus GEE, and insights provided by our method that would be missed with the GEE method. Our modelling framework is implemented in the package weightedScores within the open source statistical environment R., Comment: arXiv admin note: text overlap with arXiv:1510.06852

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 (GLM) for the outcome variable, and a working correlation among repeated measurements. In this paper, we introduce a viable competitor: the weighted scores method for GLM margins. We weight the univariate score equations using a working discretized multivariate normal model that is a proper multivariate model. Since 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 allow for correct analysis, and may change the inferential results.

A bivariate copula mixed model has been recently proposed to synthesize diagnostic test accuracy studies and it has been shown that is superior to the standard generalized linear mixed model (GLMM) 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 GLMM 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-analysing the data of two published meta-analyses. Our study suggests that there can be an improvement on trivariate GLMM 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., Comment: arXiv admin note: substantial text overlap with arXiv:1502.07505

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.

The distributional transform (DT) is amongst the computational methods used for estimation of high-dimensional multivariate normal copula models with discrete responses. Its advantage is that the likelihood can be derived conveniently under the theory for copula models with continuous margins, but there has not been a clear analysis of the adequacy of this method. We investigate the small-sample and asymptotic efficiency of the method for estimating high-dimensional multivariate normal copula models with univariate Bernoulli, Poisson, and negative binomial margins, and show that the DT approximation leads to biased estimates when there is more discretisation. For a high-dimensional discrete response, we implement a maximum simulated likelihood method, which is based on evaluating the multidimensional integrals of the likelihood with randomized quasi Monte Carlo methods. Efficiency calculations show that our method is nearly as efficient as maximum likelihood for fully specified high-dimensional multivariate normal copula models. Both methods are illustrated with spatially aggregated count data sets, and it is shown that there is a substantial gain on efficiency via the maximum simulated likelihood method., Comment: arXiv admin note: text overlap with arXiv:1304.0905

The continuous extension of a discrete random variable is amongst the computational methods used for estimation of multivariate normal copula-based models with discrete margins. Its advantage is that the likelihood can be derived conveniently under the theory for copula models with continuous margins, but there has not been a clear analysis of the adequacy of this method. We investigate the asymptotic and small-sample efficiency of two variants of the method for estimating the multivariate normal copula with univariate binary, Poisson, and negative binomial regressions, and show that they lead to biased estimates for the latent correlations, and the univariate marginal parameters that are not regression coefficients. We implement a maximum simulated likelihood method, which is based on evaluating the multidimensional integrals of the likelihood with randomized quasi Monte Carlo methods. Asymptotic and small-sample efficiency calculations show that our method is nearly as efficient as maximum likelihood for fully specified multivariate normal copula-based models. An illustrative example is given to show the use of our simulated likelihood method.