A generalized Poisson-gamma model for spatially overdispersed data

Modern disease mapping commonly uses hierarchical Bayesian methods to model overdispersion and spatial correlation. Classical random-effects based solutions include the Poisson-gamma model, which uses the conjugacy between the Poisson and gamma distributions, but which does not model spatial correlation, on the one hand, and the more advanced CAR model, which also introduces a spatial autocorrelation term but without a closed-form posterior distribution on the other. In this paper, a combined model is proposed: an alternative convolution model accounting for both overdispersion and spatial correlation in the data by combining the Poisson-gamma model with a spatially-structured normal CAR random effect. The Limburg Cancer Registry data on kidney and prostate cancer in Limburg were used to compare the conventional and new models. A simulation study confirmed results and interpretations coming from the real datasets. Relative risk maps showed that the combined model provides an intermediate between the non-patterned negative binomial and the sometimes oversmoothed CAR convolution model.     

Modelling the geographical distribution of co-infection risk from single-disease surveys

Background: The need to deliver interventions targeting multiple diseases in a cost‐effective manner calls for integrated disease control efforts. Consequently, maps are required that show where the risk of co‐infection is particularly high. Co‐infection risk is preferably estimated via Bayesian geostatistical multinomial modelling, using data from surveys screening for multiple infections simultaneously. However, only few surveys have collected this type of data. Methods: Bayesian geostatistical shared component models (allowing for covariates, disease‐specific and shared spatial and non‐spatial random effects) are proposed to model the geographical distribution and burden of co‐infection risk from single‐disease surveys. The ability of the models to capture co‐infection risk is assessed on simulated data sets based on multinomial distributions assuming light‐ and heavy‐dependent diseases, and a real data set of Schistosoma mansoni –hookworm co‐infection in the region of Man, Côte d'Ivoire. The data were restructured as if obtained from single‐disease surveys. The estimated results of co‐infection risk, together with independent and multinomial model results, were compared via different validation techniques. Results: The results showed that shared component models result in more accurate estimates of co‐infection risk than models assuming independence in settings of heavy‐dependent diseases. The shared spatial random effects are similar to the spatial co‐infection random effects of the multinomial model for heavy‐dependent data. Conclusions: In the absence of true co‐infection data geostatistical shared component models are able to estimate the spatial patterns and burden of co‐infection risk from single‐disease survey data, especially in settings of heavy‐dependent diseases. Copyright © 2011 John Wiley & Sons, Ltd.     

Disease mapping of zero-excessive mesothelioma data in Flanders

PurposeTo investigate the distribution of mesothelioma in Flanders using Bayesian disease mapping models that account for both an excess of zeros and overdispersion.MethodsThe numbers of newly diagnosed mesothelioma cases within all Flemish municipalities between 1999 and 2008 were obtained from the Belgian Cancer Registry. To deal with overdispersion, zero inflation, and geographical association, the hurdle combined model was proposed, which has three components: a Bernoulli zero-inflation mixture component to account for excess zeros, a gamma random effect to adjust for overdispersion, and a normal conditional autoregressive random effect to attribute spatial association. This model was compared with other existing methods in literature.ResultsThe results indicate that hurdle models with a random effects term accounting for extra variance in the Bernoulli zero-inflation component fit the data better than hurdle models that do not take overdispersion in the occurrence of zeros into account. Furthermore, traditional models that do not take into account excessive zeros but contain at least one random effects term that models extra variance in the counts have better fits compared to their hurdle counterparts. In other words, the extra variability, due to an excess of zeros, can be accommodated by spatially structured and/or unstructured random effects in a Poisson model such that the hurdle mixture model is not necessary.ConclusionsModels taking into account zero inflation do not always provide better fits to data with excessive zeros than less complex models. In this study, a simple conditional autoregressive model identified a cluster in mesothelioma cases near a former asbestos processing plant (Kapelle-op-den-Bos). This observation is likely linked with historical local asbestos exposures. Future research will clarify this.     

Linear models of coregionalization for multivariate lattice data: a general framework for coregionalized multivariate CAR models

We present a general coregionalization framework for developing coregionalized multivariate Gaussian conditional autoregressive (cMCAR) models for Bayesian analysis of multivariate lattice data in general and multivariate disease mapping data in particular. This framework is inclusive of cMCARs that facilitate flexible modelling of spatially structured symmetric or asymmetric cross‐variable local interactions, allowing a wide range of separable or non‐separable covariance structures, and symmetric or asymmetric cross‐covariances, to be modelled. We present a brief overview of established univariate Gaussian conditional autoregressive (CAR) models for univariate lattice data and develop coregionalized multivariate extensions. Classes of cMCARs are presented by formulating precision structures. The resulting conditional properties of the multivariate spatial models are established, which cast new light on cMCARs with richly structured covariances and cross‐covariances of different spatial ranges. The related methods are illustrated via an in‐depth Bayesian analysis of a Minnesota county‐level cancer data set. We also bring a new dimension to the traditional enterprize of Bayesian disease mapping: estimating and mapping covariances and cross‐covariances of the underlying disease risks. Maps of covariances and cross‐covariances bring to light spatial characterizations of the cMCARs and inform on spatial risk associations between areas and diseases. Copyright © 2016 John Wiley & Sons, Ltd.     

Bivariate combined linkage and association mapping of quantitative trait loci

In this paper, bivariate/multivariate variance component models are proposed for high-resolution combined linkage and association mapping of quantitative trait loci (QTL), based on combinations of pedigree and population data. Suppose that a quantitative trait locus is located in a chromosome region that exerts pleiotropic effects on multiple quantitative traits. In the region, multiple markers such as single nucleotide polymorphisms are typed. Two regression models, "genotype effect model" and "additive effect model", are proposed to model the association between the markers and the trait locus. The linkage information, i.e., recombination fractions between the QTL and the markers, is modeled in the variance and covariance matrix. By analytical formulae, we show that the "genotype effect model" can be used to model the additive and dominant effects simultaneously; the "additive effect model" only takes care of additive effect. Based on the two models, F-test statistics are proposed to test association between the QTL and markers. By analytical power analysis, we show that bivariate models can be more powerful than univariate models. For moderate-sized samples, the proposed models lead to correct type I error rates; and so the models are reasonably robust. As a practical example, the method is applied to analyze the genetic inheritance of rheumatoid arthritis for the data of The North American Rheumatoid Arthritis Consortium, Problem 2, Genetic Analysis Workshop 15, which confirms the advantage of the proposed bivariate models.     

Bivariate association analyses for the mixture of continuous and binary traits with the use of extended generalized estimating equations

Genome-wide association (GWA) study is becoming a powerful tool in deciphering genetic basis of complex human diseases/traits. Currently, the univariate analysis is the most commonly used method to identify genes associated with a certain disease/phenotype under study. A major limitation with the univariate analysis is that it may not make use of the information of multiple correlated phenotypes, which are usually measured and collected in practical studies. The multivariate analysis has proven to be a powerful approach in linkage studies of complex diseases/traits, but it has received little attention in GWA. In this study, we aim to develop a bivariate analytical method for GWA study, which can be used for a complex situation in which continuous trait and a binary trait are measured under study. Based on the modified extended generalized estimating equation (EGEE) method we proposed herein, we assessed the performance of our bivariate analyses through extensive simulations as well as real data analyses. In the study, to develop an EGEE approach for bivariate genetic analyses, we combined two different generalized linear models corresponding to phenotypic variables using a seemingly unrelated regression model. The simulation results demonstrated that our EGEE-based bivariate analytical method outperforms univariate analyses in increasing statistical power under a variety of simulation scenarios. Notably, EGEE-based bivariate analyses have consistent advantages over univariate analyses whether or not there exists a phenotypic correlation between the two traits. Our study has practical importance, as one can always use multivariate analyses as a screening tool when multiple phenotypes are available, without extra costs of statistical power and false-positive rate. Analyses on empirical GWA data further affirm the advantages of our bivariate analytical method.     

Some consequences of assuming simple patterns for the treatment effect over time in a linear mixed model

Linear mixed models are often used for the analysis of data from clinical trials with repeated quantitative outcomes. This paper considers linear mixed models where a particular form is assumed for the treatment effect, in particular constant over time or proportional to time. For simplicity, we assume no baseline covariates and complete post‐baseline measures, and we model arbitrary mean responses for the control group at each time. For the variance–covariance matrix, we consider an unstructured model, a random intercepts model and a random intercepts and slopes model. We show that the treatment effect estimator can be expressed as a weighted average of the observed time‐specific treatment effects, with weights depending on the covariance structure and the magnitude of the estimated variance components. For an assumed constant treatment effect, under the random intercepts model, all weights are equal, but in the random intercepts and slopes and the unstructured models, we show that some weights can be negative: thus, the estimated treatment effect can be negative, even if all time‐specific treatment effects are positive. Our results suggest that particular models for the treatment effect combined with particular covariance structures may result in estimated treatment effects of unexpected magnitude and/or direction. Methods are illustrated using a Parkinson's disease trial. Copyright © 2012 John Wiley & Sons, Ltd.     

A comparison of conditional autoregressive models used in Bayesian disease mapping

Disease mapping is the area of epidemiology that estimates the spatial pattern in disease risk over an extended geographical region, so that areas with elevated risk levels can be identified. Bayesian hierarchical models are typically used in this context, which represent the risk surface using a combination of available covariate data and a set of spatial random effects. These random effects are included to model any overdispersion or spatial correlation in the disease data, that has not been accounted for by the available covariate information. The random effects are typically modelled by a conditional autoregressive (CAR) prior distribution, and a number of alternative specifications have been proposed. This paper critiques four of the most common models within the CAR class, and assesses their appropriateness via a simulation study. The four models are then applied to a new study mapping cancer incidence in Greater Glasgow, Scotland, between 2001 and 2005.     

Parametric bootstrap and penalized quasi-likelihood inference in conditional autoregressive models

This paper discusses a variety of conditional autoregressive (CAR) models for mapping disease rates, beyond the usual first-order intrinsic CAR model. We illustrate the utility and scope of such models for handling different types of data structures. To encourage their routine use for map production at statistical and health agencies, a simple algorithm for fitting such models is presented. This is derived from penalized quasi-likelihood (PQL) inference which uses an analogue of best-linear unbiased estimation for the regional risk ratios and restricted maximum likelihood for the variance components. We offer the practitioner here the use of the parametric bootstrap for inference. It is more reliable than standard maximum likelihood asymptotics for inference purposes since relevant hypotheses for the mapping of rates lie on the boundary of the parameter space. We illustrate the parametric bootstrap test of the practically relevant and important simplifying hypothesis that there is no spatial autocorrelation. Although the parametric bootstrap requires computational effort, it is straightforward to implement and offers a wealth of information relating to the estimators and their properties. The proposed methodology is illustrated by analysing infant mortality in the province of British Columbia in Canada.     

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

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

A model combining excess and relative mortality for population‐based studies

Considering expected mortality provides an attractive approach to analyse mortality of population‐based cohorts of patients presenting with a chronic disease. Two classes of methods are available: either modelling the excess mortality using an additive hazard model or modelling the relative mortality using a multiplicative hazard model. Because these two models are informative to look for factors associated with mortality related to a chronic disease, we developed an alternative model modelling both the excess and the relative mortality. We generalised Andersen and Vaeth's model to fit covariates and obtain directly an estimation of the Excess Mortality Ratio and Relative Mortality Ratio for each covariate. We assessed the performances of the combined model by using simulations, and it appeared satisfactorily. We illustrate the combined model by data collected in patients presenting with end‐stage renal disease and treated by dialysis. The combined model offers the possibility of performing pure additive and multiplicative models and thus to compare their log‐likelihoods. The combined model appears useful to select one of these pure models or to conclude to the need of modelling both excess and relative mortality. In this latter case, our model enabled better describing the effect of covariates on the excess and relative mortality. Copyright © 2013 John Wiley & Sons, Ltd.     

Sample size and power calculations for correlations between bivariate longitudinal data

The analysis of a baseline predictor with a longitudinally measured outcome is well established and sample size calculations are reasonably well understood. Analysis of bivariate longitudinally measured outcomes is gaining in popularity and methods to address design issues are required. The focus in a random effects model for bivariate longitudinal outcomes is on the correlations that arise between the random effects and between the bivariate residuals. In the bivariate random effects model, we estimate the asymptotic variances of the correlations and we propose power calculations for testing and estimating the correlations. We compare asymptotic variance estimates to variance estimates obtained from simulation studies and compare our proposed power calculations for correlations on bivariate longitudinal data to power calculations for correlations on cross‐sectional data. Copyright © 2010 John Wiley & Sons, Ltd.     

Mapping disability-adjusted life years: a Bayesian hierarchical model framework for burden of disease and injury assessment

This paper presents a Bayesian disability-adjusted life year (DALY) methodology for spatial and spatiotemporal analyses of disease and/or injury burden. A Bayesian disease mapping model framework, which blends together spatial modelling, shared-component modelling (SCM), temporal modelling, ecological modelling, and non-linear modelling, is developed for small-area DALY estimation and inference. In particular, we develop a model framework that enables SCM as well as multivariate CAR modelling of non-fatal and fatal disease or injury rates and facilitates spline smoothing for non-linear modelling of temporal rate and risk trends. Using British Columbia (Canada) hospital admission-separation data and vital statistics mortality data on non-fatal and fatal road traffic injuries to male population age 20-39 for year 1991-2000 and for 84 local health areas and 16 health service delivery areas, spatial and spatiotemporal estimation and inference on years of life lost due to premature death, years lived with disability, and DALYs are presented. Fully Bayesian estimation and inference, with Markov chain Monte Carlo implementation, are illustrated. We present a methodological framework within which the DALY and the Bayesian disease mapping methodologies interface and intersect. Its development brings the relative importance of premature mortality and disability into the assessment of community health and health needs in order to provide reliable information and evidence for community-based public health surveillance and evaluation, disease and injury prevention, and resource provision.     

Disease mapping via negative binomial regression M‐quantiles

We introduce a semi‐parametric approach to ecological regression for disease mapping, based on modelling the regression M‐quantiles of a negative binomial variable. The proposed method is robust to outliers in the model covariates, including those due to measurement error, and can account for both spatial heterogeneity and spatial clustering. A simulation experiment based on the well‐known Scottish lip cancer data set is used to compare the M‐quantile modelling approach with a disease mapping approach based on a random effects model. This suggests that the M‐quantile approach leads to predicted relative risks with smaller root mean square error. The paper concludes with an illustrative application of the M‐quantile approach, mapping low birth weight incidence data for English Local Authority Districts for the years 2005–2010. Copyright © 2014 John Wiley & Sons, Ltd.     

Simultaneous estimation of parameters in the bivariate Emax model

In this paper, we explore inference in multi‐response, nonlinear models. By multi‐response, we mean models with m > 1 response variables and accordingly m relations. Each parameter/explanatory variable may appear in one or more of the relations. We study a system estimation approach for simultaneous computation and inference of the model and (co)variance parameters. For illustration, we fit a bivariate Emax model to diabetes dose‐response data. Further, the bivariate Emax model is used in a simulation study that compares the system estimation approach to equation‐by‐equation estimation. We conclude that overall, the system estimation approach performs better for the bivariate Emax model when there are dependencies among relations. The stronger the dependencies, the more we gain in precision by using system estimation rather than equation‐by‐equation estimation. Copyright © 2015 John Wiley & Sons, Ltd.     

A Bayesian semi-parametric model for colorectal cancer incidences

A Bayesian semi-parametric model is proposed to capture the interaction among demographic effects (age and gender), spatial effects (county) and temporal effects of colorectal cancer incidences simultaneously. In particular, an extension of multivariate conditionally autoregressive (CAR) processes to a partially informative Gaussian demographic spatial temporal CAR (DSTCAR) process for a spatial-temporal setting is proposed. The precision matrix of the Gaussian DSTCAR process is the Kronecker product of several components. The spatial component is modelled with a CAR prior. A pth order intrinsic autoregressive prior (IAR(p)) is implemented for the temporal component to estimate a smoothed and non-parametric temporal trend. The demographic component is modelled with a Wishart prior. Data analysis shows significant spatial correlation only exists in the age group of 50-59. Males and females in their 50s and 60s show fairly strong correlation. The hypothesis testing based on Bayes factor suggests that gender correlation cannot be ignored in this model.     

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.     

A multivariate CAR model for improving the estimation of relative risks

Disease mapping studies have been widely performed at univariate level, that is considering only one disease in the estimated models. Nonetheless, simultaneous modelling of different diseases can be a valuable tool both from the epidemiological and from the statistical point of view. In this paper we propose a model for multivariate disease mapping that generalizes the univariate conditional auto‐regressive distribution. The proposed model is proven to be an effective alternative to existing multivariate models, mainly because it overcome some restrictive hypotheses underlying models previously proposed in this context. Model performances are checked via a simulation study and via application to a case study. Copyright © 2009 John Wiley & Sons, Ltd.     

A graphical model approach to systematically missing data in meta‐analysis of observational studies

When studies in meta‐analysis include different sets of confounders, simple analyses can cause a bias (omitting confounders that are missing in certain studies) or precision loss (omitting studies with incomplete confounders, i.e. a complete‐case meta‐analysis). To overcome these types of issues, a previous study proposed modelling the high correlation between partially and fully adjusted regression coefficient estimates in a bivariate meta‐analysis. When multiple differently adjusted regression coefficient estimates are available, we propose exploiting such correlations in a graphical model. Compared with a previously suggested bivariate meta‐analysis method, such a graphical model approach is likely to reduce the number of parameters in complex missing data settings by omitting the direct relationships between some of the estimates. We propose a structure‐learning rule whose justification relies on the missingness pattern being monotone. This rule was tested using epidemiological data from a multi‐centre survey. In the analysis of risk factors for early retirement, the method showed a smaller difference from a complete data odds ratio and greater precision than a commonly used complete‐case meta‐analysis. Three real‐world applications with monotone missing patterns are provided, namely, the association between (1) the fibrinogen level and coronary heart disease, (2) the intima media thickness and vascular risk and (3) allergic asthma and depressive episodes. The proposed method allows for the inclusion of published summary data, which makes it particularly suitable for applications involving both microdata and summary data. Copyright © 2016 John Wiley & Sons, Ltd.     

The analysis of a bivariate multi-state Markov transition model for rheumatoid arthritis with an incomplete disease history.

In many long-term chronic diseases, patients pass through an observable sequence of ordered clinical states as their condition progressively worsens. Often the information on which disease state the patient is in is incompletely recorded, usually with information only available on the occasion of a clinic visit. This article describes a novel analysis of data from a clinical trial, in which several such outcome measures of disease state have been recorded simultaneously. The article is motivated by the analysis of a multi-centre double-blind placebo-controlled clinical study into the effect of continual low dose corticosteroid treatment on the progression of X-ray scores for patients with rheumatoid arthritis. Previous methods of analysis of such data have been based on an independence analysis, thus ignoring any correlation that may exist between the outcomes. This article shows that such an approach can lead to biased underestimates of the covariate effects if an independence model is used. Biased estimates of the covariate effects were found when the model was fitted to the trial data. The bivariate model was also shown to provide a significantly better fit to the data. However, the bivariate model did prove more difficult to fit, and both models demonstrated a highly significant treatment effect with comparable clinical effect.