On Bayesian shared component disease mapping and ecological regression with errors in covariates

Recent literature on Bayesian disease mapping presents shared component models (SCMs) for joint spatial modeling of two or more diseases with common risk factors. In this study, Bayesian hierarchical formulations of shared component disease mapping and ecological models are explored and developed in the context of ecological regression, taking into consideration errors in covariates. A review of multivariate disease mapping models (MultiVMs) such as the multivariate conditional autoregressive models that are also part of the more recent Bayesian disease mapping literature is presented. Some insights into the connections and distinctions between the SCM and MultiVM procedures are communicated. Important issues surrounding (appropriate) formulation of shared‐ and disease‐specific components, consideration/choice of spatial or non‐spatial random effects priors, and identification of model parameters in SCMs are explored and discussed in the context of spatial and ecological analysis of small area multivariate disease or health outcome rates and associated ecological risk factors. The methods are illustrated through an in‐depth analysis of four‐variate road traffic accident injury (RTAI) data: gender‐specific fatal and non‐fatal RTAI rates in 84 local health areas in British Columbia (Canada). Fully Bayesian inference via Markov chain Monte Carlo simulations is presented. Copyright © 2010 John Wiley & Sons, Ltd.     

Bayesian estimation of multivariate Gaussian Markov random fields with constraint

This article concerns with conditionally formulated multivariate Gaussian Markov random fields (MGMRF) for modeling multivariate local dependencies with unknown dependence parameters subject to positivity constraint. In the context of Bayesian hierarchical modeling of lattice data in general and Bayesian disease mapping in particular, analytic and simulation studies provide new insights into various approaches to posterior estimation of dependence parameters under “hard” or “soft” positivity constraint, including the well-known strictly diagonal dominance criterion and options of hierarchical priors. Hierarchical centering is examined as a means to gain computational efficiency in Bayesian estimation of multivariate generalized linear mixed effects models in the presence of spatial confounding and weakly identified model parameters. Simulated data on irregular or regular lattice, and three datasets from the multivariate and spatiotemporal disease mapping literature, are used for illustration. The present investigation also sheds light on the use of deviance information criterion for model comparison, choice, and interpretation in the context of posterior risk predictions judged by borrowing-information and bias-precision tradeoff. The article concludes with a summary discussion and directions of future work. Potential applications of MGMRF in spatial information fusion and image analysis are briefly mentioned.     

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.     

The bivariate combined model for spatial data analysis

To describe the spatial distribution of diseases, a number of methods have been proposed to model relative risks within areas. Most models use Bayesian hierarchical methods, in which one models both spatially structured and unstructured extra‐Poisson variance present in the data. For modelling a single disease, the conditional autoregressive (CAR) convolution model has been very popular. More recently, a combined model was proposed that ‘combines’ ideas from the CAR convolution model and the well‐known Poisson‐gamma model. The combined model was shown to be a good alternative to the CAR convolution model when there was a large amount of uncorrelated extra‐variance in the data. Less solutions exist for modelling two diseases simultaneously or modelling a disease in two sub‐populations simultaneously. Furthermore, existing models are typically based on the CAR convolution model. In this paper, a bivariate version of the combined model is proposed in which the unstructured heterogeneity term is split up into terms that are shared and terms that are specific to the disease or subpopulation, while spatial dependency is introduced via a univariate or multivariate Markov random field. The proposed method is illustrated by analysis of disease data in Georgia (USA) and Limburg (Belgium) and in a simulation study. We conclude that the bivariate combined model constitutes an interesting model when two diseases are possibly correlated. As the choice of the preferred model differs between data sets, we suggest to use the new and existing modelling approaches together and to choose the best model via goodness‐of‐fit statistics. Copyright © 2016 John Wiley & Sons, Ltd.     

Skew‐elliptical spatial random effect modeling for areal data with application to mapping health utilization rates

Mixed models incorporating spatially correlated random effects are often used for the analysis of areal data. In this setting, spatial smoothing is introduced at the second stage of a hierarchical framework, and this smoothing is often based on a latent Gaussian Markov random field. The Markov random field provides a computationally convenient framework for modeling spatial dependence; however, the Gaussian assumption underlying commonly used models can be overly restrictive in some applications. This can be a problem in the presence of outliers or discontinuities in the underlying spatial surface, and in such settings, models based on non‐Gaussian spatial random effects are useful. Motivated by a study examining geographic variation in the treatment of acute coronary syndrome, we develop a robust model for smoothing small‐area health service utilization rates. The model incorporates non‐Gaussian spatial random effects, and we develop a formulation for skew‐elliptical areal spatial models. We generalize the Gaussian conditional autoregressive model to the non‐Gaussian case, allowing for asymmetric skew‐elliptical marginal distributions having flexible tail behavior. The resulting new models are flexible, computationally manageable, and can be implemented in the standard Bayesian software WinBUGS. We demonstrate performance of the proposed methods and comparisons with other commonly used Gaussian and non‐Gaussian spatial prior formulations through simulation and analysis in our motivating application, mapping rates of revascularization for patients diagnosed with acute coronary syndrome in Quebec, Canada. Copyright © 2012 John Wiley & Sons, Ltd.     

A unifying modeling framework for highly multivariate disease mapping

Multivariate disease mapping refers to the joint mapping of multiple diseases from regionally aggregated data and continues to be the subject of considerable attention for biostatisticians and spatial epidemiologists. The key issue is to map multiple diseases accounting for any correlations among themselves. Recently, Martinez‐Beneito (2013) provided a unifying framework for multivariate disease mapping. While attractive in that it colligates a variety of existing statistical models for mapping multiple diseases, this and other existing approaches are computationally burdensome and preclude the multivariate analysis of moderate to large numbers of diseases. Here, we propose an alternative reformulation that accrues substantial computational benefits enabling the joint mapping of tens of diseases. Furthermore, the approach subsumes almost all existing classes of multivariate disease mapping models and offers substantial insight into the properties of statistical disease mapping models. Copyright © 2015 John Wiley & Sons, Ltd.     

Regression B-spline smoothing in Bayesian disease mapping: with an application to patient safety surveillance

In the context of Bayesian disease mapping, recent literature presents generalized linear mixed models that engender spatial smoothing. The methods assume spatially varying random effects as a route to partially pooling data and 'borrowing strength' in small-area estimation. When spatiotemporal disease rates are available for sequential risk mapping of several time periods, the 'smoothing' issue may be explored by considering spatial smoothing, temporal smoothing and spatiotemporal interaction. In this paper, these considerations are motivated and explored through development of a Bayesian semiparametric disease mapping model framework which facilitates temporal smoothing of rates and relative risks via regression B-splines with mixed-effect representation of coefficients. Specifically, we develop spatial priors such as multivariate Gaussian Markov random fields and non-spatial priors such as unstructured multivariate Gaussian distributions and illustrate how time trends in small-area relative risks may be explored by splines which vary in either a spatially structured or unstructured manner. In particular, we show that with suitable prior specifications for the random effects ensemble, small-area relative risk trends may be fit by 'spatially varying' or randomly varying B-splines. A recently developed Bayesian hierarchical model selection criterion, the deviance information criterion, is used to assess the trade-off between goodness-of-fit and smoothness and to select the number of knots. The methodological development aims to provide reliable information about the patterns (both over space and time) of disease risks and to quantify uncertainty. The study offers a disease and health outcome surveillance methodology for flexible and efficient exploration and assessment of emerging risk trends and clustering. The methods are motivated and illustrated through a Bayesian analysis of adverse medical events (also known as iatrogenic injuries) among hospitalized elderly patients in British Columbia, Canada.     

An empirical comparison of univariate and multivariate meta‐analyses for categorical outcomes

Treatment effects for multiple outcomes can be meta‐analyzed separately or jointly, but no systematic empirical comparison of the two approaches exists. From the Cochrane Library of Systematic Reviews, we identified 45 reviews, including 1473 trials and 258,675 patients, that contained two or three univariate meta‐analyses of categorical outcomes for the same interventions that could also be analyzed jointly. Eligible were meta‐analyses with at least seven trials reporting all outcomes for which the cross‐classification tables were exactly recoverable (e.g., outcomes were mutually exclusive, or one was a subset of the other). This ensured known correlation structures. Outcomes in 40 reviews had an is‐subset‐of relationship, and those in 5 were mutually exclusive. We analyzed these data with univariate and multivariate models based on discrete and approximate likelihoods. Discrete models were fit in the Bayesian framework using slightly informative priors. The summary effects for each outcome were similar with univariate and multivariate meta‐analyses (both using the approximate and discrete likelihoods); however, the multivariate model with the discrete likelihood gave smaller between‐study variance estimates, and narrower predictive intervals for new studies. When differences in the summary treatment effects were examined, the multivariate models gave similar summary estimates but considerably longer (shorter) uncertainty intervals because of positive (negative) correlation between outcome treatment effects. It is unclear whether any of the examined reviews would change their overall conclusions based on multivariate versus univariate meta‐analyses, because extra‐analytical and context‐specific considerations contribute to conclusions and, secondarily, because numerical differences were often modest. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Bayesian multivariate disease mapping and ecological regression with errors in covariates: Bayesian estimation of DALYs and ‘preventable’ DALYs

This paper presents Bayesian multivariate disease mapping and ecological regression models that take into account errors in covariates. Bayesian hierarchical formulations of multivariate disease models and covariate measurement models, with related methods of estimation and inference, are developed as an integral part of a Bayesian disability adjusted life years (DALYs) methodology for the analysis of multivariate disease or injury data and associated ecological risk factors and for small area DALYs estimation, inference, and mapping. The methodology facilitates the estimation of multivariate small area disease and injury rates and associated risk effects, evaluation of DALYs and ‘preventable’ DALYs, and identification of regions to which disease or injury prevention resources may be directed to reduce DALYs. The methodology interfaces and intersects the Bayesian disease mapping methodology and the global burden of disease framework such that the impact of disease, injury, and risk factors on population health may be evaluated to inform community health, health needs, and priority considerations for disease and injury prevention. A burden of injury study on road traffic accidents in local health areas in British Columbia, Canada, is presented as an illustrative example. Copyright © 2009 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.     

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.     

Bayesian age–period–cohort models with versatile interactions and long‐term predictions: mortality and population in Finland 1878–2050

Age–period–cohort (APC) models are widely used for studying time trends of disease incidence or mortality. Model identifiability has become less of a problem with Bayesian APC models. These models are usually based on random walk (RW1, RW2) smoothing priors. For long and complex time series and for long predicted periods, these models as such may not be adequate. We present two extensions for the APC models. First, we introduce flexible interactions between the age, period and cohort effects based on a two‐dimensional conditional autoregressive smoothing prior on the age/period plane. Our second extension uses autoregressive integrated (ARI) models to provide reasonable long‐term predictions. To illustrate the utility of our model framework, we provide stochastic predictions for the Finnish male and female population, in 2010–2050. For that, we first study and forecast all‐cause male and female mortality in Finland, 1878–2050, showing that using an interaction term is needed for fitting and interpreting the observed data. We then provide population predictions using a cohort component model, which also requires predictions for fertility and migration. As our main conclusion, ARI models have better properties for predictions than the simple RW models do, but mixing these prediction models with RW1 or RW2 smoothing priors for observed periods leads to a model that is not fully consistent. Further research with our model framework will concentrate on using a more consistent model for smoothing and prediction, such as autoregressive integrated moving average models with state‐space methods or Gaussian process priors. Copyright © 2013 John Wiley & Sons, Ltd.     

Fast subset scan for multivariate event detection

We present new subset scan methods for multivariate event detection in massive space–time datasets. We extend the recently proposed ‘fast subset scan’ framework from univariate to multivariate data, enabling computationally efficient detection of irregular space–time clusters even when the numbers of spatial locations and data streams are large. For two variants of the multivariate subset scan, we demonstrate that the scan statistic can be efficiently optimized over proximity‐constrained subsets of locations and over all subsets of the monitored data streams, enabling timely detection of emerging events and accurate characterization of the affected locations and streams. Using our new fast search algorithms, we perform an empirical comparison of the Subset Aggregation and Kulldorff multivariate subset scans on synthetic data and real‐world disease surveillance tasks, demonstrating tradeoffs between the detection and characterization performance of the two methods. Copyright © 2012 John Wiley & Sons, Ltd.     

Parameter-expanded data augmentation for analyzing correlated binary data using multivariate probit models

Data augmentation has been commonly utilized to analyze correlated binary data using multivariate probit models in Bayesian analysis. However, the identification issue in the multivariate probit models necessitates a rigorous Metropolis-Hastings algorithm for sampling a correlation matrix, which may cause slow convergence and inefficiency of Markov chains. It is well-known that the parameter-expanded data augmentation, by introducing a working/artificial parameter or parameter vector, makes an identifiable model be non-identifiable and improves the mixing and convergence of data augmentation components. Therefore, we motivate to develop efficient parameter-expanded data augmentations to analyze correlated binary data using multivariate probit models. We investigate both the identifiable and non-identifiable multivariate probit models and develop the corresponding parameter-expanded data augmentation algorithms. We point out that the approaches, based on one non-identifiable model, circumvent a Metropolis-Hastings algorithm for sampling a correlation matrix and improve the convergence and mixing of correlation parameters; the identifiable model may produce the estimated regression parameters with smaller standard errors than the non-identifiable model does. We illustrate our proposed approaches using simulation studies and through the application to a longitudinal dataset from the Six Cities study.     

A flexible parametric approach to examining spatial variation in relative survival

Most of the few published models used to obtain small‐area estimates of relative survival are based on a generalized linear model with piecewise constant hazards under a Bayesian formulation. Limitations of these models include the need to artificially split the time scale, restricted ability to include continuous covariates, and limited predictive capacity. Here, an alternative Bayesian approach is proposed: a spatial flexible parametric relative survival model. This overcomes previous limitations by combining the benefits of flexible parametric models: the smooth, well‐fitting baseline hazard functions and predictive ability, with the Bayesian benefits of robust and reliable small‐area estimates. Both spatially structured and unstructured frailty components are included. Spatial smoothing is conducted using the intrinsic conditional autoregressive prior. The model was applied to breast, colorectal, and lung cancer data from the Queensland Cancer Registry across 478 geographical areas. Advantages of this approach include the ease of including more realistic complexity, the feasibility of using individual‐level input data, and the capacity to conduct overall, cause‐specific, and relative survival analysis within the same framework. Spatial flexible parametric survival models have great potential for exploring small‐area survival inequalities, and we hope to stimulate further use of these models within wider contexts. Copyright © 2016 John Wiley & Sons, Ltd.     

Bayesian transformation cure frailty models with multivariate failure time data

We propose a class of transformation cure frailty models to accommodate a survival fraction in multivariate failure time data. Established through a general power transformation, this family of cure frailty models includes the proportional hazards and the proportional odds modeling structures as two special cases. Within the Bayesian paradigm, we obtain the joint posterior distribution and the corresponding full conditional distributions of the model parameters for the implementation of Gibbs sampling. Model selection is based on the conditional predictive ordinate statistic and deviance information criterion. As an illustration, we apply the proposed method to a real data set from dentistry.     

Approximate inference for disease mapping with sparse Gaussian processes

Gaussian process (GP) models are widely used in disease mapping as they provide a natural framework for modeling spatial correlations. Their challenges, however, lie in computational burden and memory requirements. In disease mapping models, the other difficulty is inference, which is analytically intractable due to the non‐Gaussian observation model. In this paper, we address both these challenges. We show how to efficiently build fully and partially independent conditional (FIC/PIC) sparse approximations for the GP in two‐dimensional surface, and how to conduct approximate inference using expectation propagation (EP) algorithm and Laplace approximation (LA). We also propose to combine FIC with a compactly supported covariance function to construct a computationally efficient additive model that can model long and short length‐scale spatial correlations simultaneously. The benefit of these approximations is computational. The sparse GPs speed up the computations and reduce the memory requirements. The posterior inference via EP and Laplace approximation is much faster and is practically as accurate as via Markov chain Monte Carlo. Copyright © 2010 John Wiley & Sons, Ltd.     

Adjusting power for a baseline covariate in linear models

The analysis of covariance provides a common approach to adjusting for a baseline covariate in medical research. With Gaussian errors, adding random covariates does not change either the theory or the computations of general linear model data analysis. However, adding random covariates does change the theory and computation of power analysis. Many data analysts fail to fully account for this complication in planning a study. We present our results in five parts. (i) A review of published results helps document the importance of the problem and the limitations of available methods. (ii) A taxonomy for general linear multivariate models and hypotheses allows identifying a particular problem. (iii) We describe how random covariates introduce the need to consider quantiles and conditional values of power. (iv) We provide new exact and approximate methods for power analysis of a range of multivariate models with a Gaussian baseline covariate, for both small and large samples. The new results apply to the Hotelling-Lawley test and the four tests in the "univariate" approach to repeated measures (unadjusted, Huynh-Feldt, Geisser-Greenhouse, Box). The techniques allow rapid calculation and an interactive, graphical approach to sample size choice. (v) Calculating power for a clinical trial of a treatment for increasing bone density illustrates the new methods. We particularly recommend using quantile power with a new Satterthwaite-style approximation.