Spatial Model Selection Using Bayes Factor and Ratio of Variabilities for Asthma Mortality Data

The statistical analysis of spatially correlated data has become an important scientific research topic lately. The analysis of the mortality or morbidity rates observed at different areas may help to decide if people living in certain locations are considered at higher risk than others. Once the statistical model for the data of interest has been chosen, further effort can be devoted to identifying the areas under higher risks. Many scientists, including statisticians, have tried the conditional autoregressive (CAR) model to describe the spatial autocorrelation among the observed data. This model has greater smoothing effect than the exchangeable models, such as the Poisson gamma model for spatial data. This paper focuses on comparing the two types of models using the index LG, the ratio of local to global variability. Two applications, Taiwan asthma mortality and Scotland lip cancer, are considered and the use of LG is illustrated. The estimated values for both data sets are small, implying a Poisson gamma model may be favoured over the CAR model. We discuss the implications for the two applications respectively. To evaluate the performance of the index LG, we also compute the Bayes factor, a Bayesian model selection criterion, to see which model is preferred for the two applications and simulation data. To derive the value of LG, we estimate its posterior mode based on samples derived from the BUGS program, while for Bayes factor we use the double Laplace-Metropolis method, Schwarz criterion, and a modified harmonic mean for approximations. The results of LG and Bayes factor are consistent. We conclude that LG is fairly accurate as an index for selection between Poisson gamma and CAR model. When easy and fast computation is of concern, we recommend using LG as the first and less costly index.

In spatial data analysis, the prior conditional autoregressive (CAR) model is used to express the spatial dependence on random effects from adjacent regions. This paper provides a new proposed approach regarding the development of the existing normal CAR model into a more flexible, Fernandez–Steel skew normal (FSSN) CAR model. This approach is able to capture spatial random effects that have both symmetrical and asymmetrical patterns. The FSSN CAR model is built on the basis of the normal CAR with an additional skew parameter. The FSSN distribution is able to provide good estimates for symmetry with heavy- or light-tailed and skewed-right and skewed-left data. The effects of this approach are demonstrated by establishing the FSSN distribution and FSSN CAR model in spatial data using Stan language. On the basis of the plot of the estimation results and histogram of the model error, the FSSN CAR model was shown to behave better than both models without a spatial effect and with the normal CAR model. Moreover, the smallest widely applicable information criterion (WAIC) and leave-one-out (LOO) statistical values also validate the model, as FSSN CAR is shown to be the best model used.

Ecological data often exhibit spatial pattern, which can be modeled as autocorrelation. Conditional autoregressive (CAR) and simultaneous autoregressive (SAR) models are network‐based models (also known as graphical models) specifically designed to model spatially autocorrelated data based on neighborhood relationships. We identify and discuss six different types of practical ecological inference using CAR and SAR models, including: (1) model selection, (2) spatial regression, (3) estimation of autocorrelation, (4) estimation of other connectivity parameters, (5) spatial prediction, and (6) spatial smoothing. We compare CAR and SAR models, showing their development and connection to partial correlations. Special cases, such as the intrinsic autoregressive model (IAR), are described. Conditional autoregressive and SAR models depend on weight matrices, whose practical development uses neighborhood definition and row‐standardization. Weight matrices can also include ecological covariates and connectivity structures, which we emphasize, but have been rarely used. Trends in harbor seals (Phoca vitulina) in southeastern Alaska from 463 polygons, some with missing data, are used to illustrate the six inference types. We develop a variety of weight matrices and CAR and SAR spatial regression models are fit using maximum likelihood and Bayesian methods. Profile likelihood graphs illustrate inference for covariance parameters. The same data set is used for both prediction and smoothing, and the relative merits of each are discussed. We show the nonstationary variances and correlations of a CAR model and demonstrate the effect of row‐standardization. We include several take‐home messages for CAR and SAR models, including (1) choosing between CAR and IAR models, (2) modeling ecological effects in the covariance matrix, (3) the appeal of spatial smoothing, and (4) how to handle isolated neighbors. We highlight several reasons why ecologists will want to make use of autoregressive models, both directly and in hierarchical models, and not only in explicit spatial settings, but also for more general connectivity models.

Establishing the patterns of a disease or disease mapping is very important in disease control and prevention. The level of accuracy that is achieved at this stage determines the effectiveness of control measures to be developed. Disease mapping has been widely done using the frequentist approach which is limited in that it does not consider prior probability distribution of a phenomenon. This limitation leads to lower levels of accuracy and validity. This study proposed a Bayesian Approach for mapping tuberculosis incidence in Meru County, Kenya. Correlational research design was utilized to determine association between TB cases and geographical locations where the cases were positively identified. Secondary data from the Meru County Health Records was used for this study. Spatial autocorrelation was performed to determine patterns of TB incidence. The study applied Conditional Autoregressive (CAR) model and Poisson Lognormal (PLN) model under the Bayesian Approach to model TB incidence in order to determine spatial temporal trends. Parameter estimation for the models was done using GIBBs Sampling under Markov Chain Monte-Carlo (MCMC). The two models (PLN and CAR) were compared using Deviance Information Criteria (DIC) to determine the one that had a better fit. Morans's I statistic was -0.3150 (p>0.05) meaning that there was no spatial autocorrelation for TB incidence in Meru County. Model results further indicated that there was no spacial dependence for TB incidence in Meru County. Deviance Information Criterion (DIC) values obtained were 0.22541 for CAR model and 0.56723 for PLN model meaning that CAR model had outperformed the PLN model. The study concluded that CAR model is more effective for disease mapping since it incorporates information from neighboring regions directly into the model to increase accuracy of estimates. Therefore, the study recommended use of Bayesian modelling for disease mapping as it incorporates prior information to stabilize the parameter estimates.

Conditional Autoregressive (CAR) models have been widely used in various disciplines, including epidemiological studies. The application of the CAR model in epidemiological studies is often associated with the relative risk of an infectious disease. This relative risk value can be estimated using the CAR models. Here, we evaluate four commonly used CAR models: the Intrinsic CAR, the Besag-York-Mollié CAR (BYM CAR), the BYM-modified CAR (BYM2 CAR), and the Leroux CAR (LCAR). To estimate CAR model parameters, Bayesian inference and the Integrated Nested Laplace Approximation (INLA) concept are used. The selected model was then used to model the number of dengue hemorrhagic fever (DHF) cases in Central Java Province in 2024. To support this analysis, we used 50 datasets simulated for each sample size (n), ranging from 10 to 100. The results of the study showed that of the four models compared, the best model was BYM2. This model was then used to model DHF cases in 2024 in Central Java Province. The research findings indicate the necessity of controlling population density, optimizing the role of medical personnel, and preparing for increased rainfall to curb the spread of dengue fever. Comprehensive detection and control measures through medical facilities are also required. Meanwhile, based on the coefficient of the altitude variable in the model, altitude has a positive influence on the number of dengue fever cases. Therefore, the conflicting conclusions between the model results and the medical perspective require data verification and further study of this variable.

Conditional autoregressive (CAR) distributions are used to account for spatial autocorrelation in small areal or lattice data to assess the spatial risks of diseases. The intrinsic CAR (ICAR) distribution has been primarily used as the priori distribution of spatially autocorrelated random variables in the framework of Bayesian statistics. The posterior distributions of spatial variates and unknown parameters of Bayesian ICAR models are estimated with the Markov chain Monte Carlo (MCMC) methods or integrated nested Laplace approximation (INLA), which may suffer from failures in numeric convergence. This study used the Laplace approximation, a fast computational method available in software Template Model Builder (TMB), for the maximum likelihood estimation (MLEs) of the ICAR model parameters. This study used the TMB to integrate out the latent spatial variates for the fast computations of marginal likelihood functions. This study compared the runtime and performance among the TMB, MCMC, and INLA implementations with three case studies of human diseases in the United Kingdom and the United States. The MLEs of the ICAR model with TMB were similar to those by the MCMC and INLA methods. The TMB implementation was faster than the MCMC (up to 100–200 times) and INLA (nine times) models.• This study built conditional autoregressive models in template model builder• TMB implementation was 100-200 times faster than the MCMC method• TMB implementation was also faster than Bayesian approximation with R INLA

Covid-19 cases in Indonesia occurred for the first time on 2 March 2020. By 30 September 2022, Indonesia had 158,173 Covid-19 deaths. Several studies have been done in modelling Covid-19 cases. However, research in modelling the number of Covid-19 deaths using the Bayesian Spatial Conditional Autoregressive (CAR) model is still rare. The Bayesian spatial CAR model has high flexibility in relative risk (RR) modeling. CAR models can include various types of spatial effects and can include covariates in the model. RR represents the ratio of the risk of outcome (Covid-19) in the exposed group compared to the population average (the unexposed group). This study aims to evaluate the BYM, Leroux, and Localised models with five hyperpriors, to obtain the best model for estimating the RR of Covid-19 deaths in Indonesia and to create RR maps. This study used aggregate data on Covid-19 deaths (2 March 2020 - 30 September 2022). Data on the total population and population density of each province in 2021 were also used. The best model selection is based on the lowest Watanabe Akaike Information Criterion (WAIC) and Deviance Information Criterion (DIC) values, and Modified Moran's I (MMI) residual values. The result showed that the CAR BYM model with covariates and with Inverse-Gamma IG(0.5; 0.0005) prior distribution had the lowest DIC and WAIC. As the BYM model does not converge, the model cannot be used in determining the RR of Covid-19 deaths in Indonesia. From the other three models that converge, the Bayesian CAR Leroux model without covariate with IG(0,5;0,0005) has the lowest DIC(393,76), and WAIC(400,12), and its MMI value (-0,26) is approximate to zero. Therefore, the Bayesian CAR Leroux model without covariate with IG(0,5;0,0005) is preferred. The province with the highest RR (2,76) and the lowest RR (0,22) are Yogyakarta and Papua, respectively.

A bivariate Bayes method is proposed for estimating the mortality rates of a single disease for a given population, using additional information from a second disease. The information on the two diseases is assumed to be from the same population groups or areas. The joint frequencies of deaths for the two diseases for given populations are assumed to have a bivariate Poisson distribution with joint means proportional to the population sizes. The relationship between the mortality rates of the two different diseases if formulated through the twofold conditional autoregressive (CAR) model, where spatial effects as well as indexes of spatial dependence are introduced to capture the structured clusterings among areas. This procedure is compared to a univariate hierarchical Bayes procedure that uses information from one disease only. Comparisons of two procedures are made by the optimal property, a Monte Carlo study, real data, and the Bayes factor. All of the methods that we consider demonstrate a substantial improvement in the bivariate over the univariate procedure. For analyzing male and female lung cancer data from the state of Missouri, Markov chain Monte Carlo methods are used to estimate mortality rates.

This document reviews and applies recently developed techniques for Bayesian estimation and model selection in the context of Time Series modeling for Stochastic Volatility. After the literature review on Generalized Conditional Autoregressive models, Stochastic Volatility models, and the relevant results on Markov Chain Monte Carlo methods (MCMC), an example applying such techniques is shown. The methodology is used with a series of Weekly Colombian-USA Exchange Rate on seven different mod els. The GARCH model, which uses Type-IV Pearson distribution, is favored for the selecting technique, Reversible Jump MCMC, over other models, including Stochastic Volatility Models with a Student-t distribution.

Gaussian component mixtures and CAR models in Bayesian disease mapping

Stunting continues to be a significant health issue, particularly in developing nations, with Indonesia ranking third in prevalence in Southeast Asia. This research examined the risk of stunting and influencing factors in Indonesia by implementing various Bayesian spatial conditional autoregressive (CAR) models that include covariates. A total of 750 models were run, including five different Bayesian spatial CAR models (Besag-York-Mollie (BYM), CAR Leroux and three forms of localised CAR), with 30 covariate combinations and five different hyperprior combinations for each model. The Poisson distribution was employed to model the counts of stunting cases. After a comprehensive evaluation of all model selection criteria utilized, the Bayesian localised CAR model with three covariates were preferred, either allowing up to 2 clusters with a variance hyperprior of inverse-gamma (1, 0.1) or allowing 3 clusters with a variance hyperprior of inverse-gamma (1, 0.01). Poverty and recent low birth weight (LBW) births are significantly associated with an increased risk of stunting, whereas child diet diversity is inversely related to the risk of stunting. Model results indicated that Sulawesi Barat Province has the highest risk of stunting, with DKI Jakarta Province the lowest. These areas with high stunting require interventions to reduce poverty, LBW births and increase child diet diversity.

In a latent class IRT model in which the latent classes are ordered on one dimension, the class specific response probabilities are subject to inequality constraints. The number of these inequality constraints increase dramatically with the number of response categories per item, if assumptions like monotonicity or double monotonicity of the cumulative category response functions are postulated. A Markov chain Monte Carlo method, the Gibbs sampler, can sample from the multivariate posterior distribution of the parameters under the constraints. Bayesian model selection can be done by posterior predictive checks and Bayes factors. A simulation study is done to evaluate results of the application of these methods to ordered latent class models in three realistic situations. Also, an example of the presented methods is given for existing data with polytomous items. It can be concluded that the Bayesian estimation procedure can handle the inequality constraints on the parameters very well. However, the application of Bayesian model selection methods requires more research.

Stunting remains a persistent public health issue in Indonesia, exhibiting significant spatial and temporal variation. To address this, we employed a hierarchical Bayesian spatio-temporal localized Conditional Autoregressive (CAR) model that includes a clustering component to identify risk factors and estimate relative risk (RR) across 34 provinces from 2020 to 2022. A total of 480 models were evaluated, encompassing three variants of the Bayesian spatio-temporal localized CAR model, 32 covariate combinations, and five hyperprior settings. Assuming a Poisson likelihood for stunting counts, the optimal model was estimated using Markov Chain Monte Carlo methods and included two covariates, namely the poverty rate and the incidence of low birth weight, with up to five spatial clusters. Higher poverty levels and increased prevalence of low birth weight were significantly associated with elevated stunting risk among children under five. Spatio-temporal clustering patterns and the estimated relative risks of stunting varied across Indonesian provinces from 2020 to 2022. Nusa Tenggara Timur consistently ranked among the top three provinces with the highest risk (RR = 2.421 in 2020; 2.384 in 2021; 2.676 in 2022). The highest risk was observed in Sulawesi Barat in 2022 (RR = 2.768), while DKI Jakarta consistently showed the lowest (RR = 0.004). Some key points of the article are:•Bayesian spatio-temporal models facilitate the classification of distinct area groups•The models were employed to analyze stunting patterns in Indonesia.•The inclusion of covariates influenced the number of groups identified.

The modelling of property prices has been extensively studied in econometrics, with widely used approaches including generalised linear regression and geographically weighted regression. These models commonly address local spatial correlations observed in property price data. However, despite its potential to capture spatial effects, the conditional autoregressive (CAR) model remains underutilised for this purpose. This study examines the robustness and predictive power of the CAR model, comparing it with established spatial models across three different datasets generation. An illustrative case study on Lombok house price data is also included. Simulation results showed that the CAR model demonstrates a distinct advantage, achieving lower bias and variability compared to other spatial regression models, effectively capturing neighbourhood-based spatial relationships, and exhibiting strong predictive power across various scenarios. In the Lombok case study, the CAR model outperformed other models, providing more precise estimates for property-related factors such as land size and built-up area. The results confirm that CAR’s spatial framework enables a nuanced analysis of property values across regions, enhancing accuracy in predictive models. This study also reveals the distinct strengths and limitations of each model, offering insights into their predictive accuracy and applicability across diverse real estate contexts.

Smoothing risks is one of the main goals in disease mapping as classical measures, such as standardized mortality ratios, can be extremely variable. However, smoothing risks might hinder the detection of high risk areas, since these two objectives are somewhat contradictory. Most of the work on smoothing risks and detection of high risk areas has been derived using conditional autoregressive (CAR) models. In this work, penalized splines (P-splines) models are also investigated. Confidence intervals for the log-relative risk predictor will be derived as a tool to detect high-risk areas. The performance of P-spline and CAR models will be compared in terms of smoothing (relative bias), sensitivity (ability to detect high risk areas), and specificity (ability to discard false patterns created by noise) through a simulation study based on the well-known Scottish lip cancer data.