Proportion Of Zeros Research Articles

Estimation methods for the variance of Birnbaum-Saunders distribution containing zero values with application to wind speed data in Thailand.

Thailand is currently grappling with a severe problem of air pollution, especially from small particulate matter (PM), which poses considerable threats to public health. The speed of the wind is pivotal in spreading these harmful particles across the atmosphere. Given the inherently unpredictable wind speed behavior, our focus lies in establishing the confidence interval (CI) for the variance of wind speed data. To achieve this, we will employ the delta-Birnbaum-Saunders (delta-BirSau) distribution. This statistical model allows for analyzing wind speed data and offers valuable insights into its variability and potential implications for air quality. The intervals are derived from ten different methods: generalized confidence interval (GCI), bootstrap confidence interval (BCI), generalized fiducial confidence interval (GFCI), and normal approximation (NA). Specifically, we apply GCI, BCI, and GFCI while considering the estimation of the proportion of zeros using the variance stabilized transformation (VST), Wilson, and Hannig methods. To evaluate the performance of these methods, we conduct a simulation study using Monte Carlo simulations in the R statistical software. The study assesses the coverage probabilities and average widths of the proposed confidence intervals. The simulation results reveal that GFCI based on the Wilson method is optimal for small sample sizes, GFCI based on the Hannig method excels for medium sample sizes, and GFCI based on the VST method stands out for large sample sizes. To further validate the practical application of these methods, we employ daily wind speed data from an industrial area in Prachin Buri and Rayong provinces, Thailand.

A Novel Slope-Matrix-Graph Algorithm to Analyze Compositional Microbiome Data.

Networks are widely used to represent relationships between objects, including microorganisms within ecosystems, based on high-throughput sequencing data. However, challenges arise with appropriate statistical algorithms, handling of rare taxa, excess zeros in compositional data, and interpretation. This work introduces a novel Slope-Matrix-Graph (SMG) algorithm to identify microbiome correlations primarily based on slope-based distance calculations. SMG effectively handles any proportion of zeros in compositional data and involves: (1) searching for correlated relationships (e.g., positive and negative directions of changes) based on a "target of interest" within a setting, and (2) quantifying graph changes via slope-based distances between objects. Evaluations on simulated datasets demonstrated SMG's ability to accurately cluster microbes into distinct positive/negative correlation groups, outperforming methods like Bray-Curtis and SparCC in both sensitivity and specificity. Moreover, SMG demonstrated superior accuracy in detecting differential abundance (DA) compared to ZicoSeq and ANCOM-BC2, making it a robust tool for microbiome analysis. A key advantage is SMG's natural capacity to analyze zero-inflated compositional data without transformations. Overall, this simple yet powerful algorithm holds promise for diverse microbiome analysis applications.

Zero-inflated multivariate tobit regression modeling

A frequent challenge encountered in real-world applications is data having a high proportion of zeros. Focusing on ecological abundance data, much attention has been given to zero-inflated count data. Models for non-negative continuous abundance data with an excess of zeros are rarely discussed. Work presented here considers the creation of a point mass at zero through a left-censoring approach or through a hurdle approach. We incorporate both mechanisms to capture the analog of zero-inflation for count data. Additionally, primary attention has been given to univariate zero-inflated modeling (e.g., single species), whereas data often arise jointly (e.g., a collection of species). With multivariate abundance data, a key issue is to capture dependence among the species at a site, both in terms of positive abundance as well as absence. Therefore, our contribution is a model for multivariate zero-inflated continuous data that are non-negative. Working in a Bayesian framework, we discuss the issue of separating the two sources of zeros and offer model comparison metrics for multivariate zero-inflated data. In an application, we model the total biomass for five tree species obtained from plots established in the Forest Inventory Analysis database in the Northeast region of the United States.

Patent Keyword Analysis Using Bayesian Zero-Inflated Model and Text Mining

Patent keyword analysis is used to analyze the technology keywords extracted from collected patent documents for specific technological fields. Thus, various methods related to this type of analysis have been researched in the industrial engineering fields, such as technology management and new product development. To analyze the patent document data, we have to search for patents related to the target technology and preprocess them to construct the patent–keyword matrix for statistical and machine learning algorithms. In general, a patent–keyword matrix has an extreme zero-inflated problem. This is because each keyword occupies one column even if it is included in only one document among all patent documents. General zero-inflated models have a limit at which the performance of the model deteriorates when the proportion of zeros becomes extremely large. To solve this problem, we applied a Bayesian inference to a general zero-inflated model. In this paper, we propose a patent keyword analysis using a Bayesian zero-inflated model to overcome the extreme zero-inflated problem in the patent–keyword matrix. In our experiments, we collected practical patents related to digital therapeutics technology and used the patent–keyword matrix preprocessed from them. We compared the performance of our proposed method with other comparative methods. Finally, we showed the validity and improved performance of our patent keyword analysis. We expect that our research can contribute to solving the extreme zero-inflated problem that occurs not only in patent keyword analysis, but also in various text big data analyses.

MbDecoda: a debiased approach to compositional data analysis for microbiome surveys.

Potentially pathogenic or probiotic microbes can be identified by comparing their abundance levels between healthy and diseased populations, or more broadly, by linking microbiome composition with clinical phenotypes or environmental factors. However, in microbiome studies, feature tables provide relative rather than absolute abundance of each feature in each sample, as the microbial loads of the samples and the ratios of sequencing depth to microbial load are both unknown and subject to considerable variation. Moreover, microbiome abundance data are count-valued, often over-dispersed and contain a substantial proportion of zeros. To carry out differential abundance analysis while addressing these challenges, we introduce mbDecoda, a model-based approach for debiased analysis of sparse compositions of microbiomes. mbDecoda employs a zero-inflated negative binomial model, linking mean abundance to the variable of interest through a log link function, and it accommodates the adjustment for confounding factors. To efficiently obtain maximum likelihood estimates of model parameters, an Expectation Maximization algorithm is developed. A minimum coverage interval approach is then proposed to rectify compositional bias, enabling accurate and reliable absolute abundance analysis. Through extensive simulation studies and analysis of real-world microbiome datasets, we demonstrate that mbDecoda compares favorably with state-of-the-art methods in terms of effectiveness, robustness and reproducibility.

Fractional Lindley distribution generated by time scale theory, with application to discrete-time lifetime data

ABSTRACT The fractional Lindley distribution is used to model the distribution of perturbations in count data regressions, which allow for dealing with widely dispersed data. It is obtained from the non-fractional Lindley distribution by replacing the support T = R + by T = N ∖ { 0 } and applying time scale theory, whose ambition is to unify the theories of difference equations and differential equations, integral and differential calculus, and the calculus of finite differences. It thus provides a framework for the study of dynamical systems in discrete-continuous time. Delta moments are discrete-time Laplace transforms of the frequency function of the fractional Lindley distribution. The parameter of the fractional Lindley distribution is estimated by least squares, weighted least squares, maximum likelihood, moments, and proportions. The moment estimator always exists, so that delta moments result from the nabla Laplace transform of the frequency function of the fractional Lindley distribution. The maximum likelihood estimates have the least mean-square errors. The proportion method works satisfactorily only when the mode of the distribution is null and the proportion of zeros is high. A simulation allows for quantifying the mean-square errors associated with the estimators. A count regression based on the fractional Lindley distribution with data on the total number of stays after hospital admission among U.S. residents aged 65 and over shows that the Akaike information criteria is significantly lower than with the uniform Poisson and Poisson regressions.

New discrete trigonometric distributions: estimation with application to count data

ABSTRACT In this research, we contribute to the development of original discrete distributions for count data analysis. We elaborate four new and different discrete trigonometric distributions with two parameters based on a discretization of the so-called sin-Weibull, cos-Weibull, tan-Weibull, and type II tan-Weibull distributions. We investigate the advantages of the proposed distributions and their trigonometric nature for capturing count data. In particular, some fundamental distributional properties of the discrete sin-Weibull distribution are derived. Subsequently, we turn it into a statistical model. Parameter estimation by the maximum likelihood and proportion of zeros and ones methods are then discussed. A simulation study is carried out to evaluate some frequentist properties of the developed methodology. As the study focuses mainly on the practical aspect, the applicability of the proposed models is evaluated using real-count datasets from various fields.

Modeling Zero-inflated Count Data Using Generalized Poisson and Ordinal Logistic Regression Models in Medical Research.

In medical research, the study's design and statistical methods are pivotal, as they guide interpretation and conclusion. Selecting appropriate statistical models hinges on the distribution of the outcome measure. Count data, frequently used in medical research, often exhibit over-dispersion or zero inflation. Occasionally, count data are considered ordinal (with a maximum outcome value of 5), and this calls for the application of ordinal regression models. Various models exist for analyzing over-dispersed data such as negative binomial, generalized Poisson (GP), and ordinal regression model. This study aims to examine whether the GP model is a superior alternative to the ordinal logistic regression (OLR) model, specifically in the context of zero-inflated Poisson models using both simulated and real-time data. Simulated data were generated with varied estimates of regression coefficients, sample sizes, and various proportions of zeros. The GP and OLR models were compared using fit statistics. Additionally, comparisons were made using real-time datasets. The simulated results consistently revealed lower bias and mean squared error values in the GP model compared to the OLR model. The same trend was observed in real-time datasets, with the GP model consistently demonstrating lower standard errors. Except when the sample size was 1000 and the proportions of zeros were 30% and 40%, the Bayesian information criterion consistently favored the GP model over the OLR model. This study establishes that the proposed GP model offers a more advantageous alternative to the OLR model. Moreover, the GP model facilitates easier modeling and interpretation when compared to the OLR model.

Variable Importance Measures for Multivariate Random Forests

Multivariate random forests (or MVRFs) are an extension of tree-based ensembles to examine multivariate responses. MVRF can be particularly helpful where some of the responses exhibit sparse (e.g., zero-inflated) distributions, making borrowing strength from correlated features attractive. Tree-based algorithms select features using variable importance measures (VIMs) that score each covariate based on the strength of dependence of the model on that variable. In this paper, we develop and propose new VIMs for MVRFs. Specifically, we focus on the variable’s ability to achieve split improvement, i.e., the difference in the responses between the left and right nodes obtained after splitting the parent node, for a multivariate response. Our proposed VIMs are an improvement over the default naïve VIM in existing software and allow us to investigate the strength of dependence both globally and on a per-response basis. Our simulation studies show that our proposed VIM recovers the true predictors better than naïve measures. We demonstrate usage of the VIMs for variable selection in two empirical applications; the first is on Amazon Marketplace data to predict Buy Box prices of multiple brands in a category, and the second is on ecology data to predict co-occurrence of multiple, rare bird species. A feature of both data sets is that some outcomes are sparse — exhibiting a substantial proportion of zeros or fixed values. In both cases, the proposed VIMs when used for variable screening give superior predictive accuracy over naïve measures.

PSXIV-18 Fit of the Zero-Inflated Negative Binomial Model to Analyze Fecal Egg Counts

Abstract Fecal egg count (FEC) is used as an indicator of parasite infection level in sheep. The distribution of FEC is non-Gaussian and typically overdispersed, often with an excess in zero counts. Quantifying the extent of inflation of zero counts can be difficult. Our objective was to assess the potential zero-inflation problem in FEC resulting from variation in infection with gastrointestinal nematodes by using a generalized linear model approach. The zero-inflated Poisson (ZIP) and zero-inflated negative binomial (ZINB) models are useful techniques to analyze count data with excess zeros; ZINB also handles overdispersion. The ZINB model has the potential to delineate ‘true’ zeros, in this case, animals resistant to parasitism and thereby with zero FEC, from ‘false’ zeros, animals never or minimally exposed to a parasite challenge. By distinguishing false zeros, those animals expressing parasite resistance may be more clearly identified. Two datasets on Katahdin sheep, a hair breed known to express resistance to gastrointestinal nematodes, were investigated; a smaller set (n = 3,048) with FEC and FAMACHA (FAM) scores, a subjective measure of anemia indicative of parasitism by Haemonchus contortus, a blood sucking helminth; and a larger set (n = 14,405) with FEC and a contemporary group (CG) designation, assigned based on the flock, birth year, management group, and FEC recording date of the animal. Among animals with FAM recorded, 14% had scores indicative of at least border line anemia. Amongst the 410 CG, 22% had mean FEC more than 500 egg/g, a threshold value routinely used to indicate a substantial infection level with H. contortus. For each dataset, the Poisson, Negative Binomial, ZIP, and ZINB models were fit and compared using R (pscl package) and SAS software, with the ZINB providing the best fit. In the models considered, FEC was the response variable and either FAM or CG was the explanatory variable, depending on the dataset. Despite evidence of parasite challenge, the true and false zeros could not be delineated in both data sets using these models. The estimated proportion of false zeros due to inflation did not differ from the proportion of zeros observed in the data set. Either all zeros coincided with no infection, which seems unlikely in Katahdins, or neither FAM nor CG provided sufficient information to distinguish resistant from uninfected individuals. Alternative or additional explanatory variables, such as packed cell volume or immunoglobulin concentrations indicative of parasitic infection, may be necessary to separate true from false zero FEC in sheep challenged with gastrointestinal nematodes using the ZINB model.

Bayesian Analysis of Two-Part Latent Variable Model with Mixed Data

AbstractIn analyzing semi-continuous data, two-part model is a widely appreciated tool, in which two components are enclosed to characterize the mixing proportion of zeros and the actual level of positive values in semi-continuous data. The primary interest underlying such a model is primarily to exploit the dependence of the observed covariates on the semi-continuous variables; as such, the exploitation of unobserved heterogeneity is sometimes ignored. In this paper, we extend the conventional two-part regression model to much more general situations where multiple latent factors are considered to interpret the latent heterogeneity arising from the absence of covariates. A structural equation is constructed to describe the interrelationships between the latent factors. Moreover, a general statistical analysis procedure is developed to accommodate semi-continuous, ordered and unordered data simultaneously. A procedure for parameter estimation and model assessment is developed under a Bayesian framework. Empirical results including a simulation study and a real example are presented to illustrate the proposed methodology.

Estimation of zero-inflated bivariate Poisson regression with missing covariates

Zero-inflated bivariate Poisson regression (ZIBP) models are most often applied to correlated bivariate count data that contain a large proportion of zeros. These models have been applied in various fields, such as medical research, health economics, insurance, sports, etc. Because, in practice, some of the covariates involved in ZIBP regression modeling often have missing values, we propose methods for estimating the parameters of the ZIBP regression model with missing at random covariates. Assuming that the selection probability is unknown and estimated non parametrically (by a kernel estimator), we propose inverse probability weighting (IPW) and multiple imputation (MI) methods for estimating the parameters of the ZIBP regression model with missing at random covariates. Asymptotic properties of the proposed estimators are studied under certain regularity conditions. A simulation study is conducted to evaluate the performance of the proposed methods. Finally, the practical application of the proposed methodology is illustrated with data from the health-care field.

Dealing with Zeros: Adolescent Drug Use, Perceived Disapproval, and Perceived Harm.

Because studies examining youth drug use often have data with a high proportion of zeros, they often do not meet the assumptions for univariate or linear regression analyses that are typically used. We demonstrate the use of zero-inflated negative binomial regression models to address excessive zeros in drug use frequency on perceptions of disapproval and perceived harm among middle and high school students (N = 522). We found that perceptions of parent disapproval were a better predictor of marijuana use (p = .01) than peer disapproval. Perceived harm was related to marijuana use (p = .04). Researchers should consider using zero-inflated negative binomial regression models when examining youth drug use.

Counting the Unseen: Estimation of Susceptibility Proportions in Zero-Inflated Models Using a Conditional Likelihood Approach

Zero-inflated count data models are widely used in various fields such as ecology, epidemiology, and transportation, where count data with a large proportion of zeros is prevalent. Despite their widespread use, their theoretical properties have not been extensively studied. This study aims to investigate the impact of ignoring heterogeneity in event count intensity and susceptibility probability on zero-inflated count data analysis within the zero-inflated Poisson framework. To address this issue, we propose a novel conditional likelihood approach that uses positive count data only to estimate event count intensity parameters and develop a consistent estimator for estimating the average susceptibility probability. Our approach is compared with the maximum likelihood approach, and we demonstrate our findings through a comprehensive simulation study and real data analysis. The results can also be extended to zero-inflated binomial and geometric models with similar conclusions. These findings contribute to the understanding of the theoretical properties of zero-inflated count data models and provide a practical approach to handling heterogeneity in such models.

Evaluating the performance of different Bayesian count models in modelling childhood vaccine uptake among children aged 12–23 months in Nigeria

BackgroundChoosing appropriate models for count health outcomes remains a challenge to public health researchers and the validity of the findings thereof. For count data, the mean–variance relationship and proportion of zeros is a major determinant of model choice. This study aims to compare and identify the best Bayesian count modelling technique for the number of childhood vaccine uptake in Nigeria.MethodsWe explored the performances of Poisson, negative binomial and their zero-inflated forms in the Bayesian framework using cross-sectional data pooled from the Nigeria Demographic and Health Survey conducted between 2003 and 2018. In multivariable analysis, these Bayesian models were used to identify factors associated with the number of vaccine uptake among children. Model selection was based on the -2 Log-Likelihood (-2 Log LL), Leave-One-Out Cross-Validation Information Criterion (LOOIC) and Watanabe-Akaike/Widely Applicable Information Criterion (WAIC).ResultsExploratory analysis showed the presence of excess zeros and overdispersion with a mean of 4.36 and a variance of 12.86. Observably, there was a significant increase in vaccine uptake over time. Significant factors included the mother’s age, level of education, religion, occupation, desire for last-child, place of delivery, exposure to media, birth order of the child, wealth status, number of antenatal care visits, postnatal attendance, healthcare decision maker, community poverty, community illiteracy, community unemployment, rural proportion and number of health facilities per 100,000. The zero-inflated negative binomial model was best fit with -2Log LL of -27171.47, LOOIC of 54464.2, and WAIC of 54588.0.ConclusionThe Bayesian zero-inflated negative binomial model was most appropriate to identify factors associated with the number of childhood vaccines received in Nigeria due to the presence of excess zeros and overdispersion. Improving vaccine uptake by addressing the associated risk factors should be promptly embraced.

Bayesian clustering of multiple zero-inflated outcomes.

Several applications involving counts present a large proportion of zeros (excess-of-zeros data). A popular model for such data is the hurdle model, which explicitly models the probability of a zero count, while assuming a sampling distribution on the positive integers. We consider data from multiple count processes. In this context, it is of interest to study the patterns of counts and cluster the subjects accordingly. We introduce a novel Bayesian approach to cluster multiple, possibly related, zero-inflated processes. We propose a joint model for zero-inflated counts, specifying a hurdle model for each process with a shifted Negative Binomial sampling distribution. Conditionally on the model parameters, the different processes are assumed independent, leading to a substantial reduction in the number of parameters as compared with traditional multivariate approaches. The subject-specific probabilities of zero-inflation and the parameters of the sampling distribution are flexibly modelled via an enriched finite mixture with random number of components. This induces a two-level clustering of the subjects based on the zero/non-zero patterns (outer clustering) and on the sampling distribution (inner clustering). Posterior inference is performed through tailored Markov chain Monte Carlo schemes. We demonstrate the proposed approach on an application involving the use of the messaging service WhatsApp. This article is part of the theme issue 'Bayesian inference: challenges, perspectives, and prospects'.

Territory and population attributes affect Florida scrub-jay fecundity in fire-adapted ecosystems.

Fecundity, the number of young produced by a breeding pair during a breeding season, is a primary component in evolutionary and ecological theory and applications. Fecundity can be influenced by many environmental factors and requires long-term study due to the range of variation in ecosystem dynamics. Fecundity data often include a large proportion of zeros when many pairs fail to produce any young during a breeding season due to nest failure or when all young die independently after fledging. We conducted color banding and monthly censuses of Florida scrub-jays (Aphelocoma coerulescens) across 31 years, 15 populations, and 761 territories along central Florida's Atlantic coast. We quantified how fecundity (juveniles/pair-year) was influenced by habitat quality, presence/absence of nonbreeders, population density, breeder experience, and rainfall, with a zero-inflated Bayesian hierarchical model including both a Bernoulli (e.g., brood success) and a Poisson (counts of young) submodel, and random effects for year, population, and territory. The results identified the importance of increasing "strong" quality habitat, which was a mid-successional state related to fire frequency and extent, because strong territories, and the proportion of strong territories in the overall population, influenced fecundity of breeding pairs. Populations subject to supplementary feeding also had greater fecundity. Territory size, population density, breeder experience, and rainfall surprisingly had no or small effects. Different mechanisms appeared to cause annual variation in fecundity, as estimates of random effects were not correlated between the success and count submodels. The increased fecundity for pairs with nonbreeders, compared to pairs without, identified empirical research needed to understand how the proportion of low-quality habitats influences population recovery and sustainability, because dispersal into low-quality habitats can drain nonbreeders from strong territories and decrease overall fecundity. We also describe how long-term study resulted in reversals in our understanding because of complications involving habitat quality, sociobiology, and population density.

Quantifying inherent predictability and spatial synchrony in the aphid vector Myzus persicae: field-scale patterns of abundance and regional forecasting error in the UK.

Sugar beet is threatened by virus yellows, a disease complex vectored by aphids that reduces sugar content. We present an analysis of Myzus persicae population dynamics with and without neonicotinoid seed treatment. We use 6 years' yellow water trap and field-collected aphid data and two decades of 12.2m suction-trap aphid migration data. We investigate both spatial synchrony and forecasting error to understand the structure and spatial scale of field counts and why forecasting aphid migrants lacks accuracy. Our aim is to derive statistical parameters to inform regionwide pest management strategies. Spatial synchrony, indicating the coincident change in counts across the region over time, is rarely present and is best described as stochastic. Uniquely, early season field populations in 2019 did show spatial synchrony to 90 km compared to the overall average weekly correlation length of 23 km. However, 70% of the time series were spatially heterogenous, indicating patchy between-field dynamics. Field counts lacked the same seasonal trend and did not peak in the same week. Forecasts tended to under-predict mid-season log10 counts. A strongly negative correlation between forecasting error and the proportion of zeros was shown. Field populations are unpredictable and stochastic, regardless of neonicotinoid seed treatment. This outcome presents a problem for decision-support that cannot usefully provide a single regionwide solution. Weighted permutation entropy inferred that M. persicae 12.2m suction-trap time series had moderate to low intrinsic predictability. Early warning using a migration model tended to predict counts at lower levels than observed. © 2022 The Authors. Pest Management Science published by John Wiley & Sons Ltd on behalf of Society of Chemical Industry.

Modelling malaria in southernmost provinces of Thailand: a two-step process for analysis of highly right-skewed data with a large proportion of zeros

BackgroundMalaria remains a serious health problem in the southern border provinces of Thailand. The issue areas can be identified using an appropriate statistical model. This study aimed to investigate malaria for its spatial occurrence and incidence rate in the southernmost provinces of Thailand.MethodsThe Thai Office of Disease Prevention and Control, Ministry of Public Health, provided total hospital admissions of malaria cases from 2008 to 2020, which were classified by age, gender, and sub-district of residence. Sixty-two sub-districts were excluded since they had no malaria cases. A logistic model was used to identify spatial occurrence patterns of malaria, and a log-linear regression model was employed to model the incidence rate after eliminating records with zero cases.ResultsThe overall occurrence rate was 9.8% and the overall median incidence rate was 4.3 cases per 1,000 population. Malaria occurence peaked at young adults aged 20–29, and subsequently fell with age for both sexes, whereas incidence rate increased with age for both sexes. Malaria occurrence and incidence rates fluctuated; they appeared to be on the decline. The area with the highest malaria occurrence and incidence rate was remarkably similar to the area with the highest number of malaria cases, which were mostly in Yala province's sub-districts bordering Malaysia.ConclusionsMalaria is a serious problem in forest-covered border areas. The correct policies and strategies should be concentrated in these areas, in order to address this condition.

Zero-augmented beta-prime model for multilevel semi-continuous data: a Bayesian inference

Semi-continuous data characterized by an excessive proportion of zeros and right-skewed continuous positive values appear frequently in medical research. One example would be the pharmaceutical expenditure (PE) data for which a substantial proportion of subjects investigated may report zero. Two-part mixed-effects models have been developed to analyse clustered measures of semi-continuous data from multilevel studies. In this study, we propose a new flexible two-part mixed-effects model with skew distributions for nested semi-continuous cost data under the framework of a Bayesian approach. The proposed model specification consists of two mixed-effects models linked by the correlated random effects: Part I) a model on the occurrence of positive values using a generalized logistic mixed model; and Part II) a model on the magnitude of positive values using a linear mixed model where the model errors follow skew distributions including beta-prime (BP). The proposed method is illustrated with pharmaceutical expenditure data from a multilevel observational study and the analytic results are reported by comparing potential models under different skew distributions. Simulation studies are conducted to assess the performance of the proposed model. The DIC3, LPML, WAIC, and LOO as the Bayesian model selection criteria and measures of divergence used to compare the models.