Beta-binomial Research Articles

A Bayesian beta-binomial piecewise growth mixture model for longitudinal overdispersed binomial data.

In a recent 12-week smoking cessation trial, varenicline tartrate failed to show significant improvements in enhancing end-of-treatment abstinence when compared with placebo among adolescents and young adults. The original analysis aimed to assess the average effect across the entire population using timeline followback methods, which typically involve overdispersed binomial counts. We instead propose to investigate treatment effect heterogeneity among latent classes of participants using a Bayesian beta-binomial piecewise linear growth mixture model specifically designed to address longitudinal overdispersed binomial responses. Within each class, we fit a piecewise linear beta-binomial mixed model with random changepoints for each study group to detect critical windows of treatment efficacy. Using this model, we can cluster subjects who share similar characteristics, estimate the class-specific mean abstinence trends for each study group, and quantify the treatment effect over time within each class. Our analysis identified two classes of subjects: one comprising high-abstinent individuals, typically young adults and light smokers, in which varenicline led to improved abstinence; and another comprising low-abstinent individuals for whom varenicline showed no discernible effect. These findings highlight the importance of tailoring varenicline to specific participant subgroups, thereby advancing precision medicine in smoking cessation studies.

Improving analysis of cognitive outcomes in cardiovascular trials using different statistical approaches

BackgroundThe Mini-Mental State Examination (MMSE) and the Montreal Cognitive Assessment (MoCA) questionnaires are commonly used to measure global cognition in clinical trials. Because these scales are discrete and bounded with ceiling and floor effects and highly skewed, their analysis as continuous outcomes presents challenges. Normality assumptions of linear regression models are usually violated, which may result in failure to detect associations with variables of interest.MethodsAlternative approaches to analyzing the results of these cognitive batteries include transformations (standardization, square root, or log transformation) of the scores in the multivariate linear regression (MLR) model, the use of nonlinear beta-binomial regression (which is not dependent on the assumption of normality), or Tobit regression, which adds a latent variable to account for bounded data. We aim to empirically compare the model performance of all proposed approaches using four large randomized controlled trials (ORIGIN, TRANSCEND, COMPASS, and NAVIGATE-ESUS), and using as metrics the Akaike information criterion (AIC). We also compared the treatment effects for the methods that have the same unit of measure (i.e., untransformed MLR, beta-binomial, and Tobit).ResultsThe beta-binomial consistently demonstrated superior model performance, with the lowest AIC values among nearly all the approaches considered, followed by the MLR with square root and log transformations across all four studies. Notably, in ORIGIN, a substantial AIC reduction was observed when comparing the untransformed MLR to the beta-binomial, whereas other studies had relatively small AIC reductions. The beta-binomial model also resulted in a significant treatment effect in ORIGIN, while the untransformed MLR and Tobit regression showed no significance. The other three studies had similar and insignificant treatment effects among the three approaches.ConclusionWhen analyzing discrete and bounded outcomes, such as cognitive scores, as continuous variables, a beta-binomial regression model improves model performance, avoids spurious significance, and allows for a direct interpretation of the actual cognitive measure.Trials registrationORIGIN (NCT00069784). Registered on October 1, 2003; TRANSCEND (NCT00153101). Registered on September 9, 2005; COMPASS (NCT01776424). Registered on January 24, 2013; NAVIGATE-ESUS (NCT02313909). Registered on December 8, 2014.

Assessing the hierarchical beta-binomial model as a basic information sharing tool in basket trials

ABSTRACT The majority of statistical methods to share information in basket trials are based on a Bayesian hierarchical model with a common normal distribution for the logit-transformed response rates. The methods are of varying complexity, yet they all use this basic model. Generally, complexity is an obstacle for the application in clinical trials and that includes the use of the logit-transformation. The transformation complicates the model and impedes a direct interpretation of the hyperparameters. On the other hand, there exist basket trial designs which directly work on the probability scale of the response rate which facilitates the understanding of the model for many stakeholders. In order to reduce unnecessary complexity, we considered using a hierarchical beta-binomial model instead of the transformed models. This article investigates whether this approach is a practicable alternative to the commonly applied sharing tools based on a logit-transformation of the response rates. For this purpose, we performed a systematic comparison of the two models, starting with the distributional assumptions for the response rates, continuing with the Bayesian behavior together with binomial data in an independent setting and ended with a simulation study for the hierarchical model under various data and prior scenarios. All Bayesian comparisons require equal starting points, wherefore we propose a calibration procedure to choose similar priors for the models. The evaluation of the sharing property additionally required an evaluation measure for simulation results, which we derived in this work. The conclusion of the comparison is that the hierarchical beta-binomial model is a feasible alternative basic model to share information in basket trials.

BBM: A novel beta-binomial-distribution-based biclustering algorithm for mining m6A co-methylation patterns

Studies have indicated that mining co-methylation patterns of N6-methyladenosine (m6A) can aid in understanding its functional mechanism. However, current algorithms require the conversion of IP and input data matrices into a single methylation level matrix for co-methylation pattern mining. This conversion introduces additional noise and leads to inaccurate results. To address this issue, we propose a novel biclustering algorithm, BBM, based on the beta-binomial distribution. BBM can perform biclustering analysis directly on IP and input matrices without the need for data matrix conversion. The Gibbs sampling method is used to infer parameters in BBM. Simulation studies show that BBM can accurately reproduce the three pre-planted patterns and their overlap. On real data, BBM identify two patterns. Gene Ontology enrichment analysis reveal that the sites remaining in the two patterns are enriched in different biological processes. One pattern is associated with RNA splicing, translation, transport, and localization, while the other is related to histone modification and histone H3 acetylation. These findings suggest that the two patterns are effective co-methylation patterns. In addition, GOE_Score scoring demonstrate that these two patterns are more biologically relevant than patterns mined by other current mainstream algorithms. BBM is freely available at https://github.com/cumtlzhy/BBM.

Delayed interactions in the noisy voter model through the periodic polling mechanism

We investigate the effects of delayed interactions on the stationary distribution of the noisy voter model. We assume that the delayed interactions occur through the periodic polling mechanism and replace the original instantaneous two-agent interactions. In our analysis, we require that the polling period aligns with the delay in announcing poll outcomes. As expected, when the polling period is relatively short, the model with delayed interactions is almost equivalent to the original model. As the polling period increases, oscillatory behavior emerges, but the model with delayed interactions still converges to stationary distribution. The stationary distribution resembles a Beta-binomial distribution, with its shape parameters scaling with the polling period. The observed scaling behavior is non-monotonic. Namely, the shape parameters peak at some intermediate polling period.

Determination of Single Sampling Attribute Plans Based Upon Dodge Roaming Model with Application

Bayesian sampling plans for production inspection involve using a sampling method to assess the features of the plan, under the assumption that defect rates fluctuate randomly among different production batches. This results in a likelihood distribution that can be established through experience and the quality information at hand. In this study, the parameters of a single Bayesian sampling plan were determined using the Beta-Binomial distribution, and were subsequently contrasted with parameters from other single sampling plans. Based on research findings, (Ala corporation for soft drinks) oversees the control of product quality. Since the variable fluctuates randomly between manufacturing batches, 120 batches were selected to calculate the defect rate by analyzing batch size and number of defective items. Applying Bayesian and decision-making models can lead to the development of a single sampling inspection procedure that closely approximates the actual quality level. When the decision-making model was used, the researchers discovered that the sample size was smaller and led to lower inspection costs compared to other inspection plans.

A re-analysis of about 60,000 sparse data meta-analyses suggests that using an adequate method for pooling matters.

In sparse data meta-analyses (with few trials or zero events), conventional methods may distort results. Although better-performing one-stage methods have become available in recent years, their implementation remains limited in practice. This study examines the impact of using conventional methods compared to one-stage models by re-analysing meta-analyses from the Cochrane Database of Systematic Reviews in scenarios with zero event trials and few trials. For each scenario, we computed one-stage methods (Generalised linear mixed model [GLMM], Beta-binomial model [BBM], Bayesian binomial-normal hierarchical model using a weakly informative prior [BNHM-WIP]) and compared them with conventional methods (Peto-Odds-ratio [PETO], DerSimonian-Laird method [DL] for zero event trials; DL, Paule-Mandel [PM], Restricted maximum likelihood [REML] method for few trials). While all methods showed similar treatment effect estimates, substantial variability in statistical precision emerged. Conventional methods generally resulted in smaller confidence intervals (CIs) compared to one-stage models in the zero event situation. In the few trials scenario, the CI lengths were widest for the BBM on average and significance often changed compared to the PM and REML, despite the relatively wide CIs of the latter. In agreement with simulations and guidelines for meta-analyses with zero event trials, our results suggest that one-stage models are preferable. The best model can be either selected based on the data situation or, using a method that can be used in various situations. In the few trial situation, using BBM and additionally PM or REML for sensitivity analyses appears reasonable when conservative results are desired. Overall, our results encourage careful method selection.

A robust regression model for bounded count health data.

Bounded count response data arise naturally in health applications. In general, the well-known beta-binomial regression model form the basis for analyzing this data, specially when we have overdispersed data. Little attention, however, has been given to the literature on the possibility of having extreme observations and overdispersed data. We propose in this work an extension of the beta-binomial regression model, named the beta-2-binomial regression model, which provides a rather flexible approach for fitting a regression model with a wide spectrum of bounded count response data sets under the presence of overdispersion, outliers, or excess of extreme observations. This distribution possesses more skewness and kurtosis than the beta-binomial model but preserves the same mean and variance form of the beta-binomial model. Additional properties of the beta-2-binomial distribution are derived including its behavior on the limits of its parametric space. A penalized maximum likelihood approach is considered to estimate parameters of this model and a residual analysis is included to assess departures from model assumptions as well as to detect outlier observations. Simulation studies, considering the robustness to outliers, are presented confirming that the beta-2-binomial regression model is a better robust alternative, in comparison with the binomial and beta-binomial regression models. We also found that the beta-2-binomial regression model outperformed the binomial and beta-binomial regression models in our applications of predicting liver cancer development in mice and the number of inappropriate days a patient spent in a hospital.

Task reliability index for operator performance and failure probability assessment in control room simulators

Main control room simulators are an important information source for improving the empirical basis of human reliability analysis (HRA). Both quantitative and qualitative observations provide valuable insight on the operators’ ability to successfully perform their tasks. However, relevant portions of this evidence are not used in typical approaches for estimation of human error probabilities (HEP), which are based on failure counts. With small sample sizes and the operator performance levels representative for nuclear power plants, very little information enters the probability estimates. This paper presents a novel approach to formalize the collection and use of reliability-relevant evidence from simulators. The methodology treats task reliability with the plant-centred orientation characteristic of the systems or Probabilistic Safety Assessment (PSA) perspective as well as a human-centred view of task performance based in human factors engineering. These are combined to form a Task reliability index (TRI) able to distinguish finer performance variations and consider indications of potential failures. Bayesian updating of HEP distributions is suggested by translating the TRI data into pseudo-failures and trials, preserving the convenient coupling of Beta-Binomial distributions in the Bayesian analysis. The approach is demonstrated using data from the International HRA Empirical Study and is shown to produce results that are comparable to those obtained in the empirical study, which were based on comprehensive expert analyses.

Comparison of the EQ-5D-5L and the patient-reported outcomes measurement information system preference score (PROPr) in the United States

BackgroundIn contrast to prior research, our study presents longitudinal comparisons of the EQ-5D-5L and Patient-Reported Outcomes Measurement Information System (PROMIS) preference (PROPr) scores. This fills a gap in the literature, providing a much-needed understanding of these preference-based measures and their applications in healthcare research. Furthermore, our study provides equations to estimate one measure from the other, a tool that can significantly facilitate comparisons across studies.MethodsWe administered a health survey to 4,098 KnowledgePanel® members living in the United States. A subset of 1,256 (82% response rate) with back pain also completed the six-month follow-up survey. We then conducted thorough cross-sectional and longitudinal analyses of the two measures, including product-moment correlations between scores, associations with demographic variables, and health conditions. To estimate one measure from the other, we used ordinary least squares (OLS) regression with the baseline data from the general population.ResultsThe correlation between the EQ-5D-5L and PROPr scores was 0.69, but the intraclass correlation was only 0.34 because the PROPr had lower (less positive) mean scores on the 0 (dead) to 1 (perfect health) continuum than the EQ-5D-5L. The associations between the two preference measures and demographic variables were similar at baseline. The product-moment correlation between unstandardized beta coefficients for each preference measure regressed on 22 health conditions was 0.86, reflecting similar patterns of unique associations. Correlations of change from baseline to 6 months in the two measures with retrospective perceptions of change were similar. Adjusted variance explained in OLS regressions predicting one measure from the other was 48%. On average, the predicted values were within a half-standard deviation of the observed EQ-5D-5L and PROPr scores. The beta-binomial regression model slightly improved over the OLS model in predicting the EQ-5D-5L from the PROPr but was equivalent to the OLS model in predicting the PROPr.ConclusionDespite substantial mean differences, the EQ-5D-5L and PROPr have similar cross-sectional and longitudinal associations with other variables. We provide the OLS regression equations for use in cost-effectiveness research and meta-analyses. Future studies are needed to compare these measures with different conditions and interventions to provide more information on their relative validity.

NQuack: An R package for predicting ploidal level from sequence data using site-based heterozygosity.

Traditional methods of ploidal-level estimation are tedious; using DNA sequence data for cytotype estimation is an ideal alternative. Multiple statistical approaches to leverage sequence data for ploidy inference based on site-based heterozygosity have been developed. However, these approaches may require high-coverage sequence data, use inappropriate probability distributions, or have additional statistical shortcomings that limit inference abilities. We introduce nQuack, an open-source R package that addresses the main shortcomings of current methods. nQuack performs model selection for improved ploidy predictions. Here, we implement expectation maximization algorithms with normal, beta, and beta-binomial distributions. Using extensive computer simulations that account for variability in sequencing depth, as well as real data sets, we demonstrate the utility and limitations of nQuack. Inferring ploidy based on site-based heterozygosity alone is difficult. Even though nQuack is more accurate than similar methods, we suggest caution when relying on any site-based heterozygosity method to infer ploidy.

Unassigning bacterial species for microbiome studies.

The method of 16S rRNA marker gene sequencing has fueled microbiome research and continues to be relevant. A perceived weakness of the method is that taxonomic assignments are not possible to make at the rank of species. We show that by working to rule out bacterial or archaeal species membership, we can provide an answer that is more accurate and useful. The Unassigner software operates on 16S rRNA marker gene data and computes a rule-out probability for species membership using a beta-binomial distribution. We demonstrate that our approach is accurate based on full-genome comparisons. Our method is consistent with existing approaches and dramatically improves on them based on the percentage of reads it can associate with a species in a sample. The software is available at https://github.com/PennChopMicrobiomeProgram/unassigner.IMPORTANCEWhile existing methods do not provide reliable species-level assignments for 16S rRNA marker gene data, the Unassigner software solves this problem by ruling out species membership, allowing researchers to reason at the species level.

Management and outcomes of cerebrovascular injuries after gunshot wounds to the cervical spine

Study designThis study was a multicenter retrospective analysis of cervical spine gunshot wound (GSW) patients. ObjectiveThe present study was conducted to evaluate the management and outcomes of vascular injuries following GSW involving the cervical spine. Summary of background dataGunshot wounds (GSW) injuring the cervical spine are associated with high rates of vascular injury. MethodsCharts of patients with GSW involving the cervical spine at two Level 1 trauma centers were reviewed from 2010 to 2021 for demographics, injury characteristics, management and follow-up. Statistical analysis included T tests and ANOVA for comparisons of continuous variables and chi-square testing for categorical variables, non-parametric tests were used when indicated. Beta-binomial models were used to estimate the probabilities outcomes. Bayesian regression models were utilized to compute risk ratios (RR) and their 95 % confidence intervals (CI) to enhance the inferential robustness. Results40 patients with cervical spine GSW and associated cerebrovascular injury were included in our analysis. 15 % of patients had Biffl grade (BG) V injuries, 50 % grade IV, and 35 % grade III–I. Angiography was performed in 35 % of patients. 5 of these patients (BG V–III) required endovascular treatment for pseudoaneurysm obliteration or parent vessel sacrifice. 7 patients (22 %) showed evidence of progression. 70 % of patients were placed on antiplatelet therapy for stroke prevention. Bayesian regression models with a skeptical prior for cerebral ischemia revealed a mean RR of 4.82 (95 % CI 1.02–14.48) in the BG V group, 0.75 (95 % CI 0.13–2.26) in the BG IV group, and 0.61 (95 % CI 0.06–2.01) in the combined BG III-I group. For demise the mean RR was 3.41 (95 % CI 0.58–10.65) in the BG V group and 1.69 (95 % CI 0.29–5.97) in the BG IV group. In the high BG (V, IV) group, 54.55 % of patients treated with antiplatelet therapy had complications. None of the patients that were treated with antiplatelet therapy in the low BG (III–I) group had complications. ConclusionsCervical spine GSWs are associated with high-grade vascular injuries and may require early endovascular intervention. Additionally, a high rate of injury progression was seen on follow up imaging, requiring subsequent intervention. Reintervention and demise were common and observed in high BG (V, IV) groups. The incidence of stroke was low, especially in low BG (I–III) groups, suggesting that daily aspirin prophylaxis is adequate for long-term stroke prevention.

Tilted beta regression and beta-binomial regression models: Mean and variance modeling

Tilted beta regression and beta-binomial regression models: Mean and variance modeling

A varying-coefficient model for the analysis of methylation sequencing data

DNA methylation is an important epigenetic modification involved in gene regulation. Advances in the next generation sequencing technology have enabled the retrieval of DNA methylation information at single-base-resolution. However, due to the sequencing process and the limited amount of isolated DNA, DNA-methylation-data are often noisy and sparse, which complicates the identification of differentially methylated regions (DMRs), especially when few replicates are available. We present a varying-coefficient model for detecting DMRs by using single-base-resolved methylation information. The model simultaneously smooths the methylation profiles and allows detection of DMRs, while accounting for additional covariates. The proposed model takes into account possible overdispersion by using a beta-binomial distribution. The overdispersion itself can be modeled as a function of the genomic region and explanatory variables. We illustrate the properties of the proposed model by applying it to two real-life case studies.

The Association Between Type of Insurance Plan, Out-of-Pocket Cost, and Adherence to Antihypertensive Medications in Medicare Supplement Insurance Enrollees.

Medicare supplement insurance, or Medigap, covers 21% of Medicare beneficiaries. Despite offsetting some out-of-pocket (OOP) expenses, remaining OOP costs may pose a barrier to medication adherence. This study aims to evaluate how OOP costs and insurance plan types influence medication adherence among beneficiaries covered by Medicare supplement plans. We conducted a retrospective analysis of the Merative MarketScan Medicare Supplement Database (2017-2019) in Medigap enrollees (≥65 years) with hypertension. The proportion of days covered (PDC) was a continuous measure of medication adherence and was also dichotomized (PDC ≥0.8) to quantify adequate adherence. Beta-binomial and logistic regression models were used to estimate associations between these outcomes and insurance plan type and log-transformed OOP costs, adjusting for patient characteristics. Among 27,407 patients with hypertension, the average PDC was 0.68 ± 0.31; 47.5% achieved adequate adherence. A mean $1 higher in 30-day OOP costs were associated with a 0.06 (95% confidence intervals [CIs]: -0.09 to -0.03) lower probability of adequate adherence, or a 5% (95% CI: 4%-7%) decrease in PDC. Compared with comprehensive plan enrollees, the odds of adequate adherence were lower among those with point-of-service plans (odds ratio [OR]: 0.69, 95% CI: 0.62-0.77), but higher among those with preferred provider organization (PPO) plans (OR: 1.08, 95% CI: 1.01-1.15). Moreover, the association between OOP costs and PDC was significantly greater for PPO enrollees. While Medicare supplement insurance alleviates some OOP costs, different insurance plans and remaining OOP costs influence medication adherence. Reducing patient cost-sharing may improve medication adherence.

Seeds in the guts: can seed traits explain seed survival after being digested by wild ungulates?

Plants inhabiting open landscapes are often dispersed by ungulates and are expected to be adapted to this type of dispersal through their seed traits. To find which traits help seeds survive the passage through digestion of wild ungulates, we conducted a comprehensive feeding experiment with almost forty species of plants and three species of ungulates. We fed specified numbers of seeds to the animals, collected the dung, and germinated the dung content. We explored whether seed morphological traits and seed nutrient contents are good predictors of seed survival after passage through the ungulate digestive system. We also tested how the seed survival differed after the passage through different ungulate species. To find answers, we used GLMM with beta-binomial distribution and animal and plant species as random factor, respectively. We found that species survival and germination success were negatively correlated to seed elongation and the thickness of the seed coat. Even though phylogenetically correct GLMM did not yield significant results, when we tested species from commonly represented families, separately (legumes and grasses compared to all other species) different traits had statistically significant effects. In the case of seed elongation, the effect changed direction from negative to positive when legumes and grasses were left out. Our results suggest that seed traits enabling species survival after passage through the digestive tract are strongly phylogenetically conserved and different groups of plants evolved different ways of adapting to grazing pressure and utilize it for dispersal.

A Bayesian quasi-likelihood design for identifying the minimum effective dose and maximum utility dose in dose-ranging studies.

Most existing dose-ranging study designs focus on assessing the dose-efficacy relationship and identifying the minimum effective dose. There is an increasing interest in optimizing the dose based on the benefit-risk tradeoff. We propose a Bayesian quasi-likelihood dose-ranging design that jointly considers safety and efficacy to simultaneously identify the minimum effective dose and the maximum utility dose to optimize the benefit-risk tradeoff. The binary toxicity endpoint is modeled using a beta-binomial model. The efficacy endpoint is modeled using the quasi-likelihood approach to accommodate various types of data (e.g. binary, ordinal or continuous) without imposing any parametric assumptions on the dose-response curve. Our design utilizes a utility function as a measure of benefit-risk tradeoff and adaptively assign patients to doses based on the doses' likelihood of being the minimum effective dose and maximum utility dose. The design takes a group-sequential approach. At each interim, the doses that are deemed overly toxic or futile are dropped. At the end of the trial, we use posterior probability criteria to assess the strength of the dose-response relationship for establishing the proof-of-concept. If the proof-of-concept is established, we identify the minimum effective dose and maximum utility dose. Our simulation study shows that compared with some existing designs, the Bayesian quasi-likelihood dose-ranging design is robust and yields competitive performance in establishing proof-of-concept and selecting the minimum effective dose. Moreover, it includes an additional feature for further maximum utility dose selection.

HBCR_DMR: A Hybrid Method Based on Beta-Binomial Bayesian Hierarchical Model and Combination of Ranking Method to Detect Differential Methylation Regions in Bisulfite Sequencing Data.

DNA methylation is a key epigenetic modification involved in gene regulation, contributing to both physiological and pathological conditions. For a more profound comprehension, it is essential to conduct a precise comparison of DNA methylation patterns between sample groups that represent distinct statuses. Analysis of differentially methylated regions (DMRs) using computational approaches can help uncover the precise relationships between these phenomena. This paper describes a hybrid model that combines the beta-binomial Bayesian hierarchical model with a combination of ranking methods known as HBCR_DMR. During the initial phase, we model the actual methylation proportions of the CpG sites (CpGs) within the replicates. This modeling is achieved through beta-binomial distribution, with parameters set by a group mean and a dispersion parameter. During the second stage, we establish the selection of distinguishing CpG sites based on their methylation status, employing multiple ranking techniques. Finally, we combine the ranking lists of differentially methylated CpG sites through a voting system. Our analyses, encompassing simulations and real data, reveal outstanding performance metrics, including a sensitivity of 0.72, specificity of 0.89, and an F1 score of 0.76, yielding an overall accuracy of 0.82 and an AUC of 0.94. These findings underscore HBCR_DMR's robust capacity to distinguish methylated regions, confirming its utility as a valuable tool for DNA methylation analysis.

Comments on the Bernoulli Distribution and Hilbe’s Implicit Extra-Dispersion

For decades, conventional wisdom maintained that binary 0–1 Bernoulli random variables cannot contain extra-binomial variation. Taking an unorthodox stance, Hilbe actively disagreed, especially for correlated observation instances, arguing that the universally adopted diagnostic Pearson or deviance dispersion statistics are insensitive to a variance anomaly in a binary context, and hence simply fail to detect it. However, having the intuition and insight to sense the existence of this departure from standard mathematical statistical theory, but being unable to effectively isolate it, he classified this particular over-/under-dispersion phenomenon as implicit. This paper explicitly exposes his hidden quantity by demonstrating that the variance in/deflation it represents occurs in an underlying predicted beta random variable whose real number values are rounded to their nearest integers to convert to a Bernoulli random variable, with this discretization masking any materialized extra-Bernoulli variation. In doing so, asymptotics linking the beta-binomial and Bernoulli distributions show another conventional wisdom misconception, namely a mislabeling substitution involving the quasi-Bernoulli random variable; this undeniably is not a quasi-likelihood situation. A public bell pepper disease dataset exhibiting conspicuous spatial autocorrelation furnishes empirical examples illustrating various features of this advocated proposition.