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Wolbachia are intracellular bacteria found in many species of arthropods and nematodes. They manipulate the reproduction of their arthropod hosts in various ways, may play a role in host speciation and have potential applications in biological pest control. Estimates suggest that at least 20% of all insect species are infected with Wolbachia. These estimates result from several Wolbachia screenings in which numerous species were tested for infection; however, tests were mostly performed on only one to two individuals per species. The actual percent of species infected will depend on the distribution of infection frequencies among species. We present a meta-analysis that estimates percentage of infected species based on data on the distribution of infection levels among species. We used a beta-binomial model that describes the distribution of infection frequencies of Wolbachia, shedding light on the overall infection rate as well as on the infection frequency within species. Our main findings are that (1) the proportion of Wolbachia-infected species is estimated to be 66%, and that (2) within species the infection frequency follows a 'most-or-few' infection pattern in a sense that the Wolbachia infection frequency within one species is typically either very high (>90%) or very low (<10%).

Non-Gaussian outcomes are often modeled using members of the so-called exponential family. Notorious members are the Bernoulli model for binary data, leading to logistic regression, and the Poisson model for count data, leading to Poisson regression. Two of the main reasons for extending this family are (1) the occurrence of overdispersion, meaning that the variability in the data is not adequately described by the models, which often exhibit a prescribed mean-variance link, and (2) the accommodation of hierarchical structure in the data, stemming from clustering in the data which, in turn, may result from repeatedly measuring the outcome, for various members of the same family, etc. The first issue is dealt with through a variety of overdispersion models, such as, for example, the beta-binomial model for grouped binary data and the negative-binomial model for counts. Clustering is often accommodated through the inclusion of random subject-specific effects. Though not always, one conventionally assumes such random effects to be normally distributed. While both of these phenomena may occur simultaneously, models combining them are uncommon. This paper proposes a broad class of generalized linear models accommodating overdispersion and clustering through two separate sets of random effects. We place particular emphasis on so-called conjugate random effects at the level of the mean for the first aspect and normal random effects embedded within the linear predictor for the second aspect, even though our family is more general. The binary, count and time-to-event cases are given particular emphasis. Apart from model formulation, we present an overview of estimation methods, and then settle for maximum likelihood estimation with analytic-numerical integration. Implications for the derivation of marginal correlations functions are discussed. The methodology is applied to data from a study in epileptic seizures, a clinical trial in toenail infection named onychomycosis and survival data in children with asthma.

The accurate estimation of species richness in a target region is still a statistical challenge, especially in a highly heterogeneous community. Most richness estimators have been developed based on the assumption that data are randomly sampled with replacement or that data are sampled from an infinite population. However, in reality, most sampling schemes in the field are implemented as sampling without replacement (SWOR). As such, estimators derived based on sampling with replacement may cause overestimation as the sampling fraction increases and not converge to the true richness as the sampling fraction approaches one. Sample‐based incidence data, in which the sampling unit is a plot, and only the presence or absence of a species in each chosen plot is recorded, is one of the most commonly used data types for assessing species diversity in ecological studies. In this manuscript, according to sample‐based incidence data collected through SWOR, a new richness estimator is proposed using a truncated beta‐binomial mixture model. The new estimator was obtained through the moment approach, which avoids using iterative numerical algorithms for parameter estimation and presents a closed‐form estimator as an alternative to the maximum likelihood method. Although the newly proposed method is a parametric‐based richness estimator, similar to nonparametric estimators, only the rare species in the sample (i.e. the frequencies of uniques and duplicates) are required to estimate undetected richness. Based on the hypothetical models, the statistical performances of the proposed estimator are evaluated under varying degrees of heterogeneity and different mean species detection rates. Compared to other widely used nonparametric and parametric estimators, the simulation results indicate that the proposed estimator has a smaller bias and lower root mean square error when the sampling fraction is greater than 10%, particularly in highly heterogeneous communities. In addition, one ForestGEO permanent forest plot dataset is used to evaluate and compare the proposed approach with other estimators discussed in the study. The results demonstrate that the proposed estimator, in comparison to other widely used estimators, produces less biased estimate of true richness, along with more accurate 95% confidence interval.

Background According to current official guidelines, there is no generally recommended minimum number of lymph nodes examined during surgery for cervical cancer. However, patients with few nodes removed are still common, and the prevalence of nodal invasion may be underestimated because of false‐negative findings. In this article, we introduced a statistical tool called the Nodal Staging Score (NSS), which predicts the minimum number of examined lymph nodes to confidently ensure a node‐negative status preoperatively. Methods Using the beta‐binomial model, we analyzed lymph node invasion data for 8789 patients with cervical cancer from the Surveillance, Epidemiology, and End Results database. This analysis quantified the number of lymph nodes that require assessment across various early International Federation of Gynecology and Obstetrics (FIGO) stages. We also performed univariate and multivariate Cox regression analyses to explore the prognostic significance of NSS. Results With an increased number of examined lymph nodes, the probability of missing nodal disease decreased and varied among different FIGO stages. For stages IB1–IIA, the examination of 6, 21, and 33 lymph nodes, respectively, was required to reduce the probability of missing positive nodes (i.e., 1−NSS) to less than 10%. The clinical significance of NSS was verified with prognostic information. Compared with NSS <0.90, NSS ≥0.90 was significantly associated with better overall survival for node‐negative patients. Conclusion The NSS is an auxiliary tool that not only enhances the precision of FIGO staging but also provides a statistical basis for postoperative evaluation to inform further clinical decision‐making. To our knowledge, this is the first report to make an accurate diagnosis of nodal invasion status and evaluate the minimum number of examined lymph nodes corresponding to different T stages for cervical cancer patients. Nodal staging score (NSS), as an auxiliary tool, could help surgeons to explore the real status of lymph node (LN) and assist the international federation of gynecology and obstetrics (FIGO) staging system to reflect tumor disseminating more correctly. It could also provide preoperative prediction of LN resection yields and postoperative evaluation for further clinical decision‐making.

We present a novel approach for developing summary statistics for use in approximate Bayesian computation (ABC) algorithms by using indirect inference. ABC methods are useful for posterior inference in the presence of an intractable likelihood function. In the indirect inference approach to ABC the parameters of an auxiliary model fitted to the data become the summary statistics. Although applicable to any ABC technique, we embed this approach within a sequential Monte Carlo algorithm that is completely adaptive and requires very little tuning. This methodological development was motivated by an application involving data on macroparasite population evolution modelled by a trivariate stochastic process for which there is no tractable likelihood function. The auxiliary model here is based on a beta-binomial distribution. The main objective of the analysis is to determine which parameters of the stochastic model are estimable from the observed data on mature parasite worms.

Background Patients with esophageal squamous cell carcinoma (ESCC) with lymph node metastasis may be misclassified as pN0 due to an insufficient number of lymph nodes examined (LNE). The purpose of this study was to confirm that patients with ESCC are indeed pN0 and to propose an adequate number for the correct nodal stage using the nodal staging score (NSS) developed by the beta‐binomial model. Methods A total of 1249 patients from the Surveillance, Epidemiology, and End Results (SEER) database between 2000 and 2017, and 1404 patients diagnosed with ESCC in our database between 2005 and 2018 were included. The NSS was developed to assess the probability of pN0 status based on both databases. The effectiveness of NSS was verified using survival analysis, including Kaplan–Meier curves and Cox models. Results Many patients were misclassified as pN0 based on our algorithm due to insufficient LNE. As the number of LNE increased, false‐negative findings dropped; accordingly, the NSS increased. In addition, NSS was an independent prognostic indicator for pN0 in patients with ESCC in the SEER database (hazard ratio [HR] 0.182, 95% confidence interval [CI] 0.046–0.730, p = 0.016) and our database (HR 0.215, 95% CI 0.055–0.842, p = 0.027). A certain number of nodes must be examined to achieve 90% of the NSS. Conclusions NSS could determine the probability of true pN0 status for patients, and it was sufficient in predicting survival and obtaining adequate numbers for lymphadenectomy. Probability density for some beta distributions.

Overdispersion is a common feature of models of biological data, but researchers often fail to model the excess variation driving the overdispersion, resulting in biased parameter estimates and standard errors. Quantifying and modeling overdispersion when it is present is therefore critical for robust biological inference. One means to account for overdispersion is to add an observation-level random effect (OLRE) to a model, where each data point receives a unique level of a random effect that can absorb the extra-parametric variation in the data. Although some studies have investigated the utility of OLRE to model overdispersion in Poisson count data, studies doing so for Binomial proportion data are scarce. Here I use a simulation approach to investigate the ability of both OLRE models and Beta-Binomial models to recover unbiased parameter estimates in mixed effects models of Binomial data under various degrees of overdispersion. In addition, as ecologists often fit random intercept terms to models when the random effect sample size is low (<5 levels), I investigate the performance of both model types under a range of random effect sample sizes when overdispersion is present. Simulation results revealed that the efficacy of OLRE depends on the process that generated the overdispersion; OLRE failed to cope with overdispersion generated from a Beta-Binomial mixture model, leading to biased slope and intercept estimates, but performed well for overdispersion generated by adding random noise to the linear predictor. Comparison of parameter estimates from an OLRE model with those from its corresponding Beta-Binomial model readily identified when OLRE were performing poorly due to disagreement between effect sizes, and this strategy should be employed whenever OLRE are used for Binomial data to assess their reliability. Beta-Binomial models performed well across all contexts, but showed a tendency to underestimate effect sizes when modelling non-Beta-Binomial data. Finally, both OLRE and Beta-Binomial models performed poorly when models contained <5 levels of the random intercept term, especially for estimating variance components, and this effect appeared independent of total sample size. These results suggest that OLRE are a useful tool for modelling overdispersion in Binomial data, but that they do not perform well in all circumstances and researchers should take care to verify the robustness of parameter estimates of OLRE models.

Background Meta-analyses are used to summarise the results of several studies on a specific research question. Standard methods for meta-analyses, namely inverse variance random effects models, have unfavourable properties if only very few (2 – 4) studies are available. Therefore, alternative meta-analytic methods are needed. In the case of binary data, the “common-rho” beta-binomial model has shown good results in situations with sparse data or few studies. The major concern of this model is that it ignores the fact that each treatment arm is paired with a respective control arm from the same study. Thus, the randomisation to a study arm of a specific study is disrespected, which may lead to compromised estimates of the treatment effect. The aim of this simulation study was to compare the “common-rho” beta-binomial model and several other beta-binomial models with standard meta-analyses models, including generalised linear mixed models and several inverse variance random effects models. Methods We conducted a simulation study comparing beta-binomial models and various standard meta-analysis methods. The design of the simulation aimed to consider meta-analytic situations occurring in practice. Results No method performed well in scenarios with only 2 studies in the random effects scenario. In this situation, a fixed effect model or a qualitative summary of the study results may be preferable. In scenarios with 3 or 4 studies, most methods satisfied the nominal coverage probability. The “common-rho” beta-binomial model showed the highest power under the alternative hypothesis. Conclusion The “common-rho” beta-binomial appears to be a good option for meta-analyses of very few studies. As residual concerns about the consequences of disrespecting randomisation may still exist, we recommend a sensitivity analysis with a standard meta-analysis method that respects randomisation.

Background Standard random-effects meta-analysis models for rare events exhibit significant limitations, particularly when synthesizing studies with double-zero events. While methodological advances in both frequentist and Bayesian frameworks now offer robust alternatives that bypass continuity corrections, the comparative performance of these approaches—especially between Bayesian and frequentist paradigms—remains understudied. Methods This study evaluates the performance of ten widely used meta-analysis models for binary outcomes, using the odds ratio as the effect measure. The evaluated models comprise seven frequentist and three Bayesian approaches. Simulations systematically varied key parameters, including control event rates, treatment effects, study numbers, and heterogeneity levels, to compare model performance across four metrics: percentage bias, 95% confidence/credible interval width, root mean square error, and coverage. The methods were further illustrated through applications to two published rare events meta-analyses. Results The results show that the beta-binomial model proposed by Kuss generally performed well, while the generalised estimating equations did not. In cases where heterogeneity is not large, all models tended to have a good performance except for the generalised estimating equations. When the heterogeneity is large, none of the compared models produced good performance. The Bayesian model incorporating the Beta-Hyperprior proposed by Hong et al. performed well, followed by the binomial-normal hierarchical model proposed by Bhaumik. Conclusions In summary, the beta-binomial model proposed by Kuss is recommended for rare events meta-analyses, and the Bayesian model is a promising method for pooling rare events data.

It is common practice that some or all animals in a group of animals, e.g. a herd, are tested for their health status by using a diagnostic test to investigate whether the herd is infected by a disease. Several obstacles complicate the estimation of herd prevalence on the basis of test results of the animals. First, diagnostic tests are often imperfect, resulting in a misclassification of the animal's disease status. It is well known how to correct the animal's apparent prevalence by using the diagnostic sensitivity and specificity of the animal test, but the effects on herd prevalence are less clear. Second, in practice, a herd is often defined as positive when at least one sampled animal tested positively. This definition is ambiguous and is also different from the herd prevalence that is based on having at least one diseased animal in the herd. The paper provides a discussion of these aspects and proposes a method to estimate the true herd prevalence on the basis of the health status of (all or a sample of) animals within a herd corrected for the sensitivity and specificity of the individual test, the number of animals that are tested in the herd and the uncertainty of the diagnostic test characteristics.