A new multivariate zero-adjusted Poisson model with applications to biomedicine.

Multivariate zero-inflated Bell distribution and its inference and applications

The objective of this article is to propose a comprehensive solution for analyzing multidimensional non-life claim count data that exhibits time and cross-dependence, as well as zero inflation. To achieve this, we introduce a multivariate INAR(1) model, with the innovation term characterized by either a multivariate zero-inflated Poisson distribution or a multivariate zero-inflated hurdle Poisson distribution. Additionally, our modeling framework accounts for the impact of individual and coverage-specific covariates on the mean parameters of each model, thereby facilitating the computation of customized insurance premiums based on varying risk profiles. To estimate the model parameters, we employ a novel expectation-maximization (EM) algorithm. Our model demonstrates satisfactory performance in the analysis of European motor third-party liability claim count data.

Effectiveness of sampling plans by attributes based on mixture distributions characterising microbial clustering in food

ObjectiveTo propose a more realistic model for disease cluster detection, through a modification of the spatial scan statistic to account simultaneously for inflated zeros and overdispersion.IntroductionSpatial Scan Statistics [1] usually assume Poisson or Binomial distributed data, which is not adequate in many disease surveillance scenarios. For example, small areas distant from hospitals may exhibit a smaller number of cases than expected in those simple models. Also, underreporting may occur in underdeveloped regions, due to inefficient data collection or the difficulty to access remote sites. Those factors generate excess zero case counts or overdispersion, inducing a violation of the statistical model and also increasing the type I error (false alarms). Overdispersion occurs when data variance is greater than the predicted by the used model. To accommodate it, an extra parameter must be included; in the Poisson model, one makes the variance equal to the mean.MethodsTools like the Generalized Poisson (GP) and the Double Poisson [2] may be a better option for this kind of problem, modeling separately the mean and variance, which could be easily adjusted by covariates. When excess zeros occur, the Zero Inflated Poisson (ZIP) model is used, although ZIP’s estimated parameters may be severely biased if nonzero counts are too dispersed, compared to the Poisson distribution. In this case the Inflated Zero models for the Generalized Poisson (ZIGP), Double Poisson (ZIDP) and Negative Binomial (ZINB) could be good alternatives to the joint modeling of excess zeros and overdispersion. By one hand, Zero Inflated Poisson (ZIP) models were proposed using the spatial scan statistic to deal with the excess zeros [3]. By the other hand, another spatial scan statistic was based on a Poisson-Gamma mixture model for overdispersion [4]. In this work we present a model which includes inflated zeros and overdispersion simultaneously, based on the ZIDP model. Let the parameter p indicate the zero inflation. As the the remaining parameters of the observed cases map and the parameter p are not independent, the likelihood maximization process is not straightforward; it becomes even more complicated when we include covariates in the analysis. To solve this problem we introduce a vector of latent variables in order to factorize the likelihood, and obtain a facilitator for the maximization process using the E-M (Expectation-Maximization) algorithm. We derive the formulas to maximize iteratively the likelihood, and implement a computer program using the E-M algorithm to estimate the parameters under null and alternative hypothesis. The p-value is obtained via the Fast Double Bootstrap Test [5].ResultsNumerical simulations are conducted to assess the effectiveness of the method. We present results for Hanseniasis surveillance in the Brazilian Amazon in 2010 using this technique. We obtain the most likely spatial clusters for the Poisson, ZIP, Poisson-Gamma mixture and ZIDP models and compare the results.ConclusionsThe Zero Inflated Double Poisson Spatial Scan Statistic for disease cluster detection incorporates the flexibility of previous models, accounting for inflated zeros and overdispersion simultaneously.The Hanseniasis study case map, due to excess of zero cases counts in many municipalities of the Brazilian Amazon and the presence of overdispersion, was a good benchmark to test the ZIDP model. The results obtained are easier to understand compared to each of the previous spatial scan statistic models, the Zero Inflated Poisson (ZIP) model and the Poisson-Gamma mixture model for overdispersion, taken separetely. The E-M algorithm and the Fast Double Bootstrap test are computationally efficient for this type of problem.

This study investigated the genetics of bovine tuberculosis (bTB) infectivity in Holstein-Friesian dairy cows using British national data. The analyses included cows with recorded sires from herds affected by bTB outbreaks between 2000 and 2022. Animals were considered bTB positive if they reacted positively to the skin test, had positive postmortem findings, or both. We introduced the "index case approach," based on the assumption that once the initial positively tested animals (index cases) are detected in a herd, subsequent infections (secondary cases) in the early stages of the breakdown are likely to be attributed to these animals. Genetic analysis of the number of secondary cases (NrSC) associated with a given index case was used to establish evidence of genetic variability in bTB infectivity of cattle, and derive EBV for infectivity for the sires of the index cases. Data were analyzed by employing Markov chain Monte Carlo techniques to fit generalized linear mixed models with either Poisson, zero-inflated Poisson (ZIP), hurdle Poisson, or geometric distributions. All 4 models demonstrated presence of genetic variance in cattle infectivity, with the strongest evidence provided by the ZIP and hurdle Poisson models. The hurdle Poisson model offered the most accurate and least biased predictions. Sire infectivity EBV from the Poisson, ZIP, and geometric models showed strong concordance, with pairwise correlations of 0.90 or higher. In contrast, correlations between EBV from the hurdle Poisson model and the other models ranged from 0.36 to 0.39. The association of the sire infectivity EBV with the average observed NrSC per sire and the proportion of infectious index case daughters per sire was generally moderate with correlations between 44% and 47% and 65% to 69%, respectively. Agreement among models for identifying the genetically most infectious sires was also reasonable, with 151 out of 285 sires appearing in the top 10% across models, and 122 (42.8%) also aligning with the top 10% based on observed average NrSC. Results provide novel evidence for exploitable genetic variance in bTB infectivity allowing the derivation of meaningful EBV. Based on the estimated posterior mean genetic variances obtained, reduction in infectivity by 1 genetic SD would result in a 32% to 44% decrease in the expected NrSC per index case. Further research is warranted to refine the phenotypic definition of infectivity and assess correlation with other dairy traits.

Type I multivariate zero-inflated Poisson distribution with applications

Bayesian signal detection methods, including the multiitem gamma Poisson shrinker (MGPS), assume a Poisson distribution for the number of reports. However, the database of the adverse event reporting system often has a large number of zero-count cells. A zero-inflated Poisson (ZIP) distribution can be more appropriate in this situation than a Poisson distribution. Few studies have considered ZIP-based models for Bayesian signal detection. In addition, most studies on Bayesian signal detection methods include simulation studies conducted assuming a gamma distribution for the prior. Herein, we extend the MGPS method using the ZIP model and apply various prior distributions. We evaluated the extended methods through an extensive simulation using more varied settings for the model and prior than existing methods. We varied the total number of reports, the number of true signals, the relative reporting rate, and the probability of observing a true zero. The results show that as the probability of observing a zero count increased, methods based on the ZIP model outperformed the Poisson model in most cases. We also found that using the mixture log-normal prior resulted in more conservative detection than other priors when the relative reporting rate is high. Conversely, more signals were found when using the mixture truncated normal distributions. We applied the Bayesian signal detection methods to data from the Korea Adverse Event Reporting System from 2012 to 2016.

Acceptance Sampling Plans are employed to safeguard against the inconsistent degradation of quality standards, especially when lots are of higher quality due to sophisticated technology, leading to a greater quantity of defect-free items. In this paper, a new sampling plan based on the Zero-Inflated Poisson Distribution (ZIP_F) with fuzzy parameters is proposed. As the production process improves due to technological advancements, the occurrence of defect-free items becomes more prevalent. The Zero-Inflated Poisson (ZIP_F) distribution is identified as the appropriate probability distribution for determining the number of defects in these situations. The primary aim is to develop a procedure for sampling with fuzzy parameters as a reference plan under the Zero-Inflated Poisson distribution (ZIP_F) framework. The fuzzy operating characteristic (fOC-ZIP) function of the sampling plan is assessed, and a comparative analysis is conducted with the existing sampling plan based on the Poisson distribution.

Many researchers have discussed zero-inflated univariate distributions. These univariate models are not suitable, for modeling events that involve different types of counts or defects. To model several types of defects, multivariate Poisson model is one of the appropriate model. This can further be modified to incorporate inflation at zero and we can have multivariate zero-inflated Poisson distribution. In the present article, we introduce a new Bivariate Zero Inflated Power Series Distribution and discuss inference related to the parameters involved in the model. We also discuss the inference related to Bivariate Zero Inflated Poisson Distribution. The model has been applied to a real life data. Extension to k-variate zero inflated power series distribution is also discussed.

Count Data in Medical Research: Poisson Regression and Negative Binomial Regression

Modeling accident frequencies as zero-altered probability processes: An empirical inquiry

Statistical significance varies depending on the sample size. Therefore, when the sample size is sufficient, even differences that affect the total variation very little may be statistically significant. For this reason, it is very important to report effect size measures that estimate the share of the difference between groups of samples in the total variation. This study aims to determine the most reliable effect size measures that can be used when evaluating data obtained from ecological studies. The three most popular effect size measures used in practice were compared in terms of their performance in 2700 different experimental conditions. For this purpose, random numbers generated from the multivariate Poisson distribution were used with the Monte Carlo simulation technique. As a result of the simulations, it was determined that Epsilon-squared and Omega-squared were quite unbiased estimators. Therefore, it was concluded that one of these two effect size measures should be reported in addition to the p-value when evaluating ecological studies.

: In this paper a large class of multivariate densities and frequency functions including the multivariate Poisson distribution and the compound multivariate Poisson distribution, are shown to have the decreasing in transposition property introduced by Hollander, Proschan, and Sethuraman (1977, Ann. Statist. 5, 722-733). Applications relevant to reliability are given, including applications to shock models, cumulation of damage, and component down times. (Author)

Crashes present different collision types at freeway diverge areas. The research reported in this paper applies the multivariate modeling technique to evaluate the crash risks by collision type. Three years crash data are obtained from 282 freeway exit ramps. Three types of crashes are considered [i.e., (1) rear-end, (2) sideswipe, and (3) angle collisions]. A multivariate Poisson-lognormal (MVPLN) model is estimated to jointly evaluate the impacts of explanatory variables on different collision risks. For comparison purpose, univariate negative binomial (NB) models are also estimated based on the same dataset. The results show that the MVPLN model successfully captures the correlation of latent effects among the crash counts of different collision types. Thus, the MNPLN model estimates the impacts of variables more accurately than the NB model. The MVPLN model is found outperform the NB models in predicting the crash count of each collision type. Findings of this paper can help better understand ...

Coronavirus disease 2019 (COVID-19) has spread rapidly throughout the world and has caused millions of deaths. However, the number of daily COVID-19 deaths in Thailand has been low with most daily records showing zero deaths, thereby making them fit a Zero-Inflated Poisson (ZIP) distribution. Herein, confidence intervals for the Coefficient Of Variation (CV) of a ZIP distribution are derived using four methods: the standard bootstrap (SB), percentile bootstrap (PB), Markov Chain Monte Carlo (MCMC), and the Bayesian-based highest posterior density (HPD), for which using the variance of the CV is unnecessary. We applied the methods to both simulated data and data on the number of daily COVID-19 deaths in Thailand. Both sets of results show that the SB, MCMC, and HPD methods performed better than PB for most cases in terms of coverage probability and average length. Overall, the HPD method is recommended for constructing the confidence interval for the CV of a ZIP distribution. Doi: 10.28991/esj-2021-SPER-05 Full Text: PDF