Conway-Maxwell-Poisson Research Articles

New biased estimators for the Conway–Maxwell-Poisson Model

Recent studies focus on modelling count data, which often shows overdispersion or underdispersion. The Conway–Maxwell–Poisson (COMP) regression model effectively handles such dispersion. However, multicollinearity can negatively impact the maximum likelihood estimator (MLE) by increasing variance. To address this, biased estimators like the ridge estimator have been suggested to mitigate multicollinearity effects. This research proposes a new COMP hybrid estimator to further address multicollinearity issues. Theoretical comparisons between the COMP hybrid estimator, MLE, and other COMP estimators reveal that the hybrid estimator reduces mean squared error. Monte Carlo simulations and practical applications confirm that these proposed estimators outperform both MLE and existing COMP estimators.

Proposal and appraisal of novel indicators and performance metrics for project monitoring

Project monitoring is a critical process for guaranteeing its excellence, ensuring the delivery of activities both on planned time and budget. Despite its importance, numerous discussions have arisen regarding which indicator is the most appropriate, and how and when to measure it. The primary objective of this article is to evaluate novel duration indicators based on the Earned Value Management (EVM), Earned Schedule Management (ESM) and Earned Duration Management (EDM) methodologies, and to assess new performance metrics. This involves the determination of the limits of the control charts through extensive Monte Carlo Simulations and their performance under several out-of-control scenarios. The robustness of the control charts is tested using performance metrics within a statistical control approach. The secondary objective consists of evaluating the impact of new probability distributions that describe the project activities, such as the Gamma and Conway–Maxwell–Poisson (CMP) distributions. The contributions of this study are threefold: firstly, the Duration Performance Index (DPI) from the EDM is a reliable indicator for monitoring the overall project duration. Secondly, the probability of overreaction (PO) and the accuracy (Ac) are the performance metrics that most produced robust indicators. Lastly, it is observed that different probability distributions, specifically Gamma and Conway–Maxwell–Poison (CMP) distributions, seem to interfere with the correct generation of control charts signals, although they do not significantly affect the overall duration of the entire project.

Fast Bayesian Inference for Spatial Mean-Parameterized Conway–Maxwell–Poisson Models

Count data with complex features arise in many disciplines, including ecology, agriculture, criminology, medicine, and public health. Zero inflation, spatial dependence, and non-equidispersion are common features in count data. There are currently two classes of models that allow for these features—the mode-parameterized Conway–Maxwell–Poisson (COMP) distribution and the generalized Poisson model. However both require the use of either constraints on the parameter space or a parameterization that leads to challenges in interpretability. We propose spatial mean-parameterized COMP models that retain the flexibility of these models while resolving the above issues. We use a Bayesian spatial filtering approach in order to efficiently handle high-dimensional spatial data and we use reversible-jump MCMC to automatically choose the basis vectors for spatial filtering. The COMP distribution poses two additional computational challenges—an intractable normalizing function in the likelihood and no closed-form expression for the mean. We propose a fast computational approach that addresses these challenges by, respectively, introducing an efficient auxiliary variable algorithm and pre-computing key approximations for fast likelihood evaluation. We illustrate the application of our methodology to simulated and real datasets, including Texas HPV-cancer data and US vaccine refusal data. Supplementary materials for this article are available online.

Bayesian bivariate Conway–Maxwell–Poisson regression model for correlated count data in sports

Abstract Count data play a crucial role in sports analytics, providing valuable insights into various aspects of the game. Models that accurately capture the characteristics of count data are essential for making reliable inferences. In this paper, we propose the use of the Conway–Maxwell–Poisson (CMP) model for analyzing count data in sports. The CMP model offers flexibility in modeling data with different levels of dispersion. Here we consider a bivariate CMP model that models the potential correlation between home and away scores by incorporating a random effect specification. We illustrate the advantages of the CMP model through simulations. We then analyze data from baseball and soccer games before, during, and after the COVID-19 pandemic. The performance of our proposed CMP model matches or outperforms standard Poisson and Negative Binomial models, providing a good fit and an accurate estimation of the observed effects in count data with any level of dispersion. The results highlight the robustness and flexibility of the CMP model in analyzing count data in sports, making it a suitable default choice for modeling a diverse range of count data types in sports, where the data dispersion may vary.

Compound Conway-Maxwell Poisson Gamma Distribution: Properties and Estimation

The distribution of a random sum of random events is called a compound distribution. It involves a counting (discrete) distribution to model the number of occurrences of the random event in a fixed time period and a continuous distribution to model the outcome of the random event. It has applications in the fields of actuarial sciences, meteorology etc. For example, in modelling insurance loss amounts through compound distributions, the number of claims and the claim amounts are used to calculate the total claim amount of a portfolio. The number of claims is modelled through a discrete distribution and the claim amounts are modelled through continuous distributions. Generally, Poisson distribution is used in compound models as the discrete distribution and such models are known as compound Poisson models. However, the equi-dispersion property of the Poisson distribution hinders its application in scenarios where the underlying count data is either over- or under-dispersed. In this paper, a two-parameter Poisson distribution, namely, Conway-Maxwell Poisson (CMP) distribution, which handles both over- and under-dispersed data, is considered as the counting distribution, and the corresponding compound CMP distribution is developed. Some mathematical properties of the distribution are derived and a methodology to estimate the parameters using the likelihood approach is proposed. A numerical illustration of the proposed methodology is given through a simulation study. An application of the compound CMP model is illustrated through transportation security administration (TSA) insurance claim data. Also, the estimation of the risk measures associated with the TSA claim data is discussed.

Dynamic Modeling of Spike Count Data With Conway-Maxwell Poisson Variability.

In many areas of the brain, neural spiking activity covaries with features of the external world, such as sensory stimuli or an animal's movement. Experimental findings suggest that the variability of neural activity changes over time and may provide information about the external world beyond the information provided by the average neural activity. To flexibly track time-varying neural response properties, we developed a dynamic model with Conway-Maxwell Poisson (CMP) observations. The CMP distribution can flexibly describe firing patterns that are both under- and overdispersed relative to the Poisson distribution. Here we track parameters of the CMP distribution as they vary over time. Using simulations, we show that a normal approximation can accurately track dynamics in state vectors for both the centering and shape parameters (λ and ν). We then fit our model to neural data from neurons in primary visual cortex, "place cells" in the hippocampus, and a speed-tuned neuron in the anterior pretectal nucleus. We find that this method outperforms previous dynamic models based on the Poisson distribution. The dynamic CMP model provides a flexible framework for tracking time-varying non-Poisson count data and may also have applications beyond neuroscience.

A new exponentially weighted moving average control chart to monitor count data with applications in healthcare and manufacturing

Attribute control charts are widely used to monitor count data. Many distributions are proposed to model and monitor count data. This article has developed an exponentially weighted moving average (EWMA) control chart under Generalized Conway–Maxwell–Poisson (GCOMP) distribution and named as the GCOMP-EWMA chart. The GCOMP distribution is an extension of the Conway–Maxwell–Poisson (COMP) distribution and is a longer-tailed model than the COMP distribution. The GCOMP distribution can also model the shorter tail behaviour in count data. Considering the tail behaviour, the in-control and out-of-control performance of the GCOMP-EWMA chart have been evaluated for over- and under-dispersed count data. Without using the zero-inflation property, the GCOMP-EWMA chart performs efficiently in monitoring zero-inflated count data compared to existing zero-inflated models-based control charts. Finally, illustrative examples demonstrate the practical applications of the GCOMP-EWMA charts in different fields.

Developing Safety Performance Functions for Commercial Motor Vehicle Crashes at Interchange Ramp Segments in Kentucky

Compared with roadway segments and intersections, the safety of interchange ramp segments has not been studied extensively, especially in the context of commercial motor vehicles (CMVs). The main objective of this study was to develop a safety performance function (SPF) tool for predicting CMV crashes occurring on interchange ramp segments. Four count models, including the negative binomial (NB), heterogeneous NB (HTNB), standard Conway–Maxwell–Poisson (CMP), and heterogeneous Conway–Maxwell–Poisson (HTCMP), were used and compared while fitting CMV crash-specific SPFs along interchange ramp segments in Kentucky. The HTCMP model, which is an extension of the standard CMP model, is a more flexible approach that handles both over-dispersed and under-dispersed crash data while exhibiting varying dispersion parameters. Five-year (2015 to 2019) CMV-related crashes along Kentucky’s ramp segments were used. The model comparison results showed that the HTCMP significantly outperformed the other three models in crash prediction accuracy and goodness-of-fit statistics (e.g., the Akaike information criterion, Bayesian information criterion, and McFadden’s Pseudo R-squared). The SPF model results using the HTCMP approach indicated that on-ramps (relative to off-ramps), ramp annual average daily traffic, ramp configuration, left shoulder width, ramp gore length, absence of left roadside barrier, and presence of other merging or diverging ramps within the ramp of interest were significantly associated with CMV crash frequency on ramp segments. Potential safety countermeasures were proposed, for example, increasing ramp gore length to be at least 730 ft (since this was associated with a reduction in CMV crashes on ramp segments).

Developing a two-parameter Liu estimator for the COM–Poisson regression model: Application and simulation

The Conway–Maxwell–Poisson (COMP) model is defined as a flexible count regression model used for over- and under-dispersion cases. In regression analysis, when the explanatory variables are highly correlated, this means that there is a multicollinearity problem in the model. This problem increases the standard error of maximum likelihood estimates. To manage the multicollinearity effects in the COMP model, we proposed a new modified Liu estimator based on two shrinkage parameters (k, d). To assess the performance of the proposed estimator, the mean squared error (MSE) criterion is used. The theoretical comparison of the proposed estimator with the ridge, Liu, and modified one-parameter Liu estimators is made. The Monte Carlo simulation and real data application are employed to examine the efficiency of the proposed estimator and to compare it with the ridge, Liu, and modified one-parameter Liu estimators. The results showed the superiority of the proposed estimator as it has the smallest MSE value.

Kibria–Lukman estimator for the Conway–Maxwell Poisson regression model: Simulation and applications

The Conway–Maxwell Poisson (COMP) regression model is one of the count data models to account for over– and under–dispersion. In regression analysis, when the explanatory variables are correlated, when there is multicollinearity problem, this inflates the standard error of the maximum likelihood estimates. The Kibria–Lukman estimator was provided to handle the effect of multicollinearity in the linear regression model. Therefore, we proposed to extend this estimator to the COMP model to overcome this problem in the COMP model. The proposed estimator mitigates the adverse effect of multicollinearity on the standard error of the estimates. We used the mean squared error (MSE) as the performance assessment criterion to assess the performance of the proposed estimator and others. Also, we compared the proposed estimator theoretically with other estimators (the ridge and Liu estimators). We employed a simulation study and two life applications to study the performance of the proposed estimator. The simulation study and applications results showed the superiority of the proposed estimator because the MSE of the proposed estimator is smaller than the other estimators.

New modified two-parameter Liu estimator for the Conway–Maxwell Poisson regression model

The Conway–Maxwell–Poisson (COMP) model is one of the count data regression models used in over- and underdispersion cases. Thus, COMP regression is a flexible model in the count data models. In the regression analysis, when the explanatory variables are correlated with each other, multicollinearity exists; this inflates the standard error of the maximum likelihood estimates. To handle the effect of multicollinearity, we proposed a new modified Liu estimator for the COMP regression model based on two shrinkage parameters. This estimator is proposed to reduce the effect of multicollinearity on the standard error of the estimates. To evaluate the performance of the proposed estimator, the mean squared error (MSE) criterion is employed. Theoretical comparison of the proposed estimator with existing estimators (ridge, Liu, and modified one-parameter Liu estimators) is made. The results of the simulation study and real-life application indicate the superiority of the proposed estimator.

Modeling road accident fatalities with underdispersion and zero-inflated counts.

In 2013, Thailand was ranked second in the world in road accident fatalities (RAFs), with 36.2 per 100,000 people. During the Songkran festival, which takes place during the traditional Thai New Year in April, the number of road traffic accidents (RTAs) and RAFs are markedly higher than on regular days, but few studies have investigated this issue as an effect of festivity. This study investigated the factors that contribute to RAFs using various count regression models. Data on 20,229 accidents in 2015 were collected from the Department of Disaster Prevention and Mitigation in Thailand. The Poisson and Conway-Maxwell-Poisson (CMP) distributions, and their zero-Inflated (ZI) versions were applied to fit the data. The results showed that RAFs in Thailand follow a count distribution with underdispersion and excessive zeros, which is rare. The ZICMP model marginally outperformed the CMP model, suggesting that having many zeros does not necessarily mean that the ZI model is required. The model choice depends on the question of interest, and a separate set of predictors highlights the distinct aspects of the data. Using ZICMP, road, weather, and environmental factors affected the differences in RAFs among all accidents, whereas month distinguished actual non-fatal accidents and crashes with or without deaths. As expected, actual non-fatal accidents were 2.37 times higher in April than in January. Using CMP, these variables were significant predictors of zeros and frequent deaths in each accident. The RAF average was surprisingly higher in other months than in January, except for April, which was unexpectedly lower. Thai authorities have invested considerable effort and resources to improve road safety during festival weeks to no avail. However, our study results indicate that people's risk perceptions and public awareness of RAFs are misleading. Therefore, nationwide road safety should instead be advocated by the authorities to raise society's awareness of everyday personal safety and the safety of others.

Applications of the Conway-Maxwell-Poisson Hidden Markov models for analyzing traffic accident

This paper documents the application of the Conway-Maxwell-Poisson(CMP) hidden Markov model for modelling motor vehicle crashes. The CMP distribution is a twoparameter extension of the Poisson distribution that generalizes some well-known discrete distributions(Poisson, Bernoulli and geometric). Also it leads to the generalizations of distributions derived from theses discrete distributions, that is, the binomial and negative binomial distributions. The advantage of CMP distribution is its ability to handle both under and over-dispersion through controlling one special parameter in the distribution, which makes it more flexible than Poisson distribution. We consider the data consisting of the daily number of injuries on the road in 2020 from the TAAS(Traffic Accident Analysis System). We apply CMP hidden Markov model to data, the parameters are estimated via maximim likelihood, and find that this model achieves better performance than commonly used Poisson hidden Markov model. For the decoding procedure, the Viterbi algorithm is implemented.

A New Conway Maxwell–Poisson Liu Regression Estimator—Method and Application

Poisson regression is a popular tool for modeling count data and is applied in medical sciences, engineering and others. Real data, however, are often over or underdispersed, and we cannot apply the Poisson regression. To overcome this issue, we consider a regression model based on the Conway–Maxwell Poisson (COMP) distribution. Generally, the maximum likelihood estimator is used for the estimation of unknown parameters of the COMP regression model. However, in the existence of multicollinearity, the estimates become unstable due to its high variance and standard error. To solve the issue, a new COMP Liu estimator is proposed for the COMP regression model with over-, equi-, and underdispersion. To assess the performance, we conduct a Monte Carlo simulation where mean squared error is considered as an evaluation criterion. Findings of simulation study show that the performance of our new estimator is considerably better as compared to others. Finally, an application is consider to assess the superiority of the proposed COMP Liu estimator. The simulation and application findings clearly demonstrated that the proposed estimator is superior to the maximum likelihood estimator.

Comprehensive Investigation of Commercial Motor Vehicle Crashes along Roadway Segments in Kentucky

The negative binomial (NB) model, traditionally used for safety performance function (SPF) development, suffers from a fixed over-dispersion parameter and is only valid for over-dispersed data (i.e., data exhibiting greater variance than the mean). A more flexible approach that handles over-dispersed data, under-dispersed data, and excess zero counts, in addition to exhibiting varying dispersion parameter as a function of site-specific characteristics, is the zero-inflated heterogeneous Conway–Maxwell–Poisson (ZI-HTCMP) model, which is an extension of Conway–Maxwell–Poisson (CMP)-based models. This study develops fatal + injury (FI) commercial motor vehicle (CMV) crash-specific SPFs along four roadway segment facilities in Kentucky, U.S., (urban multilane, rural multilane, urban two-lane, and rural two-lane segments). The traditional NB and newly introduced CMP-based models—ZI-HTCMP, zero-inflated Conway–Maxwell–Poisson (ZI-CMP), heterogeneous Conway–Maxwell–Poisson (HTCMP), and CMP—were compared using 14,967 CMV-related crashes on Kentucky’s road segments (between 2015 and 2019) and various roadway variables, for example, shoulder width, annual average daily traffic (AADT), and heavy vehicle percentage (HVP). From the developed SPFs, AADT and HVP >10% significantly increased FI CMV-related crashes on all four segment facilities. Various goodness-of-fit (GOF) statistics, including Akaike information criterion (AIC), mean absolute deviance (MAD), and mean square prediction error (MSPE), were used for model assessment and selection. For all four roadway facilities, CMP-based models showed better model fitting and prediction performance than the NB models. Furthermore, ZI-HTCMP was the best-fit model for urban multilane segments, which had high representation of zero-crash sites. CMP-family models could be used for effectively predicting FI CMV-related crashes (with excess zeros) on road segments.

Spatially Lagged Covariate Model with Zero Inflated Conway-Maxwell-Poisson Distribution Model for the Analysis of Pedestrian Injury Counts

Spatial dependency is important to recognize because of the mapping of pedestrian injury counts analysis. Road safety has been a major issue in contemporary societies, with road crashes incurring major human and materials costs annually worldwide. South Korea’s pedestrian traffic accident rate is the highest among the Organization for Economic Cooperation and Development (OECD) countries. In this paper, we use spatially lagged covariate model with zero inflated Conway-Maxwell-Poisson distribution model to account for spatial autocorrelation of no. of pedestrian crashes with cars. Alternatively, the Conway-Maxwell-Poisson (CMP) distribution, first proposed by Conway and Maxwell (1962) has the flexibility to handle all levels of dispersion, including underdispersion. We test spatial autocorrelation of pedestrian injury counts at 2474 sites, with several weights matrices using Moran s I statistics, under permutation scheme. Then we fit different 20 models, and finally choose the best model by the AIC and SBC values.

When is the Conway–Maxwell–Poisson distribution infinitely divisible?

An essential characteristic for a distribution to play a central role in limit theory is infinite divisibility. In this note, we prove that the Conway–Maxwell–Poisson (CMP) distribution is infinitely divisible iff it is the Poisson or geometric distribution. This explains that, despite its applications in a wide range of fields, there is no theoretical foundation for the CMP distribution to be a natural candidate for the law of small numbers.

An Improved Method for the Estimation and Comparison of Mortality Rates in Fish from Catch‐Curve Data

AbstractCatch‐curve analyses are routinely used to estimate instantaneous mortality (Z) in fish, and as the age‐frequency data are often overdispersed, the application of a variance bias‐correction factor has been recommended. The extensions of the Poisson generalized linear model (GLMPoisson) may, however, constitute a better alternative, as they model the variance (SE) in counts more adequately with their specific dispersion parameter for more accurate estimations and statistical comparisons. To test this idea, simulated age‐frequency data generated under four dispersion scenarios were analyzed according to six currently available methods and compared with the results of a GLMPoissonand five of its extensions to evaluate each method‐specific bias inZ± SE estimates. Empirical age‐frequency data from sampled WalleyeSander vitreusand Arctic CharSalvelinus alpinuspopulations in Québec, Canada, were then used to illustrate the applicability of our GLM‐based method, which relies on the behavior of Pearson residuals to assess model adequacy and an information‐theoretic approach for model selection. All analyses revealed thatZ‐estimates were generally accurate among the methods considered, except under the most likely situation of quadratic overdispersion met in ecological studies, for which only the negative binomial type 2 and the mean‐parametrized Conway–Maxwell–Poisson (CMP) extensions were adequate to estimate bothZand its SE. Linearly overdispersed data were best modeled by the negative binomial type 1 and generalized Poisson (GLMGP) extensions; the GLMCMPand GLMGPwere the most appropriate to model underdispersed data, whereas the GLMPoissonadequately modeled equi‐dispersed data, similar to the Chapman and Robson (1960) method. Statistical comparisons ofZ± SE for grouping factors, such as year or site, were correctly achieved when the most adequate and statistically supported GLMPoissonextension was applied. Altogether, the proposed GLM‐based method should help to circumvent the identified issues related to SE estimation for statistical inferences about mortality rates for fisheries management decision making.

K 1 K 2–Inflated Conway–Maxwell–Poisson Model: Bayesian Predictive Modeling with an Application in Soccer Matches

The purpose of this paper is two folds. First, to introduce a multiple inflated version of the Conway–Maxwell–Poisson model, that can be used flexibly to model count data when some values have high frequency along with over– or under–dispersion. Also, this model includes Poisson, Conway–Maxwell–Poisson (COMP), zero–inflated Poisson (ZIP), multiple–inflated Poisson, and zero–inflated Conway–Maxwell–Poisson (ZICOMP). Second, to estimate the future distribution from the multiple inflated Conway–Maxwell–Poisson model under the Kullback Leibler difference (loss) function. This model is fitted to the number of penalties scored in the Premier League’s 2019–20 season and its future distribution using Bayes and plug–in methods is estimated.

On the ridge estimation of the Conway‐Maxwell Poisson regression model with multicollinearity: Methods and applications

AbstractIn data analysis, count data modeling contributing a significant role. The Conway‐Maxwell Poisson (COMP) is one of the flexible count data models to deal over and under dispersion. In the COMP regression model, when the explanatory variables are correlated, then the maximum likelihood estimator does not give efficient results due to the large standard error (SE) of the estimates. To overcome the effect of multicollinearity, we have proposed some ridge regression estimators in the COMP regression model by introducing dispersion parameter in the context of overdispersion, equidispersion, and underdispersion. The Iterative reweighted least method is used for the estimation of ridge regression coefficients in the COMP regression model. To evaluate the performance of the proposed estimators, we use mean squared error (MSE) as the performance evaluation criteria. Theoretical comparison of the proposed estimators with the competitor estimators is made and conditions of efficiency have been derived. The proposed estimator is evaluated with the help of a simulation study and two real applications. The results of the simulation study and real applications show the superiority of the proposed estimator because the proposed estimator produces smaller MSE and SEs of the COMP regression estimates with multicollinearity.