Multivariate Negative Binomial Distribution Research Articles

Risk factors associated with SARS-CoV-2 transmission, outbreak duration, and mortality in Fraser Health acute-care settings

Background: Transmission of SARS-CoV-2 in acute-care settings affects patients, healthcare workers, and the already-burdened healthcare system. An analysis of risk factors associated with outbreak severity was conducted to inform prevention strategies. Methods: This study was a cross-sectional analysis of COVID-19 outbreaks at Fraser Health (FH) acute-care sites between March 2020 and March 2021. Outbreak severity measures included COVID-19 attack rate, outbreak duration, and 30-day case mortality. Covariates at patient, outbreak, unit level, and facility level were included (Table 1). Generalized linear models with generalized estimation equations were used for all outcome measures, with outbreak duration and 30-day case mortality using multivariate negative binomial distributions, and attack rate using Gaussian distribution. A P value of .05 indicated statistical significance. Analyseswere performed using SAS version 3.8 software, R version 4.1.0 software, and Stata version 16.0 software. Results: Between March 2020 and March 2021, 54 COVID-19 outbreaks were declared in FH acute-care sites involving 455 SARS-CoV-2–positive patients. The average outbreak duration was 23 days, the average attack rate was 28%, and the average 30-day all-cause mortality per outbreak was 2 deaths. The results of the full models are shown in Table 1. Discussion: We identified an inverse relationship between increased hand hygiene compliance during outbreaks and all 3 severity measures. Paradoxically, hand hygiene rates in the year prior to the pandemic were positively associated with duration and mortality. Increased unit age was also associated with increases in each of the severity measures. Comorbidity total factor was correlated with outbreak attack rate and duration, demonstrating the importance of individual patient characteristics in an outbreak. Conclusions: Our findings highlight the importance of hand hygiene practices during an outbreak. Additionally, it is important to understand the difficulties faced by older facilities, many of which face infrastructural challenges. This study reinforces the need to incorporate infection control standards into healthcare planning and construction.Funding: NoneDisclosures: None

A Flexible Multivariate Distribution for Correlated Count Data

Multivariate count data are often modeled via a multivariate Poisson distribution, but it contains an underlying, constraining assumption of data equi-dispersion (where its variance equals its mean). Real data are oftentimes over-dispersed and, as such, consider various advancements of a negative binomial structure. While data over-dispersion is more prevalent than under-dispersion in real data, however, examples containing under-dispersed data are surfacing with greater frequency. Thus, there is a demonstrated need for a flexible model that can accommodate both data types. We develop a multivariate Conway–Maxwell–Poisson (MCMP) distribution to serve as a flexible alternative for correlated count data that contain data dispersion. This structure contains the multivariate Poisson, multivariate geometric, and the multivariate Bernoulli distributions as special cases, and serves as a bridge distribution across these three classical models to address other levels of over- or under-dispersion. In this work, we not only derive the distributional form and statistical properties of this model, but we further address parameter estimation, establish informative hypothesis tests to detect statistically significant data dispersion and aid in model parsimony, and illustrate the distribution’s flexibility through several simulated and real-world data examples. These examples demonstrate that the MCMP distribution performs on par with the multivariate negative binomial distribution for over-dispersed data, and proves particularly beneficial in effectively representing under-dispersed data. Thus, the MCMP distribution offers an effective, unifying framework for modeling over- or under-dispersed multivariate correlated count data that do not necessarily adhere to Poisson assumptions.

Multinomial N‐mixture models for removal sampling

Multinomial -mixture models are commonly used to fit data from a removal sampling protocol. If the mixing distribution is negative binomial, the distribution of the counts does not appear to have been identified, and practitioners approximate the requisite likelihood by placing an upper bound on the embedded infinite sum. In this paper, the distribution which underpins the multinomial -mixture model with a negative binomial mixing distribution is shown to belong to the broad class of multivariate negative binomial distributions. Specifically, the likelihood can be expressed in closed form as the product of conditional and marginal likelihoods and the information matrix shown to be block diagonal. As a consequence, the nature of the maximum likelihood estimates of the unknown parameters and their attendant standard errors can be examined and tests of the hypothesis of the Poisson against the negative binomial mixing distribution formulated. In addition, appropriate multinomial -mixture models for data sets which include zero site totals can also be constructed. Two illustrative examples are provided.

Regression for doubly inflated multivariate Poisson distributions

ABSTRACTDependent multivariate count data occur in several research studies. These data can be modelled by a multivariate Poisson or Negative binomial distribution constructed using copulas. However, when some of the counts are inflated, that is, the number of observations in some cells are much larger than other cells, then the copula-based multivariate Poisson (or Negative binomial) distribution may not fit well and it is not an appropriate statistical model for the data. There is a need to modify or adjust the multivariate distribution to account for the inflated frequencies. In this article, we consider the situation where the frequencies of two cells are higher compared to the other cells and develop a doubly inflated multivariate Poisson distribution function using multivariate Gaussian copula. We also discuss procedures for regression on covariates for the doubly inflated multivariate count data. For illustrating the proposed methodologies, we present real data containing bivariate count observations with inflations in two cells. Several models and linear predictors with log link functions are considered, and we discuss maximum likelihood estimation to estimate unknown parameters of the models.

On the three-dimensional negative binomial distribution

ABSTRACTWe consider a particular generalization of the negative binomial distribution to the multivariate case obtained through a specification of the probability generating function as the negative power of a certain polynomial. The probability function itself has previously been derived for the two-dimensional case only, and inference in the multivariate negative binomial distribution has been restricted to the use of composite likelihood based on one- or two-dimensional marginals. In this article, we derive the three-dimensional probability function as a sum with all terms positive and study the range of possible parameter values. We illustrate the use of the three-dimensional distribution for modeling three correlated SAR images.

CMPH: a multivariate phase-type aggregate loss distribution

Abstract We introduce a compound multivariate distribution designed for modeling insurance losses arising from different risk sources in insurance companies. The distribution is based on a discrete-time Markov Chain and generalizes the multivariate compound negative binomial distribution, which is widely used for modeling insurance losses.We derive fundamental properties of the distribution and discuss computational aspects facilitating calculations of risk measures of the aggregate loss, as well as allocations of the aggregate loss to individual types of risk sources. Explicit formulas for the joint moment generating function and the joint moments of different loss types are derived, and recursive formulas for calculating the joint distributions given. Several special cases of particular interest are analyzed. An illustrative numerical example is provided.

Multi-Type Branching Processes Modeling of Nosocomial Epidemics

AbstractNosocomial epidemics are infectious diseases which spread among different types of susceptible individuals in a health-care facility. To model this type of epidemics, we use a multi-type branching process with a multivariate negative binomial offspring distribution. In particular, we estimate the basic reproduction number

Existence conditions of permanental and multivariate negative binomial distributions

Existence conditions of permanental distributions are deeply connected to existence conditions of multivariate negative binomial distributions. The aim of this paper is twofold. It answers several questions generated by recent works on this subject, but it also goes back to the roots of this field and fixes existing gaps in older papers concerning conditions of infinite divisibility for these distributions.

Sequential Bayesian Analysis of Multivariate Count Data

We develop a new class of dynamic multivariate Poisson count models that allow for fast online updating. We refer to this class as multivariate Poisson-scaled beta (MPSB) models. The MPSB model allows for serial dependence in count data as well as dependence with a random common environment across time series. Notable features of our model are analytic forms for state propagation, predictive likelihood densities, and sequential updating via sufficient statistics for the static model parameters. Our approach leads to a fully adapted particle learning algorithm and a new class of predictive likelihoods and marginal distributions which we refer to as the (dynamic) multivariate confluent hyper-geometric negative binomial distribution (MCHG-NB) and the dynamic multivariate negative binomial (DMNB) distribution, respectively. To illustrate our methodology, we use a simulation study and empirical data on weekly consumer non-durable goods demand.

Lancaster distributions and Markov chains with multivariate Poisson–Charlier, Meixner and Hermite–Chebycheff polynomial eigenfunctions

This paper studies new Lancaster characterizations of bivariate multivariate Poisson, negative binomial and normal distributions which have diagonal expansions in multivariate orthogonal polynomials. The characterizations extend classical Lancaster characterizations of bivariate 1-dimensional distributions. Multivariate Poisson–Charlier, Meixner and Hermite–Chebycheff orthogonal polynomials, used in the characterizations, are constructed from classical 1-dimensional orthogonal polynomials and multivariate Krawtchouk polynomials. New classes of transition functions of discrete and continuous time Markov chains with these polynomials as eigenfunctions are characterized. The characterizations obtained belong to a class of mixtures of multi-type birth and death processes with fixed multivariate Poisson or multivariate negative binomial stationary distributions.

Sarmanov family of multivariate distributions for bivariate dynamic claim counts model

To predict future claims, it is well-known that the most recent claims are more predictive than older ones. However, classic panel data models for claim counts, such as the multivariate negative binomial distribution, do not put any time weight on past claims. More complex models can be used to consider this property, but often need numerical procedures to estimate parameters. When we want to add a dependence between different claim count types, the task would be even more difficult to handle. In this paper, we propose a bivariate dynamic model for claim counts, where past claims experience of a given claim type is used to better predict the other type of claims. This new bivariate dynamic distribution for claim counts is based on random effects that come from the Sarmanov family of multivariate distributions. To obtain a proper dynamic distribution based on this kind of bivariate priors, an approximation of the posterior distribution of the random effects is proposed. The resulting model can be seen as an extension of the dynamic heterogeneity model described in Bolancé et al. (2007). We apply this model to two samples of data from a major Canadian insurance company, where we show that the proposed model is one of the best models to adjust the data. We also show that the proposed model allows more flexibility in computing predictive premiums because closed-form expressions can be easily derived for the predictive distribution, the moments and the predictive moments.

On composite likelihood estimation of a multivariate INAR(1) model

In several circumstances the collected data are counts observed in different time points, while the counts at each time point are correlated. Current models are able to account for serial correlation but usually fail to account for cross‐correlation. Motivated by the lack of appropriate tools for handling such type of data, we define a multivariate integer‐valued autoregressive process of order 1 (MINAR(1)) and examine its basic statistical properties. Apart from the general specification of the MINAR(1) process, we also study two specific parametric cases that arise under the assumptions of a multivariate Poisson and a multivariate negative binomial distribution for the innovations of the process. To overcome the computational difficulties of the maximum likelihood approach we suggest the method of composite likelihood. The performance of the two methods of estimation, that is, maximum likelihood and composite likelihood, is compared through a small simulation experiment. Extensions of the time‐invariant model to a regression model are also discussed. The proposed model is applied to a trivariate data series related to daily traffic accidents in three areas in the Netherlands.

Traditional and IS-Enabled Customer Acquisition on the Internet

Geographic variation in consumer use of Internet retailers is partly explained by variation in offline shopping costs. Explanations for geographic variation in the efficacy of different customer acquisition methods including traditional methods of offline word-of-mouth (WOM) and magazine advertising and information systems (IS)-enabled methods of online WOM and online search remain unexplored. We estimate a multivariate negative binomial distribution (NBD) model on zip code–level customer counts from a leading Internet retailer and provide new insights into factors explaining geographic variation in the success of these methods. First, we show that target customer density explains geographic variation over and above the impact due to the number of potential customers. Moreover, the effect of density is greatest for offline and online WOM acquisitions; this suggests that density contributes to contagion, connectivity, and a hypothesized “social multiplier.” Second, when senders and recipients of WOM share consumption benefits, WOM is more powerful and compelling. We find that location-based convenience benefits have stronger effects on location-dependent offline WOM acquisitions than on location-independent online WOM acquisitions. Third, acquisition channels contribute differently to the total customer pool—offline WOM acquisitions are clustered, whereas magazine acquisitions are dispersed. Finally, separate click-to-conversion data from Coremetrics.com indicates that using the model-based predictions to target specific markets delivers a twofold improvement in actual click-to-order rates. This paper was accepted by Sandra Slaughter, information systems.

Optimal Internet Media Selection

In this study we develop a method that optimally selects online media vehicles and determines the number of advertising impressions that should be purchased and then served from each chosen website. As a starting point, we apply Danaher's [Danaher, P. J. 2007. Modeling page views across multiple websites with an application to Internet reach and frequency prediction. Marketing Sci. 26(3) 422–437] multivariate negative binomial distribution (MNBD) for predicting online media exposure distributions. The MNBD is used as a component in the broader task of media selection. Rather than simply adapting previous selection methods used in traditional media, we show that the Internet poses some unique challenges. Specifically, online banner ads and other forms of online advertising are sold by methods that differ substantially from the way other media advertising is sold. We use a nonlinear optimization algorithm to solve the optimization problem and derive the optimum online media schedule. Data from an online audience measurement firm and an advertising agency are used to illustrate the speed and accuracy of our method, which is substantially quicker than using complete enumeration.

A Hybrid Pairwise Likelihood Method

A modification to the pairwise likelihood method is proposed, which aims to improve the estimation of the marginal distribution parameters. This is achieved by replacing the pairwise likelihood score equations, for estimating such parameters, by the optimal linear combinations of the marginal score functions. A further advantage of the proposed estimator of marginal parameters, over pairwise likelihood, is that it is robust to misspecification of the bivariate distributions as long as the univariate marginal distributions are correctly specified. While alternating logistic regression can be seen as a special case of the proposed method, it is shown that an existing generalization of alternating logistic regression applicable to ordinal data is not the same as and is inferior to the proposed method because it replaces certain conditional densities by pseudodensities that assume working independence. The fitting of the multivariate negative binomial distribution is another scenario involving intractable likelihood that calls for the use of pairwise likelihood methods, and the superiority of the modified method is demonstrated in a simulation study. Two examples, based on the analyses of salamander mating and patient-controlled analgesia data, demonstrate the usefulness of the proposed method. The possibility of combining optimally the pairwise, rather than marginal, scores is also considered and its difficulty and potential are discussed.

Multivariate Generalized Distributions of Order k

The study of multivariate distributions of order k, two of which are the multivariate negative binomial of order k and the multinomial of the same order, was introduced in Philippou et al. (Philippou, A. N., Antzoulakos, D. L., Tripsiannis, G. A. (1988). Multivariate distributions of order k. Statistics and Probability Letters 7(3):207–216.), and Philippou et al. (Philippou, A. N., Antzoulakos, D. L., Tripsiannis, G. A. (1990). Multivariate distributions of order k, part II. Statistics and Probability Letters 10(1):29–35.). Recently, an order k (or cluster) generalized negative binomial distribution and a multivariate negative binomial distribution were derived in Sen and Jain (Sen, K., Jain, R. (1996). Cluster generalized negative binomial distribution. In: Borthakur et al. A. C., Eds.; Probability Models and Statistics Medhi Festschrift, A. J., on the Occasion of his 70th Birthday. New Age International Publishers: New Delhi, 227–241.) and Sen and Jain (Sen, K., Jain, R. (1997). A multivariate generalized Polya-Eggenberger probability model-first passage approach. Communications in Statistics-Theory and Methods 26:871–884.), respectively. In this paper, all four distributions are generalized to a multivariate generalized negative binomial distribution of order k by means of an appropriate sampling scheme and a first passage event. This new distribution includes as special cases several known and new multivariate distributions of order k, and gives rise in the limit to multivariate generalized logarithmic, Poisson and Borel-Tanner distributions of the same order. Applications are indicated.

A new multivariate inverse polya distribution of order k

A new multivariate inverse Polya distribution of order k, type I, is derived by means of a generalized urn scheme and by compounding the multivariate negative binomial distribution of order k, type I, of Philippou, Antzoulakos and Tripsiannis (1988) with the Dirichlet distribution. It is noted that this new distribution includes as special cases a new multivariate inverse hypergeometric distribution of order k and a new multivariate negative inverse one of the same order. The mean and variance-covariance of the multivariate inverse Polya distribution of order k, type I, are derived, and two known distributions of the same order are shown to be limiting cases of it.

Microdynamics of new product purchase: A model incorporating both marketing and consumer-specific variables

The Negative Binomial-Dirichlet Model of brand choice and purchase incidence has been found to be capable of describing consumer behaviour in a large number of empirical applications. However, a major drawback of this model is the postulate of stationary market conditions. The multivariate Polya Model generalizes this approach in order to overcome the before-mentioned shortcoming. The present paper proposes a further extension: the incorporation of consumer-specific variables. This Extended Polya Model is thus suited to describe dynamic changes at the micro level of individual buyers. Amongst others a useful feature of the Extended Polya Model is its capability to account for asymmetries between brands. It is shown that the multivariate Negative Binomial Distribution Model and the Polya Model with homogeneous explanatory variables are nested within this model. This way the market analyst is provided with a general and flexible approach that allows the modelling of consumer behaviour with various degrees of complexity. The models are applied to empirical data concerning the introduction of a new brand in the German detergent market, and the results obtained are compared with regard to descriptive power and computational effort. It is found that the buyers of the new brand are more price sensitive (assessed in terms of direct and cross elasticities) than consumers of the well-established brands.

A note on the multivariate negative binomial distributions of order K

A useful recurrence relation for calculating the probabilities of the multivariare negative binomial distrtibution of order k is obtained. Analogous recurrences are also established for the type I, type II and extended multivariate negative binomial distributions of the same order. The present paper extends to the multivariate case the work of Philippou and Antzoulakos (1989) on calculating the probabilities of the multi-parameter negative binomial distribution of order k.

Multivariate Polya and Inverse Polya Distributions of Orderk

Multivariate Polya and inverse Polya distributions of order k are derived by means of generalized urn models and by compounding the type II multinomial and multivariate negative binomial distributions of order k of PHILIPPOU, ANTZOULAKOS and TRIPSIANNIS (1990, 1988), respectively, with the Dirichlet distribution. It is noted that the above two distributions include as special cases a multivariate hypergeometric distribution of order k, a negative one, an inverse one, a negative inverse one and a discrete uniform of the same order. The probability generating functions, means, variances and covariances of the new distributions are obtained and five asymptotic results are established relating them to the above-mentioned multinomial and multivariate negative binomial distributions of order k, and to the type II negative binomial and the type I multivariate Poisson distributions of order k of PHILIPPOU (1983), and PHILIPPOU, ANTZOULAKOS and TRIPSIAN-NIS (1988), respectively. Potential applications are also indicated. The present paper extends to the multivariate case the work of PHILIPPOU, TRIPSIANNIS and ANTZOULAKOS (1989) on Polya and inverse Polya distributions of order k..