Showing papers on "Negative binomial distribution published in 2005"

Random effect models for repeated measures of zero-inflated count data:

Abstract: For count responses, the situation of excess zeros (relative to what standard models allow) often occurs in biomedical and sociological applications. Modeling repeated measures of zero-inflated count data presents special challenges. This is because in addition to the problem of extra zeros, the correlation between measurements upon the same subject at different occasions needs to be taken into account. This article discusses random effect models for repeated measurements on this type of response variable. A useful model is the hurdle model with random effects, which separately handles the zero observations and the positive counts. In maximum likelihood model fitting, we consider both a normal distribution and a nonparametric approach for the random effects. A special case of the hurdle model can be used to test for zero inflation. Random effects can also be introduced in a zero-inflated Poisson or negative binomial model, but such a model may encounter fitting problems if there is zero deflation at any s...

Modeling avian abundance from replicated counts using binomial mixture models

Abstract: Abundance estimation in ecology is usually accomplished by capture–recapture, removal, or distance sampling methods. These may be hard to implement at large spatial scales. In contrast, binomial mixture models enable abundance estimation without individual identification, based simply on temporally and spatially replicated counts. Here, we evaluate mixture models using data from the national breeding bird monitoring program in Switzerland, where some 250 1-km2 quadrats are surveyed using the territory mapping method three times during each breeding season. We chose eight species with contrasting distribution (wide–narrow), abundance (high–low), and detectability (easy–difficult). Abundance was modeled as a random effect with a Poisson or negative binomial distribution, with mean affected by forest cover, elevation, and route length. Detectability was a logit-linear function of survey date, survey date-by-elevation, and sampling effort (time per transect unit). Resulting covariate effects and parameter est...

Binomial distribution based τ-leap accelerated stochastic simulation

Abstract: Recently, Gillespie introduced the τ-leap approximate, accelerated stochastic Monte Carlo method for well-mixed reacting systems [J. Chem. Phys. 115, 1716 (2001)]. In each time increment of that method, one executes a number of reaction events, selected randomly from a Poisson distribution, to enable simulation of long times. Here we introduce a binomial distribution τ-leap algorithm (abbreviated as BD-τ method). This method combines the bounded nature of the binomial distribution variable with the limiting reactant and constrained firing concepts to avoid negative populations encountered in the original τ-leap method of Gillespie for large time increments, and thus conserve mass. Simulations using prototype reaction networks show that the BD-τ method is more accurate than the original method for comparable coarse-graining in time.

BayesX: analysing Bayesian structured additive regression models

Abstract: There has been much recent interest in Bayesian inference for generalized additive and related models. The increasing popularity of Bayesian methods for these and other model classes is mainly caused by the introduction of Markov chain Monte Carlo (MCMC) simulation techniques which allow realistic modeling of complex problems. This paper describes the capabilities of the free software package BayesX for estimating regression models with structured additive predictor based on MCMC inference. The program extends the capabilities of existing software for semiparametric regression included in S-PLUS, SAS, R or Stata. Many model classes well known from the literature are special cases of the models supported by BayesX. Examples are generalized additive (mixed) models, dynamic models, varying coefficient models, geoadditive models, geographically weighted regression and models for space-time regression. BayesX supports the most common distributions for the response variable. For univariate responses these are Gaussian, Binomial, Poisson, Gamma, negative Binomial, zero inflated Poisson and zero inflated negative binomial. For multicategorical responses, both multinomial logit and probit models for unordered categories of the response as well as cumulative threshold models for ordered categories can be estimated. Moreover, BayesX allows the estimation of complex continuous time survival and hazard rate models.

Many zeros does not mean zero inflation: comparing the goodness-of-fit of parametric models to multivariate abundance data

Abstract: An important step in studying the ecology of a species is choosing a statistical model of abundance; however, there has been little general consideration of which statistical model to use. In particular, abundance data have many zeros (often 50–80 per cent of all values), and zero-inflated count distributions are often used to specifically model the high frequency of zeros in abundance data. However, in such cases it is often taken for granted that a zero-inflated model is required, and the goodness-of-fit to count distributions with and without zero inflation is not often compared for abundance data. In this article, the goodness-of-fit was compared for several marginal models of abundance in 20 multivariate datasets (a total of 1672 variables across all datasets) from different sources. Multivariate abundance data are quite commonly collected in applied ecology, and the properties of these data may differ from abundances collected in autecological studies. Goodness-of-fit was assessed using AIC values, graphs of observed vs expected proportion of zeros in a dataset, and graphs of the sample mean–variance relationship. The negative binomial model was the best fitting of the count distributions, without zero-inflation. The high frequency of zeros was well described by the systematic component of the model (i.e. at some places predicted abundance was high, while at others it was zero) and so it was rarely necessary to modify the random component of the model (i.e. fitting a zero-inflated distribution). A Gaussian model based on transformed abundances fitted data surprisingly well, and rescaled per cent cover was usually poorly fitted by a count distribution. In conclusion, results suggest that the high frequency of zeros commonly seen in multivariate abundance data is best considered to come from distributions where mean abundance is often very low (hence there are many zeros), as opposed to claiming that there are an unusually high number of zeros compared to common parametric distributions. Copyright © 2005 John Wiley & Sons, Ltd.

Negative Binomial Regression of Electric Power Outages in Hurricanes

Abstract: Hurricanes can cause extensive power outages, resulting in economic loss, business interruption, and secondary effects to other infrastructure systems. Currently, power companies are unable to accurately predict where outages will occur. Therefore, it is difficult for them to deploy repair personnel and materials, and make other emergency response decisions in advance of an event. This paper describes negative binomial regression models for the number of hurricane-related outages likely to occur in each one square kilometer grid cell and in each zip code in a region due to passage of a hurricane. The models are based on a large Geographic Information System database of outages in North and South Carolina from three hurricanes: Floyd (1999), Bonnie (1998), and Fran (1996). The most useful explanatory variables are the number of transformers in the area, the company affected, maximum gust wind speed, and a hurricane effect. Wind speeds were estimated using a calibrated hurricane wind speed model. Pseudo R-squared values and other diagnostic statistics are developed to facilitate model selection with generalized negative binomial models.

Generalized Poisson distribution: the property of mixture of Poisson and comparison with negative binomial distribution.

Abstract: We prove that the generalized Poisson distribution GP(theta, eta) (eta > or = 0) is a mixture of Poisson distributions; this is a new property for a distribution which is the topic of the book by Consul (1989). Because we find that the fits to count data of the generalized Poisson and negative binomial distributions are often similar, to understand their differences, we compare the probability mass functions and skewnesses of the generalized Poisson and negative binomial distributions with the first two moments fixed. They have slight differences in many situations, but their zero-inflated distributions, with masses at zero, means and variances fixed, can differ more. These probabilistic comparisons are helpful in selecting a better fitting distribution for modelling count data with long right tails. Through a real example of count data with large zero fraction, we illustrate how the generalized Poisson and negative binomial distributions as well as their zero-inflated distributions can be discriminated.

The role of competition and clustering in population dynamics.

Abstract: A simple argument based on the distribution of individuals amongst discrete resource sites is used to show how the form of single species population models depends on the type of competition between, and the spatial clustering of, the individuals. For scramble competition between individuals, we confirm earlier demonstrations that the Ricker model is a direct consequence of a uniform random distribution of individuals across resources. By introducing spatial clustering of individuals according to a negative binomial distribution, we are able to derive the Hassell model. Furthermore, the tent map model is seen to be a consequence of scramble competition and an ideal-free distribution of individuals. To model contest competition under different degrees of spatial clustering we derive a new three-parameter model, of which the Beverton–Holt and Skellam models are special cases, where one of the parameters relates directly to the clustering distribution. Other population models, such as the quadratic model and the theta-Ricker models, cannot be derived in our framework. Taken together our derivations of population models allows us to make a more rigorous prescription for model choice when fitting to particular datasets.

On modeling claim frequency data in general insurance with extra zeros

Abstract: In some occasions, claim frequency data in general insurance may not follow the traditional Poisson distribution and in particular they are zero-inflated. Extra dispersion appears as the number of observed zeros exceeding the number of expected zeros under the Poisson or even the negative binomial distribution assumptions. This paper presents several parametric zero-inflated count distributions, including the ZIP, ZINB, ZIGP and ZIDP, to accommodate the excess zeros for insurance claim count data. Different count distributions in the second component are considered to allow flexibility to control the distribution shape. The generalized Pearson χ2 statistic, Akaike's information criteria (AIC) and Bayesian information criteria (BIC) are used as goodness-of-fit and model selection measures. With the presence of extra zeros in a data set of automobile insurance claims, our result shows that the application of zero-inflated count data models and in particular the zero-inflated double Poisson regression model, provide a good fit to the data.

Analysis of count data using poisson regression.

Abstract: Nurses and other health researchers are often concerned with infrequently occurring, repeatable, health-related events such as number of hospitalizations, pregnancies, or visits to a health care provider. Reports on the occurrence of such discrete events take the form of non-negative integer or count data. Because the counts of infrequently occurring events tend to be non-normally distributed and highly positively skewed, the use of ordinary least squares (OLS) regression with non-transformed data has several shortcomings. Techniques such as Poisson regression and negative binomial regression may provide more appropriate alternatives for analyzing these data. The purpose of this article is to compare and contrast the use of these three methods for the analysis of infrequently occurring count data. The strengths, limitations, and special considerations of each approach are discussed. Data from the National Longitudinal Survey of Adolescent Health (AddHealth) are used for illustrative purposes.

Statistical Analysis of Efficacy in Falls Prevention Trials

Abstract: Background Many different and sometimes inappropriate statistical techniques have been used to analyze the results of randomized controlled trials of falls prevention programs for elderly people. This makes comparison of the efficacy of particular interventions difficult. Methods We used raw data from two randomized controlled trials of a home exercise program to compare the number of falls in the exercise and control groups during the trials. We developed two different survival analysis models (Andersen-Gill and marginal Cox regression) and a negative binomial regression model for each trial. These techniques a) allow for the fact that falls are frequent, recurrent events with a non-normal distribution; b) adjust for the follow-up time of individual participants; and c) allow the addition of covariates. Results In one trial, the three different statistical techniques gave surprisingly similar results for the efficacy of the intervention but, in a second trial, underlying assumptions were violated for the two Cox regression models. Negative binomial regression models were easier to use. Conclusion We recommend negative binomial regression models for evaluating the efficacy of falls prevention programs.

Alternative Risk Models for Ranking Locations for Safety Improvement

Abstract: Many types of statistical models have been proposed for estimating accident risk in transport networks, ranging from basic Poisson and negative binomial models to more complicated models, such as zero-inflated and hierarchical Bayesian models. However, little systematic effort has been devoted to comparing the performance and practical implications of these models and ranking criteria when they are used for identifying hazardous locations. This research investigates the relative performance of three alternative models: the traditional negative binomial model, the heterogeneous negative binomial model, and the Poisson lognormal model. In particular, this work focuses on the impact of the choice of two alternative prior distributions (i.e., gamma versus lognormal) and the effect of allowing variability in the dispersion parameter on the outcome of the analysis. From each model, two alternative accident estimators are computed by using the conditional mean under both marginal and posterior distributions. A sample of Canadian highway-railway intersections with an accident history of 5 years is used to calibrate and evaluate the three alternative models and the two ranking criteria. It is concluded that the choice of model assumptions and ranking criteria can lead to considerably different lists of hazardous locations.

Estimation in conditional first order autoregression with discrete support

Abstract: We consider estimation in the class of first order conditional linear autoregressive models with discrete support that are routinely used to model time series of counts. Various groups of estimators proposed in the literature are discussed: moment-based estimators; regression-based estimators; and likelihood-based estimators. Some of these have been used previously and others not. In particular, we address the performance of new types of generalized method of moments estimators and propose an exact maximum likelihood procedure valid for a Poisson marginal model using backcasting. The small sample properties of all estimators are comprehensively analyzed using simulation. Three situations are considered using data generated with: a fixed autoregressive parameter and equidispersed Poisson innovations; negative binomial innovations; and, additionally, a random autoregressive coefficient. The first set of experiments indicates that bias correction methods, not hitherto used in this context to our knowledge, are some-times needed and that likelihood-based estimators, as might be expected, perform well. The second two scenarios are representative of overdispersion. Methods designed specifically for the Poisson context now perform uniformly badly, but simple, bias-corrected, Yule-Walker and least squares estimators perform well in all cases.

Improved Base-Stock Approximations for Independent Stochastic Lead Times with Order Crossover

Abstract: Order crossovers occur when replenishment orders arrive in a sequence that is different than the one in which they were placed. Order crossovers require that optimal reorder levels be set with regard to the inventory shortfall distribution rather than the lead-time demand distribution. Assuming periodic review and independent lead times, this paper suggests simple approximations of the shortfall distribution by showing that the variance of the number of orders outstanding is bounded above by the standard deviation of lead time divided by √3. Using this bound in a normal approximation improves significantly upon the common practice of basing policies on the lead-time demand distribution. A negative binomial approximation of the shortfall, based on its exact variance, offers even greater improvement, at the cost of some additional informational and computational requirements.

Count Data Models for Trip Generation

Abstract: Count data models are established to overcome the shortcoming of linear regression model used for trip generation in conventional four step travel demand forecasting. It should be checked if there are overdispersion and excess zero responses in count data to forecast the generation of trips. The forecasted values should also be non-negative ones. The study applies to nonhome based trips at household level to perform efficient analysis on count data. The Poisson model with an assumption of equidispersion has frequently been used to analyze count data. However, if the variance of data is greater than the mean, the Poisson model tends to underestimate errors, resulting in problem in reliability. Excess zeros in data result in heterogeneity leading to biased coefficient estimates for the models. The negative binomial model and the modified count data models are established to consider overdispersion and heterogeneity to improve the reliability. The optimal model is chosen through Vuong test. Model reliability is also checked by likelihood test and accuracy of estimated value of model by Theil inequality coefficient. Finally, sensitivity analysis is performed to know the change of nonhome based trips depending on the change in socio-economic characteristics.

Modelling macroparasite aggregation using a nematode-sheep system: the Weibull distribution as an alternative to the negative binomial distribution?

Abstract: Macroparasites are almost always aggregated across their host populations, hence the Negative Binomial Distribution (NBD) with its exponent parameter k is widely used for modelling, quantifying or analysing parasite distributions. However, many studies have pointed out some drawbacks in the use of the NBD, with respect to the sensitivity of k to the mean number of parasites per host or the under-representation of the heavily infected hosts in the estimate of k. In this study, we compare the fit of the NBD with 4 other widely used distributions on observed parasitic gastrointestinal nematode distributions in their sheep host populations (11 datasets). Distributions were fitted to observed data using maximum likelihood estimator and the best fits were selected using the Akaike's Information Criterion (AIC). A simulation study was also conducted in order to assess the possible bias in parameter estimations especially in the case of small sample sizes. We found that the NBD is seldom the best fit for gastrointestinal nematode distributions. The Weibull distribution was clearly more appropriate over a very wide range of degrees of aggregation, mainly because it was more flexible in fitting the heavily infected hosts. Moreover, the Weibull distribution estimates are less sensitive to sample size. Thus, when possible, we suggest to carefully check on observed data if the NBD is appropriate before conducting any further analysis on parasite distributions.

On binomial and circular binomial distributions of orderk forl-overlapping success runs of lengthk

Abstract: The number ofl-overlapping success runs of lengthk inn trials, which was introduced and studied recently, is presently reconsidered in the Bernoulli case and two exact formulas are derived for its probability distribution function in terms of multinomial and binomial coefficients respectively. A recurrence relation concerning this distribution, as well as its mean, is also obtained. Furthermore, the number ofl-overlapping success runs of lengthk inn Bernoulli trials arranged on a circle is presently considered for the first time and its probability distribution function and mean are derived. Finally, the latter distribution is related to the first, two open problems regarding limiting distributions are stated, and numerical illustrations are given in two tables. All results are new and they unify and extend several results of various authors on binomial and circular binomial distributions of orderk.

Gibbs distributions for random partitions generated by a fragmentation process

Abstract: In this paper we study random partitions of 1,...n, where every cluster of size j can be in any of w\_j possible internal states. The Gibbs (n,k,w) distribution is obtained by sampling uniformly among such partitions with k clusters. We provide conditions on the weight sequence w allowing construction of a partition valued random process where at step k the state has the Gibbs (n,k,w) distribution, so the partition is subject to irreversible fragmentation as time evolves. For a particular one-parameter family of weight sequences w\_j, the time-reversed process is the discrete Marcus-Lushnikov coalescent process with affine collision rate K\_{i,j}=a+b(i+j) for some real numbers a and b. Under further restrictions on a and b, the fragmentation process can be realized by conditioning a Galton-Watson tree with suitable offspring distribution to have n nodes, and cutting the edges of this tree by random sampling of edges without replacement, to partition the tree into a collection of subtrees. Suitable offspring distributions include the binomial, negative binomial and Poisson distributions.

Designing Accurate Control Charts Based on the Geometric and Negative Binomial Distributions

Abstract: Attribute control charts are used effectively to monitor for process change. Their accuracy can be improved by judiciously selecting the sample size. The required sample sizes to achieve accuracy can be quite restrictive, especially when the nominal proportions of non-conforming units are quite small. The usual attribute control chart has a set sample size and the number of non-conforming units in the sample is plotted. If, instead of setting a specific sample size the number of non-conforming units is set, an alternative monitoring process is possible. Specifically, the cumulative count of conforming (CCC-r) control chart is a plot of the number of units that must be tested to find the rth non-conforming unit. These charts, based on the geometric and negative binomial distributions, are often suggested for monitoring very high quality processes. However, they can also be used very efficiently to monitor processes of lesser quality. This procedure has the potential to find process deterioration more quickly and efficiently. Xie et al. (Journal of Quality and Reliability Management 1999; 16(2):148–157) provided tables of control limits for CCC-r charts for but focused mainly on high-quality processes and the tables do not include any assessments of the risk of a false alarm or the reliability of detecting process change. In this paper, these tables are expanded for processes of lesser quality and include such assessments using the number of expected monitoring periods (average run lengths (ARLs)) to detect process change. Also included is an assessment of the risk of a false alarm, that is, a false indication of process deterioration. Such assessments were not included by Xie et al. but are essential for the quality engineer to make sound decisions. Furthermore, a hybrid of the control charts based on the binomial, geometric and negative binomial distributions is proposed to monitor for process change. Copyright © 2005 John Wiley & Sons, Ltd.

Sensitivity of Crash Models to Alternative Specifications

Abstract: This paper examines the sensitivity of policy conclusions that are derived from crash models using various specifications. Our analyses compare models specified as crash rate or population normalized models (i.e., fatalities per capita or per vehicle miles traveled) adjusted to account for serial correlation in the error term with negative binomial count models with the total number of fatalities as the dependent variable. Our analyses focus on the interpretation of key policy variables, especially the association between safety-belt laws and administrative license revocation laws on fatalities. Evaluation of statistical significance of parameters, elasticities derived from the models and total fatalities associated with changes in key variables are examined. Results suggest that negative binomial models tend to be more robust and display less variation in results than those linear regression models that account for serial correlation. From a policy perspective, we found no evidence that passage of administrative license revocation laws that automatically suspend the license of a drunk driver have been effective while laws requiring safety-belt usage have been effective. Our results suggest that providing confidence intervals on elasticity estimates and estimated fatalities would provide policy makers with greater confidence in the results of model estimates.

A Cluster Distribution As A Model For Estimating High-Order-Event Probabilities In Power Systems

Abstract: We propose the use of the cluster distribution, derived from a negative binomial probability model, to estimate the probability of high-order events in terms of number of lines outaged within a short time, useful in long-term planning and also in short-term operational defense to such events. We use this model to fit statistical data gathered for a 30-year period for North America. The model is compared to the commonly used Poisson model and the power-law model. Results indicate that the Poisson model underestimates the probability of higher-order events, whereas the power-law model overestimates it. We use the strict chi-square fitness test to compare the fitness of these three models and find that the cluster model is superior to the other two models for the data used in the study.

Bootstrap Tests for Overdispersion in a Zero-Inflated Poisson Regression Model

Abstract: SummaryRidout, Hinde, and Demetrio (2001, Biometrics57, 219–223) derived a score test for testing a zero-inflated Poisson (ZIP) regression model against zero-inflated negative binomial (ZINB) alternatives. They mentioned that the score test using the normal approximation might underestimate the nominal significance level possibly for small sample cases. To remedy this problem, a parametric bootstrap method is proposed. It is shown that the bootstrap method keeps the significance level close to the nominal one and has greater power uniformly than the existing normal approximation for testing the hypothesis.