Robust Estimates of the Negative Binomial Model

We consider robust parametric procedures for univariate discrete distributions, focusing on the negative binomial model. The procedures are based on three steps: ?First, a very robust, but possibly inefficient, estimate of the model parameters is computed. ?Second, this initial model is used to identify outliers, which are then removed from the sample. ?Third, a corrected maximum likelihood estimator is computed with the remaining observations. The final estimate inherits the breakdown point (bdp) of the initial one and its efficiency can be significantly higher. Analogous procedures were proposed in [1], [2], [5] for the continuous case. A comparison of the asymptotic bias of various estimates under point contamination points out the minimum Neyman's chi-squared disparity estimate as a good choice for the initial step. Various minimum disparity estimators were explored by Lindsay [4], who showed that the minimum Neyman's chi-squared estimate has a 50% bdp under point contamination; in addition, it is asymptotically fully efficient at the model. However, the finite sample efficiency of this estimate under the uncontaminated negative binomial model is usually much lower than 100% and the bias can be strong. We show that its performance can then be greatly improved using the three step procedure outlined above. In addition, we compare the final estimate with the procedure described in

A Note on Bayesian Estimation for the Negative-Binomial Model

2000 Mathematics Subject Classification: 62F15.

On estimation and influence diagnostics for zero-inflated negative binomial regression models

The zero-inflated negative binomial model is used to account for overdispersion detected in data that are initially analyzed under the zero-Inflated Poisson model A frequentist analysis a jackknife estimator and a non-parametric bootstrap for parameter estimation of zero-inflated negative binomial regression models are considered In addition an EM-type algorithm is developed for performing maximum likelihood estimation Then the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and some ways to perform global influence analysis are derived In order to study departures from the error assumption as well as the presence of outliers residual analysis based on the standardized Pearson residuals is discussed The relevance of the approach is illustrated with a real data set where It is shown that zero-inflated negative binomial regression models seems to fit the data better than the Poisson counterpart (C) 2010 Elsevier B V All rights reserved

A Nonstationary Negative Binomial Time Series with Time-Dependent Covariates: Enterococcus Counts in Boston Harbor

Boston Harbor has had a history of poor water quality, including contamination by enteric pathogens. We conduct a statistical analysis of data collected by the Massachusetts Water Resources Authority (MWRA) between 1996 and 2002 to evaluate the effects of court-mandated improvements in sewage treatment. Motivated by the ineffectiveness of standard Poisson mixture models and their zero-inflated counterparts, we propose a new negative binomial model for time series of Enterococcus counts in Boston Harbor, where nonstationarity and autocorrelation are modeled using a nonparametric smooth function of time in the predictor. Without further restrictions, this function is not identifiable in the presence of time-dependent covariates; consequently we use a basis orthogonal to the space spanned by the covariates and use penalized quasi-likelihood (PQL) for estimation. We conclude that Enterococcus counts were greatly reduced near the Nut Island Treatment Plant (NITP) outfalls following the transfer of wastewaters from NITP to the Deer Island Treatment Plant (DITP) and that the transfer of wastewaters from Boston Harbor to the offshore diffusers in Massachusetts Bay reduced the Enterococcus counts near the DITP outfalls.

The negative binomial-beta Weibull regression model to predict the cure of prostate cancer

In this article, for the first time, we propose the negative binomial-beta Weibull (BW) regression model for studying the recurrence of prostate cancer and to predict the cure fraction for patients with clinically localized prostate cancer treated by open radical prostatectomy. The cure model considers that a fraction of the survivors are cured of the disease. The survival function for the population of patients can be modeled by a cure parametric model using the BW distribution. We derive an explicit expansion for the moments of the recurrence time distribution for the uncured individuals. The proposed distribution can be used to model survival data when the hazard rate function is increasing, decreasing, unimodal and bathtub shaped. Another advantage is that the proposed model includes as special sub-models some of the well-known cure rate models discussed in the literature. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes. We analyze a real data set for localized prostate cancer patients after open radical prostatectomy.

A Poisson mixed model with nonnormal random effect distribution

In this paper, we propose a random intercept Poisson model in which the random effect is assumed to follow a generalized log-gamma (GLG) distribution. This random effect accommodates (or captures) the overdispersion in the counts and induces within-cluster correlation. We derive the first two moments for the marginal distribution as well as the intraclass correlation. Even though numerical integration methods are, in general, required for deriving the marginal models, we obtain the multivariate negative binomial model from a particular parameter setting of the hierarchical model. An iterative process is derived for obtaining the maximum likelihood estimates for the parameters in the multivariate negative binomial model. Residual analysis is proposed and two applications with real data are given for illustration. (C) 2011 Elsevier B.V. All rights reserved.

Statnote 37 : the negative binomial distribution

An organism living in water, and present at low density, may be distributed at random and therefore, samples taken from the water are likely to be distributed according to the Poisson distribution. The distribution of many organisms, however, is not random, individuals being either aggregated into clusters or more uniformly distributed. By fitting a Poisson distribution to data, it is only possible to test the hypothesis that an observed set of frequencies does not deviate significantly from an expected random pattern. Significant deviations from random, either as a result of increasing uniformity or aggregation, may be recognized by either rejection of the random hypothesis or by examining the variance/mean (V/M) ratio of the data. Hence, a V/M ratio not significantly different from unity indicates a random distribution, greater than unity a clustered distribution, and less then unity a regular or uniform distribution . If individual cells are clustered, however, the negative binomial distribution should provide a better description of the data. In addition, a parameter of this distribution, viz., the binomial exponent (k), may be used as a measure of the ‘intensity’ of aggregation present. Hence, this Statnote describes how to fit the negative binomial distribution to counts of a microorganism in samples taken from a freshwater environment.

Mixed Negative Binomial Distribution by Weighted Gamma Mixing Distribution

Павел Т. Стойнов - В тази работа се разглежда отрицателно биномното разпределение, известно още като разпределение на Пойа. Предполагаме, че смесващото разпределение е претеглено гама разпределение. Изведени са вероятностите в някои частни случаи. Дадени са рекурентните формули на Панжер.

A Characterization of the Negative Binomial Distribution

Only a few characterizations have been obtained in literatute for the negative binomial distribution (see Johnson et al., Chap. 5, 1992). In this article a characterization of the negative binomial distribution related to random sums is obtained which is motivated by the geometric distribution characterization given by Khalil et al. (1991). An interpretation in terms of an unreliable system is given.

Institutions and firm formation: An empirical analysis of Portuguese municipalities

A Work Project, presented as part of the requirements for the Award of a Masters Degree in Economics from the NOVA – School of Business and Economics

Under the veil of citations

Academics are often ranked on citation counts’, which is considered an adequate proxy for author's quality and reputation. This paper seeks to find what is behind a cited academic / a cited article. We constructed a rich dataset from Portuguese affiliated economists and use zero inflated negative binomial model. This procedure is appropriate for count outcomes, correcting for overdispersion and excess zeros. We also use a fixed effect poisson model to accomodate authors' unobserved heterogeneity. We analyze results in detail comparing with existing literature and making some theoretical considerations around.

Canakinumab Reduces The Risk Of Acute Gouty Arthritis Flares During Initiation Of Allopurinol Therapy: Results Of A Double-blind, Randomised Study

RATIONALE: This study assessed the efficacy and safety of canakinumab, a fully human anti-interleukin-1beta monoclonal antibody, for prophylaxis against acute gouty arthritis flares in patients initiating uratelowering therapy.METHODS: In this double-blind, double-dummy, dose-ranging study, 432 patients with gouty arthritis initiating allopurinol therapy were randomised 1:1:1:1:1:1:2 to receive: a single dose of canakinumab, 25, 50, 100, 200, or 300 mg subcutaneously (sc); four 4-weekly doses of canakinumab (50150125125 mg sc); or daily colchicine 0.5 mg orally for 16 weeks. Patients recorded details of flares in diaries. The study aimed to determine the canakinumab dose having equivalent efficacy to colchicine 0.5 mg at 16 weeks.RESULTS: A dose-response for canakinumab was not apparent with any of the four pre-defined dose-responsemodels. The estimated canakinumab dose with equivalent efficacy to colchicinewas belowthe range of doses tested.At 16 weeks, therewas a 62-72% reduction in themean number of flares per patient for canakinumab doses >50 mg vs colchicine based on a negative binomial model (rate ratio: 0.28-0.38, p50.0083), and the percentage of patients experiencing >1 flarewas significantly lower for all canakinumab doses (15- 27%) vs colchicine (44%, p<0.05). Therewas a 64-72%reduction in the risk of experiencing >1 flare for canakinumab doses >50 mg vs colchicine at 16 weeks (hazard ratio: 0.28-0.36, p50.05). The incidence of adverse events was similar across treatment groups.CONCLUSIONS: Single canakinumab doses >50 mg or four 4-weekly doses provided superior prophylaxis against flares compared with daily colchicine 0.5 mg.

Ageing Prisoners' Disease Burden: Is Being Old a Better Predictor than Time Served in Prison?

Background: The number of older prisoners entering and ageing in prison has increased in the last few decades. Ageing prisoners pose unique challenges to the prison administration as they have differentiated social, custodial and healthcare needs than prisoners who are younger and relatively healthier. Objective: The goal of this study was to explore and compare the somatic disease burden of old and young prisoners, and to examine whether it can be explained by age group and/or time served in prison. Methods: Access to prisoner medical records was granted to extract disease and demographic information of older (>50 years) and younger (≤49 years) prisoners in different Swiss prisons. Predictor variables included the age group and the time spent in prison. The dependent variable was the total number of somatic diseases as reported in the medical records. Results were analysed using descriptive statistics and a negative binomial model. Results: Data of 380 male prisoners from 13 different prisons in Switzerland reveal that the mean ages of older and younger prisoners were 58.78 and 34.26 years, respectively. On average, older prisoners have lived in prison for 5.17 years and younger prisoners for 2.49 years. The average total number of somatic diseases reported by older prisoners was 2.26 times higher than that of prisoners below 50 years of age (95% CI 1.77-2.87, p < 0.001). Conclusion: This study is the first of its kind to capture national disease data of prisoners with a goal of comparing the disease burden of older and younger prisoners. Study findings indicate that older inmates suffer from more somatic diseases and that the number of diseases increases with age group. Results clearly illustrate the poorer health conditions of those who are older, their higher healthcare burden, and raises questions related to the provision of healthcare for inmates growing old in prison. © 2014 S. Karger AG, Basel.

Efficacy Of Canakinumab (Acz885), A Fully Human Anti-Interleukin (Il)-1beta Monoclonal Antibody, In The Prevention Of Flares In Gout Patients Initiating Allopurinol Therapy

Background: Gout patients initiating urate lowering therapy have an increased risk of flares. Inflammation in gouty arthritis is induced by IL-1b. Canakinumab targets and inhibits IL-1b effectively in clinical studies. This study compared different doses of canakinumab vs colchicine in preventing flares in gout patients initiating allopurinol therapy.Methods: In this 24 week double blind study, gout patients (20-79 years) initiating allopurinol were randomized (1:1:1:1:1:1:2) to canakinumab s.c. single doses of 25, 50, 100, 200, 300 mg, or 150 mg divided in doses every 4 weeks (50+50+25+25 mg [q4wk]) or colchicine 0.5 mg p.o. daily for 16 weeks. Primary outcome was to determine the canakinumab dose giving comparable efficacy to colchicine with respect to the number of gout flares occurring during first 16 weeks. Secondary outcomes included number of patients with gout flares and C-reactive protein (CRP) levels during the first 16 weeks.Results: 432 patients were randomized and 391 (91%) completed the study. All canakinumab doses were better than colchicine in preventing flares and therefore, a canakinumab dose comparable to colchicine could not be determined. Based on a negative binomial model, all canakinumab groups, except 25 mg, reduced the flare rate ratio per patient significantly compared to colchicine group (rate ratio estimates 25 mg 0.60, 50 mg 0.34, 100 mg 0.28, 200 mg 0.37, 300 mg 0.29, q4wk 0.38; p<=0.05). The percentage of patients with flares was lower for all canakinumab groups (25 mg 27.3%, 50 mg 16.7%, 100 mg 14.8%, 200 mg 18.5%, 300 mg 15.1%, q4wk 16.7%) compared to colchicine group (44.4%). All patients taking canakinumab were significantly less likely to experience at least one gout flare than patients taking colchicine (odds ratio range [0.22 - 0.47]; p<=0.05 for all). The median baseline CRP levels were 2.86 mg/L for 25 mg, 3.42 mg/L for 50 mg, 1.76 mg/L for 100 mg, 3.66 mg/L for 200 mg, 3.21 mg/L for 300 mg, 3.23 mg/L for q4wk canakinumab groups and 2.69 mg/L for colchicine group. In all canakinumab groups with median CRP levels above the normal range at baseline, median levels declined within 15 days of treatment and were maintained at normal levels (ULN=3 mg/L) throughout the 16 week period. Adverse events (AEs) occurred in 52.7% (25 mg), 55.6% (50 mg), 51.9% (100 mg), 51.9% (200 mg), 54.7% (300 mg), and 58.5% (q4wk) of patients on canakinumab vs 53.7% of patients on colchicine. Serious AEs (SAE) were reported in 2 (3.6%; 25 mg), 2 (3.7%, 50 mg), 3 (5.6%, 100 mg), 3 (5.6%, 200 mg), 3 (5.7%, 300 mg) and 1 (1.9%, q4wk) patients on canakinumab and in 5 (4.6%) patients on colchicine. One fatal SAE (myocardial infarction, not related to study drug) occurred in colchicine group.Conclusion: In this large randomized, double-blind active controlled study of flare prevention in gout patients initiating allopurinol therapy, treatment with canakinumab led to a statistically significant reduction in flares compared with colchicine (standard of care), and was well tolerated.