Semi Varying Coefficient Zero-Inflated Generalized Poisson Regression Model

In this paper, the semi varying coefficient zero-inflated generalized Poisson model is discussed based on penalized log-likelihood. All the coefficient functions are fitted by penalized spline (P-spline), and Expectation-maximization algorithm is used to drive these estimators. The estimation approach is rapid and computationally stable. Under some mild conditions, the consistency and the asymptotic normality of these resulting estimators are given. The score test statistics about dispersion parameter is discussed based on the P-spline estimation. Both simulated and real data example are used to illustrate our proposed methods.     

Some Remarks on Testing Overdispersion in Zero-Inflated Poisson and Binomial Regression Models

This note extends the score test statistics for overdispersion in Poisson and binomial regression models (Dean, 1992 Dean , C. B. ( 1992 ). Testing for overdispersion in Poisson and binomial regression models . J. Amer. Statist. Assoc. 87 : 451 – 457 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) to the zero-inflated models. Some general results are obtained, and examples illustrate the application of the extended results.     

Zero-Inflated Poisson Regression for Longitudinal Data

Medical and public health research often involve the analysis of repeated or longitudinal count data that exhibit excess zeros such as the number of yearly doctor visits by a group of individuals over a number of years. Zero-inflated Poisson (ZIP) regression models can be used to account for excess zeros in count data. We propose an extension of the ZIP model that is appropriate for longitudinal data. Our extension includes a non stationary, observation-driven time series model based correlation structure. We discuss estimation of the model parameters and the inefficiency of the estimators when the correlation structure is mis-specified. The model's application to the analysis of health care utilization data is also discussed.     

Random Effects Modeling and the Zero-Inflated Poisson Distribution

Overdispersion due to a large proportion of zero observations in data sets is a common occurrence in many applications of many fields of research; we consider such scenarios in count panel (longitudinal) data. A well-known and widely implemented technique for handling such data is that of random effects modeling, which addresses the serial correlation inherent in panel data, as well as overdispersion. To deal with the excess zeros, a zero-inflated Poisson distribution has come to be canonical, which relaxes the equal mean-variance specification of a traditional Poisson model and allows for the larger variance characteristic of overdispersed data. A natural proposal then to approach count panel data with overdispersion due to excess zeros is to combine these two methodologies, deriving a likelihood from the resulting conditional probability. In performing simulation studies, we find that this approach in fact poses problems of identifiability. In this article, we construct and explain in full detail why a model obtained from the marriage of two classical and well-established techniques is unidentifiable and provide results of simulation studies demonstrating this effect. A discussion on alternative methodologies to resolve the problem is provided in the conclusion.     

Testing overdispersion in the zero-inflated Poisson model

The zero-inflated negative binomial (ZINB) model is used to account for commonly occurring overdispersion detected in data that are initially analyzed under the zero-inflated Poisson (ZIP) model. Tests for overdispersion (Wald test, likelihood ratio test [LRT], and score test) based on ZINB model for use in ZIP regression models have been developed. Due to similarity to the ZINB model, we consider the zero-inflated generalized Poisson (ZIGP) model as an alternate model for overdispersed zero-inflated count data. The score test has an advantage over the LRT and the Wald test in that the score test only requires that the parameter of interest be estimated under the null hypothesis. This paper proposes score tests for overdispersion based on the ZIGP model and illustrates that the derived score statistics are exactly the same as the score statistics under the ZINB model. A simulation study indicates the proposed score statistics are preferred to other tests for higher empirical power. In practice, based on the approximate mean–variance relationship in the data, the ZINB or ZIGP model can be considered, and a formal score test based on asymptotic standard normal distribution can be employed for assessing overdispersion in the ZIP model. We provide an example to illustrate the procedures for data analysis.     

Zero-inflated models and estimation in zero-inflated Poisson distribution

In this paper, we briefly overview different zero-inflated probability distributions. We compare the performance of the estimates of Poisson, Generalized Poisson, ZIP, ZIGP and ZINB models through Mean square error (MSE), bias and Standard error (SE) when the samples are generated from ZIP distribution. We propose a new estimator referred to as probability estimator (PE) of inflation parameter of ZIP distribution based on moment estimator (ME) of the mean parameter and compare its performance with ME and maximum likelihood estimator (MLE) through a simulation study. We use the PE along with ME and MLE to fit ZIP distribution to various zero-inflated datasets and observe that the results do not differ significantly. We recommend using PE in place of MLE since it is easy to calculate and the simulation study in this paper demonstrates that the PE performs as good as MLE irrespective of the sample size.     

零膨胀计数数据函数型部分变系数模型

在实际数据分析中经常会遇到零膨胀计数数据作为响应变量与函数型随机变量和随机向量作为预测变量相关联。本文考虑函数型部分变系数零膨胀模型 (FPVCZIM),模型中无穷维的斜率函数用函数型主成分基逼近,系数函数用B-样条进行拟合。通过EM 算法得到估计量,讨论其理论性质,在一些正则条件下获得了斜率函数和系数函数估计量的收敛速度。有限样本的Monte Carlo 模拟研究和真实数据分析被用来解释本文提出的方法。     

Testing the Lack-of-Fit of Zero-Inflated Poisson Regression Models

A zero-inflated Poisson regression model has been widely used for the effect of a covariate in count data containing many zeros with a linear predictor. To assess the adequacy of the linear relationship, we approximate the covariate effect with cubic B-splines. The semiparametric model parameters are estimated by maximizing the likelihood function through an expectation-maximization algorithm. A log-likelihood ratio test is then used to evaluate the adequacy of the linear relation. A simulation study is conducted to study the power performance of the test. A real example is provided to demonstrate the practical use of the methodology.     

A Multivariate Generalized Poisson Regression Model

A multivariate generalized Poisson regression model based on the multivariate generalized Poisson distribution is defined and studied. The regression model can be used to describe a count data with any type of dispersion. The model allows for both positive and negative correlation between any pair of the response variables. The parameters of the regression model are estimated by using the maximum likelihood method. Some test statistics are discussed, and two numerical data sets are used to illustrate the applications of the multivariate count data regression model.     

A Comparative Study of Observation- and Parameter-driven Zero-inflated Poisson Models for Longitudinal Count Data

Longitudinal count data with excessive zeros frequently occur in social, biological, medical, and health research. To model such data, zero-inflated Poisson (ZIP) models are commonly used, after separating zero and positive responses. As longitudinal count responses are likely to be serially correlated, such separation may destroy the underlying serial correlation structure. To overcome this problem recently observation- and parameter-driven modelling approaches have been proposed. In the observation-driven model, the response at a specific time point is modelled through the responses at previous time points after incorporating serial correlation. One limitation of the observation-driven model is that it fails to accommodate the presence of any possible over-dispersion, which frequently occurs in the count responses. This limitation is overcome in a parameter-driven model, where the serial correlation is captured through the latent process using random effects. We compare the results obtained by the two models. A quasi-likelihood approach has been developed to estimate the model parameters. The methodology is illustrated with analysis of two real life datasets. To examine model performance the models are also compared through a simulation study.     

Tests for Detecting Overdispersion in the Positive Poisson Regression Model

This article derives score tests for extra-Poisson variation in the positive or truncated-at-zero Poisson regression model against truncated-at-zero negative binomial family alternatives. It also develops size-corrected tests of overdispersion that are expected to improve their small-sample properties. Further, small-sample performance of the tests is investigated by means of Monte Carlo experiments. As an illustration, the proposed tests are applied to a model of strikes in U.S. manufacturing. The proposed tests have an interpretation as conditional moment tests and require only the positive Poisson model to be estimated. It is shown that most of the tests for overdispersion in the regular Poisson model given in the econometric and statistical literature can be obtained as special cases of the tests developed in this article. Monte Carlo experiments indicate that the size correction, based on the asymptotic expansions of the score function, is effective in improving the accuracy of the size and power of the tests in small samples.     

Score Tests for Zero-Inflation in Overdispersed Count Data

The negative binomial (NB) model and the generalized Poisson (GP) model are common alternatives to Poisson models when overdispersion is present in the data. Having accounted for initial overdispersion, we may require further investigation as to whether there is evidence for zero-inflation in the data. Two score statistics are derived from the GP model for testing zero-inflation. These statistics, unlike Wald-type test statistics, do not require that we fit the more complex zero-inflated overdispersed models to evaluate zero-inflation. A simulation study illustrates that the developed score statistics reasonably follow a χ² distribution and maintain the nominal level. Extensive simulation results also indicate the power behavior is different for including a continuous variable than a binary variable in the zero-inflation (ZI) part of the model. These differences are the basis from which suggestions are provided for real data analysis. Two practical examples are presented in this article. Results from these examples along with practical experience lead us to suggest performing the developed score test before fitting a zero-inflated NB model to the data.     

Determination of Single Sampling Plans by Attributes Under the Conditions of Zero-Inflated Poisson Distribution

When the manufacturing process is well monitored, occurrence of nondefects would be a frequent event in sampling inspection. The appropriate probability distribution of the number of defects is a zero-inflated Poisson (ZIP) distribution. In this article, determination of single sampling plans (SSPs) by attributes using unity values is considered, when the number of defects follows a ZIP distribution. The operating characteristic (OC) function of the sampling plan is derived. Plan parameters are obtained for some sets of values of (p₁, α, p₂, β). Numerical illustrations are given to describe the determination of SSP under ZIP distribution and to study its performance in comparison with Poisson SSP.     

The Poisson Generalized Linear Failure Rate Model

We formulate a new cure rate survival model by assuming that the number of competing causes of the event of interest has the Poisson distribution, and the time to this event has the generalized linear failure rate distribution. A new distribution to analyze lifetime data is defined from the proposed cure rate model, and its quantile function as well as a general expansion for the moments is derived. We estimate the parameters of the model with cure rate in the presence of covariates for censored observations using maximum likelihood and derive the observed information matrix. We obtain the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform global influence analysis. The usefulness of the proposed cure rate survival model is illustrated in an application to real data.     

Automatic Smoothing for Poisson Regression

Abstract

Adaptive choice of smoothing parameters for nonparametric Poisson regression (O'Sullivan et al., 1986 O'Sullivan , F. , Yandell , B. S. , Raynor , W. J., Jr. ( 1986 ). Automatic smoothing of regression functions in generalized linear models . J. Amer. Statist. Assoc. 81 : 96 – 103 . [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) is considered in this article. A computable approximation of the unbiased risk estimate (AUBR) for Poisson regression is introduced. This approximation can be used to automatically tune the smoothing parameter for the penalized likelihood estimator. An alternative choice is the generalized approximate cross validation (GACV) proposed by Xiang and Wahba (1996 Xiang , D. , Wahba , G. ( 1996 ). A generalized approximate cross validation for smoothing splines with non-Gaussian data . Statist. Sinica 6 (3): 675 – 692 .[Web of Science ®] , [Google Scholar]). Although GACV enjoys a great success in practice when applying for nonparametric logisitic regression, its performance for Poisson regression is not clear. Numerical simulations have been conducted to evaluate the GACV and AUBR based tuning methods. We found that GACV has a tendency to oversmooth the data when the intensity function is small. As a consequence, we suggest tuning the smoothing parameter using AUBR in practice.     

Regression for doubly inflated multivariate Poisson distributions

Dependent 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.     

Bias-reduced maximum likelihood estimation of the zero-inflated Poisson distribution

Using the techniques developed by Subrahmaniam and Ching’anda (1978), we study the robustness to nonnormality of the linear discriminant functions. It is seen that the LDF procedure is quite robust against the likelihood ratio rule. The latter yields in all cases much smaller overall error rates; however, the disparity between the error rates of the LDF and LR procedures is not large enough to warrant the recommendation to use the more complicated LR procedure.     

Score test for testing zero-inflated Poisson regression against zero-inflated generalized Poisson alternatives

In several cases, count data often have excessive number of zero outcomes. This zero-inflated phenomenon is a specific cause of overdispersion, and zero-inflated Poisson regression model (ZIP) has been proposed for accommodating zero-inflated data. However, if the data continue to suggest additional overdispersion, zero-inflated negative binomial (ZINB) and zero-inflated generalized Poisson (ZIGP) regression models have been considered as alternatives. This study proposes the score test for testing ZIP regression model against ZIGP alternatives and proves that it is equal to the score test for testing ZIP regression model against ZINB alternatives. The advantage of using the score test over other alternative tests such as likelihood ratio and Wald is that the score test can be used to determine whether a more complex model is appropriate without fitting the more complex model. Applications of the proposed score test on several datasets are also illustrated.     

Finite-Sample Properties of the Maximum Likelihood Estimator for the Poisson Regression Model With Random Covariates

We examine the small-sample behaviour of the maximum likelihood estimator for the Poisson regression model with random covariates. Analytic expressions for the second-order bias and mean squared error are derived, and we undertake some numerical evaluations to illustrate these results for the single covariate case. The properties of the bias-adjusted maximum likelihood estimator are investigated in a Monte Carlo experiment. Correcting the estimator for its second-order bias is found to be effective in the cases considered, and we recommend its use when the Poisson regression model is estimated by maximum likelihood with small samples.