A weighted Poisson distribution and its application to cure rate models

The family of weighted Poisson distributions offers great flexibility in modeling discrete data due to its potential to capture over/under-dispersion by an appropriate selection of the weight function. In this paper, we introduce a flexible weighted Poisson distribution and further study its properties by using it in the context of cure rate modeling under a competing cause scenario. A special case of the new distribution is the COM-Poisson distribution which in turn encompasses the Bernoulli, Poisson, and geometric distributions; hence, many of the well-studied cure rate models may be seen as special cases of the proposed model. We focus on the estimation, through the maximum likelihood method, of the cured proportion and the properties of the failure time of the susceptibles/non cured individuals; a profile likelihood approach is also adopted for estimating the parameters of the weighted Poisson distribution. A Monte Carlo simulation study demonstrates the accuracy of the proposed inferential method. Finally, as an illustration, we fit the proposed model to a cutaneous melanoma data set.     

A new destructive Poisson odd log-logistic generalized half-normal cure rate model

We propose a new cure rate survival model by assuming that the initial number of competing causes of the event of interest follows a Poisson distribution and the time to event has the odd log-logistic generalized half-normal distribution. This survival model describes a realistic interpretation for the biological mechanism of the event of interest. We estimate the model parameters using maximum likelihood. For different sample sizes, various simulation scenarios are performed. We propose the diagnostics and residual analysis to verify the model assumptions. The potentiality of the new cure rate model is illustrated by means of a real data.     

On bootstrap testing inference in cure rate models

Survival models deal with the time until the occurrence of an event of interest. However, in some situations the event may not occur in part of the studied population. The fraction of the population that will never experience the event of interest is generally called cure rate. Models that consider this fact (cure rate models) have been extensively studied in the literature. Hypothesis testing on the parameters of these models can be performed based on likelihood ratio, gradient, score or Wald statistics. Critical values of these tests are obtained through approximations that are valid in large samples and may result in size distortion in small or moderate sample sizes. In this sense, this paper proposes bootstrap corrections to the four mentioned tests and bootstrap Bartlett correction for the likelihood ratio statistic in the Weibull promotion time model. Besides, we present an algorithm for bootstrap resampling when the data presents cure fraction and right censoring time (random and non-informative). Simulation studies are conducted to compare the finite sample performances of the corrected tests. The numerical evidence favours the corrected tests we propose. We also present an application in an actual data set.     

An EM type estimation procedure for the destructive exponentially weighted Poisson regression cure model under generalized gamma lifetime

In this paper, we assume the number of competing causes to follow an exponentially weighted Poisson distribution. By assuming the initial number of competing causes can undergo destruction and that the population of interest has a cure fraction, we develop the EM algorithm for the determination of the MLEs of the model parameters of such a general cure model. This model is more flexible than the promotion time cure model and also provides an interesting and realistic interpretation of the biological mechanism of the occurrence of an event of interest. Instead of assuming a particular parametric distribution for the lifetime, we assume the lifetime to belong to the wider class of generalized gamma distribution. This allows us to carry out a model discrimination to select a parsimonious lifetime distribution that provides the best fit to the data. Within the EM framework, a two-way profile likelihood approach is proposed to estimate the shape parameters. An extensive Monte Carlo simulation study is carried out to demonstrate the performance of the proposed estimation method. Model discrimination is carried out by means of the likelihood ratio test and information-based methods. Finally, a data on melanoma is analyzed for illustrative purpose.     

A class of correlated weighted Poisson processes

In this paper, we propose new classes of correlated Poisson processes and correlated weighted Poisson processes on the interval [0,1], which generalize the class of weighted Poisson processes defined by Balakrishnan and Kozubowski (2008), by incorporating a dependence structure between the standard uniform variables used in the construction. In this manner, we obtain another process that we refer to as correlated weighted Poisson process. Various properties of this process such as marginal and joint distributions, stationarity of the increments, moments, and the covariance function, are studied. The results are then illustrated through some examples, which include processes with length-biased Poisson, exponentially weighted Poisson, negative binomial, and COM-Poisson distributions.     

Likelihood inference based on EM algorithm for the destructive length-biased Poisson cure rate model with Weibull lifetime

In this article, we consider the destructive length-biased Poisson cure rate model, proposed by Rodrigues et al., that presents a realistic and interesting interpretation of the biological mechanism for the recurrence of tumor in a competing causes scenario. Assuming the lifetime to follow the Weibull distribution and censoring mechanism to be non-informative, the necessary steps of the EM algorithm for the determination of the MLEs of the model parameters are developed here based on right censored data. The standard errors of the MLEs are obtained by inverting the observed information matrix. A simulation study is then carried out to examine the method of inference developed here. Finally, the proposed methodology is illustrated with a real melanoma dataset.     

Mixture cure rate models with accelerated failures and nonparametric form of covariate effects

Two-component mixture cure rate model is popular in cure rate data analysis with the proportional hazards and accelerated failure time (AFT) models being the major competitors for modelling the latency component. [Wang, L., Du, P., and Liang, H. (2012), ‘Two-Component Mixture Cure Rate Model with Spline Estimated Nonparametric Components’, Biometrics, 68, 726–735] first proposed a nonparametric mixture cure rate model where the latency component assumes proportional hazards with nonparametric covariate effects in the relative risk. Here we consider a mixture cure rate model where the latency component assumes AFTs with nonparametric covariate effects in the acceleration factor. Besides the more direct physical interpretation than the proportional hazards, our model has an additional scalar parameter which adds more complication to the computational algorithm as well as the asymptotic theory. We develop a penalised EM algorithm for estimation together with confidence intervals derived from the Louis formula. Asymptotic convergence rates of the parameter estimates are established. Simulations and the application to a melanoma study shows the advantages of our new method.     

Semiparametric methods for survival data with measurement error under additive hazards cure rate models

Lifetime Data Analysis - It is well established that measurement error has drastically negative impact on data analysis. It can not only bias parameter estimates but may also cause loss of power...     

D-optimal designs for Poisson regression models

Most of the current research on optimal experimental designs for generalized linear models focuses on logistic regression models. In this paper, D-optimal designs for Poisson regression models are discussed. For the one-variable first-order Poisson regression model, it has been found that the D-optimal design, in terms of effective dose levels, is independent of the model parameters. However, it is not the case for more complicated models. We investigate how the D-optimal designs depend on the model parameters for the one-variable second-order model and two-variable interaction model. The performance of some “standard” designs that appeal to practitioners is also studied.     

Exponentially weighted moving average charts for correlated multivariate Poisson processes

In this article, we study exponentially weighted moving average (EWMA) control schemes to monitor the multivariate Poisson distribution with a general covariance structure, so that the practitioner can simultaneously monitor multiple correlated attribute processes more effectively. The statistical performance of the charts is assessed in terms of the run length properties and compared against other mainstream attribute control schemes. The application of the proposed methods to real-life and simulated datasets is demonstrated.     

Analysis of Poisson varying-coefficient models with autoregression

In the regression analysis of time series of event counts, it is of interest to account for serial dependence that is likely to be present among such data as well as a nonlinear interaction between the expected event counts and predictors as a function of some underlying variables. We thus develop a Poisson autoregressive varying-coefficient model, which introduces autocorrelation through a latent process and allows regression coefficients to nonparametrically vary as a function of the underlying variables. The nonparametric functions for varying regression coefficients are estimated with data-driven basis selection, thereby avoiding overfitting and adapting to curvature variation. An efficient posterior sampling scheme is devised to analyse the proposed model. The proposed methodology is illustrated using simulated data and daily homicide data in Cali, Colombia.     

Statistical inference concerning weighted Poisson rates under some natural order restrictions

A conditional Poisson model is proposed for situations where one wants to compare the rate parameters for several populations, adjusting for a 'size' parameter and the random e ects of the subjects. Sometimes the physical nature of the problem makes it logical to consider some order restrictions on the rate parameters. The rate parameters are estimated and tested for under such order restrictions. This investigation is motivated by a real-life example on butterfly behavior, and the estimation and tests of hypotheses are illustrated over this data set.     

Generalized log-gamma regression models with cure fraction

In this paper, the generalized log-gamma regression model is modified to allow the possibility that long-term survivors may be present in the data. This modification leads to a generalized log-gamma regression model with a cure rate, encompassing, as special cases, the log-exponential, log-Weibull and log-normal regression models with a cure rate typically used to model such data. The models attempt to simultaneously estimate the effects of explanatory variables on the timing acceleration/deceleration of a given event and the surviving fraction, that is, the proportion of the population for which the event never occurs. The normal curvatures of local influence are derived under some usual perturbation schemes and two martingale-type residuals are proposed to assess departures from the generalized log-gamma error assumption as well as to detect outlying observations. Finally, a data set from the medical area is analyzed.     

Bivariate regression models based on compound Poisson distribution

Based on a compound Poisson distribution, new bivariate regression models are introduced and studied. The parameters of the bivariate regression models are estimated by using the maximum likelihood method. Two applications on real datasets are presented to illustrate the models. The results show that these models are compatible to other bivariate Poisson models.     

A simulation comparison on spatial Poisson mixed models

The statistical methods for analyzing spatial count data have often been based on random fields so that a latent variable can be used to specify the spatial dependence. In this article, we introduce two frequentist approaches for estimating the parameters of model-based spatial count variables. The comparison has been carried out by a simulation study. The performance is also evaluated using a real dataset and also by the simulation study. The simulation results show that the maximum likelihood estimator appears to be with the better sampling properties.     

General mixed Poisson regression models with varying dispersion

A general class of mixed Poisson regression models is introduced. This class is based on a mixing between the Poisson distribution and a distribution belonging to the exponential family. With this, we unified some overdispersed models which have been studied separately, such as negative binomial and Poisson inverse gaussian models. We consider a regression structure for both the mean and dispersion parameters of the mixed Poisson models, thus extending, and in some cases correcting, some previous models considered in the literature. An expectation–maximization (EM) algorithm is proposed for estimation of the parameters and some diagnostic measures, based on the EM algorithm, are considered. We also obtain an explicit expression for the observed information matrix. An empirical illustration is presented in order to show the performance of our class of mixed Poisson models. This paper contains a Supplementary Material.     

Zero-inflated compound Poisson distributions in integer-valued GARCH models

In this paper we introduce a wide class of integer-valued stochastic processes that allows to take into consideration, simultaneously, relevant characteristics observed in count data namely zero inflation, overdispersion and conditional heteroscedasticity. This class includes, in particular, the compound Poisson, the zero-inflated Poisson and the zero-inflated negative binomial INGARCH models, recently proposed in literature. The main probabilistic analysis of this class of processes is here developed. Precisely, first- and second-order stationarity conditions are derived, the autocorrelation function is deduced and the strict stationarity is established in a large subclass. We also analyse in a particular model the existence of higher-order moments and deduce the explicit form for the first four cumulants, as well as its skewness and kurtosis.     

Bayesian inference for Poisson and multinomial log-linear models

Categorical data frequently arise in applications in the Social Sciences. In such applications, the class of log-linear models, based on either a Poisson or (product) multinomial response distribution, is a flexible model class for inference and prediction. In this paper we consider the Bayesian analysis of both Poisson and multinomial log-linear models. It is often convenient to model multinomial or product multinomial data as observations of independent Poisson variables. For multinomial data, Lindley (1964) [20] showed that this approach leads to valid Bayesian posterior inferences when the prior density for the Poisson cell means factorises in a particular way. We develop this result to provide a general framework for the analysis of multinomial or product multinomial data using a Poisson log-linear model. Valid finite population inferences are also available, which can be particularly important in modelling social data. We then focus particular attention on multivariate normal prior distributions for the log-linear model parameters. Here, an improper prior distribution for certain Poisson model parameters is required for valid multinomial analysis, and we derive conditions under which the resulting posterior distribution is proper. We also consider the construction of prior distributions across models, and for model parameters, when uncertainty exists about the appropriate form of the model. We present classes of Poisson and multinomial models, invariant under certain natural groups of permutations of the cells. We demonstrate that, if prior belief concerning the model parameters is also invariant, as is the case in a ‘reference’ analysis, then the choice of prior distribution is considerably restricted. The analysis of multivariate categorical data in the form of a contingency table is considered in detail. We illustrate the methods with two examples.