Prediction Performance of Correlated Error Procedures for Spatial Counts

Abstract

Few guidelines exist for the application of geostatistical methods to spatial counts and the prediction to unsampled areas is an important aspect of experimental field research. The prediction performances of kriging and a correlated errors Poisson model are compared through simulation. Counts with a known spatial covariance structure are generated in an investigation involving several factors: area size, overall mean, range of correlation, spatial covariance function, and the presence of trend. The correlated errors Poisson model generally gives superior prediction performance when an exponential covariance structure is used.     

A Simple Distribution for the Sum of Correlated,Exchangeable Binary Data

This article describes a generalization of the binomial distribution. The closed form probability function for the probability of k successes out of n correlated, exchangeable Bernoulli trials depends on the number of trials and its two parameters: the common success probability and the common correlation. The distribution is derived under the assumption that the common correlation between all pairs of Bernoulli trials remains unchanged conditional on successes in all completed trials. The distribution was developed to model bond defaults but may be suited to biostatistical applications involving clusters of binary data encountered in repeated measurements or toxicity studies of families of organisms. Maximum likelihood estimates for the parameters of the distribution are found for a set of binary data from a developmental toxicity study on litters of mice.     

Weighted Multiple Testing Correction for Correlated Binary Endpoints

Multiple binary endpoints often occur in clinical trials and are usually correlated. Many multiple testing adjustment methods have been proposed to control familywise type I error rates. However, most of them disregard the correlation among the endpoints, for example, the commonly used Bonferroni correction, Bonferroni fixed-sequence (BFS) procedure, and its extension, the alpha-exhaustive fallback (AEF). Extending BFS by taking into account correlations among endpoints, Huque and Alosh proposed a flexible fixed-sequence (FFS) testing method, but this FFS method faces computational difficulty when there are four or more endpoints and the power of the first hypothesis does not depend on the correlations among endpoints. In dealing with these issues, Xie proposed a weighted multiple testing correction (WMTC) for correlated continuous endpoints and showed that the proposed method can easily handle hundreds of endpoints by using the R package and has higher power for testing the first hypothesis compared with the FFS and AEF methods. Since WMTC depends on the joint distribution of the endpoints, it is not clear whether WMTC still keeps those advantages when correlated binary endpoints are used. In this article, we evaluated the statistical power of WMTC method for correlated binary endpoints in comparison with the FFS, the AEF, the prospective alpha allocation scheme (PAAS), and the weighted Holm-Bonferroni methods. Furthermore the WMTC method and others are illustrated on a real dataset examining the circumstance of homicide in New York City.     

Hierarchical Bayesian Models for the Estimation of Correlated Effects in Multilevel Data: A Simulation Study to Assess Model Performance

In this article, we aim at assessing hierarchical Bayesian modeling for the analysis of multiple exposures and highly correlated effects in a multilevel setting. We exploit an artificial data set to apply our method and show the gains in the final estimates of the crucial parameters. As a motivating example to simulate data, we consider a real prospective cohort study designed to investigate the association of dietary exposures with the occurrence of colon-rectum cancer in a multilevel framework, where, e.g., individuals have been enrolled from different countries or cities. We rely on the presence of some additional information suitable to mediate the final effects of the exposures and to be arranged in a level-2 regression to model similarities among the parameters of interest (e.g., data on the nutrient compositions for each dietary item).     

Pseudo-Likelihood Methodology for Hierarchical Count Data

Generalized Estimating Equations (GEE) are a widespread tool for modeling correlated data, based on properly formulating a marginal regression function, combined with working assumptions about the correlation function. Should interest be placed in addition on the correlation function, then, apart from second-order GEE, pseudo-likelihood (PL) also provides an attractive alternative, especially in its pairwise form, where the covariance between each pair of the response vector is modeled as well. An elegant PL approach is formulated in this paper, based on a flexible bivariate Poisson model. The performance of the PL-method is studied, relative to GEE, using simulations. Data on repeated counts of epileptic seizures in a two-arm clinical trial are analyzed. A macro has been developed by the authors and made available on their web pages.     

Simultaneous Confidence Intervals for the Mean of Multivariate Poisson Distribution: A Comparison

Haibing (2009) proposed a procedure for successive comparisons between ordered treatment effects in one-way layout and showed that the proposed procedure has greater power than the procedure proposed by Lee and Spurrier (1995). Critical constants required for the proposed procedure were estimated using Monte Carlo simulation and few values of the constants were tabulated which limit the applications of the proposed procedure. In this article, a numerical method, using recursive integration methodology, is discussed to compute the critical constants which work efficiently for a large number of treatments and extensive values of critical constants are tabulated for the use of practitioners. Power comparisons of Haibing's and Lee and Spurrier's procedure is also discussed.     

A Simple Derivation of Deletion Diagnostic Results for the General Linear Model with Correlated Errors

A general, simple and intuitive derivation is provided for diagnostics associated with the deletion of arbitrary subsets for the linear model with general covariance structure. These are seen to be most simply expressed, even for the well-studied case of independent and identically distributed data, in terms of a residual known variously as the conditional residual, the deletion prediction residual and the cross-validation residual. Particularly simple specializations arise when the subsets are of size 1 and of size 2, but the method is easy to apply for all subsets and to conditional deletions.     

Economic Design of Cumulative Sum Control Charts for Monitoring a Process with Correlated Samples

The underlying assumption for the design of control charts is the measurements within a sample are independently distributed. However, there are many situations where the uncorrelation assumption may be unacceptable in practice. In this paper, the economic design of cumulative sum (CUSUM) control chart for correlated data within a sample is developed. The genetic algorithm is applied to find the optimal design parameters of the CUSUM control chart by minimizing the cost function. An illustrative example is given. A sensitivity analysis is then conducted to evaluate the effects of cost parameters, process parameters, and correlation coefficient on the economic design.     

A control chart for multivariate Poisson distribution using repetitive sampling

Control charts using repetitive group sampling have attracted a great deal of attention during the last few years. In the present article, we attempt to develop a control chart for the multivariate Poisson distribution using the repetitive group sampling scheme. In the proposed control chart, the monitoring statistic from the multivariate Poisson distribution has been used for the quick detection of the deteriorated process to avoid losses. The control coefficients have been estimated using the specified in-control average run lengths. The procedure of the proposed control chart has been explained by using the real-world example and a simulated data set. It has been observed that the proposed control chart is an efficient development for the quick detection of the nonrandom change in the manufacturing process.     

Testing for a Unit Root in a Time Series With a Changing Mean: Corrections and Extensions

This article explores the relation between nonexponential waiting times between events and the distribution of the number of events in a fixed time interval. It is shown that within this framework the frequently observed phenomenon of overdispersion—that is, a variance that exceeds the mean—is caused by a decreasing hazard function of the waiting times, whereas an increasing hazard function leads to underdispersion. Using the assumption of iid gamma-distributed waiting times, a new count-data model is derived. Its use is illustrated in two applications, the number of births and the number of doctor consultations.     

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.     

A new generalization of the gamma distribution with application to negatively skewed survival data

We propose a three-parameter distribution referred to as the reflected- shifted-truncated gamma (RSTG) distribution to model negatively skewed data. Various properties of the proposed distribution are derived. The estimation of the model parameters is approached by maximum likelihood methods and the observed information matrix is derived. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation for both small and large samples. Using information theoretic criteria, we compare the RSTG distribution to the exponential, generalized F, generalized gamma, Gompertz, log-logistic, lognormal, Rayleigh, and Weibull distributions in three negatively skewed real datasets.     

A bivariate Sarmanov regression model for count data with generalised Poisson marginals

We present a bivariate regression model for count data that allows for positive as well as negative correlation of the response variables. The covariance structure is based on the Sarmanov distribution and consists of a product of generalised Poisson marginals and a factor that depends on particular functions of the response variables. The closed form of the probability function is derived by means of the moment-generating function. The model is applied to a large real dataset on health care demand. Its performance is compared with alternative models presented in the literature. We find that our model is significantly better than or at least equivalent to the benchmark models. It gives insights into influences on the variance of the response variables.     

The REBMIX Algorithm for the Multivariate Finite Mixture Estimation

The article extends the REBMIX to multivariate data. Random variables may follow normal, lognormal, or Weibull parametric families and should be independent within components. The initial weights and component parameters are not required. Preprocessing of observations folows the histogram, Parzen window, or k-nearest neighbor approach. The number of components, weights, and component parameters are gained iteratively by using information measures of the distance, such as the total of positive relative deviations and the information criterion. The number of classes or the number of the nearest neighbors can be optimized, as well. The REBMIX software is available on http://www.fs.uni-lj.si/lavek.     

A Bayesian model for ranking hazardous road sites

Summary.  Road safety has recently become a major concern in most modern societies. The identification of sites that are more dangerous than others (black spots) can help in better scheduling road safety policies. This paper proposes a methodology for ranking sites according to their level of hazard. The model is innovative in at least two respects. Firstly, it makes use of all relevant information per accident location, including the total number of accidents and the number of fatalities, as well as the number of slight and serious injuries. Secondly, the model includes the use of a cost function to rank the sites with respect to their total expected cost to society. Bayesian estimation for the model via a Markov chain Monte Carlo approach is proposed. Accident data from 519 intersections in Leuven (Belgium) are used to illustrate the methodology proposed. Furthermore, different cost functions are used to show the effect of the proposed method on the use of different costs per type of injury.     

Simultaneous Confidence Intervals for the Ratios of Marginal Means of a Multivariate Poisson Distribution

This article computes simultaneous confidence intervals for the ratios of marginal means of a multivariate Poisson distribution. For this, we propose a lognormal approximation technique and a bootstrap method. We demonstrate advantages of the proposed methods over existing ones through a simulation study. To illustrate their applicability to real-world problems, we apply the proposed methods to US data on infectious diseases.     

A Latent Process Model for Temporal Extremes

This paper presents a hierarchical approach to modelling extremes of a stationary time series. The procedure comprises two stages. In the first stage, exceedances over a high threshold are modelled through a generalized Pareto distribution, which is represented as a mixture of an exponential variable with a Gamma distributed rate parameter. In the second stage, a latent Gamma process is embedded inside the exponential distribution in order to induce temporal dependence among exceedances. Unlike other hierarchical extreme‐value models, this version has marginal distributions that belong to the generalized Pareto family, so that the classical extreme‐value paradigm is respected. In addition, analytical developments show that different choices of the underlying Gamma process can lead to different degrees of temporal dependence of extremes, including asymptotic independence. The model is tested through a simulation study in a Markov chain setting and used for the analysis of two datasets, one environmental and one financial. In both cases, a good flexibility in capturing different types of tail behaviour is obtained.     

Robust Poisson likelihood estimation for frailty Cox models: A simulation study

Frailty models can be fit as mixed-effects Poisson models after transforming time-to-event data to the Poisson model framework. We assess, through simulations, the robustness of Poisson likelihood estimation for Cox proportional hazards models with log-normal frailties under misspecified frailty distribution. The log-gamma and Laplace distributions were used as true distributions for frailties on a natural log scale. Factors such as the magnitude of heterogeneity, censoring rate, number and sizes of groups were explored. In the simulations, the Poisson modeling approach that assumes log-normally distributed frailties provided accurate estimates of within- and between-group fixed effects even under a misspecified frailty distribution. Non-robust estimation of variance components was observed in the situations of substantial heterogeneity, large event rates, or high data dimensions.     

Discrete distributions in the extended FGM family: The p.g.f. approach

In this article the probability generating functions of the extended Farlie–Gumbel–Morgenstern family for discrete distributions are derived. Using the probability generating function approach various properties are examined, the expressions for probabilities, moments, and the form of the conditional distributions are obtained. Bivariate version of the geometric and Poisson distributions are used as illustrative examples. Their covariance structure and estimation of parameters for a data set are briefly discussed. A new copula is also introduced.     

Bayesian methods for generalized linear models with covariates missing at random

The authors propose methods for Bayesian inference for generalized linear models with missing covariate data. They specify a parametric distribution for the covariates that is written as a sequence of one‐dimensional conditional distributions. They propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. They examine the properties of the proposed prior and resulting posterior distributions. They also present a Bayesian criterion for comparing various models, and a calibration is derived for it. A detailed simulation is conducted and two real data sets are examined to demonstrate the methodology.