Analysis of Rates Using a Generalized Poisson Regression Model

In this paper a generalization of the Poisson regression model indexed by a shape parameter is proposed for the analysis of life table and follow-up data with concomitant variables. The model is suitable for analysis of extra-Poisson variation data. The model is used to fit the survival data given in Holford (1980). The model parameters, the hazard and survival functions are estimated by the method of maximum likelihood. The results obtained from this study seem to be comparable to those obtained by Chen (1988). Approximate tests of the dispersion and goodness-of-fit of the data to the model are also discussed.     

Extended Poisson Process Modelling and Analysis of Count Data

It is shown that any discrete distribution with non-negative support has a representation in terms of an extended Poisson process (or pure birth process). A particular extension of the simple Poisson process is proposed: one that admits a variety of distributions; the equations for such processes may be readily solved numerically. An analytical approximation for the solution is given, leading to approximate mean-variance relationships. The resulting distributions are then applied to analyses of some biological data-sets.     

Symmetric Bernoulli Distributions and Generalized Binomial Distributions

The generalized binomial distribution is defined as the distribution of a sum of symmetrically distributed Bernoulli random variates. Several two-parameter families of generalized binomial distributions have received attention in the literature, including the Polya urn model, the correlated binomial model and the latent variable model. Some properties and limitations of the three distributions are described. An algorithm for maximum likelihood estimation for two-parameter generalized binomial distributions is proposed. The Polya urn model and the latent variable model were found to provide good fits to sub-binomial data given by Parkes. An extension of the latent variable model to incorporate heterogeneous response probabilities is discussed.     

Bayesian Estimation in a Generalized Negative Binomial Distribution

A generalized negative binomial (GNB) distribution was introduced by JAIN and CONSUL (1971) and was modified by NELSON (1975). The probability function of the distribution is defined by the function p(x; m, β, θ)= θ^x (1 - θ)^m+βx—x for x=0, 1, …, and zero otherwise, where m>0, 0<θ<1 and β=0 or 1≦β<θ^?1. The Bayes estimators for a number of parametric functions of θ when m and β are known are derived. The prior information on θ may be given by a beta distribution, B(a, b), to which no subjective significance is attached. It has been illustrated that the parameters in the prior distribution can be assigned by a computer. Comparisons are made of the Bayes estimate of P(X=k) to the corresponding ML estimate and the MVU estimate for any given sample to the order n^?1 for different values of k..     

Seemingly Unrelated Generalized Linear Models

By using deviance standardized residuals, the seemingly unrelated regression estimation procedure is extended to generalized linear models, and fitted by an iterative procedure. The matrix of cross products of standardized residuals is asymptotically multivariate normal, and can be used for further multivariate analyses and for hypothesis testing.     

Two Characteristic Properties of the Generalized Negative Binomial Distribution

Infinite divisibility condition and a relation between the mean and variance are used to characterize the generalized negative binomial distribution.     

A Rapid and Efficient Estimation Procedure for the Negative Binomial Distribution

This paper presents a new noniterative procedure for estimating the parameters of a negative binomial distribution. The procedure uses the first moment equation and an equation based on the weighted sample mean, with weights ω_x∝ α^z. The selection of a value for α is examined. A simulation study has been carried out and also the method has been applied to the 35 data sets analysed by Martin and Katti (1965, Biometrics) in order to compare it with the method of moments and with the method of maximum likelihood (ML). We conclude that the new procedure has greater relative efficiency than the method of moments; it gives estimates which are consistently close to ML and are easy to calculate.     

Point and Interval Estimation of the Population Size Using a Zero‐Truncated Negative Binomial Regression Model

This paper presents the zero‐truncated negative binomial regression model to estimate the population size in the presence of a single registration file. The model is an alternative to the zero‐truncated Poisson regression model and it may be useful if the data are overdispersed due to unobserved heterogeneity. Horvitz–Thompson point and interval estimates for the population size are derived, and the performance of these estimators is evaluated in a simulation study. To illustrate the model, the size of the population of opiate users in the city of Rotterdam is estimated. In comparison to the Poisson model, the zero‐truncated negative binomial regression model fits these data better and yields a substantially higher population size estimate. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

生长曲线参数估计的一种新方法-优化回归组合法

在现有文献研究的基础上,对生长曲线参数估计问题又作了进一步研究,给出了生长曲线参数估计的一种新方法优化回归组合法,该方法创造性地将最优化方法与回归方法结合在一起,利用最优化理论中的区间搜索和一维搜索,可以得到一系列c^＊值,利用回归方法可求得与其相对应的一系列a和b的值．当c取最优值c时,a和b便得到最优值a^＊和b^＊经示例计算表明,这种参数估计法具有较高的精度,     

Comparison and Evaluation of Three Methods of Estimation of the Johnson Sb Distribution

Thre methods of estimating the parameters of the Johnson S₆ distribution were tested by simulation. The maximum likelihood method, the method based on percentiles of a sample and the method based on moments of a transformed random variable were taken into consideration. Many sets of samples were generated differing in sizes and in the actual values of parameters, whereupon the parameters were estimated by the three methods. It was proved that if the sample is small or the skewness of the distribution is considerable, the maximum likelihood estimates can assume preposterous values. The method based on moments is recommended due to its simplicity and to the fact that the estimates, though usually biased, never assume absurd values.     

利用次序统计量估计种群的平均寿命

本文讨论威布尔分布Ｗ（α,λ）中次序统计量的分布;建立了用次序统计量对威布尔分布进行参数估计,进而估计种群平均寿命和方差的方法,该方法可广泛用于可靠性统计产品平均寿命和方差的估计。     

Comparing Stochastically Restricted Linear Estimators in a Regression Model

Conditions for superiority of the minimum dispersion estimator over another with respect to the covariance matrix are derived when the vector parameter of a regression model is subject to competing stochastic restrictions. The restrictions may also consist both of a deterministic part and a stochastic part.     

Generalized nonlinear models for pharmacokinetic data

Phase I trials to study the pharmacokinetic properties of a new drug generally involve a restricted number of healthy volunteers. Because of the nature of the group involved in such studies, the appropriate distributional assumptions are not always obvious. These model assumptions include the actual distribution but also the ways in which the dispersion of responses is allowed to vary over time and the fact that small concentrations of a substance are not easily detectable and hence are left censored. We propose that a reasonably wide class of generalized nonlinear models allowing for left censoring be considered now that this is feasible with current computer power and sophisticated statistical packages. These modelling strategies are applied to a Phase I study of the drug flosequinan and its metabolite. This drug was developed for the treatment of heart failure. Because the metabolite also exhibits an active pharmacologic effect, study of both the parent drug and the metabolite is of interest.     

Parameter estimation for the distribution of single cell lag times

In Quantitative Microbial Risk Assessment, it is vital to understand how lag times of individual cells are distributed over a bacterial population. Such identified distributions can be used to predict the time by which, in a growth-supporting environment, a few pathogenic cells can multiply to a poisoning concentration level.We model the lag time of a single cell, inoculated into a new environment, by the delay of the growth function characterizing the generated subpopulation. We introduce an easy-to-implement procedure, based on the method of moments, to estimate the parameters of the distribution of single cell lag times. The advantage of the method is especially apparent for cases where the initial number of cells is small and random, and the culture is detectable only in the exponential growth phase.     

An Estimation Procedure for the Joint Distribution of Spatial Direction and Thickness of Flat Bodies Using Vertical Sections Part I: Theoretical Considerations

In soil micromorphology fissures are considered in vertical sections. To get information about the properties of the soil the joint distribution of spatial direction and width is of interest. The fissures are mathematically generalized to flat bodies which form a stationary weighted surface process with the weight “thickness”. Because of stationarity a joint distribution of spatial direction and thickness exists in a “typical point” of the surface process. A suitable parametric family of distributions is assumed. The corresponding parameters can be estimated from measurements on the vertical sections. But on the sections only the visible thickness and the visible angle of a fissure can be measured. Therefore the joint distribution of these variables is expressed by the joint spatial distribution of spatial direction and thickness. This derived distribution depends on the same parameters. The Chi-Square method is proposed for the parameter estimation. The estimation procedure is demonstrated using the Bingham-Mardia distribution for the direction and the lognormal distribution for the thickness and by defining a way to correlate the mean thickness and the direction.     

On Generalized Log-Logistic Model for Censored Survival Data

In the analysis of survival data with parametric models, it is well known that the Weibull model is not suitable for modeling cases where the hazard rate is non-monotonic. For such cases, log-logistic model is frequently used. However, due to the symmetric property of the log-logistic model, it may be poor for the cases where the hazard rate is skewed or heavily tailed. In this paper, we suggest a generalization of the log-logistic model by introducing a shape parameter. This generalized model is then applied to fit the lung cancer data of Prentice (1973). The results seem to improve over those obtained by using the log-logistic model.     

A Dynamic Logistic Regression Model for Multiple Spell Failure Time Data

The paper deals with discrete-time regression models to analyze multistate-multiepisode failure time data. The covariate process may include fixed and external as well as internal time dependent covariates. The effects of the covariates may differ among different kinds of failures and among successive episodes. A dynamic form of the logistic regression model is investigated and maximum likelihood estimation of the regression coefficients is discussed. In the last section we give an application of the model to the analysis of survival time after breast cancer operation.     

Simulated Sampling Strategies for Nematodes Distributed According to a Negative Binomial Model

A FORTRAN computer program was developed to simulate nematode soil sampling strategies consisting of various numbers of samples per field, with each sample consisting of various numbers of soil cores. The program assumes that the nematode species involved fit a negative binomial distribution. Required input data are estimates of the mean and k values, the number of samples per field and cores per sample in the strategy to be investigated, and the number of times the simulation is to be replicated. Output consists of simulated values of the relative deviation from the mean and standard error to mean ratio, both averaged over all replications. The program was used to compare 150 simulated sampling strategies for Meloidogyne incognita, involving all combinations of two mean values (2.0 and 10.0 la.rvae/10 cm³ soil), three k values (1.35, 0.544, and 0.294), five different numbers of samples per field (1, 2, 4. 10, 20), and five different numbers of cores per sample (1, 2, 4, 10, 20). Simulations resulting from different mean values were similar, but best results were obtained with higher k values and 20 cores per sample. Relatively few 20-core samples were needed to obtain average deviations from the mean of 20-25%.