Murray Aitkin

Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm

Maximum likelihood estimation of item parameters in the marginal distribution, integrating over the distribution of ability, becomes practical when computing procedures based on an EM algorithm are used. By characterizing the ability distribution empirically, arbitrary assumptions about its form are avoided. The Em procedure is shown to apply to general item-response models lacking simple sufficient statistics for ability. This includes models with more than one latent dimension.

The Fitting of Exponential, Weibull and Extreme Value Distributions to Complex Censored Survival Data using GLIM

SUMMARY Regression models may be fitted to censored survival data by the use of exponential, Weibull and extreme value distributions in GLIM. Standard probability plotting procedures for uncensored data may be modified to allow for censoring. A simultaneous test procedure may be used to determine a minimal adequate regression model. The procedure is briefly illustrated on two sets of published cancer survival data. A RECENT paper by Kay (1977) surveys methods for the analysis of censored survival data. Kay considers the fitting of exponential and Weibull models, and Coxs distribution-free model, the assessment of the form of the survival distribution through residual plots, and the determination of relevant variables in the regression of the hazard function on explanatory variables. The purpose of the present paper is to describe the use of GLIM (Baker and Nelder, 1978) to fit exponential, Weibull or extreme value distributions, by expressing the likelihood in each case as a Poisson likelihood, with a log-linear model for the Poisson mean corresponding to the log- linear model for the hazard function. A simultaneous test procedure developed for complex cross-classifications may be applied to the reduction of the complex regression model to a parsimonious form. The procedure is illustrated on two-sample data published by Gehan (1965), and on complex data published by Prentice (1973).

Modelling variance heterogeneity in normal regression using GLIM

This paper describes and presents simple GLIM macros for the modelling of variance heterogeneity in normal regression analysis, using a log‐linear regression model for the variance. The procedure is illustrated with two examples.

Mixture Models, Outliers, and the EM Algorithm

Maximum likelihood (ML) methods are described for the identification of outliers in single sample or regression problems, based on mixture models. The EM algorithm provides a simple and easily programmed iterative solution for the ML estimates of the parameters in the models. The procedure is illustrated on three examples.

A general maximum likelihood analysis of overdispersion in generalized linear models

This paper presents an EM algorithm for maximum likelihood estimation in generalized linear models with overdispersion. The algorithm is initially derived as a form of Gaussian quadrature assuming a normal mixing distribution, but with only slight variation it can be used for a completely unknown mixing distribution, giving a straightforward method for the fully non-parametric ML estimation of this distribution. This is of value because the ML estimates of the GLM parameters may be sensitive to the specification of a parametric form for the mixing distribution. A listing of a GLIM4 algorithm for fitting the overdispersed binomial logit model is given in an appendix.A simple method is given for obtaining correct standard errors for parameter estimates when using the EM algorithm.Several examples are discussed.

Simultaneous Inference and the Choice of Variable Subsets in Multiple Regression

Simple methods are described for controlling the family error rate for simultaneous tests for the choice of subsets of predictor variables in multiple regression. The cases of fixed and random predictors are considered, and in the latter case a class of “adequate” regression equations is obtained, characterized by a lower bound on the sample multiple correlation coefficient.

A Note on the Regression Analysis of Censored Data

Maximum likelihood estimates of the parameters in linear models with censored normal responses may be simply obtained using the EM algorithm. The iterative computations required for the regression coefficients are identical to those described by Schmee and Hahn for least squares estimates, but those for the variance estimates are different. The biases of the two variance estimates are discussed.

A Reanalysis of the Stanford Heart Transplant Data

Abstract This article represents analyses of survival of patients in the Stanford Heart Transplantation Program. We model survival time as a function of patient covariates and transplant status, and compare the results obtained using various parametric representations for survival time, including the Weibull, lognormal, and piecewise exponential distributions. Pretransplant and posttransplant survival are considered separately, and the effect of transplantation on survival is examined by comparison of the separate hazard functions. Comparisons are made with previous analyses. Using the piecewise exponential models, we estimate a generally declining hazard before transplant; after transplant the hazard increases for about 60 days, then declines. The presence of heavy censoring before and after transplant means that many of the other parametric models with differing shapes for the hazard all give equally adequate fits to the data. Inferences about the effect of covariates are also relatively insensitive to ...

A Simultaneous Test Procedure for Contingency Table Models

SUMMARY A simultaneous test procedure is proposed for the choice of a parsimonious model for complex contingency tables, based on a similar procedure for the ANOVA of unbalanced cross-classifications. The procedure requires a fully hierarchical partitioning of the maximized log-likelihood, or deviance, and may be applied to log-linear models for counts, and to logistic or linear models for proportions. Several examples are given.

A hybrid EM/Gauss-Newton algorithm for maximum likelihood in mixture distributions

A faster alternative to the EM algorithm in finite mixture distributions is described, which alternates EM iterations with Gauss-Newton iterations using the observed information matrix. At the expense of modest additional analytical effort in obtaining the observed information, the hybrid algorithm reduces the computing time required and provides asymptotic standard errors at convergence. The algorithm is illustrated on the two-component normal mixture.