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Über dieses Buch

This volume of Lecture Notes in Statistics consists of the published proceedings of the first international conference to be held on the topic of generalised linear models. This conference was held from 13 - 15 September 1982 at the Polytechnic of North London and marked an important stage in the development and expansion of the GLIM system. The range of the new system, tentatively named Prism, is here outlined by Bob Baker. Further sections of the volume are devoted to more detailed descriptions of the new facilities, including information on the two different numerical methods now available. Most of the data analyses in this volume are carried out using the GLIM system but this is, of course, not necessary. There are other ways of analysing generalised linear models and Peter Green here discusses the many attractive features of APL, including its ability to analyse generalised linear models. Later sections of the volume cover other invited and contributed papers on the theory and application of generalised linear models. Included amongst these is a paper by Murray Aitkin, proposing a unified approach to statistical modelling through direct likelihood inference, and a paper by Daryl Pregibon showing how GLIM can be programmed to carry out score tests. A paper by Joe Whittaker extends the recent discussion of the relationship between conditional independence and log-linear models and John Hinde considers the introduction of an independent random variable into a linear model to allow for unexplained variation in Poisson data.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
It is now ten years since the late Robert Wedderburn and I published our paper defining generalised linear models, and the papers presented in this volume give a good indication of how the idea has been taken up and extended since then.
J. A. Nelder

Prism — An Overview

Summary
The statistical program Prism is introduced as a development of 6LIM-3. The features of GLIM-3 that made it simple and convenient for analysing generalised linear models have been consolidated in the new program. Some new language features of Prism are outlined and the data structures now supported by the System are described. Finally a brief description of the data manipulation and program control facilities of Prism is given.
R. J. Baker

GLIM4 — The New Facilities

Summary
GLIM is now one module of the new PRISM system. The opportunity has been taken to completely rewrite the program, improve the syntax and extend the range of options. These changes are described and examples of their use given. There are some incompatibilities with GLIM3.
M. R. B. Clarke

Array Manipulation in Prism

Summary
The TABULATE directive for forming tables of summary statistics and directives for performing operations on multi-dimensional arrays give Prism a powerful new facility for data manipulation.
M. Green

The Graph Module

Summary
The main features of the GRAPH module of Prism are described. Details of the main directives are given together with some examples of their use. The relationship between GRAPH and GKS is discussed.
M. Slater

AOV: The Prism Module for Analysing Designed Experiments

Summary
A brief description is given of the scope and syntax of AOV, together with an example illustrating the sort of output that can be obtained.
R. W. Payne

The APL Alternative

Summary
APL is an interactive computing language. In this paper we discuss its suitability for statistical analysis and experimentation, and in particular for maximum likelihood estimation using iteratively reweighted least squares.
P. J. Green

Direct Likelihood Inference

Summary
A unified approach to statistical modelling is proposed through direct likelihood inference. An example is given for the choice of a regression model for a complex cross-classification.
Murray Aitkin

Score Tests in GLIM with Applications

Abstract
The most common method of hypothesis testing in GLIM is the likelihood ratio method. However, in certain biostatistical application areas, score tests are more commonly used. Mantel-Haenszel chi-squared tests provide good examples. In other cases where a large number of competing models are being entertained, score tests may also be preferable for economy in computing.
We show that score tests can be computed in GLIM with the same ease as likelihood ratio tests. This allows flexibility to users which was not otherwise available. The method is applied to several examples in order to illustrate its usefulness and generality.
Daryl Pregibon

Glim Syntax and Simultaneous Tests for Graphical Log Linear Models

Summary
Within the class of hierarchical log linear models for contingency tables, conditional independence models have a special place and as they are well represented by their independence graphs they are known as graphical models. Standard GLIM model formulae are extended to give a simple representation of these models and rules are given to translate formulae between the standard and extended language. The deviances for all conditional independence models can be computed from the deviances of the elementary graphical models. These computations are employed in a simultaneous test procedure for selecting a parsimonious model from the class of all graphical models. This procedure is illustrated on a contingency table from the sociological literature classified by four response factors.
Joe Whittaker

Compound Poisson Regression Models

Summary
Count data are easily modelled in GLIM using the Poisson distribution. However, in modelling such data the counts are often aggregated over one or more factors, or important explanatory variables are unavailable and as a result the fit obtained is often poor. This paper examines a method of allowing for this unexplained variation by introducing an independent random variable into the linear model for the Poisson mean, giving a compound Poisson model for the observed data. By assuming a known form for the distribution of this random variable, in particular the normal distribution, and using a combination of numerical integration, the EM algorithm and iteratively reweighted least squares, maximum likelihood estimates can be obtained for the parameters. Macros for implementing this technique are presented and its use is illustrated with several examples.
John Hinde

Some Aspects of Parametric Link Functions

Summary
In generalised linear models the mean of each observation is related to its linear predictor via the link function. When the link function is known exactly the maximum likelihood estimators of the parameters in the linear predictor can be found by an iterative weighted least squares algorithm (Neider and Wedderburn.1972). We show how the algorithm is modified to allow for the estimation of the parameters in models fitted with parametric link functions. Two illustrative examples are given and some wider aspects of the method discussed.
A. Scallan

A Model for a Binary Response with Misclassifications

Summary
Observations on a binary response may be subject to misclassification. A linear logit model for the true binary response is specified and estimated jointly with the error probabilities for the two types of misclassification. The model is illustrated using a subset of the well-known coal-miners data. The problem is formulated as an incomplete data problem and the EM-algorithm and GLIM is used for estimation. The connection to latent structure models is discussed.
Anders Ekholm, Juni Palmgren

Loglinear Models with Composite Link Functions in Genetics

Summary
Loglinear models with linear composite link functions constitute a very flexible class of models which can accommodate a wide range of estimation problems for probability models in genetics. It is shown how GLIM-3 may be used to obtain estimates of gene frequencies, inbreeding coefficients and other parameters and their asymptotic covari-ances and standard errors. Examples are taken from medical genetics and evolutionary biology.
R. Burn

Use of the Complementary Log-Log Function to Describe Dose-Response Relationships in Insecticide Evaluation Field Trials

Summary
Generalised linear modelling techniques are used to provide a standard method for the analysis of data from insecticide trials. The complementary log-log function provides very well-fitting dose-response curves (details to he published elsewhere). The offset facility allows a constant term to be incorporated in the model to adjust for the base level of insect attack which varies from experiment to experiment.
Each insecticide dose was replicated three times. By considering dose both as a variable and a factor the residual deviance can be partitioned into a “lack of fit” and a “pure error” term thus overcoming the problems of interpreting the absolute residual deviance which is related to the parameter value. The partitioned deviances can then be used to assess the goodness of fit of the line.
Kathleen Phelps

GLIM for Preference

Summary
Many types of multiple comparisons data can be described by multiplicative models, consequently GLIM can be used to fit log-linear models to such data, and to produce not only estimates of the parameters of interest, but also standard errors and tests of significance. Moreover as the data consists of counts the adequacy of the model can be tested in an objective way. Three examples from the literature are analysed, fitting the basic Bradley-Terry model and a modified Bradley-Terry model with ties and order effects. The parametrisation used in the case of ties is claimed to be more convenient than that of Fienberg. A test of linearity is also included.
C. D. Sinclair

A GLM FOR Estimating Probabilities in Retrospective Case-Control Studies

Summary
A simple GLM for estimating probabilities of disease incidence across subgroups of a population from retrospective case-control studies is proposed. The method is illustrated by an application of GLIM to a cross classification table from a study of cot deaths in Lambeth.
M. Slater, R. D. Wiggins

Backmatter

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