Applied Statistics

11Generalised Linear Models

11.1 Introduction

Like in analysis of variance (ANOVA) or regression analysis a generalised linear model (GLM) describes the relation between a random regressand (response variable) y and a vector x^T = (x₀, … , x_k) of regressor variables influencing it, and is a flexible generalisation of ordinary linear regression allowing for regressands that have error distribution models other than a normal one. The GLM generalises linear regression by writing the linear model to be related to the regressand via a link function of the corresponding exponential family and by allowing the magnitude of the variance of each measurement to be a function of its predicted value.

Possibly the first who introduced a GLM was Rasch (1960). The Rasch model is a psychometric model for analysing categorical data, such as answers to questions on a reading assessment or questionnaire responses, as a function of the trade‐off between (i) the respondent's abilities, attitudes or personality traits and (ii) the item difficulty. For example, we may use it to estimate a student's reading ability, or the extremity of a person's attitude to capital punishment from responses on a questionnaire. In addition to psychometrics and educational research, the Rasch model and its extensions are used in other areas, including the health profession and market research because of their general applicability. The mathematical theory underlying Rasch models is a special case of a GLM. Specifically, ...