Analysis of covariance and standardization as instances of prediction

In this paper, prediction provides the basis for unifying the procedures of covariance adjustment and standardization. Analysis of covariance is a method of forming predictions from a linear model; it is used when qualitative effects are to be studied and the effects of continuous variables are to be adjusted for. An essential feature is the division into effects of interest and effects for which adjustment is required. Covariates may also be qualitative: as such, they are used implicitly in experimental designs with blocks, where treatment effects are adjusted for the effect of blocks. The technique of standardization is well-known in epidemiology and demography as a method of adjusting explicitly for qualitative effects. The same division of effects applies when an analysis that uses generalized linear models is summarized. Two distinct types of prediction, which give identical results in classical linear models, are available: prediction may be conditional on a fixed value of a covariate, or marginal on a distribution of values such as the distribution in the set of data being analysed. Prediction methods are illustrated by the analysis of a table of proportions by use of a logit model.