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Prediction for small areas introduces an important application of sample survey inference, where domain sample sizes are too small to allow domain-specific inference. Typically, these domains are spatially defined, and so are referred to as small areas. Direct and indirect estimation for small areas is discussed, with the latter based on characterising the distribution of the survey variable via a linear mixed model. The empirical best linear unbiased predictor is developed, as are estimates of its mean squared error. An alternative approach, which conditions on differences between the areas, is used to motivate a domain-type linear estimator, the model-based direct estimator, as well as an estimator of its conditional mean squared error. The extension of the indirect approach to where the survey variable can be modelled via a generalised linear mixed model is sketched. The chapter concludes with a discussion of recent developments in small area inference

This chapter describes the simplest possible model for a finite population: the homogeneous population model. It is appropriate when there is no auxiliary information that can distinguish between different population units. The homogeneous population model assumes equal expected value and variance for the variable of interest for all population units. Values from different units are assumed to be independent although this is relaxed in the last section of the chapter. The empirical best and best linear unbiased predictor of a population total are derived under the model. Inference, sample design and sample size calculation are also discussed. The most appropriate design for this kind of population is usually simple random sampling without replacement. The urn model (also known as the hypergeometric model), a special case of the homogeneous population model, is also discussed.

This chapter provides an overview of parametrizing quantitative covariate effects, that is, describing how variables (covariates) influence the outcome variable, be that disease odds, rates, or some quantitative measurement. It begins by differentiating between predictions and contrasts. When reporting rate ratios between two groups or the odds ratio of a disease associated with a certain difference in exposure, one is using contrasts of the outcome variable between different values of a covariate to describe the effect. Thus, one uses ratios or differences. But when reporting the mortality rate in, say, 60-year-old males, one is making a prediction of the outcome. This requires a set of values for all covariates in a model. When describing the effects of covariates by a model, one normally uses a linear predictor. The chapter then discusses the prediction of a single rate; categorical variables; the task of modelling the effect of quantitative variables; quantitative predictors; and quantitative interactions.

This chapter revisits a regression analysis to explore the normal least squares assumption of approximately equal variance. It also considers some of the data transformations that can be used to achieve this. A linear regression of transformed data is compared with the generalized linear model equivalent that avoids transformation by using a link function and non-normal distributions. Generalized linear models based on maximum likelihood use a link function to model the mean (in this case a square-root link) and a variance function to model the variability (in this case the gamma distribution where the variance increases as the square of the mean). The Box–Cox family of transformations is explained in detail.