By P. McCullagh, J. A. Nelder (auth.)
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Extra resources for Generalized Linear Models
Consider, for example, such a factor A together with a continuous covariate X and model 1Ji = (Xi +ßx. SYST'EMATIC COMPONENT (LINEAR PREDICTOR) 39 Here the slope /l is assumed independent of the grouping factor but the intercept varies with it. Factors, as we shall call qualitative covariates, occur in many contexts : in designed experiments, for instance, factors may define the block number in which a given experimental unit (plot) lies, or may constitute a treatment factor such as variety in a variety trial; in a survey, factors might include region, sex, mari tal status, and so on.
Note that we are restricting ourselves to the subclass oflinear models where (a) there is only one error component and (b) the errors are 35 36 CONTINUOUS DATA WITH CONSTANT VARIANCE assumed independent. We now consider in more detail the random and systematic components of the model. 2 Error structure The error component of classical linear models is defined by the Normal (or Gaussian) distribution. 2) so that the Ys have equal variances and are uncorrelated. The assumption ofNormality, although important as the basis of an exact small-sample theory, need not be relied upon in large samples: the centrallimit theorem offers protection from all but the most extreme distributional deviations but there is some loss of efficiency (Cox and Hinkley, 1968).
The distribution is symmetrical with mode, mean and median all at p, and 0" (the standard deviation) is the horizontal distance between the mean and the point of inflection. The log-likelihood function for a single observation with known variance is a parabola, with a maximum at y and constant second derivative -1/0"2. The Normal distribution is relevant primarily to measurements of continuous quantities, though it is sometimes used as an approximation for discrete measurements. It is frequently used to model data, such as weights, which though continuous are essentially positive, although the distribution itself covers the whole realline.