What does 's' denote in describing a General Linear Model (GLM) and a note on Generalized Linear Models in SPSS?

The GLM terminology is described [http://www.fil.ion.ucl.ac.uk/~mgray/ here] in relation to using SPM.

Examples of GLMs include linear regressions and analysis of variance and are of form.

Y = XB + error
or, in words,
Response = Prediction + residual

s is, therefore, the residual standard deviation which, for example, corresponds to the square root of the mean square error term in an analysis of variance.

Note: The use of Mean Square Error is equivalent to the 'Deviance' Scale Parameter Method using the default linear model (for a continuous response) under the 'Estimation' tab in the Generalized Linear Model (under 'analyze' in SPSS). The default setting for the Scale Parameter method in the 'Generalized Linear Model' is actually 'Log-likelihood' which yields a statistic which has a critical value following a chi-square distribution. This statistic is not usually used directly for regression/anova models with continuous responses but is incorporated as a denominator in a F ratio. It is, however, used directly for categorical responses which is why we tend to quote chi-square values, rather than F values, when assessing the influence of predictor variables on group responses. The 'Scaled Parameter' represents the formula used for assessing the error sums of squares for the model and can take various forms which are all outputted by SPSS by default in the 'Goodness of Fit' box.

Some of these terms used by SPSS are explained further here:

Scaled Pearson = $$\sum_text{i} \frac{(\mbox{i-th residual}^text{2})}{\mbox{i-th predicted value}}$$ which is usually used for group outcome.

Log-likelihood Chi-square = 2 (difference in log-likelihoods with and without predictors) where Log-Likelihood = -n/2 ln(2 Pi s-squared) - n/2 which is usually used for group outcome.

Deviance = Pearson Chi-square = $$\sum_text{i} (\mbox{i-th residual}^text{2})$$ which is usually used in continuous outcome.