= Understanding the role of the intercept when fitting an ANOVA =
The default option (under Model) in SPSS UNIVARIATE (between subjects ANOVA) is to include the intercept and, indeed, the intercept should be included in ANOVA models. The intercept represents the overall (or grand) mean of the data and is usually a nuisance parameter in that it is not of interest in itself (testing only if the overall mean is zero) but is there to make sure our other model terms (such as group effects) test what they should be testing.
To see this consider the regression (GLM) which is being fitted. Suppose we have a single between subjects group factor, i, with two levels, whose means we wish to compare.
Then we may write the below:
Expected value of outcome = intercept + alpha i where i equals 0 if the observation is from group 1 or 0 if it is from group 2 and alpha is the group regression estimate.
It follows
group one sample mean = intercept + alpha
group two sample mean = intercept
so alpha = the difference in the two group sample means which is the group effect as we would expect.
Now let's drop the intercept term:
group one sample mean = alpha
group two sample mean = 0
so now alpha = the group one sample mean which corresponds to a one-sample t-test of whether the group one mean is equal to zero. Since alpha no longer compares the two group means it no longer represents the group 'effect'.