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Let us continue with the fat rat example, but now suppose that how insulin level depends on size varies with sex. An appropriate model is then µi = α + βj + γk + δjk if rat i is rat size level j and sex k, where the δjk terms are the parameters for the interaction of rat size and sex. Writing this model out in full it is clear that it is spectacularly unidentifiable: µ1 µ2 µ3 µ4 µ5 µ6 µ7 µ8 µ9 = 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 1 0 1 1 0 0 1 1 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 α β0 β1 β2 γ0 γ1 δ00 δ01 δ10 δ11 δ20 δ21 .
These results are of largely historical and theoretical interest: they should not be used for computational purposes, and derivation of the distributional results is much more difficult if one starts from these formulae. 8 The Gauss Markov Theorem: what’s special about least squares? How good are least squares estimators? In particular, might it be possible to find better estimators, in the sense of having lower variance while still being unbiased? ¶ A few programs still fit models by solution of XT Xβ ˆ = XT y, but this is less computationally stable than the rotation method described here, although it is a bit faster.
The problem with r 2 is that it always increases when a new predictor variable is added to the model, no-matter how useless that variable is for prediction. Part of the reason for this is that the variance estimates used to calculate r 2 are biased in a way that tends to inflate r 2 . If unbiased estimators are used we get the adjusted r 2 2 = 1− radj ˆ2i /(n − p) . (yi − y¯)2 /(n − 1) 2 A high value of radj indicates that the model is doing well at explaining the variability in the response variable.