4 Model adequacy checking

Every test and interval of the last two sections rested on assumptions:

(i)

the relationship between \(y\) and the regressors is (approximately) linear, so that \(E(\mathbf {y}) = \mathbf {X}\boldsymbol \beta \) for the regressors we have used;

(ii)

the errors have mean zero and the same variance \(\sigma ^2\) (A1, A2);

(iii)

the errors are uncorrelated (A3);

(iv)

the errors are normally distributed (A4).

None of these can be confirmed by the summary statistics of a fit. A large \(R^2\) and a significant \(F\) statistic say that the model explains a lot of the variation in \(y\); they say nothing about whether the explanation has the right form, and they are perfectly compatible with a badly wrong model. Gross violations of the assumptions can make the fitted model unstable, and the \(t\), \(F\) and interval statements of the previous sections can then be seriously wrong.

The information about the errors is in the residuals. A residual is what the model failed to explain, so if the model is right the residuals should look like the errors: centred on zero, with no pattern, constant spread, roughly normal. This section develops the residuals and their scaled versions, the plots that reveal departures from the assumptions, the treatment of outlying observations, a measure of predictive performance, and a formal test of whether a straight line (or plane) is the right shape at all.

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