Residuals with One Degree of Freedom

It is recommended practice to analyze the residuals from any model-fitting procedure.However, one occasionally finds a linear model fitted with only one residual degree of freedom (df), an exercise that cannot, in general, be recommended.We point out that doing standard residual analysis on such residuals can be misleading, as both authors discovered on different occasions.For such residuals, the standardized residuals are always ± 1 or indeterminate; the correlation between any two residuals that are not identically zero, independent of Y, is ± 1; and, in repeated experiments with the same X matrix, the same residual will always be the biggest, and always by the same relative amount.We use data analyzed by R.A. Fisher for illustration.