Robustness to non-normality of regression tests

A number of statistical procedures involve the comparison of a ‘regression’ mean square with a ‘residual’ mean square using the normal-theory F distribution for reference. The use of the procedure for the analysis of actual data implies that the distribution of the mean-square ratio is insensitive to moderate non-normality. Many investigators, in particular Pearson (1931), Geary (1947), Gayen (1950), have considered the sensitivity of this distribution to parent non-normality for important special cases and a very general investigation was carried out by David & Johnson (1951a, b). The principal object of this paper is to demonstrate the overriding influence which the numerical values of the regression variables have in deciding sensitivity to non-normality and to demonstrate the essential nature of this dependency. We first obtain a simple approximation to the distribution of the regression F statistic in the non-normal case. This shows that it is ‘the extent of non-normality’ in the regression variables (the x 's), which determines sensitivity to non-normality in the observations (the y 's). Our results are illustrated for certain familiar special cases. In particular the well-known robustness of the analysis of variance test to compare means of equal-sized groups and the notorious lack of robustness of the test to compare two estimates of variance from independent samples are discussed in this context. We finally show that it is possible to choose the regression variables so that, to the order of approximation we employ, non-normality in the y 's is without effect on the distribution of the test statistic. Our results demonstrate the effect which the choice of experimental design has in deciding robustness to non-normality.