Printed in Oreat Britain Robustness to non-normality of regression testst

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 distribu-tion 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'a), which determines sensitivity to non-normality in the observations (the y'a). 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 th