Probability of Directional Errors with Disordinal ( Qualitative ) Interaction

In a factorial design with two or more factors, there is nonzero interaction when the differences among the levels of one factor vary with levels of other factors. The interaction is disordinal or qualitative with respect to a specific factor, say A, if the difference between at least two levels of A is positive for some and negative for some levels of the other factors. Using standard methods of analysis, there is a potentially large probability of drawing incorrect conclusions about the signs of differences in the presence of disordinal interaction. The maximum probability of such incorrect conclusions, or directional errors, is derived for two-factor designs in which the factor of interest has two levels and the number of levels of the other factor varies.