Difficulties may arise in factorial analysis of variance whenever the number of cases per cell are not strictly equal; then the conventional interaction test has a special status. Unless this status is defensible, there may be alternative and equally justifiable tests of the same experimental hypotheses which have strikingly different outcomes. An illustration was simulated from random data involving African American and white students, administered an achievement test by an African American or a white teacher. It was supposed that the investigator hypothesized that students would perform better when tests were administered by the same race. Scores were adjusted for all African Americans, to reflect substantial effects for this treatment. Arranging the data according to this hypothesis resulted in examiner race as the only significant effect; students scored higher with an African American teacher. The anticipated main effect, the relationship between student's and teacher's race, was not significant. If another investigator tried to replicate this experiment, it would be crucial to investigate the reported effect of the race of the tester. Assuming identical results as the first research, scores would be arranged differently for an analysis of variance, but would result in a different conclusion. The significant effect would be the relationship between races, while the tester's race would now be insignificant. In this 2 X 2 factorial design, the problem arises from the selection of the main versus the interaction effects. The difference between the computation of main effect and interaction sums of squares may affect test results whenever the cell frequencies are not equal, even though they are proportional. Paper presented at Midwestern Psychological Association Chicago, May 4, 1963.