The Effect of Nonorthogonality on the Dependence ofFRatios Sharing a Common Denominator

Abstract

Abstract The problem of testing two or more hypotheses about cell means in an analysis of experimental data using F ratios sharing a common denominator mean squared error has been considered by Ghosh (1955), Ramachandran (1956), Krishnaiah (1964), and Hurlburt and Spiegel (1976). Hurlburt and Spiegel considered the case in which the numerator sums of squares of two F statistics are independent and the denominator sums of squares identical. In this article we extend Hurlburt and Spiegel's results to allow dependence between the numerator sums of squares. This can occur when testing row and column effects in nonorthogonal designs, with the degree of dependence varying with the degree of nonorthogonality. The Type I error in testing differences in row effects conditioned on rejecting no difference in column effects is a function of the degrees of freedom and the extent to which the design is nonorthogonal. These relations can be partially explained in terms of the correlation coefficient between the numerato...