There are times when the hypothesis H0: b ¼ 0 is not rejected, but covariance still
provides a more powerful test of H0: a1 ¼ a2 ¼ ¼ aa than would a comparable ANOVA
of the y variable. If the experimenter has reason to suspect that a sizable portion of the
variability in y is attributable to a covariate x, the experiment should be designed and data
collected with covariance analysis in mind. The worst that can happen is the loss of one degree
of freedom attributable to a nonsignificant b. But even with that loss, MS
0
e( y) may
still be sufficiently smaller than MSe( y) for covariance analysis to be more powerful than
ANOVA.