Numerous designs using analysis of variance (ANOVA) to test ordinal hypotheses were assessed using a Monte Carlo simulation. Each statistic was computed on each of over 10,000 random samples drawn from a variety of population conditions. The number of groups, population variance, and patterns of population means were varied. In the non-null patterns, the power of the alpha-corrected tests was extremely low. Extremely low power was also evident for both the uncorrected and the no significance test for ordering technique. The linear trend test exhibited the highest power in almost every fully monotonic circumstance. The tau and monotonic trend tests also had high power in most fully monotonic population means instances, but they varied as to which was more powerful. Of the tests for disconfirmations of monotonicity, the curvilinearity test for detecting inversions or ties was too often significant in monotonic but non-linear cases. Combined tests were preferred when they had high acceptance values (power) for the fully monotonic cases, but low acceptance values when disconfirmations were present. Ultimately, the choice of technique comes down to the balance between power when the ordinal hypothesis is correct versus spurious acceptance of the ordinal hypothesis when it is slightly in error. Sweep tests, particularly the linear trend test, have the needed power, but lack the ability to alone detect disconfirmations of monotonicity. The tests of disconfirmation that were evaluated showed far too many false positives to warrant wide acceptance. Six data tables are included, and the Statistical Analysis System program used to run the Monte Carlo study and the program used to calculate taus are outlined. (TJH)