Technometrics · 1974 · 148 citations · 0 references
Natural SciencesMathematicsEconometricsExperimental DesignStatisticsHypothesis TestingInferential StatisticsPower AnalysisRobust StatisticsStatistical AnalysisStatistical InferenceAsymptotic RobustnessCritical RegionRobust Analysis Of VarianceRobust Hypothesis TestingSmall-sample PropertiesTest StatisticsVariance EstimationVariances
Four statistics for testing equality of population means are compared on robustness to heteroscedasticity, power, and critical‑region overlap. The study evaluates the ANOVA F‑statistic, a modified F with the same numerator but altered denominator, and Welch and James statistics that differ mainly in their critical‑value approximations. Welch’s critical values better approximate small‑sample behavior than James’s, and both Welch’s and the modified F are robust to unequal variances; when variances are equal, the modified F’s critical region more closely matches ANOVA, while the choice between Welch and modified F depends on mean magnitudes and standard errors.
Four statistics which may be used to test the equality of population means are com-pared with respect to their robustness under heteroscedasticity, their power, and the overlap of their critical regions. The four are: the ANOVA F-statistic; a modified F which has the same numerator as the ANOVA but an altered denominator; and two similar statistics proposed by Welch and James which differ primarily in their approximations for their critical values. The critical values proposed by Welch are a better approximation for small sample sizes than that proposed by James. Both Welch's statistic and the modified F are robust under the inequality of variances. The choice between them depends upon the magnitude of the means and their standard errors. When the population variances are equal, the critical region of the modified F more closely approximates that of the ANOVA than does Welch's.