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In statistics, one purpose for the analysis of variance (ANOVA) is to analyze differences in means between groups. The test statistic, ''F'', assumes independence of observations, homogeneous variances, and population normality. ANOVA on ranks is a statistic designed for situations when the normality assumption has been violated. ==Logic of the ''F'' test on means== The ''F'' statistic is a ratio of a numerator to a denominator. Consider randomly selected subjects that are subsequently randomly assigned to groups A, B, and C. Under the truth of the null hypothesis, the variability (or sum of squares) of scores on some dependent variable will be the same within each group. When divided by the degrees of freedom (i.e., based on the number of subjects per group), the denominator of the ''F'' ratio is obtained. Treat the mean for each group as a score, and compute the variability (again, the sum of squares) of those three scores. When divided by its degrees of freedom (i.e., based on the number of groups), the numerator of the F ratio is obtained. Under the truth of the null hypothesis, the sampling distribution of the F ratio depends on the degrees of freedom for the numerator and the denominator. Model a treatment applied to group A by increasing every score by X. (This model maintains the underlying assumption of homogeneous variances. In practice it is rare – if not impossible – for an increase of X in a group mean to occur via an increase of each member's score by X.) This will shift the distribution X units in the positive direction, but will not have any impact on the variability within the group. However, the variability between the three groups' mean scores will now increase. If the resulting F ratio raises the value to such an extent that it exceeds the threshold of what constitutes a rare event (called the Alpha level), the Anova F test is said to reject the null hypothesis of equal means between the three groups, in favor of the alternative hypothesis that at least one of the groups has a larger mean (which in this example, is group A). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「ANOVA on ranks」の詳細全文を読む スポンサード リンク
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