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Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case can be covered by a t-test (Gosset, 1908). When there are only two means to compare, the t-test and the F-test are equivalent; the relation between ANOVA and t is given by F = t 2.
The one factor model can be thought of as a generalization of the two sample t-test. That is, the two sample t-test is a test of the hypothesis that two population means are equal. The one factor ANOVA tests the hypothesis that k population means are equal. The standard ANOVA assumes that the errors (i.e., residuals) are normally distributed.
Factorial ANOVA is used when there is more than one factor. Repeated measures ANOVA is used when the same subjects are used for each factor (e.g., in a longitudinal study). Multivariate analysis of variance (MANOVA) is used when there is more than one response variable.
[5] [page needed] The main difference between the sum of squares of the within-subject factors and between-subject factors is that within-subject factors have an interaction factor. More specifically, the total sum of squares in a regular one-way ANOVA would consist of two parts: variance due to treatment or condition (SS between-subjects ) and ...
The formula for the one-way ANOVA F-test statistic is =, or =. The "explained variance", or "between-group variability" is = (¯ ¯) / where ¯ denotes the sample mean in the i-th group, is the number of observations in the i-th group, ¯ denotes the overall mean of the data, and denotes the number of groups.
The parametric equivalent of the Kruskal–Wallis test is the one-way analysis of variance (ANOVA). A significant Kruskal–Wallis test indicates that at least one sample stochastically dominates one other sample. The test does not identify where this stochastic dominance occurs or for how many pairs of groups stochastic dominance obtains.
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Tukey's HSD and Scheffé's procedure are one-step procedures and can be done without the omnibus F having to be significant. They are "a posteriori" tests, but in this case, "a posteriori" means "without prior knowledge", as in "without specific hypotheses." On the other hand, Fisher's Least Significant Difference test is a two-step procedure.
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