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In statistics, one-way analysis of variance (or one-way ANOVA) is a technique to compare whether two or more samples' means are significantly different (using the F distribution). This analysis of variance technique requires a numeric response variable "Y" and a single explanatory variable "X", hence "one-way". [1]
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.
When a one-way ANOVA is performed, samples are assumed to have been drawn from distributions with equal variance. If this assumption is not valid, the resulting F-test is invalid. The Brown–Forsythe test statistic is the F statistic resulting from an ordinary one-way analysis of variance on the absolute deviations of the groups or treatments ...
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.
When using this kind of design for a binary response, one instead uses the Cochran's Q test. The Sign test (with a two-sided alternative) is equivalent to a Friedman test on two groups. Kendall's W is a normalization of the Friedman statistic between 0 {\textstyle 0} and 1 {\textstyle 1} .
Another omnibus test we can find in ANOVA is the F test for testing one of the ANOVA assumptions: the equality of variance between groups. In One-Way ANOVA, for example, the hypotheses tested by omnibus F test are: H0: μ 1 =μ 2 =....= μ k. H1: at least one pair μ j ≠μ j'
The main statistical tests available are Independent and Paired t-tests, Wilcoxon signed ranks, Mann–Whitney U, Pearson's chi squared, Kruskal Wallis H, one-way ANOVA, Spearman's R, and Pearson's R. Nested tables can be produced with row and column percentages, totals, standard deviation, mean, median, lower and upper quartiles, and sum.
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. [56] When there are only two means to compare, the t-test and the ANOVA F-test are equivalent; the relation between ANOVA and t is given by F = t 2.