Statistics (university) · Part 07
Comparing categories and many groups
Chi-square tests for counts, ANOVA for several means, ranks and effect size.
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- Chi-square goodness of fitThis test asks whether observed counts match the counts a model predicts. The statistic is the sum of (observed - expected) squared / expected, with k - 1 degrees of freedom for k categories.
- Expected countsIn a two-way table, the expected count for a cell is (row total x column total) / grand total. It is what you would see if the two variables were unrelated.
- Chi-square test of independenceThis test checks whether two categorical variables are related. The null says they are independent. Compute the sum of (O - E) squared / E over all cells, with (rows - 1) x (columns - 1) degrees of freedom.
- One-way ANOVAAnalysis of variance (ANOVA) compares the means of three or more groups at once. It checks whether the variation between group means is large compared with the variation inside the groups.
- F statisticThe F statistic is the mean square between groups divided by the mean square within groups. A mean square is a sum of squares divided by its degrees of freedom. A large F means the group means differ more than chance explains.
- Post-hoc comparisonsAfter a significant ANOVA, a post-hoc test such as Tukey's finds which pairs of groups differ. It adjusts for the many comparisons so the overall false alarm rate stays near 5 percent.
- Rank-based testsRank-based (nonparametric) tests replace the data with ranks, so they do not need a normal shape. The Mann-Whitney test is the rank version of the two-sample t test.
- Effect sizeEffect size measures how big a difference is, not just whether it is detectable. Cohen's d is the difference in means divided by the pooled standard deviation. Around 0.2 is small, 0.5 medium and 0.8 large.