Scheffé Test

Tukey's HSD is designed exclusively for pairwise comparisons and is more powerful than Scheffé in that setting. Scheffé's method covers all possible linear contrasts — including complex ones involving combinations of multiple groups — and is therefore more conservative (wider intervals, higher threshold) when applied to simple pairwise differences. If you only need pairwise tests, Tukey is preferred; if you need complex or unplanned contrasts, Scheffé is the appropriate choice.

By convention and logical necessity, yes. The post-hoc test is intended to explore which differences drive a significant omnibus result. Applying Scheffé without a prior significant F-test turns it into an exploratory fishing expedition and undermines the error-rate guarantees. Most textbooks and software documentation require a significant ANOVA as a prerequisite.

Yes. The formula for the test statistic and simultaneous confidence intervals explicitly accommodates different group sizes n_i, making Scheffé valid for unbalanced designs — an advantage over some other post-hoc procedures that assume equal sample sizes.

A contrast is any linear combination of group means whose coefficients sum to zero. A pairwise comparison (e.g., μ₁ − μ₂) is the simplest case with coefficients +1, −1, and 0 for the remaining groups. A complex contrast might compare the average of two groups against a third (e.g., ½μ₁ + ½μ₂ − μ₃). Scheffé's method controls the error rate for all such contrasts simultaneously, whereas most other post-hoc tests only address pairwise contrasts.