SchefféTests

SchefféTests refer to Scheffé's method for multiple comparisons in the context of analysis of variance (ANOVA) and the general linear model. Named after Henry Scheffé, this approach provides a conservative way to test all linear combinations of group means (all contrasts) while controlling the familywise error rate.

The core idea is to construct a simultaneous confidence region for the vector of group means and, from this region, derive tests and confidence intervals for any linear combination of means. Concretely, after fitting an ANOVA and obtaining the mean square error (MSE) as an estimate of σ^2, Scheffé’s method tests a contrast or a set of contrasts by using the F distribution with (k−1) and N−k degrees of freedom, where k is the number of groups and N is the total sample size. The method yields a critical value based on (k−1)F_{k−1,N−k,α}, which defines a broad, simultaneous tolerance for all possible linear combinations. A contrast is declared significant if the corresponding statistic exceeds this Scheffé critical value. Equivalent formulations present simultaneous confidence intervals for all linear combinations: Lμ lies in Lμ_hat ± sqrt((k−1)F_{k−1,N−k,α}) sqrt(Var(Lμ_hat)) for any L.

Scheffé tests are very general and valid under mild assumptions (normal errors, independence, and correct model specification), and they accommodate unequal group sizes. However, they are also notably conservative, often offering less power for specific, planned comparisons compared with procedures tailored to particular contrasts (such as Tukey or Bonferroni methods). Scheffé’s method thus excels when the goal is to draw inferences about a wide range of potential linear combinations rather than a few predefined comparisons.