Scheffe's Method

A Scheffe's Method is a post-hoc multiple comparison procedure for estimating of all possible contrasts.

• See: Post Hoc Analysis, Rodger's Method, Analysis of Variance, Multiple Comparisons Problem, Tukey–Kramer Method, Bonferroni Correction, Fisher's Least Significant Difference.

---

---

References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Scheffé's_method Retrieved 2016-08-28
  • In statistics, Scheffé's method, named after the American statistician Henry Scheffé, is a method for adjusting significance levels in a linear regression analysis to account for multiple comparisons. It is particularly useful in analysis of variance (a special case of regression analysis), and in constructing simultaneous confidence bands for regressions involving basis functions.

Scheffé's method is a single-step multiple comparison procedure which applies to the set of estimates of all possible contrasts among the factor level means, not just the pairwise differences considered by the Tukey–Kramer method.

2016

• (PSU online courses, 2016) ⇒ 3.3 - Multiple Comparisons. (n.d.). Retrieved August 28, 2016, from https://onlinecourses.science.psu.edu/stat503/node/15 Copyright: 2016, The Pennsylvania State University.
  • Scheffé's method for investigating all possible contrasts of the means corresponds exactly to the F-test in the following sense. If the F-test rejects the null hypothesis at level α, then there exists at least one contrast which would be rejected using the Scheffé procedure at level α . Therefore, Scheffé provides α level protection against rejecting the null hypothesis when it is true, regardless of how many contrasts of the means are tested.

2015

• (Seltman, 2015) ⇒ Seltman, H. J. (2015). “Experimental design and analysis", Chapter 13. Online at: http://www.stat.cmu.edu/~hseltman/309/Book/chapter13.pdf
  • This is a very general, but conservative procedure. It is applicable for the family of all possible contrasts! One way to express the procedure is to consider the usual uncorrected t-test for a contrast of interest. Square the t-statistic to get an F-statistic. Instead of the usual F-critical value for the overall null hypothesis, often written as [math]\displaystyle{ F(1−\alpha, k−1, N −k) }[/math], the penalized critical F value for a post-hoc contrast is [math]\displaystyle{ (k − 1)F(1 − \alpha, k − 1, N − k) }[/math]. Here, N is the total sample size for a one-way ANOVA, and N − k is the degrees of freedom in the estimate of [math]\displaystyle{ \sigma^2 }[/math].

1985

• (Kennedy & Bush, 1985) ⇒ Kennedy, J. J., & Bush, A. J. (1985). An introduction to the design and analysis of experiments in behavioral research. University Press of America. ISBN: 0-8191-4806-7
  • [...] The statistical reasoning underlying the Scheffe method appears rather complex. Approached intuitively, however, the Scheffe can be viewed as a method which: a) converts a t-ratio approach to individual comparisons to an F-test approach and b) reduces the rejection region of the F-distribution, with a concomitant loss of power, to accommodate all conceivable comparisons without exceeding the desired hypothesiswise error rate.

---