Scheffé’s Method for Comparing All Contrasts

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 analyzing variance (a special case of regression analysis) and in constructing simultaneous confidence bands for regressions involving basis functions. The Scheffé method is a one-step multiple comparison procedure that is applied to the set of estimates of all possible contrasts between the means of the factor levels, not just the pairwise differences considered by the Tukey-Kramer method. It works on principles similar to the Working-Hotelling procedure for estimating mean responses in the regression, which is applied to all possible levels of factors.Scheffé’s method applies to the set of estimates of all possible contrasts between the means of the factor levels, not just to the pairwise differences considered by Tukey’s method. It works on principles similar to the Working-Hotelling procedure for estimating mean responses in the regression, which is applied to the set of all possible factor levels.Scheffe’s test (also called Scheffe’s procedure or Scheffe’s method) is a post-hoc test used in variance analysis. It is named after the American statistician Henry Scheffe. After you have performed ANOVA and obtained a significant F-statistic (that is, you have rejected the null hypothesis that the means are the same), run the Sheffe test to find out which pairs of means are significant. Scheffe’s test corrects alpha for simple and complex mean comparisons. Comparisons of complex means involve the simultaneous comparison of more than one pair of means. Of the three average comparative tests you can perform (the other two are Fisher’s LSD and Tukey’s HSD). The Scheffe test is the most flexible, but it is also the test with the lowest statistical power. Deciding which test to perform largely depends on the comparisons you are interested in:If you just want to do pairwise comparisons, do the Tukey procedure as you will have a tighter confidence interval.If you want to compare all possible pairs of simple and complex means, run Scheffe’s test, as you will have a tighter confidence interval. Scheffé’s method is applied to the set of estimates of all possible contrasts between the means of the factor levels, not just to the pairwise differences considered by Tukey’s method. An arbitrary test is defined by Technically, there are countless contrasts. The simultaneous confidence coefficient is exactly 1 a, regardless of whether the sample size at the factor level is equal or unequal.

Some Special Properties of the Scheffé Method

As long as ANOVA rejects the null hypothesis, Scheffe’s method will find

• If the null hypothesis of the means of the equal treatment level is rejected during an ANOVA, the corresponding Scheffé multiple comparison procedure will find at least one test (of all possible contrasts) significant. In other words, at least one test has a confidence interval that does not include zero. However, this contrast may not be of great interest to the analyst.
• As stated earlier, there are an infinite number of contrasts and the vast majority have no practical value for the analyst.
• However, we can define a normalized maximum contrast. By normalized we mean: the observed value of the contrast divided by its standard error.

Denoting Scheffé significance in a table.

• Frequently, superscript letters are used to indicate which values are significantly different using the Scheffé method. For example, when mean values of variables that have been analyzed using an ANOVA are presented in a table, they are assigned a different letter superscript based on a Scheffé contrast. Values that are not significantly different based on the post-hoc Scheffé contrast will have the same superscript and values that are significantly different will have different superscripts would mean that the first and second variables both differ from the third variable but not each other because they are both assigned the superscript.

Comparison with the Tukey–Kramer method.

• If only a fixed number of pairwise comparisons are to be made, the Tukey Method will result in a more precise confidence interval. In the general case when many or all contrasts might be of interest, the Scheffé method is more appropriate and will give narrower confidence intervals in the case of a large number of comparisons.

Conclusion

• The Scheffé test is one of the oldest multiple comparison procedures in use today. It is important to recognize that it is a frequently misused procedure and that it is also a valuable test when used as Henry Scheffé intended it. Unlike competitors such as Tukey’s Honestly Significant Difference test, the Scheffé test is specifically designed for the situation in which post hoc comparisons involve more than pairwise differences. For example, it could be used to compare the mean of two groups to the mean of two other groups on the basis of interesting differences that appeared after the data had been collected.

References:-

• Scheffé, H., 1953. A method for judging all contrasts in the analysis of variance. Biometrika, 40(1–2), pp.87–110.
• https://www.itl.nist.gov/div898/handbook/prc/section4/prc472.htm