Scheffé's method

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.

The method

Let μ₁, ..., μ_r be the means of some variable in r disjoint populations.

An arbitrary contrast is defined by

C = \sum_{i=1}^r c_i\mu_i

where

\sum_{i=1}^r c_i = 0.

If μ₁, ..., μ_r are all equal to each other, then all contrasts among them are 0. Otherwise, some contrasts differ from 0.

Technically there are infinitely many contrasts. The simultaneous confidence coefficient is exactly 1 − α, whether the factor level sample sizes are equal or unequal. (Usually only a finite number of comparisons are of interest. In this case, Scheffé's method is typically quite conservative, and the experimental error rate will generally be much smaller than α.)^[1]^[2]

We estimate C by

\hat{C} = \sum_{i=1}^r c_i\bar{Y}_i

for which the estimated variance is

s_{\hat{C}}^2 = \hat{\sigma}_e^2\sum_{i=1}^r \frac{c_i^2}{n_i},

where

n_i is the size of the sample taken from the ith population (the one whose mean is μ_i), and
$\hat{\sigma}_e^2$ is the estimated variance of the errors.

It can be shown that the probability is 1 − α that all confidence limits of the type

\hat{C}\pm\,s_\hat{C}\sqrt{\left(r-1\right)F_{\alpha;r-1;N-r}}

are simultaneously correct, where as usual N is the size of the whole population.

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 (i.e. 15a, 17a, 34b 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 "a").

Comparison with the Tukey–Kramer method

If only a fixed number of pairwise comparisons are to be made, the Tukey–Kramer 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.

References

↑ Maxwell, Scott E.; Delaney, Harold D. (2004). Designing Experiments and Analyzing Data: A Model Comparison. Lawrence Erlbaum Associates. pp. 217–218. ISBN 0-8058-3718-3.
↑ Milliken, George A.; Johnson, Dallas E. (1993). Analysis of Messy Data. CRC Press. pp. 35–36. ISBN 0-412-99081-4.

Bohrer, Robert (1967). "On Sharpening Scheffé Bounds". Journal of the Royal Statistical Society. Series B 29 (1): 110–114. JSTOR 2984571.
Scheffé, H. (1999) [1959]. The Analysis of Variance. New York: Wiley. ISBN 0-471-34505-9.

External links

Scheffé's method

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