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Concept
Scheffé's method
Summary
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. It works on similar principles as the Working–Hotelling procedure for estimating mean responses in regression, which applies to the set of all possible factor levels.
Let μ1, ..., μr be the means of some variable in r disjoint populations.
An arbitrary contrast is defined by
where
If μ1, ..., μr 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 family-wise error rate (experimental error rate) will generally be much smaller than α.)
We estimate C by
for which the estimated variance is
where
ni is the size of the sample taken from the ith population (the one whose mean is μi), and
is the estimated variance of the errors.
It can be shown that the probability is 1 − α that all confidence limits of the type
are simultaneously correct, where as usual N is the size of the whole population. Draper and Smith, in their 'Applied Regression Analysis' (see references), indicate that 'r' should be in the equation in place of 'r-1'. The slip with 'r-1' is a result of failing to allow for the additional effect of the constant term in many regressions.
Official source
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