7. Product and Process Comparisons
7.4. Comparisons based on data from more than two processes
7.4.7. How can we make multiple comparisons?

7.4.7.2.

Scheffe's method

Scheffe's method tests all possible contrasts at the same time

Scheffé's method applies to the set of estimates of all possible contrasts among the factor level means, not just the pairwise differences considered by Tukey's method.

Definition of contrast

An arbitrary contrast is defined by

where

Infinite number of contrasts

Technically there is an infinite number of contrasts. The simultaneous confidence coefficient is exactly 1-, whether the factor level sample sizes are equal or unequal.

Estimate and variance for C

As was described earlier, we estimate C by: for which the estimated variance is:

Simultaneous confidence interval

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

are correct simultaneously.

Scheffe method example

Contrasts to estimate

We wish to estimate, in our previous experiment, the following contrasts

and construct 95 percent confidence intervals for them.

Compute the point estimates of the individual contrasts

The point estimates are:

Compute the point estimate and variance of C

Applying the formulas above we obtain in both cases: and

where = 1.331 was computed in our previous example. The standard error = .5158 (square root of .2661).

Scheffe confidence interval

For a confidence coefficient of 95 percent and degrees of freedom in the numerator of r - 1 = 4 - 1 = 3, and in the denominator of 20 - 4 = 16, we have:

The confidence limits for C₁ are -.5 ± 3.12(.5158) = -.5 ± 1.608, and for C₂ they are .34 ± 1.608.

The desired simultaneous 95 percent confidence intervals are

-2.108 C₁ 1.108
-1.268 C₂ 1.948

Comparison to confidence interval for a single contrast

Recall that when we constructed a confidence interval for a single contrast, we found the 95 percent confidence interval:

-1.594 C 0.594

As expected, the Scheffé confidence interval procedure that generates simultaneous intervals for all contrasts is considerabley wider.

Comparison of Scheffé's Method with Tukey's Method

Tukey preferred when only pairwise comparisons are of interest

If only pairwise comparisons are to be made, the Tukey method will result in a narrower confidence limit, which is preferable.

Consider for example the comparison between ₃ and ₁.

Tukey:    1.13 < ₃ - ₁ < 5.31
Scheffé:  0.95 < ₃ - ₁ < 5.49

which gives Tukey's method the edge.

The normalized contrast, using sums, for the Scheffé method is 4.413, which is close to the maximum contrast.

Scheffe preferred when many contrasts are of interest

In the general case when many or all contrasts might be of interest, the Scheffé method tends to give narrower confidence limits and is therefore the preferred method.