Product and Process Comparisons
7.4. Comparisons based on data from more than two processes
7.4.7. How can we make multiple comparisons?
|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
|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:
|Simultaneous confidence interval||
It can be shown that the probability is
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:
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
-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:
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.
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.|