Dear Deborah and others,
The Gläscher & Gitelman tutorial is cited frequently, but it is very misleading, likely resulting in invalid statistics. Leaving aside issues about error terms, it is problematic how they deal with factor B, for which they expect a linear increase from level B1 across B2 to B3 (page 4). This holds throughout the tutorial, also for the section on "Two groups of subjects". Unfortunately they do not explicitely state this in that section, thus people might think the contrast vectors are appropriate for *any* within-subject factor with three levels. Usually we do not have any assumptions about the levels (it could be something like neutral, positive, negative emotion). For a design with two within-subject factors A (two levels) and B (three levels, no further assumptions about the B levels) the design matrix should include the two main effects, factor subject and the interaction, resulting in something like this:
A1 A2 B1 B2 B3 A1B1 A1B2 A1B3 A2B1 A2B2 A2B3 S1 ... Sn
Testing for main effect A requires a contrast vector
[1 -1 0 0 0 1/3 1/3 1/3 -1/3 -1/3 -1/3 0 ... 0] -> as stated on page 5
Testing for main effect B requires a contrast vector with two rows
[0 0 1 -1 0 1/2 -1/2 0 1/2 -1/2 0 0 ... 0;
0 0 0 1 -1 0 1/2 -1/2 0 1/2 -1/2 0 ... 0] -> in contrast to what is stated on page 5
Testing for the interaction A x B would require a contrast vector with two rows
[0 0 0 0 0 1 -1 0 -1 1 0 0 ... 0;
0 0 0 0 0 0 1 -1 0 1 -1 0 ... 0] -> in contrast to what is stated on page 5
In addition, their B1 < B2 < B3 is NOT a linear increase at all, it's just a strict increase. For a strict increase we would have to show that B2 > B1 AND B3 > B2, or reformulated, B2 - B1 > 0 AND B3 - B2 > 0. This logical AND does *not* combine to (B2 - B1) + (B3 - B2) > 0 + 0 though (reformulated B3 - B1 > 0, maybe that's why they came up with ignoring B2).
A linear increase requires the differences B3 - B2 and B2 - B1 to be equal, meaning B3 - B2 = B2 - B1, or reformulated B3 - 2*B2 + B1 = 0. None of the reported contrasts in the tutorial tests for that, instead they ignore B2 and go with B3 - B1 (which doesn't even test for a strict increase B1 < B2 < B3, see above). In addition, testing for linear increases requires equivalence testing, as B3 - B2 = B2 - B1 has to be proven. No significant findings for a contrast reflecting B3 - 2*B2 + B1 does NOT mean the two differences are equal.
Best,
Helmut
|