number is indexed by “Order (i)” in the Table 12.3. These critical values are presented in
the right-hand column above. They were taken, with interpolation, from Dunn’s tables for
c 5 iand 35 degrees of freedom. For example, the critical value of 2.35 corresponds to the
entry in Dunn’s tables for c 5 2 and df 5 35. For the smallest , the critical value came
from the standard Student t distribution (Appendix t).
From this table you can see that the test on the complex contrast S-M, Mc-M versus
M-S, M-M, S-S required a of 2.64 or above to reject. Because was 9.95, the differ-
ence was significant. The next largest was 6.72 for Mc-M versus M-M, and that was also
significant, exceeding the critical value of 2.52. The contrast S-S versus M-S is tested as if
there were only two contrasts in the set, and thus must exceed 2.35 for significance.
Again this test is significant. If it had not been, we would have stopped at this point. But
because it is, we continue and test M-M versus S-S, which is not significant. Because of
the increased power of Holm’s test over the Bonferroni t test, we have rejected one null hy-
pothesis (S-S versus M-S) that was not rejected by the Bonferroni.Larzelere and Mulaik Test
Larzelere and Mulaik (1977) proposed a test equivalent to Holm’s test, but their primary
interest was in using that test to control FWwhen examining a large set of correlation coef-
ficients. As you might suspect, something that controls error rates in one situation will tend
to control them in another. (When you are testing all possible correlation coefficients in a
correlation matrix, it is conceptually the same as testing all possible pairwise differences in
a set of means. This would mean that perhaps we really should consider the Larzelere and
Mulaik test as being a post hoc test, and discuss it in that section of the chapter. But since it
is essentially the same as the Holm procedure, I am discussing it here.)
I will consider the Larzelere and Mulaik test with respect to correlation coefficients
rather than the analysis of variance, because such an example will prove useful to those
who conduct research that yields large numbers of such coefficients. Normally this section
should go in Chapter 9, but the underlying logic had not yet been developed when we dis-
cussed correlations, so it needed to wait until here. But if you do a lot of correlational work,
this is an important test to know. As you will see when you look at the calculations, the test
would be applied in the same way whenever you have a number of test statistics with their
associated probability values. (Your test statistic could be t, F, , or any other test statis-
tic, just so long as you can calculate its probability under the null.) If you had never heard
of Larzelere and Mulaik, you could still accomplish the same thing with Holm’s test. How-
ever, the different calculational approach is instructive. It is worth noting that when these
tests are applied in an analysis of variance setting we usually have a small number of com-
parisons. However, when they are used in a regression/correlation setting, we commonly
test all pairwise correlations.
Compas, Howell, Phares, Williams, and Giunta (1989) investigated the relationship
among daily stressors, parental levels of psychological symptoms, and adolescent behaviorx^2t¿t¿t¿ H 0 t¿t¿12.3 A Priori Comparisons 381Table 12.3 Application of Holm’s test to the morphine data
Contrast Order (i) tt
M-M vs. S-S 1 2.03
S-S vs. M-S 2 2.35*
Mc-M vs. M-M 3 2.52*
S-M, Mc-M vs. M-S, M-M, S-S 4 2.64**p , .05t¿= 2 9.95t¿= 2 6.72t¿= 2 2.47t¿= 2 0.35¿ ¿crit