Multiple Comparison Procedures - Analysis of Variance Designs - page 122

The value of q can also be directly calculated from the ANOVA sum-
mary table and a knowledge of the values of the means. Hays (1981) gives
the formula as follows:

Y Largest − Y Smallest

MS S / A
n

= .

q (7.3)

In the above formula, Y Largest and Y Smallest are the values of the largest and
smallest means, respectively, MS S/A is the within-groups mean square from
the ANOVA summary table, and n is the size of each group (assuming
equal group sizes); with unequal group sizes, Hays (1981) suggests using
the harmonic (weighted) mean of the groups (see Section 17.5).
To find your critical value - the mean difference needed in order to
assert that the means differ significantly - you engage in three relatively
simple calculations (Hays, 1981):

Divide the within-groups mean square (from the ANOVA sum-
mary table) by n, thesizeofthegroupsifthegroupsizesareequal;
with unequal group sizes, use the harmonic mean of the group
sizes.
Take the square root of this quantity.
Multiply the result by q to obtain the so-called critical difference (the
minimum difference between means that yields statistical signifi-
cance at your alpha level).

This basic approach has given rise to several variations. The TukeyHon-
estly Significant Difference (HSD) test, for example, applies the computed
critical difference to all pairs of means. On the other hand, the Student-
Newman-Keuls (S-N-K) procedure applies the above critical difference
to the pair of means that are most different. If that difference is significant,
it steps down to look at the difference between the next lowest (or highest)
mean and the one most different from it. The value of r , the range param-
eter, is reduced by 1 (in our study time example, the original value of r was
5; it would now be 4) and the process recomputed. This variation does not
doagoodjobofprotectingagainstTypeIerror,andsomeauthors(e.g.,
Keppel et al., 1992) recommend against using the S-N-K procedure for
that reason.

7.8.2.3 Studentized MaximumModulus Distribution
The Studentized maximum modulus distribution is related to the Student
range statistic and is based on research that Sid ak (1967) published in
working with variations of the t test. Whereas the range statistic distri-
bution is based on the presumption of equal sample sizes in the groups,
the maximum modulus distribution applies to unequal group sizes. Thus,