Statistical Methods for Psychology

values for each F, they still have a role to play, and her table can be found in the appendix

of this book.)

For the Bonferroni test on pairwise comparisons (i.e., comparing one mean with one

other mean), define

and evaluate against the critical value of taken from Dunn’s tables in Appendix.

Notice that we still use the standard formula for t. The only difference between and a

standard t is the tables used in their evaluation. With unequal sample sizes but homoge-

neous variances, replace the ns in the leftmost equation with and. With heterogeneity

of variance, see the solution by Games and Howell later in this chapter.

To write a general expression that allows us to test any comparison of means, pairwise

or not, we can express in terms of linear contrasts.

and

This represents the most general form for the Bonferroni t, and it can be shown that if

cis anylinear combination (not necessarily even a linear contrast, requiring ), the

FWwith ccomparisons is at most a(Dunn, 1961).^7 To put it most simply, the Bonferroni t

runs a regular t test but evaluates the result against a modified critical value of t that has

been chosen so as to limit FW.

I would offer one word of caution when it comes to the Bonferroni test and variations

on it. These tests are appropriate when you have a limited number of planned contrasts,

whether they be pairwise or complex. However SPSS and SAS offer the Bonferroni test

only with pairwise post hoc tests, for which it is usually inappropriate. If you want to apply

such a correction to a planned set of contrasts, you need to specify those contrasts and then

evaluate significance on your own in relation to ac. And to specify those contrast coeffi-

cients you will need to use Compare Means/One-way ANOVAand not the univariate

procedure. In SAS you will need to use a contrast statement with Proc GLM.

A variation on the Bonferroni procedure was proposed by Sidák (1967). His testˇ

is based on the multiplicative inequality and evaluates at

. (This is often called the Dunn-Sidák test.ˇ ) A comparison of the
power of the two tests shows only very minor differences in favor of the Sidák approach,ˇ
and we will stick with the Bonferroni test because of its much wider use. Many computer
software programs, however, provide this test. For four comparisons, the Sidák approachˇ
would test each comparison at level,
whereas the Bonferroni approach would test at ac 5 .05 4 5 .0125. You can see that
there is not a lot of difference in power.
When we considered linear contrasts earlier in this section, we ran four comparisons,
which had an FWof nearly .20. (Our test of each of those contrasts involved an Fstatistic
but, because each contrast involves 1 df, we can go from t to Fand vice versa by means of
the relationship .) If we wish to run those same comparisons but to keep FWat a
maximum of .05 instead of 4 3 (.05) 5 .20, we can use the Bonferroni t test. In each case,

t= 2 F

> >

a¿= 12 (12a)^1 >^4 = 12 .95.25=0.0127

a¿= 12 (12a)^1 >c

p(FW)... 12 (12a)c t¿

>

gaj= 0

t¿=

c

B

aa

2

jMSerror

n

c= aajXj

t¿

ni nj

t¿

t¿ t¿ t¿

t¿=

Xi 2 Xj

B

MSerror

n

1

MSerror

n

=

Xi 2 Xj

B

2 MSerror

n

378 Chapter 12 Multiple Comparisons Among Treatment Means

(^7) Note the similarity between the right side of the equation and our earlier formula for F with linear contrasts. The
resemblance is not accidental; one is just the square of the other.
Dunn-Sidák testˇ