Statistical Methods for Psychology

Define

Then

This statistic ( ) is approximately distributed as Fon k– 1 and degrees of freedom,

where

Obviously these formulae are messy, but they are not impossible to use. If you collect

all of the terms (such as ) first and then work systematically through the problem, you

should have no difficulty. (Formulae like this are actually very easy to implement if you

have access to any spreadsheet program.) When you have only two groups, it is probably

easier to fall back on a ttest with heterogeneous variances, using the approach (also attrib-

utable to Welch) taken in Chapter 7.

But!

I have shown how one can deal with heterogeneous variances so as to make an analysis of

variance test on group means robust to violations of homogeneity assumptions. However,

I must reiterate a point I made in Chapter 7. The fact that we have tests such as that by

Welch does not make the heterogeneous variances go away—it just protects the analysis of

variance on the means. Heterogeneity of variance is itself a legitimate finding. In this par-

ticular case it would appear that there are a group of people for whom cognitive/behavior

therapy is unusually effective, causing the gains in that group to become somewhat bi-

modal. That is important to notice. But even for the rest of that group the therapy is at least

reasonably effective. If we were to truncate the data for weight gains greater than 10

pounds, thus removing those participants who scored unusually well under cognitive/

behavior therapy, the resulting Fwould still be significant (F(2, 52) 5 4.71, p,.05).

A description of these results would be incomplete without at least some mention of the

unusually large variance in the cognitive/behavior therapy condition.

11.9 Transformations

In the preceding section we considered one approach to the problem of heterogeneity of

variance—calculate on the heterogeneous data and evaluate it against the usual Fdistri-

bution on an adjusted number of degrees of freedom. This procedure has been shown to

work well when samples are drawn from normal populations. But little is known about its

behavior with nonnormal populations. An alternative approach is to transform the data to a

form that yields homogeneous variances and then run a standard analysis of variance on

F–

wk

df¿=

k^221

(^3) aa

1

nk 21

ba 12

wk

awk

b

2

F– df¿

F–=

awk(Xk^2 X

¿

.)

2

k 21

11

2(k 2 2)

k^221 a

a

1

nk 21

ba 12

wk

awk

b

2

X.¿= a

wkXk

awk

wk=

nk

s^2 k

336 Chapter 11 Simple Analysis of Variance