ANOVA Assumptions - Analysis of Variance Designs - page 59

scores and then subjected to an analysis of variance. Obtaining a statis-
tically nonsignificant outcome (i.e., failing to reject the null hypothesis)
is indicative of equal variances or homogeneity. Note that the subscript i
denotes the ordinal position of each case or participant within each group
or level of the independent variable (participant number 1, 2, 3, etc.). The
j subscript denotes the level or group designation for the independent
variable (level 1, 2, 3, etc.).
The Levene and Brown and Forsythe statistics differ in how the absolute
difference scores are computed. The Levene test uses an absolute deviation
score from the group mean:

Z ij = Y ij − Y j . (5.3)

The Brown-Forsythe procedure creates a difference score based on each
original score's deviation from the group median:

Z ij = Y ij − Mdn j . (5.4)

Both of these procedures are commonly used and available in SPSS and
SAS; they will be demonstrated in Sections 5.5.5 and 5.6.5. Although
both procedures are widely used in the research literature, we agree with
the recommendation of Keppel and Wickens (2004), who note that “the
Brown-Forsythe procedure is slightly more robust” (p. 151). Specifi-
cally, the Brown-Forsythe test should be used when a distribution is
skewed.

5.4.4 SOLUTIONS FOR VIOLATIONS OF HOMOGENEITY OF VARIANCE
There are two common approaches to addressing heterogeneity of vari-
ance. The first solution is to use a more stringent alpha level, typically the
.025 level. Keppel and Wickens (2004), in summarizing the results of their
own Monte Carlo simulations, note that

the Type I error will be kept below the 5 percent level by setting 025. If your

concern is to keep the Type I error under control, then a conservative halving
of

α = .

is the fastest and simplest way to eliminate concerns about heterogeneity.
(p. 152)

α

Thesecondtypeofremedyforheterogeneityofvarianceistotransform
the original scores. Keppel and Wickens (2004) recommend this approach
when the dependent variable consists of reaction time, frequency counts,
or proportional measurements. Square root, log, and arcsine transforma-
tions can pull or compress a distribution to reduce heterogeneity and also
normalize the distribution. SPSS and SAS can easily apply transformations
to data. We recommend judicious use of these facilities, as transformed
data are sometimes difficult to explain to a reader who is familiar with the
original distribution under study.