Important Assumptions in ANOVA
A
more vexing problem in practice is the assumption of equal variances.
There is often evidence that this assumption is violated (remember the standard
deviation squared is the variance, so large differences in standard deviations
among groups is evidence of a problem). In fact, the major statistical
software packages provide tests for this assumption. The most common are
the Bartlett-Box test, Hartley's test, and Levene's test. Calculating
these statistics is beyond the scope of the course, but interpretation is
relatively straight-forward. They are available in the major statistical
packages. For these tests, the null hypothesis is that variances of
the individual cells are equal. When the null is rejected,
the variances are unequal, and the assumption is not met for
ANOVA. However, these tests tend to be very powerful, and hence it is
probably not cause for real concern until the null is rejected at less than the
.001 level (sig or p < .001), especially when sample size is large.
When
variances among groups are unequal, they are referred to as heterogenous,
and it is said that the homogeneity of variance assumption is
violated. When this is the case, there are ways of correcting the
analysis. We will now consider two approaches for testing the omnibus
null when the homogeneity of variance assumption is untenable. In both of
these tests, modifications of the calculation of MSbg and/or MSwg are made,
and, most importantly, the critical values of F are adjusted by adjusting the df
for MSwg.
These
methods are mathematically tedious, but not incomprehensible (make sure you
understand the difference- "tedious" would take a whole day to do and
would make you bored or angry, "incomprehensible" is where you
couldn't do it even if you had the time). Make sure you know what you
would have to do to perform these methods, even though you do not plan to
(there are ways of testing to see if you have done this!). There is
nothing in the formulas below you have not seen before. It is just n, N,
s (standard deviation), a (number of groups), etc. The first method is the
Brown-Forsythe method (Brown & Forsythe, 1974). In this method, the
MSwg is modified to yield a special F statistic (F*). The F* value is
then evaluated at a special denominator df value (df*).
The second is the Welch method (Welch, 1951). Here, a special statistic named W is computed, and it is evaluated against the F distribution at a specially computed denominator df value:
Post hoc Analyses with Heterogenous Variances
References