Important Assumptions in ANOVA

** robust** to violations of this assumption, provided the
violations are not too severe. The best way to assess deviation from
normality is to simply examine a histogram. The issue of normality has
been covered well in previous courses.

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

When
variances among groups are unequal, they are referred to as ** heterogenous**,
and it is said that the

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

*Biometrics, 30*, 719-724.

*Journal of the American Statistical Association, 75*,
796-800.

*Journal of
Educational Statistics, 1*, 113-125.

*Designing experiments and analyzing
data: A model comparison perspective.* Mahwah, N.J.: Lawrence Erlbaum.

*Biometrika,
38*, 330-336.