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SECTION 2.6 EFFECTS OF FAILURE TO MEET ASSUMPTIONS

Studies by Pearson (1931) and Norton, as cited by Lindguist (1953),
indicate that the F distribution is relatively unaffected by lack of symmetry
of treatment populations. It is also relatively unaffected by kurtosis except
in extreme cases of very leptokurtic or platykurtic populations. For the
fixed-effečts model, an experimenter need not be concerned if the k popula-
tions exhibit a moderate departure from the normal distribution provided
that the k populations are homogeneous in form, for example, all treatment
populations positively skewed and slightly leptokurtic. In general, unless
the departure from normality is so extreme that it can be readily detected
by visual inspection of the data, the departure will have little effect on the
probability associated with the test of significance. It may be possible to
transform nonnormally distributed scores so as to achieve normality,
under conditions described in Section 2.7.

ASSUMPTION OF HOMOGENEITY OF
POPULATION-ERROR VARIANCES

The F distribution is robust with respect to violation of the assump-
tion of homogeneity of population-error variances provided that the num-
ber of observations in the samples is egual (Cochran, 1947; Norton as
cited by Lindguist, 1953). However, for samples of unegual size, violation

"of the homogeneity assumption can have a marked effect on the test of

significance. According to Box (1953, 1954a), the nature of the bias for this
latter case may be positive or negative.

Several statistics are available for testing the homogeneity assump-
tion that

H,:o) — o x: — ož — ož.
The alternative to the above nuli hypothesis is
H, : some o;'s are unegual.

A test statistic proposed by Bartlett (1937) is

2.30259 k
B — (oa oMS.na — X Wvlognesh)].
jsl
where
1.1
— jzl v; v
Cel rea '

v; — degrees of freedom for d;, v — degrees of freedom for MS,,,,, egual
to x$.,v;, 67 <— unbiased estimate of population variance for the jth
population given by