62

FUNDAMENTAL ASSUMPTIONS IN ANALYSIS OF VARIANCE

a NOC

(HB)

oi - |zx? - —— (ir - V,
1

MS,,or — ž$-162/», and k < number of variances. For values of v; Z 5,
B is approximately distributed as the x? distribution, with k — 1 degrees
of freedom. lf v; < 5, tables prepared by Merrington and Thompson
(1946) may be used.

Two other tests are computationally simpler than Bartlett's test
and provide an adeguate test of the assumption of homogeneity of variance.
The simpler of the two tests, which was proposed by Hartley (1940, 1950),
uses the statistic F,,,,,

largest of k variances Gj1argest

F max — kai
» až
smallest of k variances Gj smallest

with degrees of freedom egual to k and n — |, where k is the number of
variances and n is the number of observations within each treatment level.
The distribution of F,,,, is given in Table D.i0. The hypothesis of homo-
geneity of variance is rejected if F,,,, is greater than the tabled value for
Fax: M the ns for the treatment levels differ only slightly, the largest of
the ns can be used for purposes of determining the degrees of freedom for
this test. This procedure leads to a slight positive bias in the test, that is, in
rejecting the hypothesis of homogeneity more freguently than it should be
rejected.

The other relatively simple test of homogeneity of variance is that
proposed by Cochran (1941). This test statistic is given by

a2
O;
š C j largest

ije
x Gi
jzl

where 67; argesi is the largest of the k treatment variances and Z). ,$; is the
sum of all of the variances. The degrees of freedom for this test are egual
to k and n — | as defined for the F,,,, test. The sampling distribution of C
is given in Table D.I1.

Since the F distribution is so robust with respect to violation of the
assumption of homogeneity of error variance, it is not customary to test
this assumption routinely. Both the Hartley and the Cochran tests have
adeguate sensitivity for testing the assumption in situations where hetero-
geneity is suspected. If variances are heterogeneous, a transformation of
scores as described in Section 2.7 may produce homogeneity.

Jt should be noted that all three tests described here are sensitive
to departures from normality as well as heterogeneity of variances (Box
and Anderson, 1955). For a description of a test that is relatively insensitive
to departures from normality, see Odeh and Olds (1959).

bt

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