Basic Analysis and Graphing - SAS

156 Performing Oneway Analysis Chapter 5
Unequal Variances
Unequal Variances
When the variances across groups are not equal, the usual analysis of variance assumptions are not satisfied
and the ANOVA F test is not valid. JMP provides four tests for equality of group variances and an ANOVA
that is valid when the group sample variances are unequal. The concept behind the first three tests of equal
variances is to perform an analysis of variance on a new response variable constructed to measure the spread
in each group. The fourth test is Bartlett’s test, which is similar to the likelihood-ratio test under normal
distributions.
Note: Another method to test for unequal variances is ANOMV. See “Analysis of Means Methods” on
page 144.
Table 5.17 Descriptions of Tests for Equal Variances
O’Brien
Brown-Forsythe
Levene
Bartlett
Constructs a dependent variable so that the group means of the new variable
equal the group sample variances of the original response. An ANOVA on the
O’Brien variable is actually an ANOVA on the group sample variances (O’Brien
1979, Olejnik, and Algina 1987).
Shows the F test from an ANOVA where the response is the absolute value of the
difference of each observation and the group median (Brown and Forsythe
1974a).
Shows the F test from an ANOVA where the response is the absolute value of the
difference of each observation and the group mean (Levene 1960). The spread is
measured as z ij
= y ij
– y i (as opposed to the SAS default z
ij
2 = ( y ij
– y i ) 2 ).
Compares the weighted arithmetic average of the sample variances to the
weighted geometric average of the sample variances. The geometric average is
always less than or equal to the arithmetic average with equality holding only
when all sample variances are equal. The more variation there is among the group
variances, the more these two averages differ. A function of these two averages is
created, which approximates a χ 2 -distribution (or, in fact, an F distribution under
a certain formulation). Large values correspond to large values of the arithmetic
or geometric ratio, and therefore to widely varying group variances. Dividing the
Bartlett Chi-square test statistic by the degrees of freedom gives the F value shown
in the table. Bartlett’s test is not very robust to violations of the normality
assumption (Bartlett and Kendall 1946).
If there are only two groups tested, then a standard F test for unequal variances is also performed. The F test
is the ratio of the larger to the smaller variance estimate. The p-value from the F distribution is doubled to
make it a two-sided test.
Note: If you have specified a Block column, then the variance tests are performed on data after it has been
adjusted for the Block means.