Tests for Equality of Variances

of W. SAS can calculate W using the NORMAL option in PROC UNIVARIATE

(see Program WATTOX.SAS in Section L.4.1.1).

The Kolmogorov-Smirnov (K-S) Test is also an acceptable test for normality

for small sample sizes, provided that the probabilities developed by Lilliefors

(1967) are used (Sokal and Rohlf 1981). The SYSTAT NPAR module provides

the appropriate test, and specifically identifies the test as Lilliefors Test

(Wilkinson 1990). Other statistical packages providing K-S Tests may not use

the Lilliefors probabilities, and the package documentation should always be

checked to determine if the appropriate probabilities are provided. The chi-

square (?2) test for normality can be used for larger sample sizes (e.g., N > 50)

(Sokal and Rohlf 1981).

Tests for Equality of Variances. There are a number of tests for equality

of variances. Some of these tests are sensitive to departures from normality,

which is why a test for normality should be performed first. Bartlett's Test,

Levene's Test, and Cochran's Test (Winer 1971; Snedecor and Cochran 1989) all

have similar power for small, equal sample sizes (n = 5) (Conover, Johnson, and

Johnson 1981), and any one of these tests is adequate for the analyses in this

Appendix. Many software packages for t-tests and analysis of variance

(ANOVA) provide at least one of the tests. SAS now provides several tests for

equality of variances, including Levene=s and Bartlett=s, in the HOVTEST=

option of the MEANS statement in the GLM or ANOVA procedures. In the

absence of specific software tests for equality of variances, Levene's Test can be

performed by comparing the absolute values of residuals between treatments

using t-tests or ANOVA.

If no tests for equality of variances are included in the available statistical

software, Hartley's Fmax can easily be calculated:

Fmax = ( larger of s2 , s2 ) / ( smaller of s2 , s2 )

(L-8)

When Fmax is large, the hypothesis of equal variances is more likely to be

rejected. Fmax is a two-tailed test because it does not matter which variance is

expected to be larger. Some statistical texts provide critical values of Fmax

(Winer 1971; Gill 1978 [includes a table for unequal replication, but only for a =

0.05]; Rohlf and Sokal 1981). In the two-sample case, Hartley's Fmax is the same

as the Folded-F or FN test. The FN test is conducted automatically in the SAS

TTEST procedure.

Cochran's Test, where C = the largest variance divided by the sum of the

variances, is also simple to calculate by hand, and is somewhat more powerful

then Hartley's Fmax for small, equal sample sizes (Conover, Johnson, and Johnson

1981). However, tables of critical values of Cochran's C are not available in

most statistical texts. Winer (1971) and Dixon and Massey (1983) include a

table for Cochran's Test, but the tables are limited to tests with equal sample

sizes. Tables of critical values for tests such as Cochran's C and Hartley's Fmax

may also be restricted to one or two a levels (usually 0.05 and 0.01). Because of

the limitations of these tables, computer programs are preferred for tests of

equality of variances.

L12

Appendix L

Statistical Methods