Statistical Power.

equality of variances. If these assumptions are not violated, an LSD test is then

performed on the rankits (Conover 1980, refers to this as van der Waerden's

Test). Tests performed on rankits are robust to departures from normality and

can still be used when the normality assumption is violated. Rankits will rarely

fail tests for normality, partly because a normal distribution is imposed over the

entire data set. The rankit data may fail the test for equality of variances, but

then t-tests can be conducted for each treatment - reference comparison. If

rankit-transformed data fail normality tests, it is probably safest to use the t-tests

for unequal variances, as some tests for equality of variance are not robust when

data are nonnormal.

When rankits cannot be easily calculated, the original data may be converted

to ranks (using SAS PROC RANK, for example). Equality of variances should

be tested after the data are ranked. There is a common misconception that

nonparametric tests can be used when variances are not equal as well as when

data are not normally distributed. However, nonparametric tests are not very

robust if the variances of the ranks are not similar among treatments. Bartlett's

Test should not be used to test equality of variances of ranks, as ranks will

follow a uniform, rather than a normal distribution, and Bartlett's Test is unduly

sensitive to nonnormality. Other tests discussed in Section L.2.1.1.1, Tests for

Equality of Variances, may be used on ranks; there are also nonparametric tests

for equality of variances provided by Conover (1980).

If the variances of the ranks are not significantly different, the Conover

T-Test (Conover 1980) should be performed. This test can most easily be con-

ducted by performing an LSD test on the ranks. If the variances of ranks are

significantly unequal, a one-tailed t-test for unequal variances should be

performed (using ranks) for each treatment - reference comparison.

Dunn=s Test, as described in Hochberg and Tamhane (1987), is an acceptable

nonparametric alternative to the Conover T-Test or the LSD on rankits.

Statistical Power. Power calculations for the LSD test are the same as for

the t-test (Equation L-8), except that the degrees of freedom for t1-a and t1- are N

- k, and MSE replaces s2:

n = 2 ( t 1-α ,v + t 1-β ,v )2 (MSE/ d 2 )

(L-21)

If the z-approximation (Equation L-9 with MSE replacing s2) is used to

calculate samples size, the result will be a slight overestimate, although the

overestimation is rarely of practical importance. Finally, the minimum

significant difference should be reported for LSD tests. Note that the test is

named the Least Significant Difference because another way to conduct the test

is to compare the observed differences to the minimum significant difference.

If power (1 - ) is low because of high variability or small sample size, one

effective method of increasing power is to increase the number of reference

replicates rather than increase the sample size for each treatment. It is even

possible to increase power without increasing overall sample size by increasing

L29

Appendix L

Statistical Methods