Nonparametric Tests

Levels of a for tests of equality of variances are provided in Table L-2; these

depend upon number of replicates in a treatment (n) and allotment of replicates

among treatments (design). Relatively high a 's are recommended because the

power of the above tests for equality of variances is rather low when n is small.

Equality of variances is rejected if the probability associated with the test

statistic is less than the appropriate a . If the test for equality of variances is

significant even after transformation, the t-test for unequal (separate) variances

should be selected rather than the t-test for equal (pooled) variances.

Nonparametric Tests. Tests such as the t-test, which analyze the original

or transformed data and which rely on the properties of the normal distribution,

are referred to as parametric tests. Nonparametric tests, which do not require

that data be normally distributed, generally analyze the ranks of data, comparing

medians rather than means. The median of a sample is the middle or 50th

percentile observation when the data are ordered from smallest to largest. In

many cases, nonparametric tests can be performed simply by converting the data

to ranks or normalized ranks, and then conducting the usual parametric test

procedures on the ranks.

Nonparametric tests are useful because of their generality but may have less

statistical power than corresponding parametric tests when the parametric test

assumptions are met.

When parametric tests are not appropriate for comparisons because the

normality assumption is not met, we recommend converting the data to

normalized ranks (rankits). Rankits are simply the z-scores expected for the rank

in a normal distribution. Thus, using rankits imposes a normal distribution over

all the data, although not necessarily within each treatment. Rankits can be

obtained by ranking the data, then converting the ranks to rankits using the

following formula:

rankit = z [(rank - 0.375) / (N + 0.25)]

(L-9)

where

z = normal deviate

N = total number of observations

For example, the approximate rankit for the sixth lowest value (rank = 6) of

20 observations would be z[(6 - 0.375)/(20 + 0.25)], which is z0.278 or -0.59.

In SAS, normalized ranks or rankits can be provided in PROC RANK with

the NORMAL = BLOM option. In SYSTAT and other packages, the ranks must

be converted to rankits using the formula above (the conversion is a one-line

command). In some programs the conversion may be more difficult to make,

especially if functions to provide z-scores for any probability are not available.

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

ranks.

L13

Appendix L

Statistical Methods