Page 190 - Applied Statistics Using SPSS, STATISTICA, MATLAB and R

5 Non-Parametric Tests of Hypotheses

The tests of hypotheses presented in the previous chapter were “parametric tests”,
that is, they concerned parameters of distributions. In order to apply these tests,
certain conditions about the distributions must be verified. In practice, these tests
are applied when the sampling distributions of the data variables reasonably satisfy
the normal model.
Non-parametric tests make no assumptions regarding the distributions of the
data variables; only a few mild conditions must be satisfied when using most of
these tests. Since non-parametric tests make no assumptions about the distributions
of the data variables, they are adequate to small samples, which would demand the
distributions to be known precisely in order for a parametric test to be applied.
Furthermore, non-parametric tests often concern different hypotheses about
populations than do parametric tests. Finally, unlike parametric tests, there are non-
parametric tests that can be applied to ordinal and/or nominal data.
The use of fewer or milder conditions imposed on the distributions comes with a
price. The non-parametric tests are, in general, less powerful than their parametric
counterparts, when such a counterpart exists and is applied in identical conditions.
In order to compare the power of a test B with a test A, we can determine the
sample size needed by B, n B, in order to attain the same power as test A, using
sample size n A, and with the same level of significance. The following power-
efficiency measure of test B compared with A, η BA, is then defined:

n
η = A . 5.1
BA
n B

For many non-parametric tests (B) the power efficiency, η BA , relative to a
parametric counterpart (A) has been studied and the respective results divulged in
the literature. Surprisingly enough, the non-parametric tests often have a high
power-efficiency when compared with their parametric counterparts. For instance,
as we shall see in a later section, the Mann-Whitney test of central location, for two
independent samples, has a power-efficiency that is usually larger than 95%, when
compared with its parametric counterpart, the t test. This means that when applying
the Mann-Whitney test we usually attain the same power as the t test using a
sample size that is only 1/0.95 bigger (i.e., about 5% bigger).