Figure 3 shows similar power functions for exponential distributions,
where the Wilcoxon rank-sum test, the Wilcoxon signed-ranks test, and the
modified t test on ranks were substituted for the parametric t tests.
Comparison of the three power functions reveals that the outcome is almost
the same as in the case of the corresponding parametric tests applied to
normally distributed data. The Wilcoxon signed-ranks test was superior to
the Wilcoxon rank-sum test for paired data, while the modified t test on
ranks was slightly superior to both.
Apparently the modified t test corrected for the correlation resulting
from pairing, while at the same time the transformation to ranks
counteracted non-normality. Figures 4, 5, and 6 indicate similar outcomes
for lognormal, chi-square, half-normal, and uniform distributions, using
several sample sizes, population correlations, and significance levels. Note
that the power functions for the smaller sample sizes were more widely
separated, while convergence is evident for the larger sample sizes.
Table 8 compares Type I error probabilities when a sample
correlation is entered into equation (2) for each sample taken and when a
fixed population correlation is entered the equation for every sample. The
first section of the table, for the normal distribution, is the result of the t test
performed on scores. The remaining three sections, for non-normal
distributions, show the result of the t test on rank-transformed data. For
relatively small correlations and relatively large sample sizes, the Type I
error probabilities for both tests were about the same and close to the
nominal significance level.
SOME PRACTICAL IMPLICATIONS
For samples of size 25 or 50 from normal distributions, the modified t
test with a correction for correlation maintained Type I error rates close to
the significance level, increased power in the case of positive correlations,
and removed spurious increases in the probability of rejecting H0 in the case
of negative correlations. The power superiority of this test over the pairedsamples
t test is about what one would expect from the difference in degrees
of freedom. The difference became less marked as sample sizes increased to
100 and 400, presumably because the difference in the critical values of the
t statistic for N – 1 and 2N – 2 degrees of freedom decreases as N increases.
Nevertheless, the power of the modified test was equal to that of the pairedsamples
test for the larger sample sizes