Occasionally, the assumptions of the t-tests are seriously violated. In particular, if the type of data you have is ordinal in nature and not at least interval. On such occasions an alternative approach is to use nonparametric tests.
Nonparametric tests are also referred to as distribution-free tests. These tests have the obvious advantage of not requiring the assumption of normality or the assumption of homogeneity of variance. They compare medians rather than means and, as a result, if the data have one or two outliers, their influence is negated.
Parametric tests are preferred because, in general, they have more power.
If the two samples are not paired, Non-paired Mann-Whitney U Test can be applied.
If the two samples are paired, paired Wilcoxon Signed Rank Test can be applied. In Empower, the paired data should be organized by two variables (eg x1 x2), each observation represents one pair. If data for each pair was represented by one variable in two observation, and a pair ID variables was used for identifying the pairs, we can use Empower “Transpose multiple observations to multiple variables” function to restructure the data.
Two sides or one side:
The p-level reported with a t-test represents the probability of error associated with rejecting the hypothesis of no difference between the two group when, in fact, the hypothesis is true.
Some researchers suggest that if the difference is in the predicted direction, you can consider only one half (one "tail") of the probability distribution and thus divide the standard p-level reported with a t-test (a "two-tailed" probability) by two. Others, however, suggest that you should always report the standard, two-tailed t-test probability.
Below is the sample input window
Below is the sample output of the above:
Nonparametric two sample U test
Wilcoxon rank sum test with continuity correction
data: OZONE by MONTH.NEW
W = 127.5, p-value = 0.0001208
alternative hypothesis: true location shift is not equal to 0
Empower also output following box plot to show the distribution for each group.