I am carrying out a study in which I have $3$ groups $n= 2$, $n= 5$, $n=17$. Each group has a different inner ear abnormality, following the fitting of an ear implant I have measured their speech (ordinal data) and hearing scores (continuous data) at 1 and 2 years.
The group with only $2$ patients in must be excluded from stats I believe. For the two remaining groups, most of the data are normally distributed and most variances are similar for speech (ordinal data), but not for the hearing scores (continuous data).
I wonder if a Mann-Whitney $U$-test should be used for the following reasons:
- Ordinal data DV: although the data are normally distributed and most variances are similar, ordinal means that only a M-W can be done.
- Scale Data DV: the data are only normally distributed with homogeneity of variance in 2/3 of the measures, therefore again M-W? Or should a $t$-test be used (but doesn't a $t$-test assume homogeneity of variance)?
Best Answer
Hannah, first you answered your own question. Because the data is ordinal, you have to use the M-W test. If not for this condition, you probably could have used both tests. In many hypothesis testing situations, it is not 100% clear what test to use (you test for Normality and result is inconclusive; or you have a very large sample whereby Normality is less critical of an issue). In such undetermined situations (this happens very often), I recommend you use both tests (t test, M-W). You will notice that very often both tests pretty much give you very similar results in terms of statistical significance level. So, when both tests concur on this topic your case is solved. You can state with confidence whether you accept or reject the null hypothesis. Much fewer cases will be anbiguous when the two tests reach opposite conclusion. In this case, you drill down and look at your data very closely and make a conservative assumption as to given the nature of the data what test may be more appropriate. Sometimes, it is still unclear. In such a situation, you may decide to go with the test that gives you the more conservative result or you may decide to go with the test that reduces the type of error you are more sensitive to (Type I vs Type II error).
But, in your case because of the ordinal data, the proper test is M-W. Just for fun, you could try using the t-test and see if the two tests would come to the same conclusion... knowing that in this case, you would have to select M-W.