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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:

  1. Ordinal data DV: although the data are normally distributed and most variances are similar, ordinal means that only a M-W can be done.
  2. 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)?
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  • $\begingroup$ Welcome to the site, @Hannah. I took the liberty of editing your question to try to make it clearer. Please make sure it still says what you want it to. $\endgroup$ – gung - Reinstate Monica Apr 12 '14 at 15:38
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    $\begingroup$ I think we will need more information. Do you have measures at baseline (before implants)? What do you mean that the ordinal data are normally distributed? What do you mean that "most of the data are normally distributed"? Can you post a sample of your data, or a plot of them? $\endgroup$ – gung - Reinstate Monica Apr 12 '14 at 15:39
  • $\begingroup$ Hello Jung and Gaetan, Thank-you very much for your replies. I have measured scores for speech and language using two scales known as CAP and SIR, which are rating scales from 1-7 and 1-5. These have been measured pre-implant and post implant at 12 and 24 months for both measures (CAP & SIR). I have also measured the change in scores for both CAP and SIR from pre-implant to post implant at 12 months, and then post implant at 12 months compared to 24 months. In addition, hearing was tested before implant and at 12 and 24 months post implant. $\endgroup$ – Hannah Apr 12 '14 at 16:04
  • $\begingroup$ I used Shapiro-Wilks to test normality by group, 1, 2 and 3- (IV) groups are based on their inner ear abnormality. This showed normal distribution for most of the variables, most results were > 0.05 (DV/dependant measures). I also ran Levine's test to check for homogeneity of variance and again for most of the variables, the result was > 0.05 showing variances were similar. However for hearing thresholds only 1 variable out of 3 had homogeneity of variance. $\endgroup$ – Hannah Apr 12 '14 at 16:05
  • $\begingroup$ As Gaetan agreed and confirmed the Mann-Whitney seems appropriate for the CAP/SIR outcomes as these are ordinal. However hearing thresholds are scale continuous scale (dB HL), they are normally distributed but homogeneity of variance is violated in 2/3 variables (DV's) measured. Therefore I wondered if again Mann-Witney would help address this issue? I wonder if a t-test could be used with a cautionary approach that not all assumptions are met. $\endgroup$ – Hannah Apr 12 '14 at 16:06
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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.

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    $\begingroup$ (1) the Jarque-Bera test is based on the asymptotic distribution of the test statistic, and is not suitable for small or even not-so-small sample sizes (2) it is not a good idea to use goodness of fit tests in this situation. $\endgroup$ – Glen_b -Reinstate Monica Apr 12 '14 at 15:43
  • $\begingroup$ Glen_b, thanks for the input. I edited my response by removing my recommendation of using the Jarques-Berra test in this situation. $\endgroup$ – Sympa Apr 12 '14 at 16:38
  • $\begingroup$ Many thanks, would the Mann Whitney also be appropriate for the data I have measured which is scale. The assumptions say it is applicable to both scale and ordinal data. The issue with the t-test is that it assumes homogeneity of variances- my scale data does not meet this assumption but is normally distributed. I wonder if M-W would therefore be more robust for this outcome I am measuring? $\endgroup$ – Hannah Apr 12 '14 at 16:59
  • $\begingroup$ Hannah, as long as you can rank the values, Mann Whitney will work and be meaningful. $\endgroup$ – Sympa Apr 12 '14 at 17:20
  • $\begingroup$ If anyone is interested reading this, I have done some further reading and apparently a test called Kolmogorov-Smirnov Z (not to be confused with the similarly named test for normality) is much the same as Mann-Whitney but tends to have a better power than the M-W when your sample size is under 25! May be useful for anyone reading. $\endgroup$ – Hannah Apr 12 '14 at 18:51

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