Paired t-test equivalent for non-normal, heteroscedastic data

#1
Greetings good people,

I work on a project where a number of subjects (N = 5) with a particular disease were measured in two specific regions of their bodies (say, region1 and region2) using a specific medical instrument. Based on the pathophysiology of the disease, region1 should give consistently higher values than region2 per subject.

I would now like to test whether this is indeed the case. The way I see it, this would be perfect for a paired t-test between region1 and region2. However, the data are both non-normal and heteroscedastic (so, Wilcoxon's signed rank test is also not appropriate).

Does anyone know of an alternative? Many thanks in advance!
 
#3
Thank you hlsmith! Unfortunately, I am not allowed to post the data here. However, I suspected that a permutation would be a solution - honestly, I tried to avoid it because I am not very knowledgeable on the matter. I will try to look into it. Thank you again for your help!
 

Karabiner

TS Contributor
#4
I have never heard of the idea that for the Wilcoxon signed rank test normality of the data or heteroscedascity of the data matters.

But if you are in doubt you can perhaps use the sign test (although I don't know whether with n=5 a result p < 5% is at all possible).

Or maybe you reconsider how much sense it will make to base general statements on tests with n=5.

With kind regards

Karabiner
 
#5
Hello Karabiner! Thank you for your answer. I might say something stupid, but I thought the Wilcoxon signed rank test has similar assumptions to the Mann-Whitney U test... am I wrong? If so, the literature for the Mann-Whitney U test says that it is sensitive to departures for normality and homoscedacity if they happen simultaneously.

About the point you make with regards to generalizations with a small sample size n=5: I completely agree, and I do not plan on making any generalizations or bold claims. This is a pilot study, and the statistics will be mainly used for internal consumption. However, I have faced the same problem with larger sample sizes (e.g. N = 30), and I have so far found no answers by searching the web, so I thought it would be a good opportunity to get an expert answer/opinion.
 

Karabiner

TS Contributor
#6
I thought the Wilcoxon signed rank test has similar assumptions to the Mann-Whitney U test... am I wrong?
I do not know why you should think so, The tests are different, require different measurement levels
of the dependent variablem, and are therefore designed for different problems.
But, since the U test is for ordinal data, normality considerations or homoscedascitiy considerations do
not apply (that can only be an issue in case of interval scaled dependent variables). Insofar you are
indeed right - both tests do not require normality or homoscedasxcity.
If so, the literature for the Mann-Whitney U test says that it is sensitive to departures for normality and homoscedacity if they happen simultaneously.
That does not make much sense. Mann-Whitney is for ranked data (ordinal data), so there is no normality or
nonormality, no hetero- or homoscedascity, and of course, there are no means Only if someone has the idea to use
that test for the comparison of means for interval scaled variables, this concepts could come into play. But I
do not know whether it makes any practical sense to use a rank based test for that purpose, since the data
would have to fulfil a dozen of usually unfulfilled distributional assumptions before test could make statements
about mean differences.

About the point you make with regards to generalizations with a small sample size n=5: I completely agree, and I do not plan on making any generalizations or bold claims. This is a pilot study, and the statistics will be mainly used for internal consumption.
So you could just work with the descriptive statistics.
However, I have faced the same problem with larger sample sizes (e.g. N = 30), and I have so far found no answers by searching the web, so I thought it would be a good opportunity to get an expert answer/opinion.
If you have problems with the distribution of the differences (not of the data, of course, since
not the data are analyses here but their within-subject differences), then you could use Wilcoxon
signed rank test. For t-test, problems with normality of the differences can be ignored if
sample size is large enough (n > 30 or som central limit theorem).

With kind regards

Karabiner
 

obh

Active Member
#7
I have never heard of the idea that for the Wilcoxon signed rank test normality of the data or heteroscedascity of the data matters.

But if you are in doubt you can perhaps use the sign test (although I don't know whether with n=5 a result p < 5% is at all possible).

Or maybe you reconsider how much sense it will make to base general statements on tests with n=5.

With kind regards

Karabiner
With Wilcoxon signed rank test, you can get significant results with n=5. (I think this is the minimum)
 
#8
Thank you everyone for the valuable input! I obviously need to read more about the matter, but you have given me clear directions.

Maybe my original confusion comes from the fact that in R, which I use for my analyses, both the Wilcoxon Rank Sum and Signed Tests are run with the same function (which also refers to the Mann Whitney test).
 

ondansetron

TS Contributor
#9
Thank you everyone for the valuable input! I obviously need to read more about the matter, but you have given me clear directions.

Maybe my original confusion comes from the fact that in R, which I use for my analyses, both the Wilcoxon Rank Sum and Signed Tests are run with the same function (which also refers to the Mann Whitney test).
Mann-Whitney U test (Mann-Whitney-Wilcoxon) is the same as the Wilcoxon rank-sum test; these are for unpaired samples. The Wilcoxon signed rank test is for paired samples. Not sure if the terminology was causing confusion.