# Paired or Unpaired?

#### Abhishek Hoshing

##### New Member
I recently concluded a clinical research which compared a certain parameter of one eye of a group of individuals to the the same parameter of the other eye of the same group of individuals. My statistical knowledge says these are unpaired (Independent) samples and as the parameter measured was a non - normally distributed continuous variable of 44 individuals, I used the Mann Whitney U test to compare the medians. I submitted the paper for publication and the reviewer came back with the following..

"The authors have performed the Mann Whitney U test while looking for differences between eyes. This is an example of paired data, between related samples, and calls for the non-parametric equivalent of the paired t-test, and thus the appropriate test ought to be the Wilcoxon test. But even here with 44 pairs, a paired t-test would be deemed a superior approach and would be quite appropriate, even if the sample data are non-normal, since with samples of this size the Central Limit Theorum kicks in, and justifies using statistical tests for parametric data. "

My question is ... Is it correct to call it Paired Sample? and Why?
Does it mean I have to re do my calculations?
Is it correct to say that for samples as small as these paired T test would be better?

#### Karabiner

##### TS Contributor
You use the right eye and the left eye within the same individuals,
therefore the right eyes are not independent of the left eyes.
Whether any left eye enters the study is determined by whether the right
eye from the same individual has entered the study (and vice versa).
And of course, the characteristics of left vs. right eyes are highly
correlated within the same individuals.

By the way, in case of a dependent samples t-test not the two measurements,
but the within-person differences between them should preferably be
sampled from a normal distribution. But as the reviewer told you correctely,
this assumption is no more needed with n = 44 (if the sample size were smaller,
[n < 30 or so] possibly the "nonparametric" Wilcoxos signed rank test
would have been the better alternative).

With kind regards

Karabiner

#### Abhishek Hoshing

##### New Member
Thank you for the lucid explanation Mr Karabiner. So for this scenario I would be ok even if I did paired Sample T test. am I correct.

#### ondansetron

##### TS Contributor
Other important issue that people commonly misunderstand: the Mann-Whitney U test (wilcoxon rank sum) and the wilcoxon signed rank test (paired samples) do not compare medians, in general. Only under much more restrictive assumptions could this be true, and then I believe you have enough assumptions to use a t test anyhow.

#### ondansetron

##### TS Contributor
Thank you for the lucid explanation Mr Karabiner. So for this scenario I would be ok even if I did paired Sample T test. am I correct.
Yes, paired t test would be a good idea. Do you have other variables like age, gender, or comorbidities?

However, the reviewer is wrong in that data are "non parameteric” or “parametric”: these terms refer to larger groups of statistical tests or estimation procedures. It is incorrect to call data “parametric” or “non parametric”. The suggested approach is correct/reasonable but some of the explanation isn’t accurate.

The central limit theorem doesn’t necessarily “kick in” in a clear on or off fashion as it’s an asymptotic idea and how “large” the sample needs to be depends on how abnormal the underlying distribution is.

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