# statistical test for two populations of proportions

I performed an experiment in which I measured the equilibration of a protein across two nuclei in a binucleated cell. This equilibration is expressed in percentages or proportions. I measured this for 18 binucleated cells at two different times. I expect the equilibration to increase overtime which is what I see. What is the appropriate statistical test for this kind of data?

Timepoint 1 Timepoint 2
0.463173601 0.841653822
0.492508565 0.552802747
0.491286736 0.770470526
0.485094383 0.803155459
0.43909895  0.834093948
0.576810606 0.8631326
0.53173807  0.654547816
0.357130791 0.847670332
0.565749794 0.678070763
0.576551079 0.799976903
0.38123646  0.714282671
0.511302996 0.74826545
0.559493963 0.713703576
0.659356403 0.686216656
0.652567058 0.789869839
0.594203285 0.862964781
0.631375163 0.752228227
0.470961189 0.817424964


EDIT -> the two measurement are not on the same cell. ie I measure 18 cells at timepoint 1 and then 18 different cells at timepoint 2

• Consider isolonic regression. I will supply some references. Commented Jan 20, 2017 at 23:56

It turns out that the OP did not mention that this is not paired data. So the Mann-Whitney U test or unpaired t-testing can be used because the proportions themselves are not significantly non-normally distributed. I had worked this out as paired data. Suggest you calculate it yourself; it's simple enough.

Solving this problem as normal is good enough. Solving it as bounded fractions on $0<x<1$ is overkill.

• I should have probably mentioned that it is not the same cell I perform the measurement on. It's 18 cells at timepoint 1 and then 18 different cells at timepoint 2 (since the process of measurement destroys the sample). Does this change the tests one should use? Commented Jan 26, 2017 at 21:48
• @mistakeNot. Yes. You should not use a paired approach. Commented Jan 27, 2017 at 1:33

Wikipedia is a good source: https://en.wikipedia.org/wiki/isotonic_regression .

Two books that cover the subject are

1) Robertson, T., Wright, F. T. and Dykstra, R. L. (1988) Order Restricted Inference. Wiley.

2) Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D.(1972). Statistical Inference Under Order Restrictions: The theory of Isotonic Regression. Wiley.

You will find that this produces the closest monotonic function (increasing or decreasing) to the data.

• 1) Please check your link, spelling perhaps? I get a non-page on it. 2) I do not doubt that your answer is correct, but, is it important?
– Carl
Commented Jan 21, 2017 at 0:36
• @Carl The link works. It takes you to a wikipedia article titled Isotnic Regression. There was no spelling error I tested it myself. Commented Jan 21, 2017 at 0:39
• @Carl That is a different wikipedia article. Mine has the title "Isotonic/Regression" and yours is "Isotonic Regression". Commented Jan 21, 2017 at 0:48
• @Carl It really doesn't matter. Both articles are exactly the same except for the title I think. Commented Jan 21, 2017 at 0:53
• When I click the link in the answer, I get "Wikipedia does not have an article with this exact name". Please fix the link. Commented Jan 21, 2017 at 1:31

To test the statistical significance between your measurements at two points in time, you could try a t-test or a Mann-Whitney U test, or a simple sign test. (The direction of the difference is always in favor of the later measurement. The probability of getting this by chance is 2 to the 18th.)