# Statistical t test for unequal sample sizes (paired samples)

I am analyzing the paired samples. Patients with drug at time point 0, at time point 12 weeks, time point 24 weeks.

I would like to see the significant differences in the patients over the time points. In other words I would like to see how effective the drug is?

One problem here is I have 10 samples at zero time point, the samples have been given the drug at 12 weeks time point but two samples are missing for some reason. The same for third time point as well. My guess is I wont be able to use paired t test now because of unequal sample sizes.

Which statistical test would be appropriate for this scenario? Any help/suggestions would be much appreciated.

Here is my experimental design.  T0 T12 T24 n=10 8 8 

• The two without follow-up data will not be useful to assess change assuming it is the same two at time 12 and time 24. – mdewey Oct 25 '17 at 14:22
• You can still use the paired $t$-test. You need to define $\kappa = n_1/n_2$, which is the ratio of the two sample sets. Use this to compute the $p$-value. – Maxtron Oct 25 '17 at 14:58

If you make the assumption that some samples are missing at random, you can:

• Conduct a paired test, leaving out the data for people that you don't have

• Conduct an unpaired test for the whole data

Then you get 2 p-values, and you have to

• Multiple adjust, that is, e.g., with a Bonferroni correction. Multiply the p-values by 2, if, after that, you reject one of them, you can reject your original null-hypothesis

Issues with the approach

• If the data is not missing at random, and in the worst case, the effect that you are testing for is inverted between the paired data you have, and the paired data you don't have, your statistic will be wrong

If you don't want to make this assumption, I don't see how you can leverage the pairedness of your data, since you don't know why the data is missing. That is, you will have to conduct an unpaired test

If the three time points are important, you should use an analysis for repeated data. But your sample size is small...(see for example http://bales.faculty.ucdavis.edu/wp-content/uploads/sites/250/2015/08/Educational-and-Psychological-Measurement-2015-Muth.pdf).

An alternative would be to compare T12 to T0 on the one hand, and T24 to T0 on the other. What is the type of outcome?

• Yes, three time points are important. I tried to compare T12 to T0 separately and T24 to T0 separately and tried to see what is common and what is unique. – Prathyusha Bachali Oct 25 '17 at 16:53
• I would also like to see the genes that are differentially expressed in more than one condition. I am doing the F-test. I am not sure if I am going in correct direction or not. – Prathyusha Bachali Oct 25 '17 at 16:55