# T-test for paired differences between sample means

Suppose I have pre-treatment samples and post-treatment samples for a group of patients. Let the treatment effect on each patient be the difference between the two sample means. Now I have a new treatment and I want to test whether this new treatment has the same treatment effect, i.e. does the sample mean difference follow the same distribution of the old treatment?

I know there are paired t-tests and 2-sample t-tests, but what's the correct t-test for 2 differences between sample means? Note that the definition of treatment effect cannot be changed. Pre-treatment and post-treatment samples do not necessarily have the same variance.

EDIT: Does it make sense to perform a welch's test on the means of mean differences? Also, the sample sizes for the old and new treatment are highly unbalanced: 1000 vs 50

But, in principle, the effect size difference between the control and treatment 2 should equal the sum of the effect size difference for control vs treatment 1 and Treatment 1 vs Treatment 2 I.e.,$$E_{CT_{1}}+ E_{T_{1}T_{2}} = E_{CT_{2}}$$.