T-test for paired differences between sample means

Question

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

Answer 1

I am not certain what you mean by "t-test for two differences" but there can only be one difference between two means. Adding a second treatment would require at least one other mean for the second treatment sample (realistically, you would want to sample for the initial control or initial treatment again to limit the influence of exogenous confounding variables).

But any time I have done a new treatment after testing a control and an earlier treatment I either compared the new treatment to the control or to the prior treatment. In fact, both options are frequently used on pharmaceutical research for Placebo controlled trials and for treatment-controlled trials (the latter being where the prior accepted treatment, which has known effects, was used as the comparator/control for ethical reasons such as not denying the control group access to an effective treatment).

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}} $.

Hope that clarifies things. If it doesn't, please try to clarify what you mean by two differences between sample means. E.g., what are your control and treatmetn variables. What are you trying to measure?