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Let's say that medication A significantly increased the patients' health on average, but so did medication B (for independent sets of patients).

Obviously both medications increased the health of the patients on average, but is there a statistical test that could test whether the increase of one of the medications was significantly higher than the increase of the other medication?

For example, if medication A increased health by 7.3 (imagine a health score) and medication B increased health by 8.8. Both significant, but can we test whether 8.8 is significantly higher increase than 7.3.

My idea is create a column of increases for each medication and then run a t-test on these two columns. Is that feasible?

Are there other techniques that have a more sound foundation? I have found a statistical concept (and I tagged it) called "difference in differences"; Is that relevant here?

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  • $\begingroup$ This is a two sample t.test. Have the hypothesis as $H_0: \mu_A = \mu_B\\H_1: \mu_A<\mu_B$ and using the sample information, ie $n_i,~~ s_i^2$ You could compute the test statistic assuming equal variance or unequal variance $\endgroup$
    – Onyambu
    Commented Mar 11, 2023 at 0:16
  • $\begingroup$ @ onyambu There seem to be four groups: A before, B before, A after, and B after. How do you see this as a two-sample problem? $\endgroup$
    – Dave
    Commented Mar 11, 2023 at 3:39

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The regression for which you have included a tag works by having four groups.

  1. The group that will receive treatment A, before the treatment

  2. The group that will receive treatment B, before the treatment

  3. The group that received treatment A, after the treatment

  4. The group that received treatment B, after the treatment

The regression works by having an indicator variable for the treatment group and another indicator variable for before/after treatment. There is also an interaction between the two indicator variables, and that interaction term describes the difference between the reaction to treatment experienced by each of the two treatment groups.

In other words, depending on how you set up your indicator variables, the coefficient on interaction term will tell you exactly $\Delta A-\Delta B$, which seems to be what you want to quantify and test. The testing of this coefficient is through the usual testing of regression coefficients.

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