permutation/bootstrapping for testing difference of difference

I have a situation where I have 3 treatments/conditions A, B, C. For each condition I have multiple trials.

I compare the means of trials under each condition A, B and C. This gives me 3 values.

Then I am computing a difference A-B and A-C. I need to obtain confidence interval/error bars for the single value obtained when I do the subtraction of mean response to condition A and mean response to condition B (A-B) same goes for A-C. Essentially I want to ask is the difference seen in mean response A-B different from mean response A-C.

I know potentially some bootstrapping/permutation test is what is needed but not sure which way to do this since its not a direct comparison of mean of A vs mean of B but rather A-B compared to A-C. Help will be appreciated.

• Why aren't you just testing B vs. C? Getting the difference from A for both only changes the mean B vs. C by a constant but increases each ones' variability.
– John
Commented Aug 16, 2013 at 4:03
• What I omitted is that A-B and A-C go through a non-linear function gfp and I need a mean and confidence on the output of gfp values. Commented Aug 16, 2013 at 6:17
• Then can you please put the information that we need to answer the question ... in the question? I'd start with an explanation of whatever that comment was trying to explain. Commented Aug 16, 2013 at 6:32

1 Answer

Consider you want to test whether the population mean of A-B = the population mean of A-C

That is:

$H_0: \mu_{A-B} - \mu_{A-C} = 0$ vs its negation

But this is testing whether $(\mu_A - \mu_B) - (\mu_A - \mu_C)$ differs from 0.

Cancel out the $\mu_A$ terms leaving: $H_0: \mu_C - \mu_B = 0$

This is fairly easy to test for.