# Statistical test for the differences of the differences in two proportions

I have four data points:

Proportion A, N = N_A
Proportion B, N = N_B
Proportion C, N = N_C
Proportion D, N = N_D

I want to know if

(Proportion A - Proportion B) is significantly different to (Proportion C - Proportion D)

Does anyone know how you test for this?

I know how to do a normal proportions test but feel that that doesn't apply here as i'm looking for a test of the DIFFERENCE OF THE DIFFERENCE between two proportions?

The same method used for the Wald test for differences in proportions can be applied to your setting. The standard error of the double difference is $\sqrt{\sum_{i=1}^{4}p_{i}(1-p_{i})/n_{i}}$ which leads to a $z$-test (large-sample test assuming normality).

But beware of the scale. You would be making a strong assumption that the absolute risk scale is the scale on which the double difference might be zero if effects are irrelevant. This may not be the case. Frequently we run these types of contrasts on the logit scale and test $H_{0}$: ratio of odds ratios = 1. This is just the test for interaction in a binary logistic model.