Proportion test with null hypothesis p1=x*p2

Question

I'd like to compare the proportion of two sets of binomial data where the null hypothesis is not p1=p2, or p1-p2=x, but \begin{equation} H0: p1=x*p2 \end{equation}

I've been computing a z test as follow: \begin{equation} ztest=\frac{(p1-p2*x)}{\sqrt{\frac{p1*(1-p1)}{n1}+\frac{p2*(1-p2)}{n2}}} \end{equation}

I suspect this is not the proper way of doing this? Any help would be welcome!

Best, David

Answer 1

$Var(p2*x)=x^2*Var(p2)=x^2\frac{p2(1-p2)}{n2}$

You'd test \begin{equation} ztest=\frac{(p1-p2*x)}{\sqrt{\frac{p1*(1-p1)}{n1}+\frac{p2*x^2*(1-p2)}{n2}}} \end{equation} Where $0\leq p2*x\leq 1$

Note:- test if you have equality of Variance, then use paired-variance.