This is an implementation of the sample size calculation formula
derived by Schmoor et al. (2000) for
the following Cox proportional hazards regression in the epidemiological
studies:
$$h(t|x_1, x_2)=h_0(t)\exp(\beta_1 x_1+\beta_2 x_2 + \gamma (x_1 x_2)),$$
where both covariates \(X_1\) and \(X_2\) are binary variables.
Suppose we want to check if
the hazard ratio of the interaction effect \(X_1 X_2=1\) to \(X_1 X_2=0\) is equal to \(1\)
or is equal to \(\exp(\gamma)=\theta\).
Given the type I error rate \(\alpha\) for a two-sided test, the total
number of subjects required to achieve a power of \(1-\beta\) is
$$n=\frac{\left(z_{1-\alpha/2}+z_{1-\beta}\right)^2 G}{
[\log(\theta)]^2 \psi (1-p) p (1-\rho^2)
},$$
where \(z_{a}\) is the \(100 a\)-th percentile of the standard normal distribution, \(\psi\) is the proportion of subjects died of
the disease of interest, and
$$\rho=corr(X_1, X_2)=(p_1-p_0)\times\sqrt{\frac{q(1-q)}{p(1-p)}},$$
and
\(p=Pr(X_1=1)\), \(q=Pr(X_2=1)\), \(p_0=Pr(X_1=1|X_2=0)\),
and \(p_1=Pr(X_1=1 | X_2=1)\), and
$$G=\frac{[(1-q)(1-p_0)p_0+q(1-p_1)p_1]^2}{(1-q)q (1-p_0)p_0 (1-p_1) p_1}$$.
If \(X_1\) and \(X_2\) are uncorrelated, we have \(p_0=p_1=p\)
leading to \(1/[(1-q)q]\). For \(q=0.5\), we have \(G=4\).