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 epidemoilogical
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
$$n=\frac{\left(z_{1-\alpha/2}+z_{1-\beta}\right)^2\delta}{[\log(\theta)]^2 \psi},$$
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,
$$\delta=\frac{1}{p_{00}}+\frac{1}{p_{01}}+\frac{1}{p_{10}}
+\frac{1}{p_{11}},$$
and
\(p_{00}=Pr(X_1=0,\mbox{and}, X_2=0)\),
\(p_{01}=Pr(X_1=0,\mbox{and}, X_2=1)\),
\(p_{10}=Pr(X_1=1,\mbox{and}, X_2=0)\),
\(p_{11}=Pr(X_1=1,\mbox{and}, X_2=1)\).