This is an implementation of the power 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 power
required to detect a hazard ratio as small as \(\exp(\gamma)=\theta\) is
$$power=\Phi\left(-z_{1-\alpha/2}+\sqrt{\frac{n}{G}[\log(\theta)]^2 p (1-p) \psi (1-\rho^2)}\right),$$
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
$$G=\frac{[(1-q)(1-p_0)p_0+q(1-p_1)p_1]^2}{(1-q)q (1-p_0)p_0 (1-p_1) p_1},$$
and
\(p0=Pr(X_1=1 | X_2=0)=myc/(mya+myc)\),
\(p1=Pr(X_1=1 | X_2=1)=myd/(myb+myd)\),
\(p=Pr(X_1=1)=(myc+myd)/n_{obs}\),
\(q=Pr(X_2=1)=(myb+myd)/n_{obs}\),
\(n_{obs}=mya+myb+myc+myd\).
\(p_{00}=Pr(X_1=0,\mbox{and}, X_2=0)\),
\(p_{01}=Pr(X_1=0,\mbox{and}, X_2=1)\),
\(p_{10}=Pr(X_1=1,\mbox{and}, X_2=0)\),
\(p_{11}=Pr(X_1=1,\mbox{and}, X_2=1)\).