# powerEpiInt2: Power Calculation Testing Interaction Effect for Cox Proportional Hazards Regression

### Description

Power calculation testing interaction effect for Cox proportional hazards regression with two covariates for Epidemiological Studies. Both covariates should be binary variables. The formula takes into account the correlation between the two covariates.

### Usage

 1 powerEpiInt2(n, theta, psi, mya, myb, myc, myd, alpha = 0.05) 

### Arguments

 n total number of subjects. theta postulated hazard ratio. psi proportion of subjects died of the disease of interest. mya number of subjects taking values X_1=0 and X_2=0 obtained from a pilot study. myb number of subjects taking values X_1=0 and X_2=1 obtained from a pilot study. myc number of subjects taking values X_1=1 and X_2=0 obtained from a pilot study. myd proportion of subjects taking values X_1=1 and X_2=1 obtained from a pilot study. alpha type I error rate.

### Details

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(β_1 x_1+β_2 x_2 + γ (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(γ)=θ. Given the type I error rate α for a two-sided test, the power required to detect a hazard ratio as small as \exp(γ)=θ is

power=Φ≤ft(-z_{1-α/2}+√{\frac{n}{G}[\log(θ)]^2 p (1-p) ψ (1-ρ^2)}\right),

where ψ is the proportion of subjects died of the disease of interest, and

ρ=corr(X_1, X_2)=(p_1-p_0)\times√{\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).

### Value

The power of the test.

### References

Schmoor C., Sauerbrei W., and Schumacher M. (2000). Sample size considerations for the evaluation of prognostic factors in survival analysis. Statistics in Medicine. 19:441-452.

powerEpiInt.default0, powerEpiInt.default1
 1 2 3 4 5 6  # Example at the end of Section 4 of Schmoor et al. (2000). # mya, myb, myc, and myd are obtained from Table III on page 448 # of Schmoor et al. (2000). powerEpiInt2(n = 184, theta = 3, psi = 139 / 184, mya = 50, myb = 21, myc = 78, myd = 35, alpha = 0.05)