powerEpiInt2 {powerSurvEpi}

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

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

Arguments

n

integer. total number of subjects.

theta

numeric. postulated hazard ratio.

psi

numeric. proportion of subjects died of the disease of interest.

mya

integer. number of subjects taking values X_1=0 and X_2=0 obtained from a pilot study.

myb

integer. number of subjects taking values X_1=0 and X_2=1 obtained from a pilot study.

myc

integer. number of subjects taking values X_1=1 and X_2=0 obtained from a pilot study.

myd

integer. number of subjects taking values X_1=1 and X_2=1 obtained from a pilot study.

alpha

numeric. 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(\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).

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.

See Also

powerEpiInt.default0, powerEpiInt.default1

Examples

  # 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)