powerEpiInt.default1 {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

powerEpiInt.default1(n, 
		     theta, 
		     psi, 
		     p00, 
		     p01, 
		     p10, 
		     p11, 
		     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.

p00

numeric. proportion of subjects taking values X_1=0 and X_2=0, i.e., p_{00}=Pr(X_1=0,\mbox{and}, X_2=0).

p01

numeric. proportion of subjects taking values X_1=0 and X_2=1, i.e., p_{01}=Pr(X_1=0,\mbox{and}, X_2=1).

p10

numeric. proportion of subjects taking values X_1=1 and X_2=0, i.e., p_{10}=Pr(X_1=1,\mbox{and}, X_2=0).

p11

numeric. proportion of subjects taking values X_1=1 and X_2=1, i.e., p_{11}=Pr(X_1=1,\mbox{and}, X_2=1).

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 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 power required to detect a hazard ratio as small as \exp(\gamma)=\theta is:

power=\Phi\left(-z_{1-\alpha/2}+\sqrt{\frac{n}{\delta}[\log(\theta)]^2 \psi}\right),

where z_{a} is the 100 a-th percentile of the standard normal distribution,

\delta=\frac{1}{p_{00}}+\frac{1}{p_{01}}+\frac{1}{p_{10}} +\frac{1}{p_{11}},

\psi is the proportion of subjects died of the disease of interest, 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).

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, powerEpiInt2

Examples

  # Example at the end of Section 4 of Schmoor et al. (2000).
  # p00, p01, p10, and p11 are calculated based on Table III on page 448
  # of Schmoor et al. (2000).
  powerEpiInt.default1(n = 184, 
		       theta = 3, 
		       psi = 139 / 184,
                       p00 = 50 / 184, 
		       p01 = 21 / 184, 
		       p10 = 78 / 184, 
		       p11 = 35 / 184,
                       alpha = 0.05)