ssizeEpi {powerSurvEpi}

Sample Size Calculation for Cox Proportional Hazards Regression

Description

Sample size calculation for Cox proportional hazards regression with two covariates for Epidemiological Studies. The covariate of interest should be a binary variable. The other covariate can be either binary or non-binary. The formula takes into account competing risks and the correlation between the two covariates.

Usage

ssizeEpi(X1, 
	 X2, 
	 failureFlag, 
	 power, 
	 theta, 
	 alpha = 0.05)

Arguments

X1

numeric. a nPilot by 1 vector, where nPilot is the number of subjects in the pilot data set. This vector records the values of the covariate of interest for the nPilot subjects in the pilot study. X1 should be binary and take only two possible values: zero and one.

X2

numeric. a nPilot by 1 vector, where nPilot is the number of subjects in the pilot study. This vector records the values of the second covariate for the nPilot subjects in the pilot study. X2 can be binary or non-binary.

failureFlag

numeric. a nPilot by 1 vector of indicators indicating if a subject is failure (failureFlag=1) or alive (failureFlag=0).

power

numeric. postulated power.

theta

numeric. postulated hazard ratio.

alpha

numeric. type I error rate.

Details

This is an implementation of the sample size formula derived by Latouche et al. (2004) 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),

where the covariate X_1 is of our interest. The covariate X_1 has to be a binary variable taking two possible values: zero and one, while the covariate X_2 can be binary or continuous.

Suppose we want to check if the hazard of X_1=1 is equal to the hazard of X_1=0 or not. Equivalently, we want to check if the hazard ratio of X_1=1 to X_1=0 is equal to 1 or is equal to \exp(\beta_1)=\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}{ [\log(\theta)]^2 p (1-p) \psi (1-\rho^2)},

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).

p, \rho^2, and \psi will be estimated from a pilot study.

Value

n

the total number of subjects required.

p

the proportion that X_1 takes value one.

rho2

square of the correlation between X_1 and X_2.

psi

proportion of subjects died of the disease of interest.

Note

(1) The calculated sample size will be round up to an integer.

(2) The formula can be used to calculate sample size required for a randomized trial study by setting rho2=0.

(3) When rho2=0, the formula derived by Latouche et al. (2004) looks the same as that derived by Schoenfeld (1983). Latouche et al. (2004) pointed out that in this situation, the interpretations are different hence the two formulae are actually different. In Latouched et al. (2004), the hazard ratio \exp(\beta_1)=\theta measures the difference of effect of a covariate at two different levels on the subdistribution hazard for a particular failure, while in Schoenfeld (1983), the hazard ratio \theta measures the difference of effect on the cause-specific hazard.

References

Schoenfeld DA. (1983). Sample-size formula for the proportional-hazards regression model. Biometrics. 39:499-503.

Latouche A., Porcher R. and Chevret S. (2004). Sample size formula for proportional hazards modelling of competing risks. Statistics in Medicine. 23:3263-3274.

See Also

ssizeEpi.default

Examples

  # generate a toy pilot data set
  X1 <- c(rep(1, 39), rep(0, 61))
  set.seed(123456)
  X2 <- sample(c(0, 1), 100, replace = TRUE)
  failureFlag <- sample(c(0, 1), 100, prob = c(0.5, 0.5), replace = TRUE)

  ssizeEpi(X1 = X1, 
	   X2 = X2, 
	   failureFlag = failureFlag, 
           power = 0.80, 
	   theta = 2, 
	   alpha = 0.05)