Power for testing mediation effect in cox regression based on Vittinghoff, Sen and McCulloch's (2009) method
Description
Calculate Power for testing mediation effect in cox regression based on Vittinghoff, Sen and McCulloch's (2009) method.
Usage
1 2 3 4 5 6 7  powerMediation.VSMc.cox(n,
b2,
sigma.m,
psi,
corr.xm,
alpha = 0.05,
verbose = TRUE)

Arguments
n 
sample size. 
b2 
regression coefficient for the mediator m in the cox regression \log(λ)=\log(λ_0)+b1 x_i + b2 m_i, where λ is the hazard function and λ_0 is the baseline hazard function. 
sigma.m 
standard deviation of the mediator. 
psi 
the probability that an observation is uncensored, so that the number of event d= n * psi, where n is the sample size. 
corr.xm 
correlation between the predictor x and the mediator m. 
alpha 
type I error rate. 
verbose 
logical. 
Details
The power is for testing the null hypothesis b_2=0 versus the alternative hypothesis b_2\neq 0 for the cox regressions:
\log(λ)=\log(λ_0)+b_1 x_i + b_2 m_i,
where λ is the hazard function and λ_0 is the baseline hazard function.
Vittinghoff et al. (2009) showed that for the above cox regression, testing the mediation effect is equivalent to testing the null hypothesis H_0: b_2=0 versus the alternative hypothesis H_a: b_2\neq 0.
The full model is
\log(λ)=\log(λ_0)+b_1 x_i + b_2 m_i
The reduced model is
\log(λ)=\log(λ_0)+b_1 x_i
Vittinghoff et al. (2009) mentioned that if confounders need to be included
in both the full and reduced models, the sample size/power calculation formula
could be accommodated by redefining corr.xm
as the multiple
correlation of the mediator with the confounders as well as the predictor.
Value
power 
power for testing if b_2=0. 
delta 
b_2σ_m√{(1ρ_{xm}^2) psi} 
, where σ_m is the standard deviation of the mediator m, ρ_{xm} is the correlation between the predictor x and the mediator m, and psi is the probability that an observation is uncensored, so that the number of event d= n * psi, where n is the sample size.
Note
The test is a twosided test. Code for onesided tests will be added later.
Author(s)
Weiliang Qiu stwxq@channing.harvard.edu
References
Vittinghoff, E. and Sen, S. and McCulloch, C.E.. Sample size calculations for evaluating mediation. Statistics In Medicine. 2009;28:541557.
See Also
minEffect.VSMc.cox
,
ssMediation.VSMc.cox
Examples
1 2 3 4 5  # example in section 6 (page 547) of Vittinghoff et al. (2009).
# power = 0.7999916
powerMediation.VSMc.cox(n = 1399, b2 = log(1.5),
sigma.m = sqrt(0.25 * (1  0.25)), psi = 0.2, corr.xm = 0.3,
alpha = 0.05, verbose = TRUE)

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