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

Arguments
n 
sample size. 
b2 
regression coefficient for the mediator m in the logistic regression \log(p_i/(1p_i))=b0+b1 x_i + b2 m_i. 
sigma.m 
standard deviation of the mediator. 
p 
the marginal prevalence of the outcome. 
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 logistic regressions:
\log(p_i/(1p_i))=b0+b1 x_i + b2 m_i
Vittinghoff et al. (2009) showed that for the above logistic 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(p_i/(1p_i))=b_0+b_1 x_i + b_2 m_i
The reduced model is
\log(p_i/(1p_i))=b_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) p (1p)} 
, 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 p is the marginal prevalence of the outcome.
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.logistic
,
ssMediation.VSMc.logistic
Examples
1 2 3 4  # example in section 4 (page 545) of Vittinghoff et al. (2009).
# power = 0.8005793
powerMediation.VSMc.logistic(n = 255, b2 = log(1.5), sigma.m = 1,
p = 0.5, corr.xm = 0.5, alpha = 0.05, verbose = TRUE)

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