# powerMediation.VSMc.cox: Power for testing mediation effect in cox regression based on... In powerMediation: Power/Sample Size Calculation for Mediation Analysis

## 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. `TRUE` means printing power; `FALSE` means not printing power.

## 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√{(1-ρ_{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 two-sided test. For one-sided tests, please double the significance level. For example, you can set `alpha=0.10` to obtain one-sided test at 5% significance level.

## Author(s)

Weiliang Qiu [email protected]

## References

Vittinghoff, E. and Sen, S. and McCulloch, C.E.. Sample size calculations for evaluating mediation. Statistics In Medicine. 2009;28:541-557.

`minEffect.VSMc.cox`, `ssMediation.VSMc.cox`
 ```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) ```