powerMediation (version 0.2.4)

powerMediation.Sobel: Power for testing mediation effect (Sobel's test)

Description

Calculate power for testing mediation effect based on Sobel's test.

Usage

powerMediation.Sobel(n,  theta.1a,  lambda.a,  sigma.x,  sigma.m, rho2.mx,  sigma.epsilon,  alpha = 0.05,  verbose = TRUE)

Arguments

n

sample size.

theta.1a

regression coefficient for the predictor in the linear regression linking the predictor $x$ to the mediator $m$ ($m_i=\theta_0+\theta_{1a} x_i + e_i, e_i\sim N(0, \sigma^2_e)$).

lambda.a

regression coefficient for the mediator in the linear regression linking the predictor $x$ and the mediator $m$ to the outcome $y$ ($y_i=\gamma+\lambda_{a} m_i+ \lambda_2 x_i + \epsilon_i, \epsilon_i\sim N(0, \sigma^2_{\epsilon})$).

sigma.x

standard deviation of the predictor.

sigma.m

standard deviation of the mediator.

rho2.mx

square of the correlation between the predictor and the mediator.

sigma.epsilon

standard deviation of the random error term in the linear regression linking the predictor $x$ and the mediator $m$ to the outcome $y$ ($y_i=\gamma+\lambda_a m_i+ \lambda_2 x_i + \epsilon_i, \epsilon_i\sim N(0, \sigma^2_{\epsilon})$).

alpha

type I error.

verbose

logical. TRUE means printing power; FALSE means not printing power.

Value

power

power of the test for the parameter $\theta_{1a}\lambda_a$

delta

$\theta_1\lambda/(sd(\hat{\theta}_{1a})sd(\hat{\lambda}_a))$

Details

The power is for testing the null hypothesis $\theta_1\lambda=0$ versus the alternative hypothesis $\theta_{1a}\lambda_a\neq 0$ for the linear regressions: $$m_i=\theta_0+\theta_{1a} x_i + e_i, e_i\sim N(0, \sigma^2_e)$$ $$y_i=\gamma+\lambda_a m_i+ \lambda_2 x_i + \epsilon_i, \epsilon_i\sim N(0, \sigma^2_{\epsilon})$$

Test statistic is based on Sobel's (1982) test: $$Z=\frac{\hat{\theta}_{1a}\hat{\lambda_a}}{\hat{\sigma}_{\theta_{1a}\lambda_a}} $$ where $\hat{\sigma}_{\theta_{1a}\lambda_a}$ is the estimated standard deviation of the estimate $\hat{\theta}_{1a}\hat{\lambda_a}$ using multivariate delta method: $$\sigma_{\theta_{1a}\lambda_a}=\sqrt{\theta_{1a}^2\sigma_{\lambda_a}^2+\lambda_a^2\sigma_{\theta_{1a}}^2}$$ and $\sigma_{\theta_{1a}}^2=\sigma_e^2/(n\sigma_x^2)$ is the variance of the estimate $\hat{\theta}_{1a}$, and $\sigma_{\lambda_a}^2=\sigma_{\epsilon}^2/(n\sigma_m^2(1-\rho_{mx}^2))$ is the variance of the estimate $\hat{\lambda_a}$, $\sigma_m^2$ is the variance of the mediator $m_i$.

From the linear regression $m_i=\theta_0+\theta_{1a} x_i+e_i$, we have the relationship $\sigma_e^2=\sigma_m^2(1-\rho^2_{mx})$. Hence, we can simply the variance $\sigma_{\theta_{1a}, \lambda_a}$ to $$\sigma_{\theta_{1a}\lambda_a}=\sqrt{\theta_{1a}^2\frac{\sigma_{\epsilon}^2}{n\sigma_m^2(1-\rho_{mx}^2)}+\lambda_a^2\frac{\sigma_{m}^2(1-\rho_{mx}^2)}{n\sigma_x^2}}$$

References

Sobel, M. E. Asymptotic confidence intervals for indirect effects in structural equation models. Sociological Methodology. 1982;13:290-312.

See Also

ssMediation.Sobel, testMediation.Sobel

Examples

 powerMediation.Sobel(n=248, theta.1a=0.1701, lambda.a=0.1998, 
   sigma.x=0.57, sigma.m=0.61, rho2.mx=0.3, sigma.epsilon=0.2, 
   alpha = 0.05, verbose = TRUE)