powerMediation (version 0.2.3)

ssMediation.Sobel: Sample size for testing mediation effectd (Sobel's test)

Description

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

Usage

ssMediation.Sobel(power, 
                  theta.1a, 
                  lambda.a, 
                  sigma.x, 
                  sigma.m,
                  rho2.mx, 
                  sigma.epsilon, 
                  n.lower = 1, 
                  n.upper = 1e+30, 
                  alpha = 0.05, 
                  verbose = TRUE)

Arguments

power

power of the test.

theta.1a

regression coefficient for the predictor in the linear regression linking the predictor $x$ to the mediator $m$ ($m_i=\theta_0+\theta_1 x_i + e_i, e_i\sim N(0, \sigma^2_e)$).

lambda.a

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

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 $y$ ($y_i=\gamma+\lambda m_i+ \lambda_2 x_i + \epsilon_i, \epsilon_i\sim N(0, \sigma^2_{\epsilon})$).

n.lower

lower bound of the sample size.

n.upper

upper bound of the sample size.

alpha

type I error rate.

verbose

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

Value

• n
  sample size.
• res.uniroot
  results of optimization to find the optimal sample size.

Details

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

Test statistic is based on Sobel's (1982) test: $$Z=\frac{\hat{\theta}_1\hat{\lambda}}{\hat{\sigma}_{\theta_1\lambda}}$$ where $\hat{\sigma}_{\theta_1\lambda}$ is the estimated standard deviation of the estimate $\hat{\theta}_1\hat{\lambda}$ using multivariate delta method: $$\sigma_{\theta_1\lambda}=\sqrt{\theta_1^2\sigma_{\lambda}^2+\lambda^2\sigma_{\theta_1}^2}$$ and $\sigma_{\theta_1}^2=\sigma_e^2/(n\sigma_x^2)$ is the variance of the estimate $\hat{\theta}_1$, and $\sigma_{\lambda}=\sqrt{\sigma_{\epsilon}^2/(n\sigma_m^2(1-\rho_{mx}^2))}$ is the variance of the estimate $\hat{\lambda}$, $\sigma_m^2$ is the variance of the mediator $m_i$.

From the linear regression $m_i=\theta_0+\theta_1 x_i+e_i$, we have the relationship $\sigma_e^2=\sigma_m^2(1-\rho^2_{mx})$. Hence, we can simply the variance $\sigma_{\theta_1, \lambda}$ to $$\sigma_{\theta_1\lambda}=\sqrt{\theta_1^2\frac{\sigma_{\epsilon}^2}{n\sigma_m^2(1-\rho_{mx}^2)}+\lambda^2\frac{\sigma_{m}^2(1-\rho_{mx}^2)}{n\sigma_x^2}}$$

References

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

See Also

powerMediation.Sobel, testMediation.Sobel

Examples

ssMediation.Sobel(power=0.8, 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)