ssMediation.Sobel {powerMediation}

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,
                  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_{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.

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})).

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.

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_{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}}

Value

n

sample size.

res.uniroot

results of optimization to find the optimal 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 [e-mail]

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, sigma.epsilon=0.2, 
   alpha = 0.05, verbose = TRUE)