powerMediation.Sobel {powerMediation}R Documentation

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,
                     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 xx to the mediator mm (mi=θ0+θ1axi+ei,eiN(0,σe2)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 xx and the mediator mm to the outcome yy (yi=γ+λami+λ2xi+ϵi,ϵiN(0,σϵ2)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 xx and the mediator mm to the outcome yy (yi=γ+λami+λ2xi+ϵi,ϵiN(0,σϵ2)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.

Details

The power is for testing the null hypothesis θ1λ=0\theta_1\lambda=0 versus the alternative hypothesis θ1aλa0\theta_{1a}\lambda_a\neq 0 for the linear regressions:

mi=θ0+θ1axi+ei,eiN(0,σe2)m_i=\theta_0+\theta_{1a} x_i + e_i, e_i\sim N(0, \sigma^2_e)

yi=γ+λami+λ2xi+ϵi,ϵiN(0,σϵ2)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=θ^1aλa^σ^θ1aλaZ=\frac{\hat{\theta}_{1a}\hat{\lambda_a}}{\hat{\sigma}_{\theta_{1a}\lambda_a}}

where σ^θ1aλa\hat{\sigma}_{\theta_{1a}\lambda_a} is the estimated standard deviation of the estimate θ^1aλa^\hat{\theta}_{1a}\hat{\lambda_a} using multivariate delta method:

σθ1aλa=θ1a2σλa2+λa2σθ1a2\sigma_{\theta_{1a}\lambda_a}=\sqrt{\theta_{1a}^2\sigma_{\lambda_a}^2+\lambda_a^2\sigma_{\theta_{1a}}^2}

and σθ1a2=σe2/(nσx2)\sigma_{\theta_{1a}}^2=\sigma_e^2/(n\sigma_x^2) is the variance of the estimate θ^1a\hat{\theta}_{1a}, and σλa2=σϵ2/(nσm2(1ρmx2))\sigma_{\lambda_a}^2=\sigma_{\epsilon}^2/(n\sigma_m^2(1-\rho_{mx}^2)) is the variance of the estimate λa^\hat{\lambda_a}, σm2\sigma_m^2 is the variance of the mediator mim_i.

From the linear regression mi=θ0+θ1axi+eim_i=\theta_0+\theta_{1a} x_i+e_i, we have the relationship σe2=σm2(1ρmx2)\sigma_e^2=\sigma_m^2(1-\rho^2_{mx}). Hence, we can simply the variance σθ1a,λa\sigma_{\theta_{1a}, \lambda_a} to

σθ1aλa=θ1a2σϵ2nσm2(1ρmx2)+λa2σm2(1ρmx2)nσx2\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

power

power of the test for the parameter θ1aλa\theta_{1a}\lambda_a

delta

θ1λ/(sd(θ^1a)sd(λ^a))\theta_1\lambda/(sd(\hat{\theta}_{1a})sd(\hat{\lambda}_a))

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 stwxq@channing.harvard.edu

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

[Package powerMediation version 0.3.4 Index]