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

In powerMediation: Power/Sample Size Calculation for Mediation Analysis

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=θ_0+θ_{1a} x_i + e_i, e_i\sim N(0, σ^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=γ+λ_a m_i+ λ_2 x_i + ε_i, ε_i\sim N(0, σ^2_{ε})).
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=γ+λ_a m_i+ λ_2 x_i + ε_i, ε_i\sim N(0, σ^2_{ε})).
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 θ_1λ=0 versus the alternative hypothesis θ_{1a}λ_a\neq 0 for the linear regressions:

m_i=θ_0+θ_{1a} x_i + e_i, e_i\sim N(0, σ^2_e)

y_i=γ+λ_a m_i+ λ_2 x_i + ε_i, ε_i\sim N(0, σ^2_{ε})

Test statistic is based on Sobel's (1982) test:

Z=\frac{\hat{θ}_{1a}\hat{λ_a}}{\hat{σ}_{θ_{1a}λ_a}}

where \hat{σ}_{θ_{1a}λ_a} is the estimated standard deviation of the estimate \hat{θ}_{1a}\hat{λ_a} using multivariate delta method:

σ_{θ_{1a}λ_a}=√{θ_{1a}^2σ_{λ_a}^2+λ_a^2σ_{θ_{1a}}^2}

and σ_{θ_{1a}}^2=σ_e^2/(nσ_x^2) is the variance of the estimate \hat{θ}_{1a}, and σ_{λ_a}^2=σ_{ε}^2/(nσ_m^2(1-ρ_{mx}^2)) is the variance of the estimate \hat{λ_a}, σ_m^2 is the variance of the mediator m_i.

From the linear regression m_i=θ_0+θ_{1a} x_i+e_i, we have the relationship σ_e^2=σ_m^2(1-ρ^2_{mx}). Hence, we can simply the variance σ_{θ_{1a}, λ_a} to

σ_{θ_{1a}λ_a}=√{θ_{1a}^2\frac{σ_{ε}^2}{nσ_m^2(1-ρ_{mx}^2)}+λ_a^2\frac{σ_{m}^2(1-ρ_{mx}^2)}{nσ_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)

Example output

[1] 324.5108

powerMediation documentation built on March 24, 2021, 1:06 a.m.