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Consider a simple mediation model
mi=a0+a∗xi+emi yi=b0+b∗mi+c∗xi+eyi
where emi∼N(0,σ2em) and eyi∼N(0,σ2ey). The mediation effect is ab=a∗b.
The Sobel test statistic is
Z=ˆaˆbˆσab
where ˆσ2ab=ˆa2∗ˆσ2b+ˆb2∗ˆσ2a. From regression analysis, we have
ˆσ2a=σ2emnσ2x
ˆσ2b=σ2eynσ2m(1−ρ2mx)
Furthermore, because ˆa=ρxm∗σm/σx, we have ρxm=ˆaσx/σm and σ2em=σ2m(1−ρ2mx)=σ2m−a2σ2x. Then
ˆσ2a=σ2m−a2σ2xnσ2x
ˆσ2b=σ2eyn(σ2m−a2σ2x)
Therefore, the Sobel test depends on the sample size, the coefficients a and b, the variances of x and m as well as their correlation, and the residual variance of y denoted by ˆσ2ey as in
Z=ˆaˆb√ˆa2∗σ2eyn(σ2m−a2σ2x)+ˆb2∗σ2m−a2σ2xnσ2x
To calculate power, one need to provide information on (1) sample size, (2) coefficient a, (3) coefficient b, (4) variance of x (σ2x), (5) variance of m (σ2m), (6) error variance for y (σ2ey), and (7) the significance level α. If the power is provided, the needed sample size can also be calculated.
Among (1) sample size, (2) coefficient a, (3) coefficient b, (4) variance of x (σ2x), (5) variance of m (σ2m), (6) error variance for y (σ2ey), (7) the significance level α, and (8) power, one and only one can be left blank.
Provide the number of observations per group. Multiple sample sizes can be provided in two ways. First, multiple sample sizes can be supplied separated by white spaces, e.g., 100 150 200
will calculate power for the three sample sizes. A sequence of sample sizes can be generated using the method s:e:i
with s
denoting the starting sample size, e
as ending sample size, and i
as the interval. For example, 100:150:10
will generate a sequence 100 110 120 130 140 150
.
By default, the sample size is 100
.
The paths in the model.
The marginal variances for x and m.
The variance of the error or residual σ2ey
The significance level (Type I error rate) for power calculation withe the default 0.05
.
The power of the test.
Whether to generate the power curve.
A note (less than 200 characters) can be provided to provide basic information on the analysis.
The output lists the related information about this power analysis. The output is given as a matrix.
N Power a b varx varm varey alpha 100 0.802 0.4 0.4 1 1 1 0.05