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A variable may be considered a mediator to the extent to which it carries the influence of a given independent variable (IV) to a given dependent variable (DV). Generally speaking, mediation can be said to occur when (1) the IV significantly affects the mediator, (2) the IV significantly affects the DV in the absence of the mediator, (3) the mediator has a significant unique effect on the DV, and (4) the effect of the IV on the DV shrinks upon the addition of the mediator to the model.
Example:
This example uses the hsbdemo dataset with write as the DV, math as the iv and read as the mediator variable. That is, the model says that math influences read, which in turn influences science. This model may or may not make much substantive sense but it will allow us to to demonstrate the process of running a Sobel-Goodman test. We will do this using the sgmediation command, you can download this command using findit sgmediation.
use http://www.ats.ucla.edu/stat/data/hsbdemo, clear
sgmediation science, mv(read) iv(math)
Model with dv regressed on iv (path c)
Source | SS df MS Number of obs = 200
-------------+------------------------------ F( 1, 198) = 130.81
Model | 7760.55791 1 7760.55791 Prob > F = 0.0000
Residual | 11746.9421 198 59.3279904 R-squared = 0.3978
-------------+------------------------------ Adj R-squared = 0.3948
Total | 19507.5 199 98.0276382 Root MSE = 7.7025
------------------------------------------------------------------------------
science | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
math | .66658 .0582822 11.44 0.000 .5516466 .7815135
_cons | 16.75789 3.116229 5.38 0.000 10.61264 22.90315
------------------------------------------------------------------------------
Model with mediator regressed on iv (path a)
Source | SS df MS Number of obs = 200
-------------+------------------------------ F( 1, 198) = 154.70
Model | 9175.57065 1 9175.57065 Prob > F = 0.0000
Residual | 11743.8493 198 59.3123704 R-squared = 0.4386
-------------+------------------------------ Adj R-squared = 0.4358
Total | 20919.42 199 105.122714 Root MSE = 7.7015
------------------------------------------------------------------------------
read | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
math | .724807 .0582745 12.44 0.000 .6098887 .8397253
_cons | 14.07254 3.115819 4.52 0.000 7.928087 20.21699
------------------------------------------------------------------------------
Model with dv regressed on mediator and iv (paths b and c')
Source | SS df MS Number of obs = 200
-------------+------------------------------ F( 2, 197) = 90.27
Model | 9328.73944 2 4664.36972 Prob > F = 0.0000
Residual | 10178.7606 197 51.6688353 R-squared = 0.4782
-------------+------------------------------ Adj R-squared = 0.4729
Total | 19507.5 199 98.0276382 Root MSE = 7.1881
------------------------------------------------------------------------------
science | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
read | .3654205 .0663299 5.51 0.000 .2346128 .4962283
math | .4017207 .0725922 5.53 0.000 .2585632 .5448782
_cons | 11.6155 3.054262 3.80 0.000 5.592255 17.63875
------------------------------------------------------------------------------
Sobel-Goodman Mediation Tests
Coef Std Err Z P>|Z|
Sobel .26485934 .05258136 5.037 4.726e-07
Goodman-1 (Aroian) .26485934 .05272324 5.024 5.072e-07
Goodman-2 .26485934 .05243909 5.051 4.400e-07
Coef Std Err Z P>|Z|
a coefficient = .724807 .058274 12.4378 0
b coeffocoent = .365421 .06633 5.50914 3.6e-08
Indirect effect = .264859 .052581 5.03713 4.7e-07
Direct effect = .401721 .072592 5.53394 3.1e-08
Total effect = .66658 .058282 11.4371 0
Proportion of total effect that is mediated: .39734065
Ratio of indirect to direct effect: .65931219
Ratio of total to direct effect: 1.6593122
In this example the mediation effect of read was statistically significant with
approximately 40% of the total effect (of math on science) being mediated.
If you have concerns about the standard error for the indirect effect, you may want to bootstrap sgmediation.
bootstrap r(ind_eff) r(dir_eff), reps(1000): sgmediation science, iv(math) mv(read)
Bootstrap replications (1000)
----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5
.................................................. 50
[output omitted]
.................................................. 1000
Bootstrap results Number of obs = 200
Replications = 1000
command: sgmediation science, iv(math) mv(read)
_bs_1: r(ind_eff)
_bs_2: r(dir_eff)
------------------------------------------------------------------------------
| Observed Bootstrap Normal-based
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_bs_1 | .2648593 .0539163 4.91 0.000 .1591853 .3705334
_bs_2 | .4017207 .085701 4.69 0.000 .2337499 .5696915
------------------------------------------------------------------------------
estat bootstrap, percentile bc
Bootstrap results Number of obs = 200
Replications = 1000
command: sgmediation science, iv(math) mv(read)
_bs_1: r(ind_eff)
_bs_2: r(dir_eff)
------------------------------------------------------------------------------
| Observed Bootstrap
| Coef. Bias Std. Err. [95% Conf. Interval]
-------------+----------------------------------------------------------------
_bs_1 | .26485934 -.0018226 .05391633 .1513887 .3672891 (P)
| .1495302 .3653122 (BC)
_bs_2 | .40172068 .0011065 .08570095 .2436569 .575866 (P)
| .2474115 .5803441 (BC)
------------------------------------------------------------------------------
(P) percentile confidence interval
(BC) bias-corrected confidence intervalIf you prefer to do a residual resampling bootstrap rather than case resampling bootstrap, you can use the resboot-mediation command (findit resboot_mediation).
resboot_mediation, dv(science) mv(read) iv(math) reps(1000)
obs was 200, now 1000
Source | SS df MS Number of obs = 200
-------------+------------------------------ F( 1, 198) = 154.70
Model | 9175.57065 1 9175.57065 Prob > F = 0.0000
Residual | 11743.8493 198 59.3123704 R-squared = 0.4386
-------------+------------------------------ Adj R-squared = 0.4358
Total | 20919.42 199 105.122714 Root MSE = 7.7015
------------------------------------------------------------------------------
read | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
math | .724807 .0582745 12.44 0.000 .6098887 .8397253
_cons | 14.07254 3.115819 4.52 0.000 7.928087 20.21699
------------------------------------------------------------------------------
Source | SS df MS Number of obs = 200
-------------+------------------------------ F( 2, 197) = 90.27
Model | 9328.73944 2 4664.36972 Prob > F = 0.0000
Residual | 10178.7606 197 51.6688353 R-squared = 0.4782
-------------+------------------------------ Adj R-squared = 0.4729
Total | 19507.5 199 98.0276382 Root MSE = 7.1881
------------------------------------------------------------------------------
science | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
read | .3654205 .0663299 5.51 0.000 .2346128 .4962283
math | .4017207 .0725922 5.53 0.000 .2585632 .5448782
_cons | 11.6155 3.054262 3.80 0.000 5.592255 17.63875
------------------------------------------------------------------------------
Nonparametric resampled residual bootstrap of mediation with 1000 replications
Bootstrap
Coef Bias Std Err [95% Conf Interval]
ind eff .264859 -.006671 .125424 .033054 .51854 (P)
.054796 .556361 (BC)
dir eff .401721 .012071 .175588 .065645 .75491 (P)
.029874 .72355 (BC)
tot eff .66658 .005399 .14155 .401588 .968667 (P)
.401588 .968667 (BC)
(P) percentile confidence interval
(BC) bias-corrected confidence interval
Aroian, L.A. (1944). The probability function of the product of two normally distributed variables.
Annals of Mathematical Statistics, 18, 265-271.
Baron, R.M. & Kenny, D.A. (1986), Moderator-Mediator Variables Distinction in Social Psychological Research:
Conceptual, Strategic, and Statistical Considerations. Journal of Personality and Social Psychology, 51
(6), 1173–82.
Goodman, L.A. (1960) On the exact variance of products. Journal of the American Statistical Association,
55, 708-713.
MacKinnon, D. P. & Dwyer, J. H. (1993). Estimating mediated effects in prevention studies. Evaluation
Review, 17, 144-158.
MacKinnon, D. P., Warsi, G., & Dwyer, J. H. (1995). A simulation study of mediated effect measures.
Multivariate Behavioral Research, 30(1), 41-62.
Preacher, K. J. & Hayes, A. F. (2004). SPSS and SAS procedures for estimating indirect effects in simple
mediation models. Behavior Research Methods, Instruments, & Computers, 36(4), 717-731.
Sobel, M.E. (1982) Asymptotic confidence intervals for indirect effects in structural equation
models. Sociological Methodology, 13, 290-312.
Sobel, M.E. (1986) Some new results on indirect effects and their standard errors in
covariance structure models. Sociological Methodology, 16, 159-186.
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