How to perform Sobel-Goodman mediation tests in Stata?
The purpose of the Sobel-Goodman tests is to test whether a mediator carries the influence of an IV to a DV.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 science 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 sgmediationcommand, you can download this command using findit sgmediation.use http://www.ats.ucla.edu/stat/data/hsbdemo, clearsgmediation 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-07Goodman-1 (Aroian) .26485934 .05272324 5.024 5.072e-07Goodman-2 .26485934 .05243909 5.051 4.400e-07 Coef Std Err Z P>|Z|a coefficient = .724807 .058274 12.4378 0b coeffocoent = .365421 .06633 5.50914 3.6e-08Indirect effect = .264859 .052581 5.03713 4.7e-07 Direct effect = .401721 .072592 5.53394 3.1e-08 Total effect = .66658 .058282 11.4371 0Proportion of total effect that is mediated: .39734065Ratio of indirect to direct effect: .65931219Ratio of total to direct effect: 1.6593122In this example the mediation effect of read was statistically significant with approximately 40% of the total effect (of math onscience) being mediated.Bootstrap with case resampling
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].................................................. 1000Bootstrap 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 bcBootstrap 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 intervalBootstrap with residual resampling
If you prefer to do a residual resampling bootstrap rather than case resampling bootstrap, you can use the resboot-mediationcommand (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 intervalReferencesAroian, 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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