Welcome to the Institute for Digital Reseach and Education
Institute for Digital Reseach and Education Home
Help the Stat Consulting Group by giving a gift
Stata FAQ
How can I perform mediation with multilevel data? (Method 2)
Mediator variables are variables that sit between the independent variable and dependent variable and
mediate the effect of the IV on the DV. A model with one mediator is shown in the figure below.
The idea, in mediation analysis, is that some of the effect of the predictor variable, the IV, is transmitted to the DV through the mediator variable, the MV. And some of the effect of the IV passes directly to the DV. That portion of of the effect of the IV that passes through the MV is the indirect effect. The FAQ page How can I perform mediation with multilevel data? (Method 1) showed how to do multilevel mediation using an approach suggested by Krull & MacKinnon (2001). This page will demonstrate an alternative approach given in the 2006 paper by Bauer, Preacher & Gil. This approach combines the dependent variable and the mediator into a single stacked response variable and runs one mixed model with indicator variables for the DV and mediator to obtain all of the values needed for the analysis. We will begin by loading in a synthetic data set and reconfiguring it for our analysis. All of the variables in this example (id the cluster ID, x the predictor variable, m the mediator variable, and y the dependent variable) are at level 1 Here is how the first 16 observations look in the original dataset.
use http://www.ats.ucla.edu/stat/data/ml_sim, clear
list in 1/16, sep(8)
+-------------------------------------------+
| id x m y |
|-------------------------------------------|
1. | 1 1.5451205 .10680165 .56777601 |
2. | 1 2.275272 2.1104394 1.2061451 |
3. | 1 .7866992 .03888269 -.26127707 |
4. | 1 -.06185872 .47940645 -.75899036 |
5. | 1 .11725984 .59082413 .51907688 |
6. | 1 1.4809238 .89094498 -.63111928 |
7. | 1 .89275955 -.22767749 .1520794 |
8. | 1 .92460471 .72917853 .23463058 |
|-------------------------------------------|
9. | 2 1.0031088 -.36224011 -1.1507768 |
10. | 2 -1.1914322 -2.9658042 -3.7168343 |
11. | 2 -1.8003534 -3.6484285 -4.4695533 |
12. | 2 -1.2585149 -2.3043152 -3.2232902 |
13. | 2 -.47077056 -1.4640601 -2.6614767 |
14. | 2 -2.8150223 -2.1196492 -2.1692942 |
15. | 2 -.26598751 -.36127735 -2.1726739 |
16. | 2 -1.5617038 -2.5556008 -3.857205 |
+-------------------------------------------+summarize x m y
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
x | 800 -.1539876 1.330374 -4.151426 3.869661
m | 800 -.0247739 1.483614 -6.47837 5.012422
y | 800 -.1833981 1.66918 -8.600032 5.919008
corr x m y
(obs=800)
| x m y
-------------+---------------------------
x | 1.0000
m | 0.5404 1.0000
y | 0.5537 0.7591 1.0000
tab id
id | Freq. Percent Cum.
------------+-----------------------------------
1 | 8 1.00 1.00
2 | 8 1.00 2.00
3 | 8 1.00 3.00
[output omitted]
98 | 8 1.00 98.00
99 | 8 1.00 99.00
100 | 8 1.00 100.00
------------+-----------------------------------
Total | 800 100.00xtmixed y x || id: x, var cov(un)
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -1214.1953
Iteration 1: log likelihood = -1214.1952
Computing standard errors:
Mixed-effects ML regression Number of obs = 800
Group variable: id Number of groups = 100
Obs per group: min = 8
avg = 8.0
max = 8
Wald chi2(1) = 139.83
Log likelihood = -1214.1952 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x | .6897692 .0583307 11.83 0.000 .575443 .8040953
_cons | -.0273891 .0954593 -0.29 0.774 -.214486 .1597077
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(x) | .2138149 .0465543 .1395419 .3276209
var(_cons) | .737622 .1272275 .5260364 1.034313
cov(x,_cons) | -.0262811 .0559555 -.1359518 .0833896
-----------------------------+------------------------------------------------
var(Residual) | .822606 .0470298 .735406 .9201455
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(3) = 367.76 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference.xtmixed m x || id: x, var cov(un)
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -1113.6567
Iteration 1: log likelihood = -1113.6567
Computing standard errors:
Mixed-effects ML regression Number of obs = 800
Group variable: id Number of groups = 100
Obs per group: min = 8
avg = 8.0
max = 8
Wald chi2(1) = 175.20
Log likelihood = -1113.6567 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
m | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x | .6114761 .0461973 13.24 0.000 .5209311 .7020212
_cons | .094974 .0911449 1.04 0.297 -.0836668 .2736148
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(x) | .1147085 .0284269 .0705751 .1864401
var(_cons) | .7010477 .1165732 .5060648 .9711559
cov(x,_cons) | .0065896 .0412717 -.0743015 .0874806
-----------------------------+------------------------------------------------
var(Residual) | .6449282 .0365222 .5771756 .7206342
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(3) = 396.83 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference.xtmixed y m x || id: m x, var cov(un)
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -1016.1698
Iteration 1: log likelihood = -1016.0624
Iteration 2: log likelihood = -1016.0623
Computing standard errors:
Mixed-effects ML regression Number of obs = 800
Group variable: id Number of groups = 100
Obs per group: min = 8
avg = 8.0
max = 8
Wald chi2(2) = 311.46
Log likelihood = -1016.0623 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
m | .6252951 .04657 13.43 0.000 .5340196 .7165706
x | .2500122 .0384508 6.50 0.000 .1746501 .3253743
_cons | -.0937981 .0616632 -1.52 0.128 -.2146558 .0270596
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(m) | .1178902 .0302985 .0712383 .1950931
var(x) | .0376483 .0210683 .012572 .112742
var(_cons) | .2573218 .0525485 .1724447 .3839753
cov(m,x) | -.0018785 .0188857 -.0388937 .0351368
cov(m,_cons) | -.0060299 .0281886 -.0612786 .0492187
cov(x,_cons) | -.0307884 .0242063 -.0782318 .016655
-----------------------------+------------------------------------------------
var(Residual) | .5078589 .0307909 .4509575 .57194
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(6) = 313.19 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference.rename y z0 // rename for reshaping
generate z1=m // create z1 for reshaping
gen fid=_n // create temp id for reshaping
reshape long z, i(fid) j(s)
rename s sm // indicator for the mediator
gen sy= ~sm // indicator for the dv
list in 1/16, sep(8)
+----------------------------------------------------------+
| fid sm id x m z sy |
|----------------------------------------------------------|
1. | 1 0 1 1.5451205 .10680165 .567776 1 |
2. | 1 1 1 1.5451205 .10680165 .1068017 0 |
3. | 2 0 1 2.275272 2.1104394 1.206145 1 |
4. | 2 1 1 2.275272 2.1104394 2.110439 0 |
5. | 3 0 1 .7866992 .03888269 -.2612771 1 |
6. | 3 1 1 .7866992 .03888269 .0388827 0 |
7. | 4 0 1 -.06185872 .47940645 -.7589904 1 |
8. | 4 1 1 -.06185872 .47940645 .4794064 0 |
|----------------------------------------------------------|
9. | 5 0 1 .11725984 .59082413 .5190769 1 |
10. | 5 1 1 .11725984 .59082413 .5908241 0 |
11. | 6 0 1 1.4809238 .89094498 -.6311193 1 |
12. | 6 1 1 1.4809238 .89094498 .890945 0 |
13. | 7 0 1 .89275955 -.22767749 .1520794 1 |
14. | 7 1 1 .89275955 -.22767749 -.2276775 0 |
15. | 8 0 1 .92460471 .72917853 .2346306 1 |
16. | 8 1 1 .92460471 .72917853 .7291785 0 |
+----------------------------------------------------------+Now we can run our mixed model using xtmixed. Notice that because we include the sm and sy indicators in the model that we need to use the nocons option for both the fixed and random effects.gen smx=sm*x // a gen sym=sy*m // b gen syx=sy*x // cprime
xtmixed z sm smx sy sym syx, nocons ///
> || id: sm smx sy sym syx, nocons cov(un) ///
> || fid: sm, nocons var reml
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log restricted-likelihood = -2130.2314
Iteration 1: log restricted-likelihood = -2126.1762
Iteration 2: log restricted-likelihood = -2126.1352
Iteration 3: log restricted-likelihood = -2126.1345
Iteration 4: log restricted-likelihood = -2126.1345
Computing standard errors:
Mixed-effects REML regression Number of obs = 1600
-----------------------------------------------------------
| No. of Observations per Group
Group Variable | Groups Minimum Average Maximum
----------------+------------------------------------------
id | 100 16 16.0 16
fid | 800 2 2.0 2
-----------------------------------------------------------
Wald chi2(5) = 342.64
Log restricted-likelihood = -2126.1345 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
z | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
sm | .0932149 .0894316 1.04 0.297 -.0820678 .2684976
smx | .6118566 .0464952 13.16 0.000 .5207276 .7029856
sy | -.096852 .0619583 -1.56 0.118 -.218288 .0245839
sym | .6105632 .0455363 13.41 0.000 .5213136 .6998127
syx | .2208116 .0372474 5.93 0.000 .147808 .2938152
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(sm) | .6794348 .1131795 .49018 .9417594
var(smx) | .1204833 .0293158 .0747849 .1941063
var(sy) | .2702841 .0540872 .1825932 .400089
var(sym) | .1118722 .0291408 .0671426 .1864
var(syx) | .032437 .020045 .0096611 .1089068
cov(sm,smx) | .018161 .0410871 -.0623682 .0986902
cov(sm,sy) | .0568122 .0645235 -.0696515 .183276
cov(sm,sym) | .0093218 .0390042 -.0671251 .0857687
cov(sm,syx) | -.0066738 .0345052 -.0743028 .0609553
cov(smx,sy) | .0118807 .0292113 -.0453725 .0691338
cov(smx,sym) | .0989554 .0228226 .054224 .1436868
cov(smx,syx) | -.0214964 .0188476 -.0584371 .0154443
cov(sy,sym) | -.0042835 .0278161 -.058802 .050235
cov(sy,syx) | -.0182759 .0233301 -.0640021 .0274502
cov(sym,syx) | .005422 .017884 -.02963 .040474
-----------------------------+------------------------------------------------
fid: Identity |
var(sm) | .1377656 .0477088 .0698822 .2715911
-----------------------------+------------------------------------------------
var(Residual) | .5089653 .0306737 .4522609 .5727793
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(16) = 765.76 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference.avg_ind_eff = a*b + σaj,bj
se_avg_ind_eff = b2*Var(a) + a2*Var(b) + Var(a)*Var(b) + 2*a*b*Cov(a,b) +
Cov(a,b)2 + Var(σaj,bj)
avg_tot_eff = a*b + σaj,bj + cprime
se_avg_tot_eff = b2*Var(a) + a2*Var(b) + Var(a)*Var(b) + 2*a*b*Cov(a,b) + Cov(a,b)2 +
Var(cprime) + 2*b*Cov(a,c) + 2*a*Cov(b,c) + Var(σaj,bj)We also need the value for Var(σaj,bj), which we can obtain using the _diparm (display parameter) command. For more information on _diparm see How can I access the random effects after xtmixed? Here is how we use the _diparm command for our example.global a = _b[smx] // a path global sa = _se[smx] // se a path global b = _b[sym] // b path global sb = _se[sym] // se b path global c = _b[syx] // cprime path global sc = _se[syx] // se cprime path * covariances of fixed effects parameter estimates mat v=e(V) global covab=v[4,2] // cov(a,b) global covac=v[5,2] // cov(a,c) global covbc=v[5,4] // cov(b.c)
We are now ready to compute the values we need and to display the results. Here are the commands._diparm atr1_1_2_4 lns1_1_2 lns1_1_4, f(tanh(@1)*exp(@2+@3)) /// d((1-(tanh(@1)^2))*exp(@2+@3) tanh(@1)*exp(@2+@3) /// tanh(@1)*exp(@2+@3)) global sajbj=r(est) global seajbj=r(se)
And here are the results.display dis "a: " _col(30) $a dis "b: " _col(30) $b dis "c prime: " _col(30) $c dis "cov(ajbj): " _col(30) $sajbj dis global avg_ind_eff = $a*$b + $sajbj global avg_tot_eff = $a*$b + $sajbj + $c dis "average indirect effect: " _col(30) $avg_ind_eff global var9 = ($b)^2*($sa)^2 + ($a)^2*($sb)^2 + ($sa)^2*($sb)^2 + /// 2*$a*$b*$covab + ($covab)^2 + ($seajbj)^2 dis "standard error average indirect effect: " _col(30) sqrt($var9) dis "ratio: " _col(30) $avg_ind_eff/sqrt($var9) dis "LCL: " _col(30) $avg_ind_eff-1.96*sqrt($var9) dis "UCL: " _col(30) $avg_ind_eff+1.96*sqrt($var9) dis dis "average total effect: " _col(30) $avg_tot_eff global var10 = $var9 + ($sc)^2 + 2*$b*$covac + 2*$a*$covbc dis "standard error average total effect: " _col(30) sqrt($var10) dis "ratio: " _col(30) $avg_tot_eff/sqrt($var10) dis "LCL: " _col(30) $avg_tot_eff-1.96*sqrt($var10) dis "UCL: " _col(30) $avg_tot_eff+1.96*sqrt($var10)
a: .61185658 b: .61056318 c prime: .22081162 cov(ajbj): .09895544 average indirect effect: .47253254 standard error average indirect effect: .05333007 Ratio: 8.8605279 LCL: .36800561 UCL: .57705946 average total effect: .69334416 standard error average total effect: .05829333 ratio: 11.894057 LCL: .57908923 UCL: .80759908
References
Bauer,D.J., Preacher,K.J. & Gil,K.M. (2006) Conceptualizing and testing random indirect effects and moderated mediation in multilevel models: New procedures and recommendations. Psychological Methods, 11(2), 142-163. Krull,J.L. & MacKinnon,D.P. (2001) Multilevel modeling of individual and group level mediated effects. Multivariate Behavioral Research, 36(2), 249-277.The content of this web site should not be construed as an endorsement of any particular web site, book, or software product by the University of California.