Preacher and Hayes (2008) show how to analyze models with multiple mediators in SPSS and SAS, how can I analyze multiple mediators in Stata?

Here is the full citation:

Preacher, K.J. and Hayes, A.F. 2008. Asymptotic and resampling strategies for assessing and

comparing indirect effects in multiple mediator models. *Behavioral Research Methods*, 40, 879-891.

NOTE: If running the code on this page, please copy it all into a do-file and run all of it.

Mediator variables are variables that sit between independent variable and dependent variable and mediate the effect of the IV on the DV. A model with two mediators is shown in the figure below.

In the figure above **a1** represents the regression coefficient for the IV when the MV is regressed on
the IV while **b** is the coefficient for the MV when the DV is regressed on MV and IV. The symbol
**c’** represents the direct effect of the IV on the DV.
Generally, researchers want to determine
the indirect effect of the IV on the DV through the MV. One common way to compute the indirect
effect is by using the product of the coefficients method. This method determines the
indirect effect by multiplying the regression coefficients, for example, **a1*b1 = a1b1**.
In addition to computing the indirect
effect we also want to obtain the standard error of **a1b1**. Further, we want to be able to do
this for each of the mediator variables in the model.

Thus, we need the **a** and **b** coefficients for each of the mediator variable in the model. We
will obtain all of the necessary coefficients using the **sureg** (seemingly unrelated regression)
command as suggested by Maarten Buis on the Statalist. The general form of the **sureg**
command will look something like this:

sureg (mv1 iv)(mv2 iv)(dv mv1 mv2 mv3 iv)

## Example 1

**science** as the dv,
**math** as the iv and **read** and **write** as the two mediator variables.

We will need the coefficients for **read** on **math** and **write**
on **math** as well as the coefficients for **science** on **read** and **write** from
the equation that also includes **math**.

use http://www.ats.ucla.edu/stat/data/hsb2, clear sureg (read math)(write math)(science read write math)Seemingly unrelated regression ---------------------------------------------------------------------- Equation Obs Parms RMSE "R-sq" chi2 P ---------------------------------------------------------------------- read 200 1 7.662848 0.4386 156.26 0.0000 write 200 1 7.437294 0.3812 123.23 0.0000 science 200 3 6.983853 0.4999 199.96 0.0000 ---------------------------------------------------------------------- ------------------------------------------------------------------------------ | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- read | math | .724807 .0579824 12.50 0.000 .6111636 .8384504 _cons | 14.07254 3.100201 4.54 0.000 7.996255 20.14882 -------------+---------------------------------------------------------------- write | math | .6247082 .0562757 11.10 0.000 .5144099 .7350065 _cons | 19.88724 3.008947 6.61 0.000 13.98981 25.78467 -------------+---------------------------------------------------------------- science | read | .3015317 .0679912 4.43 0.000 .1682715 .434792 write | .2065257 .0700532 2.95 0.003 .0692239 .3438274 math | .3190094 .0759047 4.20 0.000 .170239 .4677798 _cons | 8.407353 3.160709 2.66 0.008 2.212476 14.60223 ------------------------------------------------------------------------------

Now we have all the coefficients we need to compute the indirect effect coefficients and their
standard errors. We can do this using the **nlcom** (nonlinear combination) command.
We will run **nlcom** three times: Once for each of the two
specific indirect effects for **read** and **write** and once for the total
indirect effect.

To compute an indirect direct we specify a product of coefficients. For example,
the coefficient for **read** on **math** is **[read]_b[math]** and the coefficient for
**science** on **read** is **[science]_b[read]**. Thus, the product is
**[read]_b[math]*[science]_b[read]**. To get the total indirect effect we just add the
two product terms together in the **nlcom** command.

/* indirect via read */ nlcom [read]_b[math]*[science]_b[read]_nl_1: [read]_b[math]*[science]_b[read] ------------------------------------------------------------------------------ | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- _nl_1 | .2185523 .05229 4.18 0.000 .1160659 .3210388 ------------------------------------------------------------------------------/* indirect via write */ nlcom [write]_b[math]*[science]_b[write]_nl_1: [write]_b[math]*[science]_b[write] ------------------------------------------------------------------------------ | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- _nl_1 | .1290183 .0452798 2.85 0.004 .0402715 .2177651 ------------------------------------------------------------------------------/* total indirect */ nlcom [read]_b[math]*[science]_b[read]+[write]_b[math]*[science]_b[write]_nl_1: [read]_b[math]*[science]_b[read]+[write]_b[math]*[science]_b[write] ------------------------------------------------------------------------------ | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- _nl_1 | .3475706 .0594916 5.84 0.000 .2309693 .4641719 ------------------------------------------------------------------------------

The results above suggest that each of the separate indirect effects as well as the total
indirect effect are significant. From the above results it is also possible to compute
the ratio of indirect to direct effect and the proportion due to the indirect effect.
These computations require an estimate of the direct effect, which can be found in the **sureg**
output. In this example the direct effect is given by the coefficient for **math** in the last
equation (.3190094). Here are the manual computations for the ratio of indirect to direct and the
proportion of total effect that is mediated.

/* ratio of indirect to direct */ display .3475706/.31900941.0895309/* proportion of total effect that is mediated */ display .3475706/(.3475706+.3190094).52142369

**nlcom** computes the standard errors using the delta
method which assumes that the estimates of the indirect effect are normally distributed.
For many situations this is acceptable but it does not work well for the indirect effects
which are usually positively skewed and kurtotic. Thus the z-test and p-values for
these indirect effects generally cannot be trusted. Therefore, it is recommended
that bootstrap standard errors and confidence intervals be used.

Below is a short ado-program that is called by the **bootstrap** command. It computes the
indirect effect coefficients as the product of **sureg** coefficients (as before) but
does not use the **nlcom** command since the standard errors will be computed using the
bootstrap.

**bootmm** is an rclass program that produces three return values which we have called
“indread”, “indwrite” and “indtotal.” These are the local names for each of the indirect effect
coefficients and for the total indirect effect.

We run **bootmm** with the **bootstrap** command. We
give the **bootstrap** command the names of the three return values and select options
for the number of replications and to omit printing dots after each replication.

Since we selected 5,000 replications you may need to be a bit patient depending upon the speed of your computer.

capture program drop bootmm program bootmm, rclass syntax [if] [in] sureg (read math)(write math)(science read write math) `if' `in' return scalar indread = [read]_b[math]*[science]_b[read] return scalar indwrite = [write]_b[math]*[science]_b[write] return scalar indtotal = [read]_b[math]*[science]_b[read]+ /// [write]_b[math]*[science]_b[write] end bootstrap r(indread) r(indwrite) r(indtotal), bca reps(5000) nodots: bootmmBootstrap results Number of obs = 200 Replications = 5000 command: bootmm _bs_1: r(indread) _bs_2: r(indwrite) _bs_3: r(indtotal) ------------------------------------------------------------------------------ | Observed Bootstrap Normal-based | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- _bs_1 | .2185523 .0544617 4.01 0.000 .1118094 .3252953 _bs_2 | .1290183 .0498037 2.59 0.010 .0314048 .2266318 _bs_3 | .3475706 .0653076 5.32 0.000 .2195701 .4755711 ------------------------------------------------------------------------------

We could use the bootstrap standard errors to see if the indirect effects are significant but it is usually recommended that bias-corrected or percentile confidence intervals be used instead. These confidence intervals are nonsymmetric reflecting the skewness of the sampling distribution of the product coefficients. If the confidence interval does not contain zero than the indirect effect is considered to be statistically significant.

estat boot, percentile bc bcaBootstrap results Number of obs = 200 Replications = 5000 command: bootmm _bs_1: r(indread) _bs_2: r(indwrite) _bs_3: r(indtotal) ------------------------------------------------------------------------------ | Observed Bootstrap | Coef. Bias Std. Err. [95% Conf. Interval] -------------+---------------------------------------------------------------- _bs_1 | .21855231 -.0009252 .05446169 .1116576 .3263005 (P) | .1140179 .3286456 (BC) | .1140179 .3286456 (BCa) _bs_2 | .12901828 .0009822 .04980373 .0375536 .2286579 (P) | .0375377 .22842 (BC) | .0333511 .2264691 (BCa) _bs_3 | .34757059 .000057 .0653076 .2181866 .4773324 (P) | .2209776 .4805473 (BC) | .2158857 .4752103 (BCa) ------------------------------------------------------------------------------ (P) percentile confidence interval (BC) bias-corrected confidence interval (BCa) bias-corrected and accelerated confidence interval

In this example, the total indirect effect of **math** through **read** and **write** is
significant as are the individual indirect effects.

## Example 2

What do you do if you also have control variables? You just add them to each of the equations in the
**sureg** model. Let’s say that **socst** is a covariate. Here is how the bootstrap
process would work.

capture program drop bootmm program bootmm, rclass syntax [if] [in] sureg (read math socst)(write math socst)(science read write math socst) `if' `in' return scalar indread = [read]_b[math]*[science]_b[read] return scalar indwrite = [write]_b[math]*[science]_b[write] return scalar indtotal = [read]_b[math]*[science]_b[read] + /// [write]_b[math]*[science]_b[write] end bootstrap r(indread) r(indwrite) r(indtotal), bca reps(5000) nodots: bootmmBootstrap results Number of obs = 200 Replications = 5000 command: bootmm _bs_1: r(indread) _bs_2: r(indwrite) _bs_3: r(indtotal) ------------------------------------------------------------------------------ | Observed Bootstrap Normal-based | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- _bs_1 | .1561855 .040306 3.87 0.000 .0771872 .2351837 _bs_2 | .0890589 .0352121 2.53 0.011 .0200444 .1580733 _bs_3 | .2452443 .0477817 5.13 0.000 .1515939 .3388947 ------------------------------------------------------------------------------estat boot, percentile bc bcaBootstrap results Number of obs = 200 Replications = 5000 command: bootmm _bs_1: r(indread) _bs_2: r(indwrite) _bs_3: r(indtotal) ------------------------------------------------------------------------------ | Observed Bootstrap | Coef. Bias Std. Err. [95% Conf. Interval] -------------+---------------------------------------------------------------- _bs_1 | .15618546 -.0016606 .04030598 .0784972 .2359464 (P) | .0836141 .2407494 (BC) | .0838816 .2413034 (BCa) _bs_2 | .08905886 .0000963 .0352121 .0241053 .163005 (P) | .0274379 .1664222 (BC) | .0260387 .164438 (BCa) _bs_3 | .24524432 -.0015643 .0477817 .1536668 .341307 (P) | .1581453 .3477974 (BC) | .1581453 .3477974 (BCa) ------------------------------------------------------------------------------ (P) percentile confidence interval (BC) bias-corrected confidence interval (BCa) bias-corrected and accelerated confidence interval

Although the total and individual indirect are much smaller in the model with the covariate, they are still statistically significant using the 95% confidence intervals.