How can I compare indirect effects in a multiple group model?

How can I compare indirect effects in a multiple group model?

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A multiple group model is a statistical analysis technique used to examine the effects of one or more independent variables on a dependent variable in different groups or conditions. One aspect of this analysis is comparing the indirect effects of the independent variable on the dependent variable. This is done by estimating the indirect effects separately for each group and then comparing the estimates between the groups. The indirect effects can be compared using statistical tests or by examining the size and direction of the effects in each group. This comparison allows researchers to determine if the indirect effects differ significantly between the groups, providing valuable insights into the underlying mechanisms driving the relationship between the independent and dependent variables.

How can I compare indirect effects in a multiple group model? | Stata FAQ

Consider the following mediation model run as a multiple group structural equation model (sem) with science as the final response variable, math as the independent variable and read
as the mediator variable for the four levels of grp.

use https://stats.idre.ucla.edu/stat/data/hsbanova, clear
sem (read <- math)(science <- read math), group(grp)
Endogenous variables

Observed:  read science

Exogenous variables

Observed:  math

Fitting target model:

Iteration 0:   log likelihood =  -2051.912  
Iteration 1:   log likelihood =  -2051.912  

Structural equation model                       Number of obs      =       200
Grouping variable  = grp                        Number of groups   =         4
Estimation method  = ml
Log likelihood     =  -2051.912

------------------------------------------------------------------------------
             |                 OIM
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
Structural   |
  read <-    |
    math     |
        grp1 |   .5118412   .1408644     3.63   0.000     .2357521    .7879304
        grp2 |   .4896411   .1699667     2.88   0.004     .1565124    .8227698
        grp3 |   .8001383   .1030434     7.77   0.000      .598177      1.0021
        grp4 |   .6923993   .1199016     5.77   0.000     .4573964    .9274022
    _cons    |
        grp1 |   22.44033   6.632336     3.38   0.001     9.441189    35.43947
        grp2 |   25.26262   8.593626     2.94   0.003     8.419422    42.10582
        grp3 |   10.58565   5.792941     1.83   0.068    -.7683057    21.93961
        grp4 |    17.2921   7.065407     2.45   0.014     3.444152    31.14004
  -----------+----------------------------------------------------------------
  science <- |
    read     |
        grp1 |    .370112   .1421434     2.60   0.009     .0915161    .6487079
        grp2 |   .5359768   .1138689     4.71   0.000     .3127978    .7591559
        grp3 |   .5117303   .1229149     4.16   0.000     .2708215    .7526391
        grp4 |   .0790987    .132056     0.60   0.549    -.1797262    .3379236
    math     |
        grp1 |   .5519313   .1591829     3.47   0.001     .2399386    .8639239
        grp2 |   .4858494   .1412957     3.44   0.001     .2089149    .7627839
        grp3 |   .2365737   .1389155     1.70   0.089    -.0356956    .5088431
        grp4 |    .605095   .1401508     4.32   0.000     .3304044    .8797855
    _cons    |
        grp1 |   4.500176   7.390037     0.61   0.543     -9.98403    18.98438
        grp2 |   1.473353   7.166934     0.21   0.837    -12.57358    15.52029
        grp3 |    10.8461   5.666826     1.91   0.056    -.2606705    21.95288
        grp4 |   16.15488   6.662502     2.42   0.015     3.096616    29.21314
-------------+----------------------------------------------------------------
Variance     |
  e.read     |
        grp1 |   61.51621   12.30324                      41.56705     91.0395
        grp2 |    70.4011   14.84186                      46.57259    106.4213
        grp3 |   47.59862   8.690279                      33.28028    68.07722
        grp4 |   46.68944   9.842998                      30.88657    70.57773
  e.science  |
        grp1 |   62.14594   12.42919                      41.99257    91.97146
        grp2 |   41.07736   8.659869                        27.174    62.09428
        grp3 |   43.14742   7.877605                      30.16806    61.71096
        grp4 |    36.6393   7.724243                      24.23808    55.38552
------------------------------------------------------------------------------
LR test of model vs. saturated: chi2(0)   =      0.00, Prob > chi2 =      .

Let’s look at the direct and indirect effects using estat teffects.

estat teffects
Direct effects
------------------------------------------------------------------------------
             |                 OIM
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
Structural   |
  read <-    |
    math     |
        grp1 |   .5118412   .1408644     3.63   0.000     .2357521    .7879304
        grp2 |   .4896411   .1699667     2.88   0.004     .1565124    .8227698
        grp3 |   .8001383   .1030434     7.77   0.000      .598177      1.0021
        grp4 |   .6923993   .1199016     5.77   0.000     .4573964    .9274022
  -----------+----------------------------------------------------------------
  science <- |
    read     |
        grp1 |    .370112   .1421434     2.60   0.009     .0915161    .6487079
        grp2 |   .5359768   .1138689     4.71   0.000     .3127978    .7591559
        grp3 |   .5117303   .1229149     4.16   0.000     .2708215    .7526391
        grp4 |   .0790987    .132056     0.60   0.549    -.1797262    .3379236
    math     |
        grp1 |   .5519313   .1591829     3.47   0.001     .2399386    .8639239
        grp2 |   .4858494   .1412957     3.44   0.001     .2089149    .7627839
        grp3 |   .2365737   .1389155     1.70   0.089    -.0356956    .5088431
        grp4 |    .605095   .1401508     4.32   0.000     .3304044    .8797855
------------------------------------------------------------------------------

Indirect effects
------------------------------------------------------------------------------
             |                 OIM
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
Structural   |
  read <-    |
    math     |
         [*] |          0  (no path)
  -----------+----------------------------------------------------------------
  science <- |
    read     |
         [*] |          0  (no path)
    math     |
        grp1 |   .1894386   .0895064     2.12   0.034     .0140093    .3648678
        grp2 |   .2624363   .1068059     2.46   0.014     .0531006     .471772
        grp3 |    .409455   .1115931     3.67   0.000     .1907367    .6281734
        grp4 |   .0547679    .091926     0.60   0.551    -.1254038    .2349395
------------------------------------------------------------------------------
Note: [*] identifies parameter estimates constrained to be equal across

      groups.

Total effects
------------------------------------------------------------------------------
             |                 OIM
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
Structural   |
  read <-    |
    math     |
        grp1 |   .5118412   .1408644     3.63   0.000     .2357521    .7879304
        grp2 |   .4896411   .1699667     2.88   0.004     .1565124    .8227698
        grp3 |   .8001383   .1030434     7.77   0.000      .598177      1.0021
        grp4 |   .6923993   .1199016     5.77   0.000     .4573964    .9274022
  -----------+----------------------------------------------------------------
  science <- |
    read     |
        grp1 |    .370112   .1421434     2.60   0.009     .0915161    .6487079
        grp2 |   .5359768   .1138689     4.71   0.000     .3127978    .7591559
        grp3 |   .5117303   .1229149     4.16   0.000     .2708215    .7526391
        grp4 |   .0790987    .132056     0.60   0.549    -.1797262    .3379236
    math     |
        grp1 |   .7413699   .1508776     4.91   0.000     .4456553    1.037084
        grp2 |   .7482857   .1586025     4.72   0.000     .4374305    1.059141
        grp3 |   .6460288     .11138     5.80   0.000      .427728    .8643295
        grp4 |   .6598629   .1066384     6.19   0.000     .4508554    .8688704
------------------------------------------------------------------------------

In particular, we are interested in comparing the indirect effects for group 3 with group 4 (.409455 versus .0547679). These are the largest and the smallest of the indirect effects.

There are two ways we can do this: 1) Using delta method standard errors via nlcom, or 2) using bootstrap standard errors.

Using nlcom

Before we begin using nlcom, let’s rerun sem with the coeflegend option to keep the names of the coefficients straight.

sem, coeflegend
Structural equation model                       Number of obs      =       200
Grouping variable  = grp                        Number of groups   =         4
Estimation method  = ml
Log likelihood     =  -2051.912

------------------------------------------------------------------------------
             |      Coef.  Legend
-------------+----------------------------------------------------------------
Structural   |
  read <-    |
    math     |
        grp1 |   .5118412  _b[read:1bn.grp#c.math]
        grp2 |   .4896411  _b[read:2.grp#c.math]
        grp3 |   .8001383  _b[read:3.grp#c.math]
        grp4 |   .6923993  _b[read:4.grp#c.math]
    _cons    |
        grp1 |   22.44033  _b[read:1bn.grp]
        grp2 |   25.26262  _b[read:2.grp]
        grp3 |   10.58565  _b[read:3.grp]
        grp4 |    17.2921  _b[read:4.grp]
  -----------+----------------------------------------------------------------
  science <- |
    read     |
        grp1 |    .370112  _b[science:1bn.grp#c.read]
        grp2 |   .5359768  _b[science:2.grp#c.read]
        grp3 |   .5117303  _b[science:3.grp#c.read]
        grp4 |   .0790987  _b[science:4.grp#c.read]
    math     |
        grp1 |   .5519313  _b[science:1bn.grp#c.math]
        grp2 |   .4858494  _b[science:2.grp#c.math]
        grp3 |   .2365737  _b[science:3.grp#c.math]
        grp4 |    .605095  _b[science:4.grp#c.math]
    _cons    |
        grp1 |   4.500176  _b[science:1bn.grp]
        grp2 |   1.473353  _b[science:2.grp]
        grp3 |    10.8461  _b[science:3.grp]
        grp4 |   16.15488  _b[science:4.grp]
-------------+----------------------------------------------------------------
Variance     |
  e.read     |
        grp1 |   61.51621  _b[var(e.read):1bn.grp]
        grp2 |    70.4011  _b[var(e.read):2.grp]
        grp3 |   47.59862  _b[var(e.read):3.grp]
        grp4 |   46.68944  _b[var(e.read):4.grp]
  e.science  |
        grp1 |   62.14594  _b[var(e.science):1bn.grp]
        grp2 |   41.07736  _b[var(e.science):2.grp]
        grp3 |   43.14742  _b[var(e.science):3.grp]
        grp4 |    36.6393  _b[var(e.science):4.grp]
------------------------------------------------------------------------------
LR test of model vs. saturated: chi2(0)   =      0.00, Prob > chi2 =      .

First, we will reproduce the indirect effects so that we can compare the coefficients and standard errors to those from the estat teffects command. Here is the indirect effect for group 3.

nlcom _b[read:3.grp#c.math]*_b[science:3.grp#c.read]

       _nl_1:  _b[read:3.grp#c.math]*_b[science:3.grp#c.read]

------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _nl_1 |    .409455   .1115931     3.67   0.000     .1907367    .6281734
------------------------------------------------------------------------------

And now, the indirect effect for grp 4.

nlcom _b[read:4.grp#c.math]*_b[science:4.grp#c.read]

       _nl_1:  _b[read:4.grp#c.math]*_b[science:4.grp#c.read]

------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _nl_1 |   .0547679    .091926     0.60   0.551    -.1254038    .2349395
------------------------------------------------------------------------------

Since these values agree with estat teffects we are finally, we are ready to look at the difference in the two indirect effects.

nlcom (_b[read:3.grp#c.math]*_b[science:3.grp#c.read]) - ///
      (_b[read:4.grp#c.math]*_b[science:4.grp#c.read])

       _nl_1:  (_b[read:3.grp#c.math]*_b[science:3.grp#c.read]) - (_b[read:4.grp#
> c.math]*_b[science:4.grp#c.read])

------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _nl_1 |   .3546871   .1445801     2.45   0.014     .0713154    .6380589

-----------------------------------------------------------------------------

The difference in the indirect effects is .355 and is statistically significant using
the delta method standard errors. If you are uncomfortable with the normal theory
assumptions behind the delta method, you may want to use the bootstrap method show in
the next section.

Using bootstrap

We begin by writing a program, which we are calling bootind, that computes and returns the two indirect effects and the difference between the two. This program must be saved as bootind.ado.

program bootind, rclass

  sem (read <- math)(science <- read math), group(grp)
  
  nlcom _b[read:3.grp#c.math]*_b[science:3.grp#c.read]
  return scalar ind3  = el(r(b),1,1)
  
  nlcom _b[read:4.grp#c.math]*_b[science:4.grp#c.read]
  return scalar ind4  = el(r(b),1,1)
  
  nlcom (_b[read:3.grp#c.math]*_b[science:3.grp#c.read]) - ///
        (_b[read:4.grp#c.math]*_b[science:4.grp#c.read])
  return scalar ind3v4  = el(r(b),1,1)
end

We will demonstrate bootind with 500 bootstrap replications. You will probably want to run more replications, maybe a lot more. Also, we will set the random seed so that the results are replicable.

set seed 676767

bootstrap r(ind3) r(ind4) r(ind3v4), reps(500) nodots: bootind

Bootstrap results                               Number of obs      =       200
                                                Replications       =       500

      command:  bootind
        _bs_1:  r(ind3)
        _bs_2:  r(ind4)
        _bs_3:  r(ind3v4)

------------------------------------------------------------------------------
             |   Observed   Bootstrap                         Normal-based
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _bs_1 |    .409455   .1059136     3.87   0.000     .2018682    .6170418
       _bs_2 |   .0547679   .0929578     0.59   0.556    -.1274261    .2369618
       _bs_3 |   .3546871   .1456614     2.44   0.015      .069196    .6401783
------------------------------------------------------------------------------

The confidence intervals shown above are normal-based, so next, we will obtain results
with percentile and bias-corrected confidence intervals.

estat boot, percentile bc

Bootstrap results                               Number of obs      =       200
                                                Replications       =       500

      command:  bootind
        _bs_1:  r(ind3)
        _bs_2:  r(ind4)
        _bs_3:  r(ind3v4)

------------------------------------------------------------------------------
             |    Observed               Bootstrap
             |       Coef.       Bias    Std. Err.  [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _bs_1 |   .40945504   .0040702   .10591358    .2073264   .6299226   (P)
             |                                       .2047078   .6144838  (BC)
       _bs_2 |   .05476789   .0025732   .09295781   -.1129886   .2373435   (P)
             |                                      -.1134027   .2346437  (BC)
       _bs_3 |   .35468715    .001497   .14566141    .0743807   .6392108   (P)
             |                                        .080749   .6458999  (BC)
------------------------------------------------------------------------------
(P)    percentile confidence interval
(BC)   bias-corrected confidence interval

Neither the percentile nor the bias-corrected confidence intervals for the difference
include zero, so we can conclude that the mediated effects are significantly different
between group 2 and group 4.

 

Cite this article

stats writer (2024). How can I compare indirect effects in a multiple group model?. PSYCHOLOGICAL SCALES. Retrieved from https://scales.arabpsychology.com/stats/how-can-i-compare-indirect-effects-in-a-multiple-group-model/

stats writer. "How can I compare indirect effects in a multiple group model?." PSYCHOLOGICAL SCALES, 1 Jul. 2024, https://scales.arabpsychology.com/stats/how-can-i-compare-indirect-effects-in-a-multiple-group-model/.

stats writer. "How can I compare indirect effects in a multiple group model?." PSYCHOLOGICAL SCALES, 2024. https://scales.arabpsychology.com/stats/how-can-i-compare-indirect-effects-in-a-multiple-group-model/.

stats writer (2024) 'How can I compare indirect effects in a multiple group model?', PSYCHOLOGICAL SCALES. Available at: https://scales.arabpsychology.com/stats/how-can-i-compare-indirect-effects-in-a-multiple-group-model/.

[1] stats writer, "How can I compare indirect effects in a multiple group model?," PSYCHOLOGICAL SCALES, vol. X, no. Y, ص Z-Z, July, 2024.

stats writer. How can I compare indirect effects in a multiple group model?. PSYCHOLOGICAL SCALES. 2024;vol(issue):pages.

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