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Now, what if MV1 and DV were binary variables while MV2 and IV were continuous. In that case the calculation of the indirect effects would require a combination of OLS regression along with either logit or probit models. This web page presents a Stata program, binary_mediation, that can be with multiple mediator variables in any combination of binary or continuous along with either a binary or continuous response variable. You can download binary_mediation by typing findit binary_mediation in Stata's command window and following the instructions.
Different researchers compute indirect effects using different approaches. We will compute indirect effects using the product of coefficients approach. This is fairly straight forward when all the variables are continuous. Having a combination of continuous and binary variables makes things a bit trickier.
David A. Kenny in a paper available from his website (Mediation with Dichotomous Outcomes), recommends rescaling (standardizing) coefficients before computing indirect effects. The reasoning behind this is that in OLS regression the residual variance for the model changes as variables are entered or removed from the regression equation. In logistic or probit regression, on the other hand, the residual variance is fixed. Since the residual is fixed the scaling of the coefficients varies. Computing indirect effects involves multiple models, each with different variables. In order to compare coefficients from one model to Kenny recommends standardizing the coefficients. Coefficients from OLS models are rescaled using the standard deviations of the observed variables. For logit or probit models the rescaling involves the standard deviation of the underlying latent variable for the binary variable. Once the coefficients are rescaled (standardized) the indirect effects van be computed as the product of coefficients. Nathaniel Herr has a very nice diagram on his webpage that illustrates the different scaling that occurs when both the mediator and response variables are binary.
The user written command, binary_mediation, can be used to compute indirect effects using the product of coefficients approach. The program standardizes all the coefficients for OLS, logit and probit models. The results using logit or probit, once standardized, are very similar.
Please note: binary_mediation does not compute standard errors or confidence intervals directly. You will need to use binary_mediation with the bootstrap command to obtain standard errors and confidence intervals.
For this series of example we will use the hsbdemo dataset. We will create a binary mediator hiread by dichotomizing read. We do not recommend dichotomizing continuous variables, we just want to demonstrate the process with one binary mediator. Along with hiread we will use science as a continuous mediator, ses as a continuous predictor and honors as a binary response variable.
The binary_mediation program will detect which variables are continuous and which are binary.
use http://www.ats.ucla.edu/stat/data/hsbdemo, clear
generate hiread=read>=50 /* create binary mediator */
summarize ses hiread science honors /* descriptive statistics */
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
ses | 200 2.055 .7242914 1 3
hiread | 200 .585 .4939585 0 1
science | 200 51.85 9.900891 26 74
honors | 200 .265 .4424407 0 1
binary_mediation, dv(honors) mv(hiread science) iv(ses)
Logit: hiread on iv (a1 path)
Logistic regression Number of obs = 200
LR chi2(1) = 12.40
Prob > chi2 = 0.0004
Log likelihood = -129.52516 Pseudo R2 = 0.0457
------------------------------------------------------------------------------
hiread | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
ses | .7204026 .2109932 3.41 0.001 .3068636 1.133942
_cons | -1.115341 .4465912 -2.50 0.013 -1.990643 -.2400381
------------------------------------------------------------------------------
OLS regression: science on iv (a2 path)
------------------------------------------------------------------------------
science | Coef. Std. Err. t P>|t| Beta
-------------+----------------------------------------------------------------
ses | 3.866564 .9317955 4.15 0.000 .2828553
_cons | 43.90421 2.029732 21.63 0.000 .
------------------------------------------------------------------------------
Logit: dv on iv (c path)
Logistic regression Number of obs = 200
LR chi2(1) = 7.34
Prob > chi2 = 0.0068
Log likelihood = -111.97593 Pseudo R2 = 0.0317
------------------------------------------------------------------------------
honors | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
ses | .6185825 .2344357 2.64 0.008 .159097 1.078068
_cons | -2.337778 .5417028 -4.32 0.000 -3.399496 -1.27606
------------------------------------------------------------------------------
Logit: dv on mv & iv (b & c' paths)
Logistic regression Number of obs = 200
LR chi2(3) = 51.61
Prob > chi2 = 0.0000
Log likelihood = -89.83923 Pseudo R2 = 0.2231
------------------------------------------------------------------------------
honors | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
hiread | 1.597298 .5332837 3.00 0.003 .552081 2.642515
science | .0901672 .0253211 3.56 0.000 .0405389 .1397956
ses | .2516925 .266301 0.95 0.345 -.2702479 .7736328
_cons | -7.658069 1.456197 -5.26 0.000 -10.51216 -4.803975
------------------------------------------------------------------------------
Indirect effects with binary response variable honors
indir_1 = .09141282 (hiread, binary)
indir_2 = .10582395 (science, continuous)
total indirect = .19723677
direct effect = .07639769
total effect = .27363446
c_path = .23980637
proportion of total effect mediated = .72080384
ratio of indirect to direct effect = 2.5817112
Binary models use logit regressionThe binary_mediation program does not produce any standard errors or confidence intervals on its own. We will use the bootstrap command to obtain a standard errors for the direct and indirect effects along with a 95% percentile confidence intervals. We will demonstrate the process using 500 bootstrap replications but you can set the number to anything you prefer. We recommend the percentile or biased-corrected confidence intervals over normal-based confidence intervals. You can bootstrap any of the effects found in the return list.
quietly bootstrap r(indir_1) r(indir_2) r(tot_ind) r(dir_eff) r(tot_eff), ///
reps(500): binary_mediation, dv(honors) iv(ses) mv(hiread science)
estat bootstrap, percentile bc
Bootstrap results Number of obs = 200
Replications = 499
command: binary_mediation, dv(honors) iv(ses) mv(hiread science)
_bs_1: r(indir_1)
_bs_2: r(indir_2)
_bs_3: r(tot_ind)
_bs_4: r(dir_eff)
_bs_5: r(tot_eff)
------------------------------------------------------------------------------
| Observed Bootstrap
| Coef. Bias Std. Err. [95% Conf. Interval]
-------------+----------------------------------------------------------------
_bs_1 | .09141282 -.0000552 .03717104 .0299178 .1781988 (P)
| .0333105 .1959342 (BC)
_bs_2 | .10582395 .001447 .03999136 .0421071 .1912641 (P)
| .0443143 .1973525 (BC)
_bs_3 | .19723677 .0013918 .05159597 .098798 .3049328 (P)
| .107806 .3141167 (BC)
_bs_4 | .07639769 -.0046966 .07954484 -.0831474 .2288187 (P)
| -.0747334 .2309053 (BC)
_bs_5 | .27363446 -.0033048 .09258509 .0802001 .4406839 (P)
| .0739526 .4394651 (BC)
------------------------------------------------------------------------------
(P) percentile confidence interval
(BC) bias-corrected confidence interval
Note: one or more parameters could not be estimated in 1 bootstrap replicate;
standard-error estimates include only complete replications.In looking at the bootstrap results, we see that both of the indirect effects appear to be significant (confidence interval does not contain zero) along with the total indirect effect. The direct effect, however, is not statistically significant.
For comparison purposes we will rerun binary_mediation using the probit option along with the diagram option.
binary_mediation, dv(honors) mv(hiread science) iv(ses) probit diagram
Probit: hiread on iv (a1 path)
Probit regression Number of obs = 200
LR chi2(1) = 12.37
Prob > chi2 = 0.0004
Log likelihood = -129.54145 Pseudo R2 = 0.0456
------------------------------------------------------------------------------
hiread | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
ses | .4437993 .1279512 3.47 0.001 .1930196 .6945791
_cons | -.687353 .2744143 -2.50 0.012 -1.225195 -.1495109
------------------------------------------------------------------------------
OLS regression: science on iv (a2 path)
------------------------------------------------------------------------------
science | Coef. Std. Err. t P>|t| Beta
-------------+----------------------------------------------------------------
ses | 3.866564 .9317955 4.15 0.000 .2828553
_cons | 43.90421 2.029732 21.63 0.000 .
------------------------------------------------------------------------------
Probit: dv on iv (c path)
Probit regression Number of obs = 200
LR chi2(1) = 7.05
Prob > chi2 = 0.0079
Log likelihood = -112.12049 Pseudo R2 = 0.0305
------------------------------------------------------------------------------
honors | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
ses | .3500684 .1332785 2.63 0.009 .0888474 .6112894
_cons | -1.36609 .3001514 -4.55 0.000 -1.954376 -.7778043
------------------------------------------------------------------------------
Probit: dv on mv & iv (b * c' paths)
Probit regression Number of obs = 200
LR chi2(3) = 50.99
Prob > chi2 = 0.0000
Log likelihood = -90.149337 Pseudo R2 = 0.2205
------------------------------------------------------------------------------
honors | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
hiread | .8613714 .2822841 3.05 0.002 .3081048 1.414638
science | .0510501 .0142893 3.57 0.000 .0230435 .0790566
ses | .1156009 .1543426 0.75 0.454 -.186905 .4181068
_cons | -4.250488 .7704867 -5.52 0.000 -5.760614 -2.740362
------------------------------------------------------------------------------
Indirect effects with binary response variable honors
indir_1 = .09911685 (hiread, binary)
indir_2 = .10883112 (science, continuous)
total indirect = .20794797
direct effect = .06373715
total effect = .27168512
c_path = .24577434
proportion of total effect mediated = .76540067
ratio of indirect to direct effect = 3.2625868
Binary models use probit regression
Reference Mediation Diagram
IV --- coef c --- DV
MV1
/ \
coef a1 coef b1
/ \
IV --- coef c' --- DV
\ /
coef a2 coef b2
\ /
MV2
Kenny, D. A.(2008) Mediation with Dichotomous Outcomes. Retrieved April 23, 2010 from website: http://davidakenny.net/doc/dichmed.doc .
Kenny, D. A.(2009) Mediation. Retrieved April 23, 2010 from website: http://davidakenny.net/cm/mediate.htm .
Herr, N. A. (undated) Mediation with Dichotomous Outcomes. Retrieved April 18, 2011 from website: http://www.nrhpsych.com/mediation/logmed.html .
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