# Comparing Indirect Paths and Relative Contributions

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### Sn Q. Nguyen

Jun 27, 2019, 4:48:31 PM6/27/19
to lavaan
Hi all,

I am trying to construct a mediation model with two independent variables (X1 and X2), one mediator (M), and one outcome (Y) - with the generic syntax below:

'
# regressions -
M ~ a1*X1 + a2*X2
Y ~ b*M + X1 + X2

# covariance -
X1 ~~ X2

# indirect effects -
IDE_X1 := a1*b
IDE_X2 := a2*b
'

(1.) Is it possible to compare the two indirect effects (i.e., whether one is stronger than the other)? If so, how should I go about doing so? Would it be to add 'contrast := IDE_X1 - IDE_X2' or 'contrast := IDE_X2 - IDE_1', which other posts have done to compare the indirect effects of multiple different mediators, but not of different independent variables? If there are other alternative/additional approaches to extract and assess the relative contributions between X1 versus X2, please advise.

(2.) Also, would the option of equality constraints (i.e., 'IDE_X1 == IDE_X2' or 'a1 == a2') be appropriate?

Thank you so much!

### Terrence Jorgensen

Jun 27, 2019, 5:18:42 PM6/27/19
to lavaan
(1.) Is it possible to compare the two indirect effects (i.e., whether one is stronger than the other)?

Only if they are on the same scale (e.g., 7-point Likert scale, or actual units like meters)

If so, how should I go about doing so? Would it be to add 'contrast := IDE_X1 - IDE_X2' or 'contrast := IDE_X2 - IDE_1'?

Either way you prefer to interpret it.  Same absolute value and z test.

If there are other alternative/additional approaches to extract and assess the relative contributions between X1 versus X2, please advise.

I've seen people compare standardized slopes, but that doesn't generalize as well as people expect.  A sample SD is not a "standard".  Besides, you can't (easily) get a test of the comparison that way.

If you labeled the direct effects too, and defined total effects of each predictor, then you could compare the proportion of each variable's effect that is mediated.

`M ~ a1*X1 + a2*`
`X2Y ~ b*M + c1*X1 + c2*X2`
`# indirect effects -IDE_X1 := a1*bIDE_X2 := a2*b`
`# total effectstot_x1 := IDE_X1 + c1tot_x2 := IDE_X2 + c2# proportion mediatedratio_x1 := IDE_X1 / tot_x1ratio_x2 := IDE_X2 / tot_x2## compare using a ratio (like a risk ratio)ratio12 := ratio_x1 / ratio_x2ratio21 := ratio_x2 / ratio_x1 # other direction## expressed as a percentage higher (or lower, if negative)perc12 := (ratio_12 - 1)*100`

(2.) Also, would the option of equality constraints (i.e., 'IDE_X1 == IDE_X2' or 'a1 == a2') be appropriate?

Sure.  You could also use a Wald test after fitting the unrestricted model:

`lavTestWald(fit, constraints = 'IDE_X1 == IDE_X2')`

Terrence D. Jorgensen
Assistant Professor, Methods and Statistics
Research Institute for Child Development and Education, the University of Amsterdam

### Nickname

Jun 28, 2019, 4:16:44 PM6/28/19
to lavaan
There seem to be some interesting special cases in which the indirect effect and direct effect differ in sign.

`# proportion mediatedratio_x1 := IDE_X1 / tot_x1`
`If IDE_X1 = .5 and c1 = -.25 (direct effect) then tot_x1 = .25 and ratio_x1 = 2If IDE_X1 = .5 and c1 = -.75 then tot_x1 = -.25 and ratio_x1 = -2.`
Perhaps the conclusion to be drawn is to carefully consider each constituent step before interpreting the final ratio of ratios and to think carefully about whether one is testing the desired hypothesis.  In some cases, it might be useful to test the ratio of the ratios of the indirect effect to the direct effect, in which case positive constituent ratios would indicate reinforcing effects and negative ratios would indicate counteracting effects.

Keith
------------------------
Keith A. Markus
John Jay College of Criminal Justice, CUNY
http://jjcweb.jjay.cuny.edu/kmarkus
Frontiers of Test Validity Theory: Measurement, Causation and Meaning.
http://www.routledge.com/books/details/9781841692203/