Structural Equation Modeling – Understanding Direct and Indirect Effect Contributions in SEM

structural-equation-modeling

I'm trying to understand how the direct and indirect impact on a SEM using the plspm package.
The structure path of the model is the following :

After estimating I get the following results :

  relationships    direct indirect     total
   IMAG -> SAT 0.3757397 0.000000 0.3757397
   IMAG -> LOY 0.1228504 0.225799 0.3486493
   SAT -> LOY 0.6009453 0.000000 0.6009453

My question comes when trying to decompose LOY how the effect of IMAG should be measured, by using direct and indirect or total?

For example the following are true:

LOY_fitted = Intercept + 0.6009453*SAT + 0.1228504*IMAG
SAT_fitted = Intercept + 0.3757397*IMAG

However when I try to include the indirect contribution of IMAG the result is different from the Fitted values

LOY_fitted ≠ Intercept + 0.6009453*SAT + 0.1228504*IMAG + 0.225799 *IMAG

which one would be the correct way to measure the impact of total impact of IMAG to LOY?

Code to reproduce :

library(plspm)
data(satisfaction)   
IMAG = c(0,0,0)
SAT = c(1,0,0)
LOY = c(1,1,0)
sat_path = rbind(IMAG, SAT, LOY)
sat_blocks = list(1,20,24)
sat_mod = rep("A", 3)
satpls = plspm(Data = satisfaction, path_matrix = sat_path, blocks = sat_blocks,
               modes = sat_mod,scaled = FALSE)
satpls$inner_model
satpls$effects

Best Answer

Your expansion of LOY is incorrect. The indirect effect of IMG on LOY is the effect that goes through SAT. Hence it should be included by substituting SAT by its expression in terms of IMG.

Here I have shortened the variable names to I, S, and L; and I have rounded the coefficients a bit:

$$ S = b + 0.38I, $$

and so

$$ \begin{align*} L &= a + 0.6S + 0.12I \\ &= a + 0.6(b + 0.38I) + 0.12I \\ &= (a + 0.6b) + 0.6\cdot 0.38I + 0.12 I \\ &= c + 0.228I + 0.12I \\ &= c + 0.348 I, \end{align*} $$

which is what's reported as the total effect of IMG on LOY. Note that your expansion would be correct if you dropped the LOY term as you see from the second to last line above.

We see that to get the indirect effect we simply multiply coefficients along the path from IMG to LOY via SAT: $0.6\cdot 0.38$ is the product of SAT's coefficient in the one expression and IMG's coefficient in the other.

NB that the package plspm seems to be no longer supported and has been removed from CRAN. Hence your reproducible example is hard to reproduce and I haven't tried to do so.

Related Solutions

Structural Equation Modeling – Estimating Direct and Total Effects with Regressions and SEM (lavaan)

Drawing back to the world of mediation as defined by Baron & Kenny (1986), you would typically test a mediation model using separate regression equations. According to this approach the researcher would seek to demonstrate a series of conditions, which if they held, would constitute evidence of mediation. The first condition is demonstrated by your first regression model lm(Y~X, data=Data). This shows that there is a significant overall effect of $X$ on $Y$ (side note: this is also the total effect).

The next condition to test is to show that $X$ is significantly related to $M$. This is missing from your modeling, but can be easily estimated based on the data you generated.

> summary(lm(M~X))

Call:
lm(formula = M ~ X)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.88626 -0.61401  0.00236  0.58645  2.98774 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.03715    0.10498   0.354    0.724    
X            0.47392    0.10378   4.567 1.44e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.037 on 98 degrees of freedom
Multiple R-squared:  0.1755,    Adjusted R-squared:  0.1671 
F-statistic: 20.85 on 1 and 98 DF,  p-value: 1.442e-05

Here we can see that indeed the second condition of mediation is met as $X$ significantly predicts $M$. Also note that you get the same unstandardized regression coefficient .474 in this model as you do in the SEM model above.

Following the Baron and Kenny approach the next two conditions are tested simultaneously in one final regression model. First, controlling for $X$, $M$ should significantly predict $Y$, which it does in your model above. Second, the relation between $X$ and $Y$ should be attenuated when controlling for $M$. This condition is also met according to your model above.

To calculate the significance of the mediation (i.e., the indirect effect), researchers tended to use Sobel's Test, at least when the full Baron and Kenny approach was still in vogue. To perform this test you needed to take the direct effect of $X$ on $M$ (from regression model #2) and multiply it by the direct effect of $M$ on $Y$, controlling for $X$ (to calculate significance you also needed the standard errors for these regression coefficients).

An SEM model tests all of these same conditions and can assess the significance of mediation in one step essentially. Note that all of the elements needed for a Sobel's test are in the single SEM model. Also note that all of the coefficients are essentially the same as those you calculated using the linear regression framework proposed originally by Baron and Kenny.

Perhaps the issue comes down to parlance? The total effect in a mediation model is sometimes delineated as $c$ (it is necessarily the unconditional effect of $X$ on $Y$). The direct effect of $X$ on $Y$ conditioned on $M$ is often notated as $c'$ where $c' + a*b = c$. These are terminology conventions used by researchers in reference to mediation models specifically and may not translate perfectly to all other uses of the terms "total effect" and "direct effect" in the world of statistics.

Best Answer

Related Solutions

Structural Equation Modeling – Estimating Direct and Total Effects with Regressions and SEM (lavaan)

Related Question