SEM - Direct and Indirect effect contribution

Question

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

Answer 1

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.