Interpreting lavaan SEM coefficients

Question

First time using lavaan package in R to run Structural Equation Model (SEM) analysis.

Code:

fac1 =~ a1+a3+a4+a5

fac2 =~ a2+a7+a8+a12

fac3 =~ a9+a10+a11+a14

fac4 =~ a12+a13+a15+a16

fac4 ~ fac1+fac2+fac3

fac3 ~ fac1+fac2

..... some more specifying the co-variance between items a1 to a16

Output:

Latent Variables:
               Estimate  Std.Err  Z-value  P(>|z|)   Std.lv  Std.all
fac1 =~

a1                1.000                               0.624    0.684

a3                0.848    0.112    7.589    0.000    0.529    0.568
....
....
....
fac2 =~

a12               1.000                               0.463    0.330

a2                3.764    1.290    2.918    0.004    1.742    1.691

My questions:

1. How did lavaan select a1 for fac1 and a12 for fac2 and why did it assign the values 1 as coefficients?
2. Are they a1 and a12 significant contributors to respective latent variables?
3. Is there a way to let the model estimate them or derive them without the value being set to 1?

Answer 1

When performing latent variable modeling, there is always an issue of what scale to assign the unobserved variables. After all, you have not collected data on these latent variables, so how could you possibly know their scale or their variance?

One way to solve this problem, which is the default in many SEM programs, including the lavaan package in R, is to fix the loading of the first variable for a given latent variable to 1. This has the effect of assigning the scale of that observed variable to the latent variable.

Another popular alternative is to used a standardized scale for the latent variable (i.e., mean = 0, sd = 1), in which case the loading for the first variable is freely estimated by the model. In lavaan this can be implemented as follows:

`fit<-cfa(model, data=df, std.lv=T)`

Adding std.lv=T tells lavaan to use a standardized scale for the latent variable instead of fixing a loading to 1.