1. 2nd Order CFA Hierarchical Latent Models
Kline Chapter 9
Beaujean Chapter 9

2. Naming Hierarchical: Higher-order: Bi-factor
Structured into an order or rank
Higher-order:
Describes when latent variables are structured so that latents influence other latents into levels
Bi-factor
Generally used to describe a CFA with two sets of latent variables (not hierarchical)

3. Hierarchical Models The idea of a higher order model is:
You have some latent variables that are measured by the observed variables
A portion of the variance in the latent variables can be explained by a second (set) of latent variables

4. Hierarchical Models
Therefore, we are switching out the covariances between factors and using another latent to explain them.

5. Hierarchical Models When are these used?
When there are multiple latent variables that covary with each other (and a lot)
A second set of latents explains that covariance

7. Hierarchical Models
The covariance of the first order is accounted for by the second order plus a specific factor
Specific factors are error that is not explained by the second order latents.
The higher order is thought to indirectly influence the manifest variables through the first order.

8. Identification
Remember that each portion of the model has to be identified.
The section with each latent variable has to be identified (so you need at least one loading or LV set to 1).
The section with the latents has to be identified

9. Identification You can do get over identification in a couple of ways:
Set some of the loadings in the upper portion of the model to be equal (give them the same name)
You can set the variance in the upper latent to be 1
You can set some of the error variances of the latents in the lower portion to be equal

11. Bi-Factor Models
Special type of model with two sets of latents, but they are not hierarchically structured.
Best used when:
General factor that accounts for variance in the manifest variables
Domain specific areas that are thought to influence the manifest variables

12. Bi-Factor Models
One thing to note is that the latent variables are left uncorrelated in this type of model.
This structure represents the domain specific part of the interpretation.

13. Model Differences Differences between bi-factor and hierarchical:
In hierarchical models, the second order influences the first order, while the two sets of latents in bi-factors are uncorrelated.
What does that allow you to test differently?

14. Model Differences Advantages:
Allows you to see how the first order LV influence the manifest variables separately from the other LV.
After accounting for the general LV, are the domain specific items still accounting for variance?
You can compare models with and without the domain specific areas.

15. Book Examples First, let’s fit a first order model for the WISC
If the first order model doesn’t work, then a second order isn’t appropriate
You should also check out the correlations/covariances between factors to make sure that they are even related.

16. Book Examples
In the first order model, we find that the correlations are pretty high.
That’s a good sign that maybe a second order model is appropriate.
(or that factors should be collapsed, they are not distinct).

17. Book Examples Now, let’s try a second order model.
Set the variance of the latent to 1
g=~ NA*gf + gc + gsm + gs
g~~ 1*g
Set a path to 1
g=~ gf + gc + gsm + gs

18. Book Examples Bi-factor model What does this code do?
gf =~ a*Matrix.Reasoning + a*Picture.Concepts
gsm =~ b*Digit.Span + b*Letter.Number
gs =~ c*Coding + c*Symbol.Searc

19. Book Examples Bi-factor model What does this code do? orthogonal=TRUE
Forces the latents to be uncorrelated.