# SEM: What is measurement model?

I would like to know the definition of measurement model in SEM. Is it only about the latent variables? (THat's what I seemed to see what I google.) But I remember long time ago I read somewhere that we need to include all the variables in the SEM with bi-directional arrows too. I do not know which one I should follow...

For instance, I have a model with 3 latent variables (2 predictors, 1 outcome), and one exogenous predictor, one exogenous outcome, together with age and gender that I include in the structural model. The model fit of the structural model is good. But I am stuck at the measurement model, do I need to include those exogenous variables, age and gender? It is very bad if I include them, but if I just use all the latent variables, the measurement model is also good.

Simplified syntax:

model <- specifyModel()
love -> L1, lam1, NA
love -> L2, lam2, NA
hate -> H1, lam3, NA
hate -> H2, lam4, NA
L1 <-> L1, e3, NA
L2 <-> L2, e4, NA
H2 <-> H2, e5, NA
H2 <-> H2, e6, NA
Gender <-> Gender, e1, NA
Income <-> Income, e2, NA
love <-> love, NA, 1
hate <-> hate, NA, 1
hate <-> love, lh, NA
Gender <-> Income, gi, NA
Gender <-> love, gl, NA
Gender <-> hate, gh, NA
Income <-> love, li, NA
Income <-> hate, hi, NA
#


(By the way, I don't know how to represent this "love <-> love, NA, 1" in the path diagram)

The measurement model is the part of the model that examines relationship between the latent variables and their measures. The structural model is the relationship between the latent variables.

To test the measurement model, you typically saturate the structural model, by allowing all the latents to correlate. Then any misfit is in the measurement model.

I don't think you can assess the fit of the structural model if the measurement model doesn't fit.

You say that you have an outcome, along with age and gender? These should be considered part of the structural model. (In the old days, you would have set them up as latent variables with a single indicator, and no measurement error; nowadays you don't need to do that.)

"we need to include all the variables in the SEM with bi-directional arrows too." I'm not sure what that means. Syntax or a path diagram would help.

• +1, FWIW I also didn't understand the sentence from your last paragraph. Commented May 12, 2015 at 22:17
• @PatrickCoulombe, sorry I with the those curved double-headed arrows.... Commented May 13, 2015 at 4:54
• @jeremymiles, I have added the syntax and path diagram... thanks! Commented May 13, 2015 at 4:55
• I would like to how why we need to saturate the structural model and include the exogenous variable like gender if we just would like to examines the relationship between the latent variables and their measures? gender is not a measure of the latent variables... I am not sure what it is needed.... Commented May 13, 2015 at 7:47
• That model has a saturated measurement model (I would say), but it only has two latents - in your question you say you have three. Does that model that you have presented fit? What is the model you want to test? Commented May 13, 2015 at 14:02