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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)

measurement model

ceoec
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1 Answers1

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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.

Jeremy Miles
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    +1, FWIW I also didn't understand the sentence from your last paragraph. – Patrick Coulombe May 12 '15 at 22:17
  • @PatrickCoulombe, sorry I with the those curved double-headed arrows.... – ceoec May 13 '15 at 04:54
  • @jeremymiles, I have added the syntax and path diagram... thanks! – ceoec May 13 '15 at 04:55
  • 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.... – ceoec May 13 '15 at 07:47
  • 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? – Jeremy Miles May 13 '15 at 14:02
  • I simplified the model for now -- the diagram with 3 latent variables is too messy, and it doesn't fit... – ceoec May 13 '15 at 14:30
  • I don't even know if it is not fit, or fit... it is like this: Model Chisquare = 60.25102 Df = 38 Pr(>Chisq) = 0.01223957 RMSEA index = 0.06871829 90% CI: (0.03252335, 0.100333) Bentler-Bonett NFI = 0.8948807 Tucker-Lewis NNFI = 0.9077624 Bentler CFI = 0.9550637 – ceoec May 13 '15 at 14:50
  • Those fit indices are not the fit indices of the model that you have presented. I would say that they are probably not great, but that it's a very unparsimonious model. – Jeremy Miles May 13 '15 at 15:26
  • "I would like to how why we need to saturate the structural model and include the exogenous variable like gender" - as soon as you add a non-latent variable to the model, the notion of measurement and structural model breaks down. If we want to talk about measurement and structural models, we need to decide where (say) gender is. So you make it a latent variable. Now it's part of the measurement model, and it's relationships go into the structural model. – Jeremy Miles May 13 '15 at 15:28
  • Thanks @JeremyMiles, I actually tried to remove gender from the model and use multigroupModel in sem library of R to run, the model fit is great for female. But multigroup apparently just separated my data into two groups and run the data into the model one by one.... So may be that's not the right way to control for gender... – ceoec May 13 '15 at 15:37
  • Let us continue this discussion in chat. – Jeremy Miles May 13 '15 at 15:38
  • I understand about adding gender into measurement model now, but let's say, for instance, I am very certain gender has nothing to do with Income (as I run correlations) by saturated model I still need to add the path of income and gender, and this just seems weird to me. – ceoec May 13 '15 at 15:40
  • I moved this to a chat room. I'll answer there. – Jeremy Miles May 13 '15 at 15:42