Solved – SEM: What is measurement model

structural-equation-modeling

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

Best Answer

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.

Related Solutions

Solved – In testing a structural model fit, how strongly do the measurement models fit influence the overall structural model fit

There are two questions here: (1) do you need to fit/examine a measurement model before fitting/examining a structural model? and (2) if a given measurement model exhibits poor fit, is it still worth fitting/examining a structural model.

This is essentially a matter of could v. should. Could you just skip to fitting a structural model? Yes. Should you? Probably not. One reason: measurement models help to evaluate how reasonable your theory of construct measurement is. If you skip right to a structural model and get bad model fit, it could be because your theory of construct measurement sucks, or because your theory of structural construct relations sucks. In other cases, certain measurement models need to empirically tenable before you can make accurate inferences about structural relations. For example, if you want to examine certain structural parameters between groups, or over time, you would first need to establish measurement invariance between groups or over time, respectively (Little, 2013). You then compare your initial structural models against the more restricted of your measurement invariance models. Logistically, the difference in code between your final measurement model and initial structural model is likely so trivial, that I don't see any reason why not to look at a measurement model first.
There is a method of accomplishing this goal--parcelling (see Little, Cunningham, Shahar, & Widaman, 2002, for an introduction)--though it is a somewhat controversial tactic, with vocal supporters/critics. Essentially, parceling entails grouping a set variables for a construct (let's say you have a 9-item questionnaire) into 3 aggregates (i.e., sums or averages). You then use these 3 aggregates as indicators of their respective latent variable, and repeat this process for each latent variable. With 3 indicators each, every latent variable in your model will be "just-identified", virtually guaranteeing a strong fitting measurement model. The rationale for parcelling is that it allows you to better evaluate the structural portion of your model, as essentially all model misfit will be on account of misspecified structural relations. The controversial aspect of it, is that some see it as a way of mathematically "tricking" your way into good model fit. Related to your first question, Little et al. (2002) therefore recommend first examining an un-parcelled measurement model, especially if there is not prior research strongly supporting your theorized measurement model. If, however, you're using a measurement model that has replicated consistently, you may be fine with using a parceled model right away. There are a number of other benefits/drawbacks to parcelling, and you really should just read the Little et al. (2002) paper if you're considering it--it's a pretty manageable read, even if you are new to SEM.

References

Little, T. D. (2013). Longitudinal structural equation modeling. New York, NY: Guilford Press.

Little, T. D., Cunningham, W. A., Shahar, G., & Widaman, K. F. (2002). To parcel or not to parcel: Exploring the question, weighing the merits. Structural Equation Modeling, 9, 151-173.

SEM – Differences Between a MIMIC Factor and a Composite with Indicators in Structural Equation Modeling

They're the same model.

It's useful to be able to define a latent variable as a composite outcome where that variable only has composite indicators.

If you don't have:

f1 =~ y1 + y2 + y3

You can't put:

f1 ~ x1 + x2 + x3

But you can have:

f1 <~ x1 + x2 + x3

Best Answer

Related Solutions

Solved – In testing a structural model fit, how strongly do the measurement models fit influence the overall structural model fit

SEM – Differences Between a MIMIC Factor and a Composite with Indicators in Structural Equation Modeling

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