Anonymous:Specific Question:In this Tuesday’s class, we started latent variable for discrete data with a simple example in which Y represents the success in task and X means the experience. Then we got a probit model: Y*=β0+β1 X +ε, where Y is the latent variable while X is not. And then, you showed us factor analysis with discrete data by using the outsourcing example Yij= βj Fi +εij. Then, the FA model we got from this example is Yij*= βj Fi +εij, where Yij* is a latent, continuous success propensity variable. Here, you emphasized Fi itself is a latent variable and we can not get Fi* latent variable. Yes, I agree with what you said, but what confused me is the simple example you mentioned in the beginning of the class. In that probit model: Y*=β0+β1 X +ε, the X is not a latent variable but we didn’t let the model be Y*=β0+β1 X* +ε, so why? In what kind of cases, when X is not a latent variable, we need to make it to be a latent variable X* to match the latent variable Y*?
You said "Then we got a probit model: Y*=β0+β1 X +ε, where Y is the latent variable while X is not. "
Correction: "Then we got a probit model: Y*=β0+β1 X +ε, where Y* is the latent variable while X is not."
You said "And then, you showed us factor analysis with discrete data by using the outsourcing example Yij= βj Fi +εij. "
Note: The issue was that we don't believe the model when Yij is 0/1 (why not? review the class notes to answer) but we can believe the model when modeled in terms of the continuous latent Y*ij (as you note in your next sentence.)
Then you said, "In what kind of cases, when X is not a latent variable, we need to make it to be a latent variable X* to match the latent variable Y*?"
The answer is that if X is measured without error, then there is no need to model it in terms of some underlying latent "true score". But if the "true score" is unknown, latent (like "intelligence" or "satisfaction" or "religiosity" etc.) then we will try for an SEM model to estimate the model Y*=β0+β1 X* +ε."
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General Question:
My question refers to an article from http://www2.gsu.edu/~mkteer/sem2.html. In the Structural Error part, it says “To achieve consistent parameter estimation, these error terms are assumed to be uncorrelated with the model's exogenous constructs. Violations of this assumption come about as a result of the excluded predictor problem” How to understand the excluded predictor problem here? Then it says “However, structural error terms may be modeled as being correlated with other structural error terms. Such a specification indicates that the endogenous constructs associated with those error terms share common variation that is not explained by predictor relations in the model.” My question is in the graphed model the article mentions, if there is a correlation between zeta1 and zeta2, can we say that there is a reciprocal causation between the latent variables eta1 and eta2? Probably not, because the reciprocal causation exists only between latent variables?
You said "Violations of this assumption come about as a result of the excluded predictor problem” How to understand the excluded predictor problem here? "
Suppose Y = .5 X1 + .7 X2 + e, with X1 and X2 correlated, but e uncorrelated with the X's. Now, suppose you model as a function of X1 only, Y = .5 X1 + e*. Here e* = e + .7X2 is correlated with X1.
This issue disappears when you think the model is Y = beta X1 + e**, where beta is the regression coefficient relating X1 to Y. So I don;t view the "omitted variable problem" as a concern, but others do.
Correlation itself says nothing about causation. If you allow terms to correlate, you aren't saying anything about one causing the other, or the other cuasing the first, or both.
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