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 kloud posted on Tuesday, February 26, 2019 - 3:23 pm
Dear Dr.Muthen

Im planning a simulation dealing with 2 continuous(exogenous) covariates X1 & X2, 2-binary mediators T1 & T2, and continuous outcomes Y1 & Y2. (no latent variable). To make binary mediators, I generated 2 underlying latent variables (T1*, T2*) ~ MVN(0, SIGMA). then, dichotomized these into binary ones. Variances of X1, X2, T1*, T2*, Y1, Y2 were set to be 1.

X1 with X2
T1* on X1
(then, dichotomize T1* into T1)
Y1 on T1 X1
T2* on X2 T1 Y1
(then, dichotomize T2* into T2)

1. When generating T2, there are 2 options. One is to regress T2* on X2 T1* Y1(T1* : continuous latent), and the other is to regress T2* on X2 T1 Y1(T1 : dichotomized observed). Im wondering which one is correct.

2. (applying WLSMV) I think it's necessary to decompose the covariance matrix with parameters.

ex)
T1*=a*X1+r1
Var(T1*)=1=a^2*1+Var(r1)

But dichotomization makes me confused.

ex)
Y1=b*T1+c*X1+e1
Var(Y1)=b^2*1+c^2*1+b*c*COV(T1,X1)+Var(e1)

I dichotomized T1* into T1. So I have difficulty in decomposing COV. I think that slope parameter a(T1* on X1) somehow is related to COV(T1,X1).

Thank you.
 Bengt O. Muthen posted on Wednesday, February 27, 2019 - 11:31 am
It is a substantive choice whether the mediator used as a predictor of Y should be treated as binary or as the corresponding continuous latent response variable. ML can handle only the case where the mediator is treated as observed binary, WLSMV only the case where it is latent, and Bayes both. So depending on which estimator you want to use, you have to make a choice.
 kloud posted on Sunday, March 03, 2019 - 8:03 pm
Dear. Dr. Muthen
Thank you for your answer.

Im going to apply WLSMV estimator.
In this case, as you said, I think I could treat a binary mediator as latent variable. But in case of simulation, I had to generate underlying latent variable(normal), then dichotomize it into binary one. In this case, I can not understand how I decompose covariance.

ex)
Y1=b*T1+c*X1+e1
Var(Y1)=b^2*1+c^2*1+b*c*COV(T1,X1)+Var(e1)

I dichotomized T1* into T1. So I have difficulty in decomposing COV. I think that slope parameter a(T1* on X1) somehow is related to COV(T1,X1).
 Bengt O. Muthen posted on Monday, March 04, 2019 - 5:25 pm
Is the situation that you have regress T1* on X1 and you then dichotomize T1* into T1 and use that as a predictor of Y1? And then you want to figure out the variance of Y1?
 kloud posted on Monday, March 04, 2019 - 5:59 pm
yes.

As I have seen, most papers generated a latent continuous variable, and dichotomized it into a binary variable.

When generating T2, there seem to be 2 options. One is to regress T2* on X2 T1* Y1(T1* : continuous latent), and the other is to regress T2* on X2 T1 Y1(T1 : dichotomized observed). Im wondering which one is correct.

But I'm not sure which is correct...

and moreover, I think I have to decompose var(Y1) with parameters to assign a specific number as true value. but in this case, I don't figure out what the cov(corr) changed by dichotomization.
that is, cov(T1*, X1) and cov(T1, X1)

Is this way correct to reflect the process of WLSMV?

Thank you for reading.
 Bengt O. Muthen posted on Tuesday, March 05, 2019 - 2:55 pm
T1* is regressed on X1 in WLSMV, so you work with * variables throughout. It is then easy to derive their variances and covariances.
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