## What's the relationship between a 【latent variable】and its 【indicator(s)】

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Joined: 03/10/2014

Hi, everyone! I am a rookie of SEM.
To use SEM, I have read some textbooks about it. And there are something I cannot quite sure about.

It is usual that a latent variable has several indicators, let's say three as an example.
[indicator1] <-- (latent variable)
[indicator2] <--
[indicator3] <--

the path diagram above can be represented by equations as follow:

indicator1 = a1 * latent variable + error1 ... (*)
indicator2 = a2 * latent variable + error2 ... (**)
indicator3 = a3 * latent variable + error3 ... (***)

in which a1, a2, a3 means the regression weights of indicator1, 2, 3 respectively.

Using the equations above, I can easily replace latent variable in (*) using indicator2.

So my questions are:
1、If indicator1 can be linearly represented by indicator2, does that mean there are linear relationship between these two indicators?
2、If so. I always assume that indicators are items can be measure and represent partly of correspondent latent variable, so the more independent between indicators, the better they are. If they have some kind of linear relationship, that may cause some negative effect of the Model.

How can I understand these?

Since I am a new to SEM, my opinion may itself be wrong. So please tell me If so.

Thanks! :)

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Joined: 07/31/2009

Hi and welcome to the forums!

indicator1 = a1 * latent variable + error1 ... (*)
indicator2 = a2 * latent variable + error2 ... (**)
indicator3 = a3 * latent variable + error3 ... (***)

how would you replace the latent variable in (*) using indicator2?

I would do this

indicator2 = a2 * latent variable + error2 ... (**)
indicator2 - error2 = a2 * latent variable
(indicator2 - error2)/a2 = latent variable ... (&)

Therefore substituting (&) into (*)

indicator1 = (a1/a2) * (indicator2 - error2) + error1 ... (*&)
indicator1 = (a1/a2) * indicator2 - (a1/a2) * error2 + error1

They same procedure applies to (*), (**), and (***), so they can be replaced with

indicator1 = (a1/a2) * indicator2 - (a1/a2) * error2 + error1 ... (*&)
indicator2 = (a2/a2) * indicator2 - (a2/a2) * error2 + error2 = indicator2 ... (**&)
indicator3 = (a3/a2) * indicator2 - (a3/a2) * error2 + error3 ... (***&)