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

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colin.Fu's picture
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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! :)

mhunter's picture
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Joined: 07/31/2009
More math and an answer

Hi and welcome to the forums!

Using your equations

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 ... (***&)

To answer your questions:
1. I would say there is a linear relationship between the indicators. This is another justification for the the term LISREL (LInear Sructural RELations) for SEM. In general, SEMs are linear models, just like regression, but they include latent AND manifest variables.

2. Notice from equations (*&) through (***&) that there is a linear relationship between the indicators, but it is not perfect. There is additive error in the relationship. Also, you may want to check your understanding of factor models. One way of thinking about a factor model is considering the factor to be "the thing that all the indicators have in common". In this way, you would want all the indicators to be closely related and highly correlated so that factor is what they share. Another way to think about the factor is "the factor is the thing that explains the correlations among the indicators". You would not want the indicators to be independent. If all the indicators were completely independent, there would be no common factor at all, just error.

Does that answer your question?