Question on Latent Variable Indicators

[TRM]

Hello,

I am new to SEM/lavaan and have a question about using latent variables. How can I use data with different scales to estimate a latent variable (i.e. one indicator measured in miles w/ large values (1000's) and another indicator of a different type with values up to only 5 or so)? Any links to literature on the subject of latent variables would be appreciated as well, I can't find anything that goes into the specifics of how an LV is estimated like this, though I gather this is one of the main draws of latent variables.

Thanks!

TRM

[Sunthud Pornprasertmanit]

In factor analysis, latent variables do not have any scale. Users should define their scaling of their latent variables. The methods of the scale specification could be by using marker variable, fixing the scale of factor (e.g., mean of 0 and variance of 1), or effect coding method. You can search it in all introduction to SEM book. 

However, there is a problem if you use indicators with largely different scales. I believe it is the problem of computer algorithm. The inversion of the matrix or eigenvalue decomposition could have a problem. I would recommend you to rescale some of you indicators by dividing it by an appropriate constant.

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[TRM]

So in a lavaan sem model with one latent variable and one or more other endogenous variables, should the scale of the latent variable be defined or does that not make a difference?

Thanks for your help.

[Edward Rigdon]

  Common factors generally have no natural scale, but yes, you must specify their variance directly or indirectly, or else the factor model will be. to identified.  The CFA/SEM default of fixing the first loading for each factor to 1 is the indirect way to solve this problem.  The direct way is to directly fix the factor variance to a value.  Except in some very peculiar circumstances, one or the other is necessary, for every common factor.

--Ed Rigdon

Sent from my iPad

[Harya S. Dillon]

Hi TRM,

Echo Ed.

Simplest fix is to rescale (e.g. divide by constant, 100 or 1,000) the 'large value' indicator so that all indicators that measures a Latent Variable (LV) is at a similar scale. This is just a 'numerical' trick to simplify the matrix inversion -- no theoretical imperative.

I'm guessing you are doing a 'Full SEM' (structural + CFA). The LV is measured in terms of the first indicator that has a loading of 1.
Would help if you share your model/code.

Best,
-K-

 

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From: Edward Rigdon [mailto: edward...@gmail.com]
Sent: Monday, July 28, 2014 12:14 PM (PST)
To: lav...@googlegroups.com
Subject: Re: Question on Latent Variable Indicators

[TRM]

Thanks K and Ed

You'd be right, K. This is essentially what I want to do:

'

LV =~ x1 + x2 + x3

y ~ LV

'

After some re-scaling, the means of x1, x2, x3 are 1.7, 0.3237, and 17.47. What I don't get is how will having the loading of either x1, x2, or x3 fixed to one affect the latent variable differently (or fixing the factor variance) and affect the structural part of the model? 

Thanks again

[Harya S. Dillon]

 Why not simply y ~ x1+x2+x3 ?

---

From: TRM [mailto: thomas.r...@gmail.com]
Sent: Tuesday, July 29, 2014 10:13 AM (PST)
To: lavaan

Subject: Re: Question on Latent Variable Indicators

[Edward Rigdon]

TRM—

     In a simple case of a single group and a well-fitting model, the differences are a matter of the unstandardized parameter estimates.  Remember that the formula for a simple regression coefficient is:

 

b = covariance (X,Y) / variance (X)

 

So anything that affects the variance of X or the covariance of X and Y (two factors, say) will affect the value of b.  You can also write this as:

 

b = correlation(X,Y) x stdev(Y) / stdev(X)

 

if that clarifies the impact of the scale of Y and X on b.

     So if we directly set the variance of factor X or factor Y to a different value, we also set stdev(X) or stdev(Y) and thus affect b.

     For the other approach, assume we have an exclusive reflective indicator of factor X, called x1, and the equation is:

 

x1 = 0 + loading x X + e

 

So I am setting the intercept for x1 to 0.  Otherwise, the arithmetic is more involved.  If factor X and e are orthogonal (the usual regression assumption), then we also have:

 

Variance(x1) = loading ^2 x variance(X) + variance(e)

 

If we fix this loading to 1, then it becomes:

 

Variance(x1) = variance(X) + variance(e)

 

Which also implies:

 

Variance(X) = variance(x1) - variance(e)

Now variance(x1) is fixed in the data.  So fixing the loading to 1 constraints variance(X) to be the difference between variance(x1) and variance(e).  If different indicators have different variances, or even just different error variances, then setting different loadings to 1 will lead to different variances for the factor.

 

--Ed Rigdon

 

 

From: lav...@googlegroups.com [mailto:lav...@googlegroups.com] On Behalf Of TRM
Sent: Tuesday, July 29, 2014 1:13 PM
To: lav...@googlegroups.com
Subject: Re: Question on Latent Variable Indicators

 

Thanks K and Ed

--

[Alex Schoemann]

I can't beat Ed's reply (in fact I'll probably use some of that explanation next time I teach my SEM class). I'll also point out that different methods of scale setting can affect tests of parameters (using the Wald statistic). If you're concerned about this (or even if you're not) I recommend looking at Gonzalez & Griffin (2001).

Alex

Gonzalez, R., & Griffin, D. (2001). Testing parameters in structural equation modeling: every" one" matters. Psychological Methods, 6(3), 258.

[Edward Rigdon]

     And you can go back further, to Lawley and Maxwell’s (1967) book, which highlighted the biased standard errors that can come from standardizing factors (in ML estimation), and recommended the standard fix.

 

Lawley, D.N., Maxwell, A.E. (1967), Factor analysis as a statistical method.  London:  Butterworths.

 

Some packages now incorporate the fix, but I still recommend scaling by fixing loadings.

     Roger Millsap has written about issues that arise with constraints across factors, and other issues arise with constraints across groups, as with invariance testing.

--Ed Rigdon