Non-significant variance of a latent variable despite acceptable model fit?

[Philipp Thomas]

Hi all, 

I have a comprehension question: Is it common to ignore the p-values of variances of latent variables when doing CFA/SEM? I recently stumbled about the variances in the summary() command, and I asked myself if one should take the p-values of the variances of the latent variables into consideration when evaluating model fit. In my understanding, a non-significant variance of a latent variable implicates that the variance captured by the latent variable is not substantial, thus does not reflect a psychologically meaningful process. 

However, I've never seen a person discarding a model based on non-significant variances of latent variables, even though it is possible to get a non-significant chi-square value and a non-significant variance of a latent variable within the same model. Can somebody please explain why checking the significance of variances of latent variables is not common practice?

Philipp

[Edward Rigdon]

     Surprises are always important and worth tracking down.  Since the factor variance is a free parameter, you must have scaled that factor by fixing a loading to 1.  This approach certainly could obscure the fact that a factor and its indicators are only weakly linked.  Check the R2’s for the indicators of that factor.  I think you will find that they are low—and that is a reason that people have used to doubt the value of a model.

     The relation between indicator X and factor F can be expressed as:

 

X = loading * F + e

 

If loading is fixed to 1, this becomes:

 

X = F + e

 

Implying the variance equation:

 

Var(X) = var(F) + var(e)

 

Or:

 

Var(F) = var(X) – var(e)

 

So if var(F) is small, then var(X) and var(e) must be about the same size.  In other words, very little of var(X) is being accounted for by the factor.  So check the R2’s for the indicators of the factor.

     “Significance” is affected by both the size of the parameter and sample size.  Low sample size tends estimates toward nonsignificance and also reduces the power of the chi-square test.  If your estimates lack significance, maybe this is a caution that the chi-square isn’t testing very much.

--Ed Rigdon

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[Philipp Thomas]

Dear Ed,

You are right, my R2s are pretty low. 

Despite your explanations, I still don't understand why people don't care about the significance of the variances of the latent variables. From my point of view, the variance captured by an assumed process needs to reach statistical significance. If this is not the case, how can I ensure my interpretation is not based on random variance ?

Can you please explain this in more detail?

Thanks in advance! 

Philipp

[Edward Rigdon]

Philipp—

     I can’t speak for anyone else.  For myself, “statistical significance” is overvalued.  If you want to read more, I recommend the works of Gerd Gigerenzer.  For myself, knowing the link between variance of indicator, residual variance and variance of factor, I would think that something was wrong if one of my factors had a surprisingly small variance, whether it was “significant” or not.

     In a published context, researchers might be more likely to talk about the low indicator R2s than about the variance of the factor.  Deleting a single item can change the performance of the factor model, possibly eliminating the “low variance” condition.  A researcher might prefer to delete single items in the hope of salvaging the rest, rather than discard the factor and all of its indicators.

--Ed Rigdon

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[Philipp Thomas]

Thank you for your thoughts, Ed.

Greetings, Philipp

[Philipp Thomas]

Dear Ed,

Although you have answered my question, I'm still curious what other people think of this topic. Don't think badly of me when I'm gonna post my question on SEMnet :)

Philipp