Syntax for correlation between residual variances of a latent factor and an observed variable

Rob R

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Jul 9, 2019, 3:07:40 PM7/9/19

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Hi all,

I am looking for any advice on how to include correlated residuals in a model, such that the residual variance of a latent variable is correlated with one of an observed measure.

I am aware that the double tilda (i.e. ~~) can be used to include residual covariances in a model. However, when I attempt to correlate a latent factor from the model with an observed measure that loads on to a different latent factor, I encounter an error.

Here is some sample syntax, and the exact error message that is received when running the code:

HS.model <- ' visual =~ x1 + x2 + x3

textual =~ x4 + x5 + x6

speed =~ x7 + x8 + x9

visual ~~ x4 '

fit <- cfa(HS.model, data=HolzingerSwineford1939)

Error in lav_model(lavpartable = lavpartable, lavoptions = lavoptions, :

lavaan ERROR: parameter is not defined: visual ~~ x4

If anybody is able to point out or provide advice on what syntax can be used to add the correlation between residual variances of a latent factor and an observed variable, your input would be very highly appreciated.

Thanks,

Rob

Andy Supple

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Jul 9, 2019, 3:35:14 PM7/9/19

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Hi Rob,

I'm not sure what you're trying to do makes sense. Given that visual is a latent variable, correlating it to another item for another factor is odd. If you wanted to specify them as related, you probably would just add a cross-loading so that x4 loads on visual as well. In addition, in the syntax you posted, VISUAL wouldn't have a residual because there isn't anything predicting it. Typically one would be attempting to correlate the uniquenesses/residuals of the items in a measurement model.

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Rob R

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Jul 9, 2019, 4:28:09 PM7/9/19

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Hi Andy,

Thanks for your response and your explanation. Your advice on using a cross-loading is well-taken, however I would like to provide a more complete picture of the kind of model I am aiming to construct, to see whether or not any additional insight on the matter can be gleaned.

I now understand that, at least with the example I provided, there wouldn't be a residual for the latent factor ('visual'). However, I believe that the error still occurs even when the latent factor in question is indeed being predicted by something. To get more precise, I am attempting to construct a model similar to the one seen below (source is below as well). This model is reported as including a correlation between the residual variances of Extraversion (latent factor) and attractiveness (observed variable). Is there any apparent reason as to why a measurement model such as this one could include a correlation between residual variances of a latent factor and an observed factor in a way that makes sense?

Model with ~~.png

Thanks again for your help,

Rob

Source for model:

Anusic, I., Schimmack, U., Pinkus, R. T., & Lockwood, P. (2009). The nature and structure of correlations among Big Five ratings: The halo-alpha-beta model. Journal of Personality and Social Psychology, 97(6), 1142–1156. https://doi.org/10.1037/a0017159

On Tuesday, July 9, 2019 at 3:35:14 PM UTC-4, Andy Supple wrote:

Hi Rob,

I'm not sure what you're trying to do makes sense. Given that visual is a latent variable, correlating it to another item for another factor is odd. If you wanted to specify them as related, you probably would just add a cross-loading so that x4 loads on visual as well. In addition, in the syntax you posted, VISUAL wouldn't have a residual because there isn't anything predicting it. Typically one would be attempting to correlate the uniquenesses/residuals of the items in a measurement model.

On Tue, Jul 9, 2019 at 3:07 PM Rob R <redf...@gmail.com> wrote:

Hi all,

I am looking for any advice on how to include correlated residuals in a model, such that the residual variance of a latent variable is correlated with one of an observed measure.

I am aware that the double tilda (i.e. ~~) can be used to include residual covariances in a model. However, when I attempt to correlate a latent factor from the model with an observed measure that loads on to a different latent factor, I encounter an error.

Here is some sample syntax, and the exact error message that is received when running the code:

HS.model <- ' visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9
visual ~~ x4 '
fit <- cfa(HS.model, data=HolzingerSwineford1939)

Error in lav_model(lavpartable = lavpartable, lavoptions = lavoptions, :
lavaan ERROR: parameter is not defined: visual ~~ x4

If anybody is able to point out or provide advice on what syntax can be used to add the correlation between residual variances of a latent factor and an observed variable, your input would be very highly appreciated.

Thanks,
Rob

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Edward Rigdon

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Jul 9, 2019, 6:04:22 PM7/9/19

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Rob--

You *can* do this, but I also have the graest reservations about the model you picture, because I believe there will be numerous equivalent models, making it hard to make a case for this model in comparison.

However, the key to doing what you want is to convert the observed variable residual to a factor, then correlate the new factor with the other residual variance. Here is your syntax:

HS.model <- ' visual =~ x1 + x2 + x3

textual =~ x4 + x5 + x6

speed =~ x7 + x8 + x9

visual ~~ x4 '

Create a new factor named RES. Force the residual variance of x4 to 0. Let x4 load on RES, with a fixed loading of 1. Make RES orthogonal to all other factors, as a residual should be. Then RES must be the residual variance of x4. Then enable the covariance:

HS.model <- ' visual =~ x1 + x2 + x3

textual =~ x4 + x5 + x6

speed =~ x7 + x8 + x9

RES =~ 1*x4 # create the new factor with loading 1 on x4

x4 ~~ 0*x4 # fix x4's residual variance to 0

textual + speed ~~ 0*RES # Res orthogonal to other factors

visual ~~ RES '

Run the model with visual ~~ 0*RES, instead, and results should be identical to those for the original model for the HS9 data.

You may need to provide a starting value for the variance of RES, or maybe not.

--Ed Rigdon

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Rob R

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Jul 9, 2019, 8:37:51 PM7/9/19

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Hi Ed,

After applying your advice to my own syntax, the model is able to be fit with the desired parameter included.