Correlation type between continuous latent residuals and dichotomous manifest variables in WLSMV/theta

[Johanna Hartung]

Dear lavaan community,

I have a question regarding the correlation between a residual variance and an external, manifest variable.

In our model, all manifest variables are dichotomous. We created pseudo-latent factors for the residuals and correlate these with a dichotomous manifest variable. We use WLSMV and theta parameterization. If I understand CFA for binary indicators correctly, the residual variance is expected to be approximately normally distributed.

My question is: what type of correlation gets calculated in lavaan for the continuous latent residual and the dichotomous manifest variable? Is it a biserial correlation, as would be expected with two manifest variables?

Thank you in advance,

Johanna

 

Here is a short example of our model syntax:

model <- ‘

f =~ a_1 + a_2 + a_4 + b_1 + b_3 + b_4

f ~~ 1*f

a_1~~0*a_1

a_2~~0*a_2

a_4~~0*a_4

b_1~~0*b_1

b_3~~0*b_3

b_4~~0*b_4

 

resid_a_1=~ 1*a_1

resid_a_2=~ 1*a_2

resid_a_4=~ 1*a_4

resid_b_1=~ 1*b_1

resid_b_3=~ 1*b_3

resid_b_4=~ 1*b_4

 

resid_a_1~~resid_a_1

resid_a_2~~resid_a_2

resid_a_4~~resid_a_4

resid_b_1~~resid_b_1

resid_b_3~~resid_b_3

resid_b_4~~resid_b_4

 

m_1~~resid_a_1 + resid_a_2 + resid_a_4 + resid_b_1 + resid_b_3 + resid_b_4’

model_fit <- lavaan(model=model, data = dat, ordered = T, estimator = "WLSMV", orthogonal = T, parameterization = "theta")) # estimate model

[Terrence Jorgensen]

the correlation between a residual variance and an external, manifest variable.

By “external”, do you mean “exogenous”?

If the residuals of the continuous variable were observed, it would be a point-biserial correlation, so I guess it is analogous to that. Because the residuals are the manifest indicators, given the factor “f”, you could call this a semi-partial point-biserial correlation (because f is partialed out of its indicators, but not out of the exogenous binary variable).

We created pseudo-latent factors for the residuals

Not sure why you think that is necessary.  

We use WLSMV and theta parameterization. If I understand CFA for binary indicators correctly, the residual variance is expected to be approximately normally distributed.

The residuals are assumed to be normally distributed. That is the likelihood used to estimate the polychoric correlations that are used as input data. 

[Johanna Hartung]

Thank you for your response! 

the correlation between a residual variance and an external, manifest variable.

By “external”, do you mean “exogenous”?

Yes, I meant exogenous. Thank you for the clarification.   

We created pseudo-latent factors for the residuals

Not sure why you think that is necessary.  

We tried modeling the "semi-partial point-biserial correlations" without the pseudo-latent factors for the residuals. However, the correlations in the output were the same as the manifest tetrachoric correlations. It seems that the pseudo-latent factors are necessary to actually use the residuals rather than just the manifest indicators.