Is LMS supported for latent–binary moderator interactions?

[maria-jal]

Hi there,

thank you for this great package! I have a question regarding moderation involving an observed binary variable (0/1) and a latent variable.

Is method = "lms" supported for latent × observed (binary) interactions in the current implementation?

I would be grateful for any advice on how to handle this within the package.

Kind regards,
Maria

[Kss2k]

Hi,

In general, the LMS approach is only implemented for models with all continuous data. There are available options for handling categorical binary variables, treating them as functions of latent response variables, but this doesn't always make sense for exogenous/independent variables.

When we have an interaction between a latent variable and a binary categorical variable, it is often more practical to measure the interaction effect via a (linear) multigroup model. Consider this hypothetical model, where we have an interaction between sex and X on Y

library(lavaan)
library(modsem)

# here we just create some toy data from the oneInt dataset
data <- oneInt
data$sex <- as.integer(oneInt$z1 > mean(oneInt$z1))

model <- '
  X =~ x1 + x2 + x3
  Y =~ y1 + y2 + y3
  Y ~ c(x_male, x_female) * X
  interaction := x_female - x_male
'

fit <- sem(model, data, group = "sex")
summary(fit)
#> ...
#> Defined Parameters:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>     interaction       1.124    0.060   18.753    0.000

Alternatively you could use it like a normal dummy variable in a regression. This does of course violate the distributional assumptions of the different estimators in modsem, but the estimators should be relatively robust to these kinds of violations. Indeed, we see that we get almost the exact same results using sex as a dummy variable, using both the LMS and QML approaches.

dummy_model <- '
  X =~ x1 + x2 + x3
  Y =~ y1 + y2 + y3
  Y ~ X + sex + X:sex
'

fit_dummy_lms <- modsem(dummy_model, data, method = "lms", robust.se = TRUE)
summary(fit_dummy_lms)
#> Regressions:
#>                 Estimate  Std.Error  z.value  P(>|z|)
#>  Y ~           
#>    X               0.146      0.043    3.368    0.001
#>    sex             0.896      0.059   15.118    0.000
#>    X:sex           1.122      0.066   17.134    0.000

fit_dummy_qml <- modsem(dummy_model, data, method = "qml", robust.se = TRUE)
summary(fit_dummy_qml)
#> Regressions:
#>                  Estimate  Std.Error  z.value  P(>|z|)
#>   Y ~           
#>     X               0.145      0.043    3.357    0.001
#>     sex             0.896      0.059   15.106    0.000
#>     X:sex           1.123      0.066   17.143    0.000

[Kss2k]

Thanks, this error seems like a bug, caused by modsem() trying to find observed indicators for change21, when specifying the intercepts. I've uploaded a fix, so you can try with the latest development version.

devtools::install_github("kss2k/modsem")

That said, I think there are some other problems with the model.

Warning: modsem->checkCovEtaXi():  
   Covariances between exogenous and endogenous variables are not available, 
   and will be ignored! The problematic covariances are: T1~~IS_T1, 
   T1~~IS_STRE3, T2~~IS_T1, T2~~IS_STRE3, change21~~IS_T1, change21~~IS_STRE3

Note that T2 is treated as an endogenous variable because it is a lower order latent variable (i.e., a function of change21)

change21 =~ 1*T2

The (co-)variance structure of the endogenous and exogenous latent variables are stored in different matrices in the LMS approach, and there is thus no room for overlapping (residual) covariances between them.

You seem to be constraining all of the covariances to zero, and it shouldn't matter unless you free the covariances later.

T1 ~~ 0*IS_T1 + 0*IS_STRE3
T2 ~~ 0*IS_T1 + 0*IS_STRE3
change21 ~~ 0*IS_T1 + 0*IS_STRE3

[maria-jal]

Thanks a lot for the clarification and for implementing the fix.

I think I now better understand the issue regarding the covariance structure in the LMS approach and the distinction between endogenous and exogenous latent variables.

What still confuses me a bit is how similar latent change score models with indicator-specific factors and latent interactions are handled internally in Mplus, since these types of specifications seem to be estimable there as well, at least in theory.

For now, I will probably continue with the dblcent approach (res.cov.method = "none"), which seems to run much more stably in my case.

[Kss2k]

The Mplus implementation of the LMS approach is significantly different from Klein & Moosbrugger (2000), and thus also the modsem implementation. Mplus is not open source, so it is hard to tell what happens under the hood.

I'm working on allowing the estimation of these latent covariances in the LMS approach, by adding a third matrix. You can try it out on your model. I'm not sure it will yield a more stable model though.

devtools::install_github("kss2k/modsem")

Best,
Kjell

[maria-jal]

Hi,

yes, that is true. Thank you for the update and for sharing the development version.

I tested the updated implementation in the context of latent change score models combined with LMS interactions. I experience some estimation instability. The MLR estimator does not seem to be supported in the LMS setting. Without the estimator = "MLR", the algorithm terminates at starting values, and in successful runs I receive warnings related to non–positive definite latent covariance matrices.

In addition, I observe that the estimated covariance structure differs from what is obtained in the corresponding SEM specification (e.g., in lavaan), in particular with respect to the covariances between manifest exogenous variables (e.g., group indicators) and the indicator-specific (IS) latent factors.

In contrast, the same latent change score specification converges when using the double-centering approach for the latent interaction.

Given the deterministic structure of the latent change score model (i.e., T2 being fully decomposed into T1 and the latent change factor), I am not yet sure whether these warnings reflect a genuine identification problem or are a consequence of the model specification itself.

Best,
Maria

[Kss2k]

I see,

While the MLR estimator isn't supported in the estimator argument for modsem(..., method = "lms"), you can get the equivalent using modsem(..., method = "lms", robust.se = TRUE). The MLR estimator in lavaan (and Mplus) doesn't (at least in theory) affect the estimation of the parameters. The MLR estimator only provides robust standard errors, and scaled statistic. When passing robust.se=TRUE you get Huber-White robust standard errors, as well as scaled test statistic for the H0 model.

The LMS approach doesn't minimize the discrepancy between the observed and implied covariance matrix, the same way that a linear SEM does, so you wouldn't expect to get the same model implied covariance structure. Though it should be concerning if the differences are too large.

Best,
Kjell