group equality constraint

[Dimitrios Zacharatos]

I am trying to figure out the semantics of group equality constraint in lavaan syntax.

the lavaan::cfa function allows to define group equality constraints using the group.equal argument.

According to package manual:

group.equal

A vector of character strings. Only used in a multiple group analysis. Can be one or more of the following:"loadings", "intercepts", "means", "regressions", "residuals" or "covariances", specifying the pattern of equality constraints across multiple groups.

As I understand it

1. loadings can be interpreted as the beta coefficients of a regression equation i.e. the lambda matrix

2. intercepts can be interpreted as the alpha coefficient of a regression equation

3. means

4. regressions

5. residuals 

6. covariances for some reason does not work perhaps I should report it as a bug I am not sure

Question

5. residuals: does this refer to the off diagonal of the theta θ (error) matrix or it refers to both the diagonals and the off diagonal elements of the theta θ (error) matrix?

4. regressions: I have no idea what this means can anyone provide a suggestion regarding this?

3. means: does this refer to the mu matrix? however what is the difference between this and the intercepts situation profited that an intercept can be interpreted as the mean of one variable when the other variable is 0

[Terrence Jorgensen]

According to package manual:

group.equal

A vector of character strings. Only used in a multiple group analysis. Can be one or more of the following:"loadings", "intercepts", "means", "regressions", "residuals" or "covariances", specifying the pattern of equality constraints across multiple groups.

Make sure you consult the latest version: http://lavaan.ugent.be/development.html

The options are:

"loadings", "intercepts", "means", "thresholds", "regressions", "residuals", "residual.covariances", "lv.variances" or "lv.covariances"

Note that these labels are meant to apply to common-factor (CFA) models, as shortcuts for testing measurement invariance.  They can be applied to path analyses and other types of SEM as well, but the labels correspond to parameter as they are interpreted in CFA.

1. loadings can be interpreted as the beta coefficients of a regression equation i.e. the lambda matrix

Correct, the lambdas, in LISREL notation, not the betas

 

2. intercepts can be interpreted as the alpha coefficient of a regression equation

of observed variables, yes (the tau vector in LISREL, or nu in Mplus and lavaan).  In CFA, observed variables are (only) endogenous indicators, but this refers to intercepts of observed variables in any type of model.

5. residuals: does this refer to the off diagonal of the theta θ (error) matrix or it refers to both the diagonals and the off diagonal elements of the theta θ (error) matrix?

Only the diagonal elements of theta.  "residual.covariances" refers (only) to off-diagonal elements of theta (regardless of the type of model; i.e., it does not constrain residual (co)variances of latent variables in structural regression models, for example).  Constraining latent (co)variances is similarly accomplished using "lv.variances" or "lv.covariances".

4. regressions: I have no idea what this means can anyone provide a suggestion regarding this?

Regressions among latent variables (the Beta matrix in LISREL)

 

3. means: does this refer to the mu matrix? however what is the difference between this and the intercepts situation profited that an intercept can be interpreted as the mean of one variable when the other variable is 0

This always refers to intercepts of latent variables (the alpha vector in LISREL notation).  In CFA, latent variables are exogenous, so their intercepts are means.  But in any type of SEM, this will constrain the latent intercepts.  Means (mu) are not model parameters, so they cannot be constrained without some clever syntax reproducing the model-implied means as user-defined parameters, then constraining those.

Terrence D. Jorgensen

Postdoctoral Researcher, Methods and Statistics

Research Institute for Child Development and Education, the University of Amsterdam

UvA web page: http://www.uva.nl/profile/t.d.jorgensen

[Dimitrios Zacharatos]

Thank you so much for the response. This is brilliant.