(Residual) Variances Interpretation

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Arushi Kapoor

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Feb 24, 2022, 3:15:49 AM2/24/22
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Greetings to the lavaan community,

This is the first time I am conducting research and utilizing lavaan package for conducting CFA and SEM. I have used WLSMV estimator and cfa and sem functions for my model.  I had some confusion regarding the interpretation of (residual) variances and it would be of great help if somebody could clear my doubts.

  1. Why does the Variance section for my model does not have p values for the observed variables? The p values are displayed for the latent variables only. I had observed in articles by Dr. Rosseel that some models computed results that did not have p values, while some had and I could not detect the reasoning for that difference.
  2. Can I display the variances of the observed variables in the path diagrams along with the variances for latent variables, even though I do not know if the variances for the observed variables are significant or not.
  3. What is the difference in the interpretations of the unstandardized v/s standardized variances. (Like for the covariance section, the estimates (unstandardized) are covariances, whereas the standardized ones showcase the correlation between factors/latent variables). For all the observed variables the unstandardized and standardized values are the same, but they are different for the latent variables which have p values with them.
  4. I will be reporting the standardized values in the path diagram so if anyone could kindly tell me its interpretation. How to interpret the values of 1 for each latent variable in CFA and exogenous variables in SEM (I read that it happens for the standardized values) and the standardized values of variances for endogenous latent variables (which are displaying different values in unstandardized v/s standardized for SEMoutput).
Here is a glimpse of my model's output in the variance section for 
CFA: 
 .D1               0.288                               0.288    0.288
   .D2                0.221                               0.221    0.221
   .D3           0.809                               0.809    0.809
   .D4           0.956                               0.956    0.956
   .D5            0.960                               0.960    0.960
    L1                0.713    0.043   16.488    0.000    1.000    1.000
    L2                0.652    0.052   12.563    0.000    1.000    1.000
    L3                0.200    0.058    3.439    0.001    1.000    1.000
    L4              0.230    0.060    3.811    0.000    1.000    1.000
    L5             0.712    0.115    6.175    0.000    1.000    1.000

SEM:
   .D1              0.295                               0.295    0.295
   .D2                0.214                               0.214    0.214
   .D3           0.808                               0.808    0.808
   .D4           0.956                               0.956    0.956
   .D5            0.960                               0.960    0.960
   .L1                0.474    0.042   11.254    0.000    0.665    0.665
    L2                0.653    0.053   12.444    0.000    1.000    1.000
    L3                0.208    0.060    3.475    0.001    1.000    1.000
   .L4              0.219    0.058    3.758    0.000    0.928    0.928
    L5             0.705    0.115    6.133    0.000    1.000    1.000

Thank you in advance!

Terrence Jorgensen

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Mar 9, 2022, 9:14:34 AM3/9/22
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  1. Why does the Variance section for my model does not have p values for the observed variables?
Because those are not estimated parameters (by default) when the observed variables are categorical. 

  1. Can I display the variances of the observed variables in the path diagrams along with the variances for latent variables, even though I do not know if the variances for the observed variables are significant or not.
Yes.  Most people have no hypotheses about the residual variances, so a hypothesis test is irrelevant. 

  1. What is the difference in the interpretations of the unstandardized v/s standardized variances. (Like for the covariance section, the estimates (unstandardized) are covariances, whereas the standardized ones showcase the correlation between factors/latent variables). For all the observed variables the unstandardized and standardized values are the same, but they are different for the latent variables which have p values with them.
The "std.all" solution rescales such that everything is in units of SD, rather than the original units of the variables.  For categorical data, the units of the latent responses are arbitrary anyway (see linked paper above), and are identical under the default parameterization="delta" when setting std.lv=TRUE.
  1. I will be reporting the standardized values in the path diagram so if anyone could kindly tell me its interpretation.
1 minus the standardized solution's residual variance is R-squared.  Standardizing an exogenous variance just sets it to 1, nothing to interpret.

Terrence D. Jorgensen
Assistant Professor, Methods and Statistics
Research Institute for Child Development and Education, the University of Amsterdam

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