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update product-term's variance when correlation=TRUE #427

@TDJorgensen

Description

@TDJorgensen

Suppose a product term is included in the model syntax:

mod <- ' y5 ~ x1 + y1 + x1:y1 '

When analyzing a correlation matrix (or sample-standardized raw data), we can set correlation=TRUE to ensure correct _SE_s (with a GLS estimator):

fit <- sem(mod, data = scale(PoliticalDemocracy),
           correlation = TRUE, fixed.x = FALSE)

In this case, the product x1:y1 will not have unit SD, but that is how the model constrains it.

summary(fit)

Variances:
                   Estimate  Std.Err  z-value  P(>|z|)
   .y5                0.360                          
    x1                1.000                          
    y1                1.000                          
    x1:y1             1.000                          

The product's SD should be a function of the summary statistics of x1 and y1.
Likewise, the covariance of x1 and y1 with x1:y1 is also a function of the x1 and y1 summary stats. But because this only works with fixed.x=FALSE, I guess those formulas wouldn't be applied here

Covariances:
                   Estimate  Std.Err  z-value  P(>|z|)
  x1 ~~                                              
    y1                0.382    0.099    3.848    0.000
    x1:y1             0.292    0.106    2.742    0.006
  y1 ~~                                              
    x1:y1             0.238    0.110    2.167    0.030

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