Eta_t, given random innovations PreviousNext
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 Dan Berry posted on Friday, October 13, 2017 - 12:25 pm
Hello. I was hoping that someone might be able to help me with the substantive interpretation of the estimated mean of the covariance between to random innovations in a DSEM. (e.g., the mean of eta_t on slide 48 of Hopkins Day 1).

Specifically, Iím trying to figure out the analog of the WP residual covariance in a multilevel vector AR (per slide 27), but when the innovations are allowed to vary randomly between individuals (per slide 48).

For instance, if Iíve constrained the loadings for eta_t to be positive 1 (based on theory), would this be the antilog of the mean for eta_t? If I wanted the standardized version of the mean covariance, would I antilog the (across cluster) standardized mean for et_t?

Thanks in advance for any thoughts.
Best,
-db
 Tihomir Asparouhov posted on Wednesday, October 18, 2017 - 9:51 am
1. There are two correlations residual and not residual. The full correlation average across cluster is available with the residual(cluster) output command option

RESIDUAL OUTPUT
ESTIMATED MODEL
AVERAGE CLUSTER ESTIMATES
Correlations
DAYPA DAYNA DAYPA&1 DAYNA&1
________ ________ ________ ________
DAYPA 1.000
DAYNA -0.280 1.000

2. The residual correlation has to be computed for each cluster separately and then averaged across cluster. Probably the easiest way to get it is using the stand(cluster) output option and look at the STD results for each cluster. For example

STD Standardization
Posterior One-Tailed 95% C.I.
Estimate S.D. P-Value Lower 2.5% Upper 2.5% Significance
COV BY
DAYPA 0.070 0.169 0.000 0.045 0.775 *
DAYNA -0.070 0.169 0.000 -0.775 -0.045 *

LOGCOV |
COV 1.000 0.000 0.000 1.000 1.000

LOGVARPA |
DAYPA 0.049 0.165 0.000 0.035 0.729 *

LOGVARNA |
DAYNA 0.004 0.137 0.000 0.002 0.459 *


resid correl = -0.070*0.070/(sqrt(0.004+0.070*0.070)*sqrt(0.049+0.070*0.070)) =-0.22

3. While the above computation is generally ok, it is still a shortcut. The most accurate computation for the residual correlation involves saving the plausible values for each
LOGVARPA, LOGVARNA and LOGCOV for each cluster. Then the residual correlation estimate would be (for each cluster computed separately)
the average across plausible values of
exp(LOGCOV)/ (sqrt( exp(LOGVARPA)+exp(LOGCOV))*sqrt( exp(LOGVARNA )+exp(LOGCOV)))
Then this has to be further averaged across clusters.
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