Correction of correlation and covariance between simple concepts
We have derived for a simple concept that $q_{i}= r_{i}v_{i}$ and $m_{i} = r_{i}\mu_{i}$. Therefore, we can formulate for two simple concepts the following model:
For this model the observed correlation is
$\rho_{_{Y_{1}Y_{2}}}=q_{_{Y_{1}}}\rho_{_{F_{1}F_{2}}}q_{_{Y_{2}}}+m_{_{Y_{1}}}m_{_{Y_{2}}}$
and it follows that the correlation between the latent variables is
$\rho_{_{F_{1}F_{2}}}=\frac{(\rho_{_{Y_{1}Y_{2}}}-m_{_{Y_{1}}}m_{_{Y_{2}}})}{q_{_{Y_{1}}}q_{_{Y_{2}}}}$
For the covariance of the unstandardized variables $f_{1}$ and $f_{2}$ denoted by $\sigma_{_{f_{1}f_{2}}}$ hold that
$\sigma_{_{f_{1}f_{2}}}=\sigma_{_{f_{1}}} \rho_{_{F_{1}F_{2}}} \sigma_{_{f_{2}}}$
and for the covariance of the unstandardized variable $f_{1}$
$\sigma_{_{f_{1}f_{1}}}=q^2_{_{i}} \sigma_{_{y_{1}y_{1}}}$
and for the covariance of the unstandardized variable $f_{2}$
$\sigma_{_{f_{2}f_{2}}}=q^2_{_{i}} \sigma_{_{y_{2}y_{2}}}$
The approach to estimate a causal model after correcting the correlation or covariance matrix for measurement errors has been in detail illustrated in the European Social Survey Edunet:
De Castellarnau A. and W.E.Saris (2016) A simple procedure to correct for measurement error in survey research. Edunet, ESS, chapters 1-6. (http://essedunet.nsd.uib.no/cms/topics/measurement/4)
$\rho_{_{Y_{1}Y_{2}}}=q_{_{Y_{1}}}\rho_{_{F_{1}F_{2}}}q_{_{Y_{2}}}+m_{_{Y_{1}}}m_{_{Y_{2}}}$
and it follows that the correlation between the latent variables is
$\rho_{_{F_{1}F_{2}}}=\frac{(\rho_{_{Y_{1}Y_{2}}}-m_{_{Y_{1}}}m_{_{Y_{2}}})}{q_{_{Y_{1}}}q_{_{Y_{2}}}}$
For the covariance of the unstandardized variables $f_{1}$ and $f_{2}$ denoted by $\sigma_{_{f_{1}f_{2}}}$ hold that
$\sigma_{_{f_{1}f_{2}}}=\sigma_{_{f_{1}}} \rho_{_{F_{1}F_{2}}} \sigma_{_{f_{2}}}$
and for the covariance of the unstandardized variable $f_{1}$
$\sigma_{_{f_{1}f_{1}}}=q^2_{_{i}} \sigma_{_{y_{1}y_{1}}}$
and for the covariance of the unstandardized variable $f_{2}$
$\sigma_{_{f_{2}f_{2}}}=q^2_{_{i}} \sigma_{_{y_{2}y_{2}}}$
The approach to estimate a causal model after correcting the correlation or covariance matrix for measurement errors has been in detail illustrated in the European Social Survey Edunet:
De Castellarnau A. and W.E.Saris (2016) A simple procedure to correct for measurement error in survey research. Edunet, ESS, chapters 1-6. (http://essedunet.nsd.uib.no/cms/topics/measurement/4)
