## Correction of correlation and covariance between simple concepts

We have derived for a simple concept that $q_{ij}= r_{ij}v_{ij}$ and $\mu_{ij} = r_{ij}m_{ij}$. Therefore we can formulate for two simple concepts the following model:

for this model we get

$\rho_{(Y_{1}Y_{2})}=q_{Y_{1}}\rho_{(F_{1}F_{2})}q_{Y_{2}}+\mu_{Y_{1}}\mu_{Y_{2}}$

and follows that

$\rho_{(F_{1}F_{2})}=\frac{(\rho_{(Y_{1}Y_{2})}-\mu_{Y_{1}}\mu_{Y_{2}})}{q_{Y_{1}}q_{Y_{2}}}$

for the covariance of the unstandardized variables $F_{1}$ and $F_{2}$ denoted by $\rho_{F_{1}F_{2}}$ hold that

$\sigma_{(f_{1}f_{2})}=\sigma_{f_{1}} \rho_{(F_{1}F_{2})} \sigma_{f_{1}}$

where $\sigma_{F_{i}}=q_{i} \sigma_{Y_{i}}$

$\rho_{(Y_{1}Y_{2})}=q_{Y_{1}}\rho_{(F_{1}F_{2})}q_{Y_{2}}+\mu_{Y_{1}}\mu_{Y_{2}}$

and follows that

$\rho_{(F_{1}F_{2})}=\frac{(\rho_{(Y_{1}Y_{2})}-\mu_{Y_{1}}\mu_{Y_{2}})}{q_{Y_{1}}q_{Y_{2}}}$

for the covariance of the unstandardized variables $F_{1}$ and $F_{2}$ denoted by $\rho_{F_{1}F_{2}}$ hold that

$\sigma_{(f_{1}f_{2})}=\sigma_{f_{1}} \rho_{(F_{1}F_{2})} \sigma_{f_{1}}$

where $\sigma_{F_{i}}=q_{i} \sigma_{Y_{i}}$

This 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 4-7. http://essedunet.nsd.uib.no/cms/topics/measurement/4/