## Correcting the correlation and covariance between a complex and a simple concept

The model to correct the correlations between a measure of a composite score ($F_{cs}$) and the measures of a simple concept ($F_{a}$) which were measured with the same method ($M$):

The simplified model of a complex concept and an additional simple concept where all the variables are standardized.

As we have seen for a similar model it can be proven that

$\rho_{_{Y_{cs}Y_{a}}}= q_{_{Y_{cs}}}q_{_{Y_{a}}}\rho_{_{F_{cs}F_{a}}}+ m_{_{Y_{cs}}}m_{_{Y_{a}}}$

Which shows:

$\rho_{_{_{F_{cs}F_{a}}}}=\frac{(\rho_{Y_{cs}Y_{a}}-m_{Y_{cs}}m_{Y_{a}})}{q_{_{Y_{cs}}}q_{_{Y_{a}}}}$

$q_{_{Y_{cs}}}$ , $m_{_{Y_{cs}}}$ , $q_{_{Y_{a}}}$ , and $m_{_{Y_{a}}}$ can be derived as shown before.

To derive the covariance between these two observed variables corrected for measurement error, we denote the unstandardized latent complex concept by $f_{cs}$

We have shown before that for any simple concept $f_{a}$ holds

$\sigma_{f_{a}{f_{a}}} = q_{a}^{2} \sigma_{y_{a}{y_{a}}}$

and for any complex concept $f_{cs}$

$\sigma_{f_{cs}f_{cs}} = q_{Y_{cs}}^{2} \sigma_{y_{cs}{y_{cs}}}$

then it follows that

$\sigma_{_{f_{cs}f_{a}}}=\sigma_{_{f_{cs}}} \rho_{_{F_{cs}F_{a}}} \sigma_{_{f_{a}}}$

The covariance corrected for measurement error is equal to the corrected correlations multiplied with the standard deviations of the latent variables corrected for measurement error.

$\rho_{_{Y_{cs}Y_{a}}}= q_{_{Y_{cs}}}q_{_{Y_{a}}}\rho_{_{F_{cs}F_{a}}}+ m_{_{Y_{cs}}}m_{_{Y_{a}}}$

Which shows:

$\rho_{_{_{F_{cs}F_{a}}}}=\frac{(\rho_{Y_{cs}Y_{a}}-m_{Y_{cs}}m_{Y_{a}})}{q_{_{Y_{cs}}}q_{_{Y_{a}}}}$

$q_{_{Y_{cs}}}$ , $m_{_{Y_{cs}}}$ , $q_{_{Y_{a}}}$ , and $m_{_{Y_{a}}}$ can be derived as shown before.

To derive the covariance between these two observed variables corrected for measurement error, we denote the unstandardized latent complex concept by $f_{cs}$

*and its unstandardized measure by the variable $y_{cs} = \sum_{i} w_{i}y_{i}$ with variance $\sigma_{y_{cs}{y_{cs}}}$ while the unstandardized simple concept $f_{a}$ with variance $\sigma_{f_{a}{f_{a}}}$*

We have shown before that for any simple concept $f_{a}$ holds

$\sigma_{f_{a}{f_{a}}} = q_{a}^{2} \sigma_{y_{a}{y_{a}}}$

and for any complex concept $f_{cs}$

$\sigma_{f_{cs}f_{cs}} = q_{Y_{cs}}^{2} \sigma_{y_{cs}{y_{cs}}}$

then it follows that

$\sigma_{_{f_{cs}f_{a}}}=\sigma_{_{f_{cs}}} \rho_{_{F_{cs}F_{a}}} \sigma_{_{f_{a}}}$

The covariance corrected for measurement error is equal to the corrected correlations multiplied with the standard deviations of the latent variables corrected for measurement error.