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

Next, we discuss the model where a complex concept measured by a composite score $Y_{cs}$ shares the same method with an additional simple concept ($F_{a}$). Since we have derived in the previous section the effect of the concept on interest ($F$) and the method $M$ on the computed composite score of the complex variable $F$ we can simplified this model as presented in the Figure below.

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

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

Which shows:

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

In order to compute the covariance between a complex concept and a simple concept the correlation between the two variables has to be multiplied with the standard deviations of these variables corrected for measurement error. This means that

$\sigma_{FF_{a}}=\sigma_{F} \rho_{FF_{a}} \sigma_{F_{a}}$

for any complex concept the standard deviation is equal to $q_{Y_{cs}}\sigma_{Y_{cs}}$ which was derived before.

For $F_{a}$ the standard deviation has been derived to be equal to $q_{a}\sigma_{Y_{a}}$

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

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

Which shows:

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

In order to compute the covariance between a complex concept and a simple concept the correlation between the two variables has to be multiplied with the standard deviations of these variables corrected for measurement error. This means that

$\sigma_{FF_{a}}=\sigma_{F} \rho_{FF_{a}} \sigma_{F_{a}}$

for any complex concept the standard deviation is equal to $q_{Y_{cs}}\sigma_{Y_{cs}}$ which was derived before.

For $F_{a}$ the standard deviation has been derived to be equal to $q_{a}\sigma_{Y_{a}}$