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 Ycs shares the same method with an additional simple concept (Fa). Since we have derived in the a 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.
As we have seen for a similar model it can be proven that
ρYcsYa = qYcs qa ρFFa + mYcsma
ρFFa = (ρYcsYa - mYcs ma ) / qYcs qa
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
σFFa = σF  ρFFa  σFa
For any complex concept the standard deviation is equal to qYcsσYcs which was derived before.
For Fa the standard deviation has been derived to be equal to qa σYa