## 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 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

ρ_{YcsYa} = q_{Ycs} q_{a} ρ_{FFa} + m_{Ycs}m_{a}

Which shows:

ρ_{FFa} = (ρ_{YcsYa} - m_{Ycs} m_{a} ) / q_{Ycs} 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

σ_{FFa} = σ_{F} ρ_{FFa} σ_{Fa}

For any complex concept the standard deviation is equal to q_{Ycs}σ_{Ycs} which was derived before.

For F_{a} the standard deviation has been derived to be equal to q_{a} σ_{Ya}