Correcting the correlation and covariance between two complex concepts
It may happen that in a study not only one complex concept is used but two or more are used. In this section we will discuss how in this case the correction of the correlation and covariance matrices can be performed. We start with the specification of the standardized model with two complex concepts sharing the same method.
Given what has been derived for complex concepts and their composites scores, we can reduce the model to the one presented in the figure below.
Given what has been derived for complex concepts and their composites scores, we can reduce the model to the one presented in the figure below.
The simplified measurement model for two complex concepts where all variables are standardized
Because for both complex concepts the quality coefficients are known and this figure is now not different from the situation discussed for two simple concepts, we can refer to the derivation there and conclude for the correlation between the two concepts that the correlation will be:
$\rho_{_{F_{1}F_{2}}}=\frac{(\rho_{_{Y_{1cs}Y_{2cs}}}-m_{_{Y_{1cs}}}m_{_{Y_{2cs}}})}{q_{_{Y_{1cs}}}q_{_{Y_{2cs}}}}$
All $q_{_{Y_{cs}}}$ and $m_{_{Y_{cs}}}$ coefficients can be derived as shown before.
To derive the covariance between two complex concepts corrected for measurement errors we use the same notation as before: the latent complex concept are denoted by $f$ and its unstandardized measure by the variable $y_{cs} = \sum_{i} w_{i}y_{i}$ with variance $\sigma_{y_{cs}{y_{cs}}}$.
We have shown before for any latent complex concept that
$\sigma_{ff} = q_{Y_{cs}}^{2} \sigma_{y_{cs}{y_{cs}}}$
Then it follows that
$\sigma_{_{f_{1}f_{2}}}=\sigma_{_{f_{1}}} \rho_{_{F_{1}F_{2}}} \sigma_{_{f_{2}}}$
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.
Because all these coefficients have been derived before in terms of the quality of the simple questions we can compute these correlations and covariances for any possible number of simple and complex concepts, for any type of complex concept and for all correlations and covariances between the observed variables which are the measures for the concepts of interest in a study.
$\rho_{_{F_{1}F_{2}}}=\frac{(\rho_{_{Y_{1cs}Y_{2cs}}}-m_{_{Y_{1cs}}}m_{_{Y_{2cs}}})}{q_{_{Y_{1cs}}}q_{_{Y_{2cs}}}}$
All $q_{_{Y_{cs}}}$ and $m_{_{Y_{cs}}}$ coefficients can be derived as shown before.
To derive the covariance between two complex concepts corrected for measurement errors we use the same notation as before: the latent complex concept are denoted by $f$ and its unstandardized measure by the variable $y_{cs} = \sum_{i} w_{i}y_{i}$ with variance $\sigma_{y_{cs}{y_{cs}}}$.
We have shown before for any latent complex concept that
$\sigma_{ff} = q_{Y_{cs}}^{2} \sigma_{y_{cs}{y_{cs}}}$
Then it follows that
$\sigma_{_{f_{1}f_{2}}}=\sigma_{_{f_{1}}} \rho_{_{F_{1}F_{2}}} \sigma_{_{f_{2}}}$
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.
Because all these coefficients have been derived before in terms of the quality of the simple questions we can compute these correlations and covariances for any possible number of simple and complex concepts, for any type of complex concept and for all correlations and covariances between the observed variables which are the measures for the concepts of interest in a study.
