## The quality of a simple sum score

Examples of these complex concepts are variables like the “total time spend at the Media” based on the sum of the time spend on the different media. Concepts with a similar structure are “total household income” and “total expenditure of a household”.

In the case that the complex concept is just the sum score of different measures of simple concepts which may be correlated, the model for this complex concept can be represented by the following model:

In the case that the complex concept is just the sum score of different measures of simple concepts which may be correlated, the model for this complex concept can be represented by the following model:

The model represents a measurement model for a complex concept with three indicators all measured with the same method $M$ and with random measurement errors $e_{i}$.

Assuming that all variables are standardized, it can be shown that the general equation for $Y_{i}$ is:

$Y_{i}=q_{i}F_{i}+r_{i}\mu_{j}M+e_{i}^{'}$ for $i= 1,2,...,k$

where $q_{i}=r_{i}v_{i}$

The unstandardized composite score is denoted by $y_{cs}$.

$y_{cs} = \sum_{i=1}^{k}w_{i}Y_{i}$ with variance $\sigma_{y_{cs}y_{cs}}$ and standard deviation $\sigma_{y_{cs}}$

In order to standardize $y_{cs}$ the weights are divided by $\sigma_{y_{cs}}$.

$Y_{CS}= \sum_{i=1}^{k}\frac{w_{i}}{\sigma _{Y_{cs}}}Y_{i} =\sum_{i=1}^{k}\frac{w_{i}}{\sigma _{Y_{cs}}}(q_{i}F_{i}+r_{i}\mu_{i}M+e_{i}^{'})$

$Y_{CS}= \sum_{i=1}^{k}\frac{w_{i}}{\sigma _{Y_{cs}}}q_{i}F_{i}+\sum_{i=1}^{k}\frac{w_{i}}{\sigma _{Y_{cs}}}r_{i}\mu_{i}M+\sum_{i=1}^{k}\frac{w_{i}}{\sigma _{Y_{cs}}} e_{i}^{'}$

It has been derived by Anna DeCastellarnau and Willem Saris (2020) that the quality of the standardized composite score is $q^2_{Y_{CS}}=\sum_{i=1}^{k}(\frac{w_{i}}{\sigma _{Y_{cs}}}q_{i})^2+2\sum_{i\lt j}\frac{w_{i}}{\sigma _{Y_{cs}}}(\rho_{Y_{i}Y_{j}}-r_{i}\mu_{i} r_{j}\mu_{j})\frac{w_{j}}{\sigma _{Y_{cs}}}$

This term is equal to the variance of the standardized composite score corrected for measurement error.

The method effect in the composite score is equal to

$m^2_{Y_{CS}}=(\sum_{i=1}^{k}\frac{w_{i}}{\sigma _{Y_{cs}}}r_{i}\mu_{i})^2$

The corrected variance for the unstandardized composite score ($\sigma_{ff}$) can be obtained by multiplying the coefficient $q_{Y_{cs}}^2$ with the observed variance of the unstandardized composite score ($\sigma_{y_{cs}y_{cs}}$)

$\sigma_{ff}=q^2_{Y_{cs}}\sigma_{y_{cs}y_{cs}}$

The proofs of these results can be found in:

DeCastellarnau, Anna and Willem Saris (2020) "Correcting correlation and covariance matrices for measurement errors before using SEM" (obtainable from SRF).

Assuming that all variables are standardized, it can be shown that the general equation for $Y_{i}$ is:

$Y_{i}=q_{i}F_{i}+r_{i}\mu_{j}M+e_{i}^{'}$ for $i= 1,2,...,k$

where $q_{i}=r_{i}v_{i}$

The unstandardized composite score is denoted by $y_{cs}$.

$y_{cs} = \sum_{i=1}^{k}w_{i}Y_{i}$ with variance $\sigma_{y_{cs}y_{cs}}$ and standard deviation $\sigma_{y_{cs}}$

In order to standardize $y_{cs}$ the weights are divided by $\sigma_{y_{cs}}$.

$Y_{CS}= \sum_{i=1}^{k}\frac{w_{i}}{\sigma _{Y_{cs}}}Y_{i} =\sum_{i=1}^{k}\frac{w_{i}}{\sigma _{Y_{cs}}}(q_{i}F_{i}+r_{i}\mu_{i}M+e_{i}^{'})$

$Y_{CS}= \sum_{i=1}^{k}\frac{w_{i}}{\sigma _{Y_{cs}}}q_{i}F_{i}+\sum_{i=1}^{k}\frac{w_{i}}{\sigma _{Y_{cs}}}r_{i}\mu_{i}M+\sum_{i=1}^{k}\frac{w_{i}}{\sigma _{Y_{cs}}} e_{i}^{'}$

It has been derived by Anna DeCastellarnau and Willem Saris (2020) that the quality of the standardized composite score is $q^2_{Y_{CS}}=\sum_{i=1}^{k}(\frac{w_{i}}{\sigma _{Y_{cs}}}q_{i})^2+2\sum_{i\lt j}\frac{w_{i}}{\sigma _{Y_{cs}}}(\rho_{Y_{i}Y_{j}}-r_{i}\mu_{i} r_{j}\mu_{j})\frac{w_{j}}{\sigma _{Y_{cs}}}$

This term is equal to the variance of the standardized composite score corrected for measurement error.

The method effect in the composite score is equal to

$m^2_{Y_{CS}}=(\sum_{i=1}^{k}\frac{w_{i}}{\sigma _{Y_{cs}}}r_{i}\mu_{i})^2$

The corrected variance for the unstandardized composite score ($\sigma_{ff}$) can be obtained by multiplying the coefficient $q_{Y_{cs}}^2$ with the observed variance of the unstandardized composite score ($\sigma_{y_{cs}y_{cs}}$)

$\sigma_{ff}=q^2_{Y_{cs}}\sigma_{y_{cs}y_{cs}}$

The proofs of these results can be found in:

DeCastellarnau, Anna and Willem Saris (2020) "Correcting correlation and covariance matrices for measurement errors before using SEM" (obtainable from SRF).