The Quality of Composite Scores of Concepts with Reflective Indicators
From SQP, or otherwise, one can get information about the quality coefficients ri, vi, mi, for all three or any number of single questions. That means that in the model only the effects of F on its indicators and the weights are unknown. However the weights can be chosen to be 1 or unequal to 1 that is up to the researcher. Lawley and Maxwel (1971) have suggested an approach to determine the weights that maximize the quality (correlation) of the composite score for the complex concept. Given the weights the values of the composite score can be computed for all cases and the variance and standard deviation of this variables are also known. In order to take care that the composite score has a variance of 1 like all other variables, the chosen weights have to be divided by sycs which is the standard deviation of the composite score before standardization.We have also shown before that on the basis of the correlations between the observed variables Y1, Y2, Y3 the correlations between the latent variables F1, F2, F3 can be obtained.
On the basis of these correlations the effects λi between the concept of interest and the indicators can be estimated if there are at least 3 indicators. If there are only two questions used the λ1 and λ2 can only be estimated assuming that these two coefficients are equal. Because the indicators will not all prefectly represent the complex variables of interest, we introduce also unique components (u1, u2, u3) for the indicators. They are another source of errors.
Given this specification of the model it follows that
Yi = ri vi λi F+ ri mi M+ ri vi ui + ei for all i
Ycs = ∑i Yi = ∑i ri vi λi F + ∑i ri mi M + ∑i (ri vi ui + ei) over all i
Ycs = qYcs F+ mYcs M+ ∑i (ri vi ui + ei) over all i
where qYcs = ∑i ri vi λi which is the effect of F on Ycs
and mYcs = ∑i ri mi which is the systematic method effect on Ycs
The quality of the composite score as measure of F is equal to qYcs2