Correction of correlation and covariance between simple concepts
We have derived for a simple concept that qij = rij vij and ɱij = rij mij. Therefore we can formulate for two simple concepts the following model:
For this model we get
ρ(y1,y2) = q1 ρ(f1,f2) q2 + m1 m2
and follows that
ρ(f1,f2)= [ρ(y1,y2) – m1 m2]/ q1 q2
For the covariance of the unstandardized variables f1 and f2 denoted by σf1f2 hold that
σf1f2=σf1 ρf1f2 σf2
where σfi = qi σyi
This approach to estimate a causal model after correcting the correlation or covariance matrix for measurement errors has been in detail illustrated in the European Social Survey Edunet:
De Castellarnau A. and W.E.Saris (2016) A simple procedure to correct for measurement error in survey research. Edunet, ESS, chapters 4-7. http://essedunet.nsd.uib.no/cms/topics/measurement/4/