Subjects have 5 measures on each eye. DA is an area measurement and the C's are counts of area size categores. The C's can be linearly combined to closely estimate DA. The idea is that DA and the C's are indicators of the true area. (It's far easier to count size categories using a template than it is to try to make an accurate area measurement.) I've attached a pdf of the path diagram (OS = left and OD = right, the one latent variable is shown as OS when it should be OD.) The goal for this part is to just estimate the correlations between the "true" area and the indicators. (I use another SEM to calibrate measured area from estimated area (where the estimate is based on the counts) using data from both eyes which shows that there is little if any bias between the approaches.) Typically I constrain the betas and the sigmas to be the same for both eyes. I standardized the data so that each observed variable has a mean of zero and sd of 1. I constrained the latent variables to have sigma's equal to one. My understanding is in this situation, the beta's are then also standardized so that they are correlations.
When I run this, I get good estimates that have reasonably narrow confidence intervals. There is no indication of any problems. The rho between eyes is estimated to be about 0.7 (again with a very narrow ci) which is very typical for ophthalmic data. What is strange is that the beta for DA is estimated as 1.0000 (95% CI: 0.9418 to 1.0657, note upper bound for correlation can't really be higher than 1) and the sigma for the error is estimated as 0.0000 (0.0000 to 0.0400). I interpret this to mean that DA (measured area) in this model is close to being perfectly accurate with very little imprecision as an estimate of the true area (I understand that the latent variables are hypothetical and we never really know what the "true" values are).
I was thinking this result (getting a beta of 1 for DA) occurred because DA is nearly identical to a linear combination of the counts, but I'm not sure if this really explains the result. Although, if I drop out one or more of the counts, the beta for DA drops below 1 and the sigma for DA is above 0.
I've also fit a similar model using the covariance matrix instead of the correlation matrix. I fix the beta for DA to be 1 and let the sigma's for the latent variables be free. This would enable me to calibrate each count to DA, for example. The sigma for DA is again 0 with a narrow confidence interval. The results agree as you would expect with the model based on correlation.
I believe my results are correct and the models reasonable but I would just like to be able to explain them. I would appreciate the insight of others who are much more knowledgeable than I am. Thanks.
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