Hello,
I am conducting a Confirmatory Factorial Analysis with AMOS (two factors as latent variables, each one with three items as indicators) and when I checked latent correlations, I noticed that the correlation between the two latent unobserved variables is .62. However when I conduct a simple correlation analyses in spss between the two facors (each one calculated as the mean of the three items) the correlations is much lower (.48). Why do I observe such huge difference between latent variables in the CFA and the same variables constructed as the mean of the same items in SPSS? Any explanation and help would be much appreciated.
All the best,
Esther
First, I wouldn't say that the difference between .48 and .62 is "huge." But yes, it is noticeable. Here are a few possible explanations:
- the CFA model likely provides a more thorough and "cleaner" correction for measurement error in each item than does the aggregation to a simple mean score. Note that the mean scores of the items still contain measurement error (unreliability; although less than each individual item given that it is an aggregate of multiple items), whereas the factors inn CFA should not. In other words, the mean scores are observed variables, whereas the factors are latent variables. The CFA model provides a correction for attenuation of the correlation that is more complete than the observed item means approach.
- the mean score approach implies equal weighting of the items, whereas you may have estimated different item loadings in CFA (implying different weights). That could make a difference for the estimated correlations as well.
- there could be misspecification in the CFA model that leads to an overcorrection. The question here is how well the model fits and whether the items within each factor are truly unidimensional measures.
Generally speaking, I would prefer the CFA approach as long as it is correctly specified because it provides a cleaner correction for measurement error.
I agree with Christian Geiser that CFA gives errors-free correlation. However, I have not experienced this much difference from .62 to .48. In most cases, it provides only a tiny difference. One reason I go for SPSS correlation is that CFA correlation does not tell Significant (p-value) if the correlation is significant or not (when required). At the end, both the approaches are correct.
Muhammad Anwar In most (if not all) SEM software programs, you should be able to get a p value for a latent correlation as part of the completely standardized solution for a CFA model.
Christian Geiser Interesting to know it. I have the 21st version and perhaps the latest one 25th could give the p-value.
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