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I have been explained this technique in class. However, I did not understand some stuff.

The professor said that a model to be estimated needs to be at least identified. Identified meaning to be when Number of observations > Number of parameters. In CFA, number of observations are the number of elements of the covariance matrix under the diagonal + the variances. Number of parameters are the number of loadings we do not constraint to 0 in the Loading Matrix + the variances of unique factors (errors) + Variances of the factors.

However, something is not working here.

My model has 12 variables and 5 factors. The first factor affects the 3 first variables, the next factor the following 3 variables, and the rest of factors are affecting 2 variables each one.

For example, a model with 78 observations from the covariance matrix (as I defined above), and 39 parameters (12 loadings, 12 error terms, 10 correlation factors, 5 factor variances) cannot be estimated in R in Lavaan. 78 > 39 so I don't understand what is happening here.

However, if I fix the 5 variances of the factors to 1, everything is computed. So estimating 34 parameters instead of 39 is working, but the number of observations is sufficiently high to estimate whatever model I pursue.

Furthermore, in case it is not possible, what is the point of fixing loadings of the rest of factors to 0? I could let that all variables are affected by all factors, to see some unexpected relationship...

Thank so much

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78 > 39 so I don't understand what is happening here. However, if I fix the 5 variances of the factors to 1, everything is computed

$df \ge 0$ is a necessary, but not sufficient, condition for identification. An identified model has a unique solution that minimizes data–model discrepancy. When your SEM includes latent factors, their lack of (location or) scale requires an additional per factor to sufficiently identify the model. Otherwise, there are infinite combinations of parameter estimates that would yield identical model-implied summary statistics. All introductory SEM/CFA textbooks must explain this at some point, e.g.:

  • Brown, T. A. (2015). Confirmatory factor analysis for applied research. Guilford.

A few popular options are available to give unobserved common factors an arbitrary (location and) scale, one of which is the method you used: fixing their variances to 1 (and means to 0, when modeling mean structure). Since the scale is arbitrary anyway, this scaling method allows us to interpret the factors as though they were $z$ scores. Other options (equally arbitrary) are explained in this excellent paper, which also provides ample discussion about how interpretations are(n't) affected by different scaling methods:

  • Klößner, S., & Klopp, E. (2019). Explaining constraint interaction: How to interpret estimated model parameters under alternative scaling methods. Structural Equation Modeling, 26(1), 143–155. https://doi.org/10.1080/10705511.2018.1517356
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  • $\begingroup$ Thanks so much! Parameter constrain and interpretation seems a tough topic, but it has clarified some of my doubts $\endgroup$ Oct 30, 2021 at 10:42

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