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Let's say we have one latent factor L and three observed variables X1, X2 and X3. From the variance/covariance matrix of X1, X2, X3, we create this latent factor that explains the most out of the communalities between the X variables. So we compute the loading of each variable to the factor (three coefficients estimated). Then we estimate the variances of each variable (as I have read in other places, the error variance in the case of the X variables).

The main question is:

  • Why must we estimate (i.e. calculate or approximate an unknown) the variance of observed variables? We already have their variances in the original var-cov matrix and as far as I understand we have not modified the input data (i.e. we have only added variables into the equation, but not modified the variables we started with).

And the secondary questions (which most probably will be answered with the main answer) would be:

  • If it is called the error variance, why do we expect the estimated variance to be close or equal to the original variance of the variables (as a mean of validating our model, as DataCamp suggests)? Shouldn't it be the original variance minus the variance explained by the factor?

I have not been able to find an answer to the following question (nor anyone else having the same question) so I assume that it has an evident answer that just escapes me.

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  • $\begingroup$ You estimate the error variance. And if the factor does a good job explaining the items, the error variances will be quite small. The values you expect to be close to the original model are not the error variances but the model-implied variance covariance matrix. $\endgroup$
    – Heteroskedastic Jim
    Commented Oct 28, 2018 at 2:37
  • $\begingroup$ But if the model is estimating the original variances and also the error variances, my remaining degrees of freedom do not fit. In the example above, if you tell me that besides the loadings of each variable to the L factor (3 df - 1 df used for the scaling), we estimate the variances of each variable (4 df) and the error variance of each variable (4 df). Which is far beyond the degrees of freedom the input gives us. And even then it still does not answer why the model-implied variance of observed variables is different from the input var-matrix (what is being modified in the original data?) $\endgroup$
    – Kuku
    Commented Oct 28, 2018 at 23:39

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I think it's best to develop this using a full example. Imagine a unidimensional CFA with four items. There are 6 pieces of information $(4\times 3)/2$ if we work with the correlation matrix, there are 10 if we work with the covariance matrix. We'll use the covariance matrix.

If we estimate the CFA and fix the latent variable variance to 1, we will estimate 4 factor loadings, and 4 error variances, so 8 statistics. Hence, we will be left with 2 degrees of freedom.

From the factor loadings, we can calculate the model-implied covariance between any two items. Multiplying the loading on item 1 by the loading on item 2 will recover the model-implied covariance between item 1 and 2. It will differ from the actual covariance of both items.

Squaring an item's factor loadings plus the error variance will return the item's original variance. You do not estimate both error variance and original variance.

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  • $\begingroup$ Yes thanks, after coming back into it I realize I was mainly struggling with definition of terms. $\endgroup$
    – Kuku
    Commented Jun 16, 2019 at 23:39

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