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Can one do a confirmatory factor analysis (CFA) with items on different response scales? Let's say $50$ items have a Likert response scale from $1$ to $4$, and the second group of items have a Likert response scale from $1$ to $5$.

From my understanding, for exploratory factor analysis (EFA), this is not a problem, especially if I estimate the EFA on a correlation matrix. Am I right to assume this?

I am not quite sure about the CFA, whether this is okay. Any pitfalls, considerations, most responses I find online just beat around the bush and I can't make my mind up based on them.

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CFA is fine with items using different scales. Factor analysis in your case isn't even an analysis of variables, but is an eigendecomposition of the correlation matrix of the variables with the diagonal identity vector replaces with communality (shared variance) estimates, so the distribution of individual variables is largely irrelevant.

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  • $\begingroup$ I am struggling to understand your explanation for why the distribution of the individual variables is largely irrelevant. The eigendecomposition of the estimated correlation matrix has a distribution which should depend on the distribution of the input random variables. Your answer seems to suggest the influence of the input distribution is small. Can you explain why? $\endgroup$
    – Galen
    Commented Jun 8, 2023 at 20:26
  • $\begingroup$ An extreme example would be if the input variables have a joint Cauchy distribution. The second moments would not exist, and therefore the sample correlation matrix would not be estimating a corresponding population correlation matrix. $\endgroup$
    – Galen
    Commented Jun 8, 2023 at 20:31
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Yes, just like you can fit a confirmatory factor analysis (CFA) to continuous data with different ranges, you can fit a CFA model to ordinal data with a different number of response options.

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