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I have a data set with two items that potentially measure the same latent variable: personal well-being. Can it be useful to run a factor analysis (Principal Axis Factoring Analysis) on them, or does a factor analysis require more items?

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  • $\begingroup$ Theoretically , to study what is (exploratory) FA and how it works - yes of course. (In some of my answers about FA on this site I show pictures with two variables and a factor, and that would be helpful.) Practically, to construct factors - no reason. Two variables can have maximum 1 common factor defining and (reflecting in) their single correlation values. So, if the correlation is large enough for you taste, just declare there is a factor behing them both. $\endgroup$
    – ttnphns
    Dec 5 '21 at 19:56
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With two items (variables) it's not likely that FA or PCA would be helpful. In fact, aren't there usually multiple items required in order to derive a score for most of the scales (anxiety, depression, worry, etc.), based on the weighted sum of numerous e.g. Likert scores?

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I have difficulty seeing how factor analysis of two items would tell you anything new, above and beyond the correlation (or covariance) between the two variables.

The factor model will perfectly reproduce the correlation (or covariance) between the items. You would have to fix the loadings to equality to be able to estimate the model (or the variance of the latent factor). Then you can identify the scale of the latent factor in one of two ways:

  1. Fix the variance of the latent factor to 1. The loadings will equal the square root of the correlation between x1 and x2. So no need to perform factor analysis, you can just compute the correlation between x1 and x2 and you're done.

  2. Fix the loadings to 1. The variance of the factor will now equal the squared correlation between x1 and x2. So no need to perform factor analysis, you can just compute the correlation between x1 and x2 and you're done.

A similar result applies when you choose to factor analyse the (co)variances, save for multiplication of the loadings or factor variance by the standard deviations of the item scores, and you will reproduce the covariance instead of the correlation between x1 and x2.

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