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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
    Commented Dec 5, 2021 at 19:56

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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, because with two indicators you have to specify the loadings to be equal. 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 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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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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Below are a variety of models that one might consider, with or without a latent variable. Only the simple regression models will be identified, while the others are under-identified. To use these other models, you will have to introduce constraints.

Simple Regression

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Latent Factor

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Latent Factor & Covariance

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Latent Factor & Regression

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  • $\begingroup$ Graphs are very nice! But are any of these models, except for the first two regression models, identified without additional restrictions? Any difference in fit will be due to a-priori restrictions applied by the researcher, sample statistics cannot discern between better and worse fit here. Most likely, all models will have perfect fit. E.g., the latent factor model will require fixing the variance of eta to a constant (eg, 1) and restricting loadings to be equal. The loadings are then the square root of the corr between x1 and x2. Indistinguishable from the two simple regression models. $\endgroup$ Commented Dec 6, 2021 at 1:41

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