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I've checked two textbooks about EFA, one is the seminal work of Gorsuch (1974, 'Factor Analysis') and the other 'Exploratory and Confirmatory Factor Analysis' by Thompson (2004). Both described determination of number of factors to be extracted as a stage before choosing factor extraction method in the EFA decision sequence. But I'm wondering that all criteria for determination of the number of the factors (e.g. Kaiser criterion, Scree test, etc) need some sort of factor extraction to be done before. So are these authors wrong ? or am I missing something ?

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    $\begingroup$ The criteria (such as Kettell's scree plot, Kaiser's rule etc.) are based on PCA's eigenvalues, not the eigenvalues or extraction sum-of-squared loadings that you get at factor extraction. So, you do PCA first and decide on the number of factors. Then you proceed to FA. Tre problem of the number of factors has nothing to do with choosing this or that extraction method. $\endgroup$
    – ttnphns
    May 7, 2014 at 6:17
  • $\begingroup$ Thanks @ttnphns. This would really clarify the problem, but can you give any references mentioning this idea ? I appreciate it. $\endgroup$
    – Ehsan88
    May 7, 2014 at 7:46
  • $\begingroup$ The Kaiser rule and PCA are both also applied to common factor analysis in the lit. See for one example, Timmerman, M. E. and Lorenzo-Seva, U. (2011). Dimensionality assessment of ordered polytomous items with parallel analysis. Psychological Methods, 16(2):209. $\endgroup$
    – Alexis
    May 7, 2014 at 13:08
  • $\begingroup$ Kaiser's observation was based on Guttman's (1954) earlier interpretation of the lower bound on the rank of data in PCA (i.e. number of dimensions or components) as eigenvalues $\ge 1$, and rank of data in CFA (i.1. number of dimensions or common factors) as eigenvalues $\ge 0$. Despite the widespread use and/or fame of the Kaiser rule, Kaiser himself found the performance of this 'rule' to be "This is not a very delightful result."Kaiser (1960) $\endgroup$
    – Alexis
    May 7, 2014 at 16:07

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Ehsan: eigenvalues, upon which the Kaiser rule, parallel analysis, and other empirical factor-retention criteria are based (empirical factor retention decisions being data-driven, rather than driven by preferences regarding number of factors, or theoretical motivations, which may inform other retention approaches), can be extracted initially using principal components or common factor analysis (or iterated factor analysis), and a retention decision made about the underlying dimensionality (number of components or number of common factors). Subsequent to a retention decision, full-on factor analysis (rotation, extracted factors using ML factor analysis, interpretation of eigenvectors, etc.) can take place. The authors you cite are fully correct.

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  • $\begingroup$ I see this answer as confused a bit. It sounds that, in your view, principal components or common factor analysis are both equivalently a preliminary step, and then Subsequently, full-on factor analysis (rotation, extracted factors using ML factor analysis... takes place. So, common FA is not full-on thus, and is different from ML FA? $\endgroup$
    – ttnphns
    May 7, 2014 at 11:21
  • $\begingroup$ I do not see it as confused. The view I am presenting is common in the methods lit, and reflects desires to treat the number of factors as fixed and independent of subsequent rotation, interpretation, etc. PCA and CFA are not equivalent, and different perspectives hold sway as to the appropriateness of either in selecting #factors actually used. Last two questions: empirical retention decisions are not full on, and the use of MLFA either requires the retention decision a priori, or entails different motivating assumptions about retention. $\endgroup$
    – Alexis
    May 7, 2014 at 13:04

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