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Date:         Thu, 29 Sep 2005 11:34:21 -0500
Reply-To:     Bruce Thompson <bruce-thompson@tamu.edu>
Sender:       "SPSSX(r) Discussion" <SPSSX-L@LISTSERV.UGA.EDU>
From:         Bruce Thompson <bruce-thompson@tamu.edu>
Subject:      PCA and Rotation
Content-Type: text/plain; charset="us-ascii"

This is discussed in some detail in my EFA/CFA book (pp. 53-55): http://www.apa.org/books/4316025.html

Two factors affect the congruence of PCA and PFA for a given dataset: (1) how close to 1's the measured variable reliabilities are (assuming you factoring variable, and not people, or time), and (2) the rank of the factored matrix (e.g., Pearson r matrix), because the proportional contribution of the diagonal gets smaller as the rank of this matrix gets larger.

FYI, issue #3 of volume 27 (1992) of _Multivariate Behavioral Research_ was all about these issues, and related heated controversy.

---------------------------- Original Message ---------------------------- Subject: Re: PCA and Rotation From: "Art Kendall" <Art@DrKendall.org> Date: Wed, September 28, 2005 10:26 am To: SPSSX-L@LISTSERV.UGA.EDU --------------------------------------------------------------------------

PCA and PFA are different varieties of FA. It is more a matter of terminology than substantial debate. PCA and PFA are specific ways of FA i.e., of representing a larger set of variables with a smaller set of variables, with "little" loss of meaningful variance.

The reason for the Kaiser criterion for the maximum number of factors to initially extract is that when 1's are on the diagonal, an eigenvalue of one is one item variable's worth of the total variance. If there are 100 variables the sum of eigenvalues is 100. If you are doing data REDUCTION, why would you be interested in a new variable that accounted for that little of the variance. In other words, for purposes of computation you need to tell the computer when to stop. The Kaiser criterion says something like "there is no reasonable way I would be interested in more factors than this." When there are communality (reliability) estimates on the diagonal, the meaning of 1.00 eigenvalue is similar, a very small proportion of the variance. Very seldom would the stopping rule for initial extraction give the same number of factors that would finally be retained.

Parallel analysis is one way to estimate the number of factors there would be with the same number of cases and variables but all of the apparent correlation being consistent with that which would occur from completely random processes.

Art

---------------------------- Original Message ---------------------------- Subject: Re: PCA and Rotation From: "Art Kendall" <Art@DrKendall.org> Date: Wed, September 28, 2005 10:26 am To: SPSSX-L@LISTSERV.UGA.EDU --------------------------------------------------------------------------

PCA and PFA are different varieties of FA. It is more a matter of terminology than substantial debate. PCA and PFA are specific ways of FA i.e., of representing a larger set of variables with a smaller set of variables, with "little" loss of meaningful variance.

The reason for the Kaiser criterion for the maximum number of factors to initially extract is that when 1's are on the diagonal, an eigenvalue of one is one item variable's worth of the total variance. If there are 100 variables the sum of eigenvalues is 100. If you are doing data REDUCTION, why would you be interested in a new variable that accounted for that little of the variance. In other words, for purposes of computation you need to tell the computer when to stop. The Kaiser criterion says something like "there is no reasonable way I would be interested in more factors than this." When there are communality (reliability) estimates on the diagonal, the meaning of 1.00 eigenvalue is similar, a very small proportion of the variance. Very seldom would the stopping rule for initial extraction give the same number of factors that would finally be retained.

Parallel analysis is one way to estimate the number of factors there would be with the same number of cases and variables but all of the apparent correlation being consistent with that which would occur from completely random processes.

Art

Bruce Thompson

"Exploratory and Confirmatory Factor Analysis" http://www.apa.org/books/4316025.html "Score Reliability" http://www.sagepub.com/printerfriendly.aspx?pid=7958&ptype=B

http://www.coe.tamu.edu/~bthompson


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