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Date:         Tue, 20 Sep 2005 14:16:52 -0300
Reply-To:     Hector Maletta <hmaletta@fibertel.com.ar>
Sender:       "SPSSX(r) Discussion" <SPSSX-L@LISTSERV.UGA.EDU>
From:         Hector Maletta <hmaletta@fibertel.com.ar>
Subject:      Re: PCA and Rotation
Comments: To: "Swank, Paul R" <Paul.R.Swank@uth.tmc.edu>
In-Reply-To:  <6250203B042D8349A920AA61038309364B5382@UTHEVS3.mail.uthouston.edu>
Content-Type: text/plain; charset="iso-8859-1"

I have not read Preacher and MacCallum's article. However, some remarks:

1. From n variables one can extract up to n factors. If one does, and uses PCA, one explains 100% of total variance in the n variables. With other methods, n factors explain 100% of the commonality. If one extracts only some of the factors, one explains only part of total variance or total commonality.

2. There is no objective criterion to stop extracting factors. One can stop after one factor, two, three, 14 or 28 with equal theoretical justification. One can aim at explaining 20% of total variance, or 50%, or 90%, depending on goals and circumstances. Therefore there is no under- or over-factoring as such, only relative to some standard of comparison. And the decision to extract fewer or more numerous factors can be made with any of the factoring methods, not just with PCA.

3. One criterion for determining the number of factors is stopping at the last factor whose eigenvalue is >1, on the fragile grounds that factors with eigenvalues >1 explain more variance than any observed variable (whose "eigenvalue" is 1 by convention). This criterion is essentially arbitrary, and has been criticized many times. For different purposes you may want fewer or more factors.

4. Another criterion is the so-called scree curve: stop when the eigenvalue of a new factor differs only little from the preceding one (i.e. when the slope of the eigenvalues curve becomes flatter). This is still less justifiable than the above, but still widely used.

5. In practical work one often needs (or has theoretical grounds to seek) only the main one or just a few factors, and the extraction stops when the analyst seems fit. In some specific applications one is trying to capture as much as possible of total variance (or total commonality as the case might be), and so one wants to use more factors. Since factors are just shorthand constructs for the intercorrelation of variables, it is all in the hands of the analyst, and one may take any of these decisions without fear of divine retribution.

Hector

> -----Original Message----- > From: SPSSX(r) Discussion [mailto:SPSSX-L@LISTSERV.UGA.EDU] > On Behalf Of Swank, Paul R > Sent: Tuesday, September 20, 2005 1:17 PM > To: SPSSX-L@LISTSERV.UGA.EDU > Subject: Re: PCA and Rotation > > I highly recommend reading : > > Preacher, K. J., & MacCallum, R. C. (2003). Repairing Tom > Swift's electric factor analysis machine. Understanding > Statistics, 2(1), 13-32. > > It describes the "little jiffy", principal components > analysis followed by a varimax rotation, as the most commonly > used, yet incorrect, factoring procedure. It is uncorrect > because PCA will typically underfactor, especially if using > Kaiser's eigenvalue > 1 criterion, it does not attempt to > eliminate specific variance, and most of the time, factors > will have some correaltion. > > > Paul R. Swank, Ph.D. > Professor, Developmental Pediatrics > Director of Research, Center for Improving the Readiness of > Children for Learning and Education (C.I.R.C.L.E.) Medical > School UT Health Science Center at Houston > > -----Original Message----- > From: SPSSX(r) Discussion [mailto:SPSSX-L@LISTSERV.UGA.EDU] > On Behalf Of Luis O. > Sent: Tuesday, September 20, 2005 8:53 AM > To: SPSSX-L@LISTSERV.UGA.EDU > Subject: Re: PCA and Rotation > > Hector, thank you for your clear explanations. > > Your comments bring me to the ultimate question: when we use > SPSS's "Factor Analysis" with the "Principal Component" > method and VARIMAX rotation. are we conducting FA or PCA? > > I realized that some researchers are arguing that SPSS's > "Factor Analysis" > with the default function of "Principal component" is causing > a great confusion in distinguishing between FA and PCA. > Because of this, my stat friends recommend not to choose > principal componet method in FA. > Furthermore, their explanations are accidentally similar to what Paul > suggests: the use of oblique rotation. > > Luis > > > >From: "Hector Maletta" <hmaletta@fibertel.com.ar> > >To: "'Luis O.'" <soka28806@hotmail.com> > >Subject: RE: PCA and Rotation > >Date: Tue, 20 Sep 2005 08:26:07 -0300 > > > >Luis, > >I do not think your friends are right. Historically rotation > arose with > >principal axes, but nothing hinders rotation for principal > components. > >Both PCA and PA are methods of data reduction. They represent or > >summarize n variables through k<n factors or components. > Your friends > >are probably thinking that if you apply PCA to identify one dominant > >factor underlying all your varibles, and explaining by > itself a large > >portion of total variance, it is pointless to rotate it. But > PCA is not > >restricted to that situation, and often two or more factors > are needed > >to explain a large part of variance. Those factors may be rotated to > >make the structure more clear, i.e. to associate each factor more > >closely with some group of variables that have a common meaning (e.g. > >various linguistic tests associated mainly with one factor or > >component). > > > >By the way, factor and component are the same. The vocabulary just > >reflects how they were used by their introductors in the > past, nearly 100 years ago. > > > >Hector > > > > > > > -----Original Message----- > > > From: Luis O. [mailto:soka28806@hotmail.com] > > > Sent: Tuesday, September 20, 2005 5:43 AM > > > To: hmaletta@fibertel.com.ar > > > Subject: Re: PCA and Rotation > > > > > > Paul and Hector, thank you very much for your quick reply. I > > > sincerely appreciaed your explanations. > > > > > > I want to ask one more question to Hector. Based on your > > > explanations, you seem to be saying that in BOTH cases (PCA and > > > Principal Axes), we can rotate the principal components (although > > > you switched the word to "factor" in the later > > > paragraphs) in order to "put one factor closer to one set of > > > observed variables, and far from others, and the opposite for > > > another factor". Is this correct? I am asking this > question, because > > > this is precisely the doubt I had. Some of my stat > friends say that > > > in PCA the rotation should not be used, because it is a data > > > reduction procedure (as Paul explains), and provides a > best linear > > > combination of the variance. > > > Furthermore, they are making a point in that if we rotate the > > > principal components in PCA, they are no longer "principal" > > > components. > > > > > > I would appreciate your further clarification. > > > > > > Luis > > > > > > > > > >From: Hector Maletta <hmaletta@fibertel.com.ar> > > > >Reply-To: Hector Maletta <hmaletta@fibertel.com.ar> > > > >To: SPSSX-L@LISTSERV.UGA.EDU > > > >Subject: Re: PCA and Rotation > > > >Date: Mon, 19 Sep 2005 20:54:50 -0300 > > > > > > > >Some clarification: > > > >PCA is designed to maximize the first underlying factor, > > > maximizing its > > > >contribution, and (besides) when it extracts all > possible factors > > > >(number of factor equal to the number of variables) it > > > accounts for the > > > >entire variance of the variables. Other methods of extraction do > > > >not maximize the first, and besides they try to explain only the > > > >COMMON variance, leaving aside a portion of variance that is > > > >regarded as unique to each observed variable. > > > >Historically, PCA was used first to identify a single factor > > > underlying > > > >a set of related measures, supposedly measuring all the > same trait > > > >(intelligence, as it was), while Principal Axes was used > > > based on the > > > >theory that instead of a single Intelligence factor there were > > > >underlying "intelligences" for varios "faculties of the > > > mind" such as > > > >linguistic, graphic or mathematical ability. > > > >In BOTH cases, the position of the coordinate system on > which the > > > >underlying factors are measured is essentially arbitrary. By > > > rotation, > > > >the analyst may be able to put one factor closer to one set > > > of observed > > > >variables, and far from others, and the opposite for > another factor. > > > >This make happen by chance in the initial extraction, but > > > >ordinarily doesn't, so rotation is used in order to get that nice > > > characteristic > > > >of a factor linked to sets of interrelated variables. > For instance, > > > >using the same IQ example, by rotation one may get one > > > factor strongly > > > >associated with several linguistic tests, and only > weakly related > > > >to other tests, while another factor is strongly related to > > > mathematical > > > >tests and weakly to other tests, so one intuitively > calls the first > > > >factor "linguistic" and the second "mathematical". > > > >These different factors may be independe from each other, or > > > >correlated. It is perfectly possible that being good in language > > > >implies being good also in math, and so at least to some > > > degree, some > > > >correlation between linguistic and math factors is only to > > > be expected. > > > >Initial patterns of extraction ordinarily extract > factors that are > > > >orthogonal or uncorrelated to each other, because each > successive > > > >factor is extracted on the unexplained residuals left by the > > > preceding > > > >ones, but rotation can position the factor axes in ways that > > > imply they > > > >are correlated to each other. > > > >Rotation preserving the independence of factors is called > > > >orthogonal rotation. Rotation allowing them to be correlated to > > > >each other is called oblique rotation. > > > >So in Paul Swank responde there is a (probably involuntary) > > > confusion > > > >at the > > > >end: oblique rotation yields correlated factors (though > it does not > > > >FORCE them to be correlated) while orthogonal rotation methods > > > >FORCE rotated factors to be uncorrelated to each other. > > > > > > > >Hector > > > > > > > > > > > > > -----Original Message----- > > > > > From: SPSSX(r) Discussion > [mailto:SPSSX-L@LISTSERV.UGA.EDU] On > > > > > Behalf Of Swank, Paul R > > > > > Sent: Monday, September 19, 2005 8:26 PM > > > > > To: SPSSX-L@LISTSERV.UGA.EDU > > > > > Subject: Re: PCA and Rotation > > > > > > > > > > Principal components is a data reduction procedure, > not a way to > > > > > identify interpretable factors. To do the latter, use > > > principal axes > > > > > analysis or some other factor algorithm that targets > > > common factors. > > > > > It makes sense to rotate these since you are intreseted in > > > > > inpretable factors. However, I suggest an oblique > > > rotation to ensure > > > > > that the factors are no correlated before forcing them to be. > > > > > > > > > > > > > > > Paul R. Swank, Ph.D. > > > > > Professor, Developmental Pediatrics Director of > Research, Center > > > > > for Improving the Readiness > > > of Children > > > > > for Learning and Education (C.I.R.C.L.E.) Medical School > > > UT Health > > > > > Science Center at Houston > > > > > > > > > > -----Original Message----- > > > > > From: SPSSX(r) Discussion > [mailto:SPSSX-L@LISTSERV.UGA.EDU] On > > > > > Behalf Of Luis O. > > > > > Sent: Monday, September 19, 2005 5:14 PM > > > > > To: SPSSX-L@LISTSERV.UGA.EDU > > > > > Subject: PCA and Rotation > > > > > > > > > > Dear List Members, > > > > > > > > > > I am new in the list and want to ask a very basic > > > question regarding > > > > > the principal component analysis. I was doing some > > > analysis by using > > > > > principal component method and VArimax rotation. However, > > > one of my > > > > > friends told me that the stat book says that we should not > > > > > rotate principal components. > > > > > That is, the principal component analysis should not > rotate the > > > > > soluations, because, by theory, it produces a unique > > > soluation. On > > > > > the other hand, when I read some SPSS manuals, they > > > usually tell you > > > > > to use the principal component method with some > rotation method. > > > > > Which is correct? > > > > > > > > > > Luis O. Benavent > > > > > Benavnet Talps Research > > > > > Gandía, Spain > > > > > > > > > > __________ Información de NOD32 1.1222 (20050919) __________ > > > > > > > > > > Este mensaje ha sido analizado con NOD32 Antivirus System > > > > > http://www.nod32.com > > > > > > > > > > > > > > > > > > > > > > __________ Información de NOD32 1.1224 (20050920) __________ > > > > > > Este mensaje ha sido analizado con NOD32 Antivirus System > > > http://www.nod32.com > > > > > > > > > > __________ Información de NOD32 1.1224 (20050920) __________ > > Este mensaje ha sido analizado con NOD32 Antivirus System > http://www.nod32.com > >


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