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Date:         Tue, 20 Sep 2005 19:36:42 -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: "Luis O." <soka28806@hotmail.com>
In-Reply-To:  <BAY107-F431FCDCA09F2E06DB8E9AED950@phx.gbl>
Content-Type: text/plain; charset="iso-8859-1"

Luis, Some authors think different than others. As I told you before, rotating PCA is usual and may be legitimate, though for certain purposes it may make no sense.

Hector

> -----Original Message----- > From: SPSSX(r) Discussion [mailto:SPSSX-L@LISTSERV.UGA.EDU] > On Behalf Of Luis O. > Sent: Tuesday, September 20, 2005 7:19 PM > To: SPSSX-L@LISTSERV.UGA.EDU > Subject: Re: PCA and Rotation > > Hi Hector, Paul, Manny, Stanley and William, > > Thank you for your help. Manny and William, I have just > downloaded the papers you suggested, and will read them > carefully this week. > > What I have learned through this debate is that there is a > considerable debate over this issue, and it seems to me that > there is a consensus that PCA and FA are two different > methods in terms of the underlining statistical theory. > > Still, Paul's latest comment makes me a little puzzled: "If > all I want to do is account for the maximum variance with the > smallest number of indepedent components, then PCA with > variamax rotation is okay". So if this is the case, PCA with > the rotation is acceptable after all. That is, all depend > upon what you plan to do in your research and analysis. Also, > if you are not interested in latent structure of the > variables, PCA is indeed a usuful method. Paul, please > correct my statement if I misunderstand your explanation. > > Another point. In most of "classic" marketing research books, > such as Multivariate Data Analysis by Hair, Anderson, Tatham > and Blanck (Prentice > Hall) explains that FA can be classified into two methods, > CFA and PCA, and in BOTH cases, we can rotate the results > (they even show a table of "VARIMAX-rotated component > analysis factor matrix" on p. 126). Furthermore, Barbara G > Tabachinick and Linda S Fidell provides a similar explanation > in their Using Multivariate Statistics (4th ed.): "Principal > component extraction with varimax rotation through SAS FACTOR > is used in an initial run to estimate the number of factors > from eigenvalues" (p. 633). > > If all the explanations I have learned in this discussion are > true, it is rather surprising that the authors teach such > procedure which is not considered as "correct" by many theorists. > > Luis > > > > >From: "Swank, Paul R" <Paul.R.Swank@uth.tmc.edu> > >Reply-To: "Swank, Paul R" <Paul.R.Swank@uth.tmc.edu> > >To: SPSSX-L@LISTSERV.UGA.EDU > >Subject: Re: PCA and Rotation > >Date: Tue, 20 Sep 2005 13:54:48 -0500 > > > >PCA does not distinguish between common and specific factor variance > >and does not explicity model the error variance. In studies > where the > >factor structure is known, PCA with the eigenvalue > 1 rule > will often > >incorrectly identify the number of factors present. Thirdly, > rotation > >is used for interpretation of the factors. Why rotate if not > to interpret the factors? > >Or components in the case of PCA. But since the components are not > >common factors since they can contain specific variance, why try to > >interpret them. If one wants to examine the underlying > structure of the > >data to find the dimensions represented, then one should > look at common > >factors with principal axes or maximum likelihood methods. > Finally, in > >the areas of biomedical and social sciences in which I work, > there are > >rarely uncorrelated factors. In fact there is a saying that all > >psychological variables are correlated at least .30. This of > course is > >a rank generalization but it is true that many variables, and ther! > > efore factors, are correlated. If the factors are not > correlated then > >the oblique solution will be similar to the orthogonal one > so what is > >lost by doing the oblique solution in the first place? > > > >Considering the variance accounted for by the components is a > >characteristic of data reduction, not interpretation. If all > I want to > >do is account for the maximum variance with the smallest number of > >indepedent components, then PCA with variamax rotation is > okay. But if > >one is attempting to understand the underlying dimensions then the > >percentage of variance accounted for is less applicable. Of more > >interest is accounting for all of the common factor > variance. Then one > >can eliminate factors that do not make sense or are > otherwise not interpreatble. > > > > > >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: Hector Maletta [mailto:hmaletta@fibertel.com.ar] > >Sent: Tuesday, September 20, 2005 11:17 AM > >To: Swank, Paul R; SPSSX-L@LISTSERV.UGA.EDU > >Subject: RE: PCA and Rotation > > > >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 > > > > > > > > __________ Información de NOD32 1.1226 (20050920) __________ > > Este mensaje ha sido analizado con NOD32 Antivirus System > http://www.nod32.com > >


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