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Procedures of Exploratory Factor AnalysisLet's work our way up the right-hand side of the "SEM Pyra id of Success!" exa ining how the Pearsoncorrelation gi es rise to exploratory factor analysis #EFA$%

Starting out with a fairly large set of aria&les #usually single ite s$! EFA will arrange the aria&les intosu&sets! where the aria&les within each su&set are strongly correlated with each other% hese su&sets areorgani(ed along axes #the plural of "axis!" not the "axe" like a hatchet$%

)ou could ha e a one-factor #one-di ensional$ solution! in which case all the aria&les will &e capa&le of &eing located along a single line #e%g%! across fro left to right! with low scores to the left and high scoresto the right$% *r there could &e a two-factor #two-di ensional$ solution! where the axes are across and up-and-down% hree-factor #three-di ensional$ solutions are harder to descri&e er&ally! so let's look ata picture % hese exa ples hold only as long as the axes are orthogonal #at +,-degree angles$ to eachother #which denotes co pletely un correlated factors$! an issue to which we'll return% Solutions can alsoexceed three factors! &ut we cannot isuali(e four spatial di ensions #at least can't$%

n conducting factor analyses with a progra such as SPSS! there are three main steps ! at each of whicha decision has to &e ade.

#/$ *ne ust first decide what extraction ethod to use #i%e%! how to "pull out" the di ensions$% he two &est-known approaches are Principal Axis Factoring #PAF0 also known as co on factor analysis$and Principal Components Analysis #P1A$% here's only one difference! co putationally! &etweenPAF and P1A! as descri&ed in this docu ent ! yet so e authors portray the two techni2ues as &eing erydifferent #further! P1A is technically not a for of factor analysis! &ut any researchers treat it as such$%

#3$ Second! one ust decide how many factors to retain% here is no a&solute! definiti e answer to this2uestion% here are arious tests! including the 4aiser 1riterion #how any factors or co ponentsha e eigen alues greater than or e2ual to /%,,$ and Scree est #an "el&ow cur e!" where one looks fordrop-off in the si(es of the eigen alues$%

he &ook Does Measurement Measure Up? ! &y 5ohn 6enshaw! addresses the indeter inacy of factoranalysis in the context of intelligence testing as follows. "Statistical analyses of intelligence test data%%%ha e &een perfor ed for a long ti e% 7i en the sa e set of data! one can ake a con incing! statisticallysound argu ent for a single! o erriding intelligence #so eti es called the g factor$ or an e2ually soundargu ent for ultiple intelligences% n Fra es of Mind! 6oward 7ardner argues that 'when it co es tothe interpretation of intelligence testing! we are faced with an issue of taste or preference rather than oneon which scientific closure is likely to &e reached' " #p% +8$%

#9$ he axes fro the original solution will not necessarily co e close to sets of data points #looselyspeaking! it's like the &est-fitting line in a correlational plot$% he axes can &e rotated to put the into &etter align ent with the data points% he third decision! therefore! in ol es the choice of rotation

ethod% wo classes of rotation ethods are orthogonal #as descri&ed a&o e$ and oblique #in which theaxes are free to intersect at other than +,-degree angles! which allows the factors to &e correlated witheach other$% Mathworks has a we& docu ent on factor rotation ! including a nice color-coded depiction oforthogonal and o&li2ue rotation% #As of 5anuary 3,/8! the graphics do not show up in the Mathworksdocu ent0 howe er! had pre iously sa ed a copy of the factor-rotation diagra ! which reproduce

&elow%$

http://en.wikipedia.org/wiki/Cartesian_coordinate_systemhttp://en.wikipedia.org/wiki/Cartesian_coordinate_systemhttp://www.uic.edu/classes/epsy/epsy546/Lecture%204%20---%20notes%20on%20PRINCIPAL%20COMPONENTS%20ANALYSIS%20AND%20FACTOR%20ANALYSIS1.pdfhttp://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectorshttp://documentation.statsoft.com/STATISTICAHelp.aspx?path=glossary/GlossaryTwo/S/ScreePlotScreeTesthttp://documentation.statsoft.com/STATISTICAHelp.aspx?path=glossary/GlossaryTwo/S/ScreePlotScreeTesthttp://documentation.statsoft.com/STATISTICAHelp.aspx?path=glossary/GlossaryTwo/S/ScreePlotScreeTesthttp://jhupbooks.press.jhu.edu/ecom/MasterServlet/GetItemDetailsHandler?iN=9780801883750&qty=1&viewMode=1&loggedIN=false&JavaScript=yhttp://jhupbooks.press.jhu.edu/ecom/MasterServlet/GetItemDetailsHandler?iN=9780801883750&qty=1&viewMode=1&loggedIN=false&JavaScript=yhttp://www.infed.org/thinkers/gardner.htmhttp://www.infed.org/thinkers/gardner.htmhttp://www.infed.org/thinkers/gardner.htmhttp://reifman-sem.blogspot.com/2008/01/in-tomorrows-class-well-be-covering.htmlhttp://reifman-sem.blogspot.com/2008/01/in-tomorrows-class-well-be-covering.htmlhttp://www.mathworks.com/products/demos/statistics/factorandemo.htmlhttp://www.mathworks.com/products/demos/statistics/factorandemo.htmlhttp://www.uic.edu/classes/epsy/epsy546/Lecture%204%20---%20notes%20on%20PRINCIPAL%20COMPONENTS%20ANALYSIS%20AND%20FACTOR%20ANALYSIS1.pdfhttp://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectorshttp://documentation.statsoft.com/STATISTICAHelp.aspx?path=glossary/GlossaryTwo/S/ScreePlotScreeTesthttp://jhupbooks.press.jhu.edu/ecom/MasterServlet/GetItemDetailsHandler?iN=9780801883750&qty=1&viewMode=1&loggedIN=false&JavaScript=yhttp://www.infed.org/thinkers/gardner.htmhttp://reifman-sem.blogspot.com/2008/01/in-tomorrows-class-well-be-covering.htmlhttp://www.mathworks.com/products/demos/statistics/factorandemo.htmlhttp://en.wikipedia.org/wiki/Cartesian_coordinate_system

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uncorrelated in empirical studies# -ather, researchers should use o$lique rotation methods#

As we'll see! %B3 ? te 3@ >%8, ? te 9@%%%%%%%$ or Cust add the ite s up withe2ual #or "unit"$ weighting # te / te 3 te 9$%

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he &ook Does Measurement Measure Up? ! &y 5ohn 6enshaw! pro ides a concise su ary of howo&ser a&le! easura&le anifestations are used to infer underlying! uno&ser a&le propensities. "Aristotleand others ha e contrasted the o&ser ed &eha ior of an indi idual with the underlying capacity on whichthat &eha ior depended% ntelligence! as one of those underlying capacities! is an a&ility that ay or aynot always &e o&ser ed in e eryday life% his underlying capacity ust &e deduced fro o&ser ed &eha iors" #p% +3$