Common Factor Analysis• “World View” of PC vs. CF• Choosing between PC and CF• PAF -- most common kind of CF• Communality & Communality Estimation• Common Factor ScoresWorld View of PC AnalysesPC analysis is based on a very simple “world view”• We measure variables• The goal of factoring is data reduction • determine the # of kinds of information in the variables • build a PC for each• R holds the relationships between the variables• PCs are composite variables computed from linear combinations of the measured variablesWorld View of CF AnalysesCF is based on a somewhat more complicated and “causal” world view• Any domain (e.g., intelligence, personality) has some set of “latent constructs”• A person’s “values” on these “latent constructs” causes their scores on any measured variable(s)• any variable has two parts• “common part” -- caused by values of the latent constructs”• “unique part” -- not related to any latent construct (“error”)World View of CF Analyses, cont• the goal of factoring is to reveal the number and identify of these “latent constructs”• R must be “adjusted” to represent the relationships between portions of the variables that are produced by the “latent constructs”• represent the correlations between the “common” parts of the variables• CFs are linear combinations of the “common” parts of the measured variables that capture the underlying constructs”Example of CF world view“latent constructs”IQ Math Ability Reading Skill Social Skills“measures”adding, subtraction, multiplication vocabulary, reading speed, reading comprehension politeness, listening skills, sharing skillsEach measure is “produced” by a weighted combination of the latent constructs, plus something unique to that measure . . .adding = .5*IQ +.8*Math + 0*Reading + 0*Social + Uasubtraction = .5*IQ +.8*Math + 0*Reading + 0*Social + Usvocabulary = .5*IQ + 0*Math + .8*Reading + 0*Social + Uvpoliteness = .4*IQ + 0*Math + 0*Reading +.8*Social + UpExample of CF world view, contWhen we factor these, we might find something likeCF1 CF2 CF3 CF4adding .4 .6subtraction .4 .6multiplication .4 .6vocabulary .4 .6reading speed .4 .6reading comp .4 .6politeness .3 .6listening skills .3 .6sharing skills .3 .6Name each “latent construct” that was revealed by this analysisPrincipal Axis Analysis“Principal” again refers to the extraction process• each successive factor is orthogonal and accounts for the maximum available covariance among the variables“Axis” tells us that the factors are extracted from a “reduced” correlation matrix• diagonals < 1.00 • diagonals = the estimated “communality” of each variable• reflecting that not all of the variance of that variable is “produced” by the set of “latent variables”• So, factors extracted from the “reduced” R will reveal the latent variablesWhich model to choose -- PC or PAF ?Traditionally...PC is used for “psychometric” purposes• reduction of collinear predictor sets• examination of the structure of “scoring systems”• consideration of scales and sub-scales• works with full R because composites will be computed from original variable scores not “common parts”CF is used for “theoretical” purposes• identification of “underlying constructs”• number and identity of “basic elements of behavior”• The basis for “latent class” analyses of many kinds• both measurement & structural models• works with reduced R because it hold the “meaningful” part of the variables and their interrelationshipsThe researcher selects the procedure based on their purpose for the factor analysis !!Communality & Its EstimationThe communality of a variable is the proportion of that variable’s variance that is produced by the common factors underlying the set of variablesCommon Estimations• α (reliability coefficient) -- only the reliable part of the variable can be common• largest r (or r2) with another in the set -- at least that much is shared with other variables•R2predicting that variable from all the others -- tells how much is shared with other variablesNote how the definition shifts from “variance shared with the latent constructs” to “variance shared with the other variables in the set” !!Communality & Its Estimation: How SPSS does it…Step 1: Perform a PC analysis• extract # PCs from the full R matrix Step 2: Perform 1stPAF Iteration•Use R2predicting each variable from others -- put in diagonal of R• extract same # PAFs from that reduced R matrix• compute (output) variable communalitiesStep 3: Perform 2ndPAF Iteration• use variable (output) communalities from last PAF step as estimated (input) communalities -- put in diagonals of R • extract same # PAFs from that reduced R matrix• compute (output) variable communalities• Compare estimated (input) and computed (output) variable communalitiesAdditional Steps: Iterate to convergence of estimated (input) & computed (output) variable communalitiesCommunality & Its Estimation How SPSS does it…, contHuh?!!?The idea is pretty simple (and elegant) …• If the communality estimates are correct, then they will be returned from the factor analysis !• So, start with a “best guess” of the communalities, and iterate until the estimates are stable Note: This takes advantages of the “self-correcting” nature of this iterative process • the initial estimates have very little effect on the final communalities (R2really easy to calculate)• starting with the PC communalities tends to work quicklyNote: This process assumes the latent constructs are adequately represented by the variable set !!Problems estimating communalities in a CF analysis“failure to converge”• usually this can be solved by increasing the number of iterations allowed (=1000)“Heywood case” Æ λ > 1.00• During iteration communality estimates can become larger than 1.00• However no more than “all” of a variable’s variance can be common variance!• Usual solutions…• Use the solution from the previous iteration • Drop the offending variable • If other variables are “threatening to Heywood” consider aggregating them together into a single variableCommon Factor Scores• The “problem” is that common factors can only be computed as combinations of the “common parts” of the