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Common Factors versus Components: Principals and Principles, Errors and Misconceptions Keith F. Widaman University of California at Davis Presented at conference Factor Analysis at 100 L. L. Thurstone Psychometric Lab, University of North Carolina at Chapel Hill, May 2004

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2 Goal of the Talk Flip rendition –(With apologies to Will) I come not to praise principal components, but to bury them –Thus, we might inter the procedure beside its creator More serious – To outline several key assumptions, usually implicit, of the simpler principal components approach –Compare and contrast common factor analysis and principal component analysis

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3 Organization of the Talk Principals –Major figures/events –Important dimensions – factors/components Principles –To organize our thinking –Lead to methods to evaluate procedures Errors –Structures of residuals –Unclear presentations Misconceptions

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4 Principal Individuals & Contributions Spearman (1904) –First conceptualization of the nature of a common factor – the element in common to two or more indicators (preferably three or more) –Stressed presence of two classes of factors – general (with one member) and specific (with a potentially infinite number) –Key: Based evaluation of empirical evidence on the tetrad difference criterion (i.e., on patterns in correlations among manifest variables) with no consideration of diagonal

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5 Principal Individuals & Contributions Thomson (1916) –Early recognition of elusiveness of theory – data connection –Single common factor implies hierarchical pattern of correlations, but so does an opposite conceptualization –Key for this talk: Focus was still on the patterns displayed by off-diagonal correlation. Diagonal elements were of no interest or importance

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6 Principal Individuals & Contributions Thurstone (1931) –First foray into factor analysis –Devised a center of gravity method for estimation of loadings –Led to centroid method –Key: Again, diagonal values explicitly disregarded

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7 Principal Individuals & Contributions Hotelling (1933) –Proposed method of principal components –Method of estimation Least squares Decomposition of all of the variance of the manifest variables into dimensions that are: (a)orthogonal (b)conditionally variance maximized –Key 1: Left unities on diagonal –Key 2: Interpreted unrotated solution

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8 Principal Individuals & Contributions Thurstone (1935) – The Vectors of Mind –It is a fundamental criterion for a valid method of isolating primary abilities that the weights of the primary abilities for a test must remain invariant when it is moved from one test battery to another test battery. –If this criterion is not fulfilled, the psychological description of a test will evidently be as variable as the arbitrarily chosen batteries into which the test may be placed. Under such conditions no stable identification of primary mental abilities can be expected.

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9 Principal Individuals & Contributions Thurstone (1935) –This implies invariant factorial description of a test (a) across batteries and (b) across populations –Again, diagonal values explicitly disregarded –Developed rationale for necessity for rotation –Contra Hotelling: Unities on diagonal – imply manifest variables are perfectly reliably Need for # dimensions = # manifest variables No rotation! This appears, to me, to be the most important criticism of Hotelling by Thurstone.

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10 Principal Individuals & Contributions McCloy, Metheny, & Knott (1938) –Published in Psychometrika –Sought to compare Common FA (Thurstones method) vs. Principal Components Analysis (Hotellings method) –Perhaps the first comparison of the two methods

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11 Principal Individuals & Contributions Thomson (1939) –Clear statement of the differing aims of Common factor analysis – to explain the off- diagonal correlations among manifest variables Principal component analysis – to re-represent the manifest variables in a mathematically efficient manner

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12 Principal Individuals & Contributions Guttman (1955, 1958) –Developed lower bounds for the number of factors –Weakest lower bound was number of factors with eigenvalues greater than or equal to unity With unities on diagonal With population data –Other bounds used other diagonal elements (e.g., strongest lower bound used SMCs), but these did not work as well

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13 Principal Individuals & Contributions Kaiser (1960, 1971) –Described the origin of the Little Jiffy Principal components Retain components with eigenvalues >= 1.0 Rotate using varimax –Later modifications – Little Jiffy Mark IV – offered important improvements, but were not followed

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14 Principles – Mislaid or Forgotten Principle 1: Common factor analysis and principal component analysis have different goals – à la Thomson (1939) –Common factor analysis – to explain the off-diagonal correlations among manifest variables –Principal component analysis – to re-represent the original variables in a mathematically efficient manner (a) in reduced dimensionality, or (b) using orthogonal, conditionally variance maximized way

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15 Principles – Mislaid or Forgotten Principle 2: Common factor analysis was as much a theory of manifest variables as a theory of latent variables –Spearman – doctrine of the indifference of the indicator, so any manifest variable was a more-or-less good indicator of g –Thurstone – test ones theory by developing new variables as differing mixtures of factors and then attempt to verify presumptions –Today, focus seems largely on the latent variables –Forgetting about manifest variables can be problematic

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16 Principles – Mislaid or Forgotten Principle 3: Invariance of the psychological/ mathematical description of manifest variables is a fundamental issue –It is a fundamental criterion for a valid method of isolating primary abilities that the weights of the primary abilities for a test must remain invariant when it is moved from one test battery to another test battery –Much work on measurement & factorial invariance –But, only similarities between common factors and principal components are stressed; differences are not emphasized

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17 Principles – Mislaid or Forgotten Principle 4: Know data and model –Should know relation between data and model –Should know all assumptions (even implicit) of model –Frequently told: information in correlation matrix is difficult to discern so, dont look at data run it through FA or PCA interpret the results –This is not justifiable!

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18 Common FA & Principal CA Models Common Factor Analysis –R = FF + U 2 = PΦP + U 2 –where R is (p x p) correlation matrix among manifest vars F is a (p x k) unrotated factor matrix, with loadings of p manifest variables on k factors U 2 is a (p x p) matrix (diagonal) of unique factor variances P is a (p x k) rotated factor matrix, with loadings of p manifest variables on k rotated factors Φ is a (k x k) matrix of covariances among factors (may be I, usually diag = I)

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19 Common FA & Principal CA Models Principal Component Analysis –R = F c F c = P c Φ c P c –R = F c F c + GG = P c Φ c P c + GG –R = F c F c + Δ = P c Φ c P c + Δ –where F c, P c, & Φ c have same order as like-named matrices for CFA, but with c subscript to denote PCA G is a (p x [p-k]) matrix of loadings of p manifest variables on the (p-k) discarded components Δ (= GG) is a (p x p) matrix of covariances among residuals

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20 Present Day: Advice to Practicing Scientist Velicer & Jackson (1990): CFA vs. PCA –Four practical issues Similarity between solutions Issues related to # of dimensions to retain Improper solutions in CFA Differences in computational efficiency –Three theoretical issues Factorial indeterminacy in CFA, not PCA CFA can be used in exploratory and confirmatory modes, PCA only exploratory CFA is latent procedure, PCA is manifest

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21 Present Day: Advice to Practicing Scientist Goldberg & Digman (1994) and Goldberg & Velicer (in press): CFA vs. PCA –Results from CFA and PCA are so similar that differences are unimportant –If differences are large, data are not well-structured enough for either type of analysis –Use factor to refer to factors and components –Aim is to explain correlations among manifest vars

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22 Present Day: Quantitative Approaches Recent paper in Psychometrika (Ogasawara, 2003) –Based work on oblique factors & components with: Equal number of indicators per dimension Independent cluster solution Sphericity (equal error variances), hence equal factor loadings –Derived expression for SEs (standard errors) for factor and component loadings and intercorrelations –SEs for PCA estimates were smaller than those for CFA estimates, implying greater stability of (i.e., lower variability around) population estimates

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23 An Apocryphal Example Researcher wanted to develop a new inventory to assess three cognitive traits Knew to collect data in at least two initial, derivation samples Use exploratory procedures to verify initial, a priori hypotheses Then, move on to confirmatory techniques So, Sample 1, N = 1600, and 8 manifest variables 3 Components explain 51% of total variance

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24 Oblique Components, Sample 1 VariableFac 1Fac 2Fac 3. h 2. V – V – V – N N2– N3– – S1– S2– Fac 11.0 Fac Fac

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25 Orthogonal Components, Sample 1 VariableFac 1Fac 2Fac 3. h 2. V V V N N N3– – S S Fac 11.0 Fac Fac

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26 An Apocryphal Example After confirming a priori hypotheses in Sample 1, the researcher collected data from Sample 2 –Same manifest variables –Sampled from the same general population –Same mathematical approach – principal components followed by oblique and orthogonal rotation –Got same results! Decided to test the theory in Sample 3 – using replicate and extend approach –Major change: Switch to Confirmatory Factor Analysis

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27 Confirmatory Factor Analysis, Sample 3 Variable Fac 1 Fac 2 Fac 3. θ 2. V12.50 (.18) V23.00 (.21) V33.50 (.25) N (.13) N (.14) N (.16) S (.44) S (.50) Fac 11.0 Fac 2.50 (.04)1.0 Fac 3.50 (.10).50 (.10)1.0

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28 Fully Standardized Solution, Sample 3 Variable Fac 1 Fac 2 Fac 3. h 2. V V V N N N S S Fac 11.0 Fac Fac

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29 Oblique Component Solution, Sample 3 VariableFac 1Fac 2Fac 3. h 2. V – V – V – N N2– N3– – S1– S2– Fac 11.0 Fac Fac

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30 An Early Comparison McCloy, Metheny, & Knott (1938) –Published in Psychometrika –Sought to compare Common FA (Thurstones method) vs. Principal Components Analysis (Hotellings) –Stated that Principal Components can be rotated –So, both techniques are different means to same end –Principal difference: Thurstone inserts largest correlation in row in the diagonal of each residual matrix Hotelling begins with unities and stays with residual values in each residual matrix

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31 Hypothetical Factor Matrix (McCloy et al.) VariableFac 1Fac 2Fac 3. h

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32 Rotated Factor Matrix (McCloy et al.) VariableFac 1Fac 2Fac 3. h – – – – – – –

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33 Rotated Component Matrix (McCloy et al.) VariableFac 1Fac 2Fac 3. h – – – – – –.094– – –

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34 An Early Comparison McCloy, Metheny, & Knott (1938) –Argued that: both CFA and PCA were means to same end both led to similar pattern of loadings, but Thurstones method was more accurate (Δh 2 =.056) than Hotellings (Δh 2 =.125) – [but these were average absolute differences] I averaged signed differences, and Thurstones method was much accurate (Δh 2 = -.013) than Hotellings (Δh 2 =.120)

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35 An Early Comparison McCloy, Metheny, & Knott (1938) –Found similar pattern of high and low loadings from PCA and CFA –But, they found (but did not stress) that PCA led to decidedly higher loadings Tukey (1969) –Amount, as well as direction, is vital –For any science to advance, we must pay attention to quantitative variation, not just qualitative

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36 Regularity Conditions or Phenomena Relations between population values of P and R Features of eigenvalues Covariances among residuals –Need a theory of errors –Recount my first exposure … –Should have to acknowledge (predict? live with?) the patterns in residuals

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37 Practicing Scientists vs. Statisticians Interesting dimension along which researchers fall: PracticingStatisticians scientists (Dark side) use CFA prefer PCA use regressionwarn of probs analysiserrors in vars

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38 Practicing Scientists vs. Statisticians At first seems odd –Practicing scientist prefers CFA (which partials out errors of measurement and specific variance) Regression analysis – despite the implicit assumption of perfect measurement –Statistician prefers To warn of ill-effects of errors in variables on results of regression analysis PCA (despite lack of attention to measurement error), perhaps due to elegant, reduced rank representation

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39 Practicing Scientists vs. Statisticians On second thought, is rational: –Practicing scientist prefers Assumptions that residuals (in CFA or regression analysis) are independent, uncorrelated, normally distributed –Statistician prefers To try to circumvent (or solve) problem of errors in variables in regression To relegate errors in variables problems in PCA to that part of solution (GG) that is orthogonal to the retained part, thereby circumventing (or solving) this problem

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40 Regularity Conditions or Phenomena In Common Factor Analysis, –Char. of correlations Char. of variables1:1 In Principal Component Analysis, –Char. of correlations Char. of variables1:1 (??) –Char. of correlations ~[ ] Char. of variablesmany:1

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41 Manifest Correlations Var V1V2V3 V4 V5 V6 V V V V V V

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42 Eigenvalues, Loadings, and Explained Variance Var EVP1P2h 2 EV c P c 1P c 2h c 2 V V V V4 V5 V6 P P2

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43 Residual Covariances: CFA Var V1V2V3 V4 V5 V6 V V V V4 V5 V6 Covs below diag., corrs above diag.

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44 Residual Covariances: PCA Var V1V2V3 V4 V5 V6 V V V V4 V5 V6 Covs below diag., corrs above diag.

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45 Eigenvalues, Loadings, and Explained Variance Var EVP1P2h 2 EV c P c 1P c 2h c 2 V V V V V V P P2

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46 Residual Covariances: CFA Var V1V2V3 V4 V5 V6 V V V V V V Covs below diag., corrs above diag.

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47 Residual Covariances: PCA Var V1V2V3 V4 V5 V6 V V V V V V Covs below diag., corrs above diag.

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48 Regularity Conditions or Phenomena In Common Factor Analysis, –If (a) the model fits in the population, (b) there is one factor, and (c) communalities are estimated optimally, –Single non-zero eigenvalue –Factor loadings and residual variances for first three variables are unaffected by addition of 3 identical variables –Residuals = specific + error variance –Residual matrix is diagonal

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49 Regularity Conditions or Phenomena In Principal Component Analysis, –If (a) the common factor model fits in the population, (b) there is one factor, and (c) unities are retained on the main diagonal, –Single large eigenvalue, plus (p – 1) identical, smaller eigenvalues –Residual component matrix G is independent of the space defined by F c –But, residual covariance matrix is clearly non-diagonal –And, (a) population component loadings and (b) residual variances and covariances vary as a function of number of manifest variables!

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50 Manifest Correlations Var V1V2V3 V4 V5 V6 V V V V V V

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51 Eigenvalues, Loadings, and Explained Variance Var EVP1P2h 2 EV c P c 1P c 2h c 2 V V V V4 V5 V6 P P2

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52 Residual Covariances: CFA Var V1V2V3 V4 V5 V6 V V V V4 V5 V6 Covs below diag., corrs above diag.

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53 Residual Covariances: PCA Var V1V2V3 V4 V5 V6 V V V V4 V5 V6 Covs below diag., corrs above diag.

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54 Eigenvalues, Loadings, and Explained Variance Var EVP1P2h 2 EV c P c 1P c 2h c 2 V V V V V V P P2

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55 Residual Covariances: CFA Var V1V2V3 V4 V5 V6 V V V V V V Covs below diag., corrs above diag.

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56 Residual Covariances: PCA Var V1V2V3 V4 V5 V6 V V V V V V Covs below diag., corrs above diag.

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57 Regularity Conditions or Phenomena So, the difference between population parameters from CFA and PCA diverge more: –(a) the fewer the number of indicators per dimension, and –(b) the lower the true communality But, some regularities still seem to hold (although these vary with the number of indicators) –regular estimates of loadings –regular magnitude of residual covariance –regular form of eigenvalue structure

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58 Regularity Conditions or Phenomena But, what if we have variation in loadings?

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59 Manifest Correlations Var V1V2V3 V4 V5 V6 V V V V V V

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60 Eigenvalues, Loadings, and Explained Variance Var EVP1P2h 2 EV c P c 1P c 2h c 2 V V V V V V P P2

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61 Residual Covariances: CFA Var V1V2V3 V4 V5 V6 V V V V V V Covs below diag., corrs above diag.

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62 Residual Covariances: PCA Var V1V2V3 V4 V5 V6 V V V V V V Covs below diag., corrs above diag.

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63 Regularity Conditions or Phenomena So, with variation in loadings One piece of approximate stability –regular estimates of loadings But, sacrifice –regular magnitude of residual covariance –regular form of eigenvalue structure

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64 Regularity Conditions or Phenomena But, what if we have multiple factors? Lets start with –(a) equal loadings –(b) orthogonal factors

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65 Eigenvalues, Loadings, and Explained Variance Var EVP1P2h 2 EV c P c 1P c 2h c 2 V V V V V V P P

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66 Residual Covariances: PCA Var V1V2V3 V4 V5 V6 V V V V V V Covs below diag., corrs above diag.

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67 Regularity Conditions or Phenomena So, strange result: –Same factor inflation as with 1-factor, 3 indicators –Same within-factor residual covariances as for 1-factor, 3 indicators –But, between-factor residual covariances = 0! Lets go to –(a) equal loadings, but –(b) oblique factors

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68 Eigenvalues, Loadings, and Explained Variance Var EVP1P2h 2 EV c P c 1P c 2h c 2 V V V V V V P P

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69 Residual Covariances: PCA Var V1V2V3 V4 V5 V6 V V V V V V Covs below diag., corrs above diag.

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70 Regularity Conditions or Phenomena So, strange result: –Same factor inflation as with 1-factor, 3 indicators –Reduced correlation between factors –But, residual covariances matrix is identical! Lets go to –(a) unequal loadings, and –(b) orthogonal factors

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71 Eigenvalues, Loadings, and Explained Variance Var EVP1P2h 2 EV c P c 1P c 2h c 2 V V V V V V P P

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72 Residual Covariances: PCA Var V1V2V3 V4 V5 V6 V V V V V V Covs below diag., corrs above diag.

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73 Regularity Conditions or Phenomena So, strange result: –Different factor inflation than with 1-factor, 3 indicators –Reduced correlation between factors –But, residual covariances matrix has unequal covariances and correlations among residuals, but between-factor covariances = 0! Lets go to –(a) unequal loadings, and –(b) oblique factors

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74 Eigenvalues, Loadings, and Explained Variance Var EVP1P2h 2 EV c P c 1P c 2h c 2 V V V V V V P P

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75 Residual Covariances: PCA Var V1V2V3 V4 V5 V6 V V V V V V Covs below diag., corrs above diag.

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76 Regularity Conditions or Phenomena So, strange result: –Extremely different factor inflation than with 1-factor, 3 indicators –Largest loading is now UNderrepresented –Very different population factor loadings (.8,.6, &.4) have very similar component loadings –Now, between-factor covariances are not zero, and some are positive!

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77 R from Component Parameters All the preceding from a CFA view: –Develop parameters from a CF model –Analyze using CFA and PCA –CFA procedures recover parameters –PCA procedures exhibit failings or anomalies –So What? What else could you expect? Challenge (to me): –Generate data from a PC model –Analyze using CFA and PCA –PCA should recover parameters, CFA should exhibit problems and/or anomalies

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78 R from Component Parameters Difficult to do Leads to –Impractical, unacceptable outcomes, from the point of view of the practicing scientist –Crucial indeterminacies with the PCA model

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79 R from Component Parameters Impractical, unacceptable outcomes, from the point of view of the practicing scientist

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80 Manifest Correlations Var V1V2V3 V4 V5 V6 V V V V4 V5 V6 First principal component has 3 loadings of.8 First principal factor has 3 loadings of (.46) 1/2, or about.67

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81 Manifest Correlations Var V1V2V3 V4 V5 V6 V V V V V V First principal component has 6 loadings of.8 First principal factor has 6 loadings of (.568) 1/2, or about.75 But, one would have to alter the first 3 tests, as their population correlations are altered

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82 R from Component Parameters Crucial indeterminacies with the PCA model –Consider case of well-identified CFA model: 6 manifest variables loading on a single factor –One could easily construct the population matrix as FF + uniquenesses to ensure diag(R) = I –With 6 manifest variables, 6(7)/2 = 21 unique elements of covariance matrix –12 parameter estimates –therefore 9 df

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83 R from Component Parameters Crucial indeterminacies with the PCA model –Consider now 6 manifest variables with defined loadings on first PC –To estimate the correlation matrix, must come up with the remaining 5 PCs –A start: [Fc | G] [Fc | G] = diag, so orthogonality constraint yields 6(5)/2 = 15 equations –Sum of squares across rows = 1, so 6 more equations –In short, 15 equations, but 30 unknowns (loadings of 6 variables on the 5 components in G) –Therefore, an infinite # of R matrices will lead to the stated first PC

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84 R from Component Parameters Crucial indeterminacies with the PCA model –Related to the Ledermann number, but in reverse –For example, with 10 manifest variables, one can minimally overdetermine no more than 6 factors (so use 6 or fewer factors) –But, here, one must specify at least 6 components (to ensure more equations than unknowns) to ensure a unique R –If fewer than 6 components are specified, an infinite number of solutions for R can be found

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85 Conclusions: CFA CFA factor models may not hold in the population But, if they do (in a theoretical population): –The notion of a population factor loading is realistic –The population factor loading is unaffected by presence of other variables, as long as the battery contains the same factors –In one-factor case, loadings can vary from 0 to 1 (provided reflection of variables is possible) –This generalizes to the case of multiple factors

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86 Conclusions: CFA CFA factor models may not hold in the population But, if they do: –Residual (i.e., unique) variances are uncorrelated –Magnitude of unique variance for a given variable is unaffected by other variables in the analysis

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87 Conclusions: PCA PCA factor models cannot hold in the population (because all variables have measurement error) Moreover: –The notion of the population component loading for a particular manifest variable is meaningless –The population component loading is affected strongly by presence of other variables –SEs for component loadings have no interpretation –In the one-component case, component loadings can only vary from (1/m) 1/2 to 1, where m is the number of indicators for the dimension –Generalizes to multiple component case

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88 Conclusions: PCA PCA factor models cannot hold in the population (because all variables have measurement error) Moreover: –Residual variables are correlated, often in unpredictable and seemingly haphazard fashion –Magnitude of unique variance and covariances for a given manifest variable are affected by other variables in the analysis

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89 Conclusions: PCA PCA factor models cannot hold in the population (because all variables have measurement error) Moreover: –Finally, generating data from a PC model leads either to Impractical, unacceptable outcomes Indeterminacies in the parameter – R relations

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