Factor Analysis ( 因素分析 ) Kaiping Grace Yao National Taiwan University

“ Factor Analysis ” in MEDLINE

Classification on FA Exploratory factor analysis (EFA, 探索 性因素分析 ) Confirmatory factor analysis (CFA, 驗證 性因素分析 )

Exploratory factor analysis (EFA) Steps: 1.Communalities ( 估計共通值 ) 2.Number of Factors ( 決定因素的數目 ) 3.Estimation ( 估計因素負荷量 ) 4.Rotations ( 對因素做轉軸 ) 5.Interpretation ( 對結果作解釋 ) 6.Reports ( 報告 )

Confirmatory factor analysis (CFA) Steps: 1.Model specification ( 確立因素結構關係 ) 2.Identification ( 對參數找出單一組解 ) 3.Estimation ( 參數的估計 ) 4.Evaluation ( 檢驗資料與假設模式間的適合度 ) ＊ Structural Equation Model (SEM, 結構方程模式 )

Purposes of EFA Developing a measurement scale ： *construct validity (factor validity) *select items Test other hypotheses ：

Data Reduction Methods Exploratory factor analysis ( 探索性因素分析 ) Cluster analysis ( 群聚分析 ) Multidimensional scaling ( 多元尺度分析 )

EFA History Spearman ’ s two-factor theory (1904, 1923, 1927) Holzinger & Swineford ’ s bi-factor model (1937) Thurstone (1921, 1931) ： multiple factors Guilford (1956, 1967): Sternberg (1985, 1986): Gardner (1986):

Spearman ’ s two-factor theory g 1234 S1S1 S2S2 S3S3 S4S4 g factor specific factor

Holzinger & Swineford ’ s bi-factor model (1937)

EFA Under the assumption of having measurement error, EFA is to use fewer factors to interpret the correlations among many variables. Factor pattern matrix （ 4 vars ， 2 factors ） F1F1 F2F2 X1X1 λ 11 λ 12 X2X2 λ 21 λ 22 X3X3 λ 31 λ 32 X4X4 λ 41 λ 42

Model of EFA i =1,…,P variables j =1,…,N subjects k =1,…,M factors (M << P) λ ik (PxM) factor loadings F kj (MxN) factor scores E ij (PxN) errors Observed score ＝ Σ(factor loadings)x(factor score)+error assumptions ： ε(F)=0; Cov(F)=I; ε(E)=0; Cov(E)=diagonal matrix; Cov(F,E)=0

EFA method r kk’ =φ kk’ = corr. between factor k & k’ = root mean square when no correlation among factors  == the goal of EFA is to minimize this value

F1F1 F2F2 X1X1 X2X2 X3X3 X4X4 λ 11 λ 21 λ 31 λ 41 λ 12 λ 22 λ 32 λ 42 E 1 E 2 E 3 E 4 φ 21 Est. Corr(X 1, X 2 ) =

Example of FA Holzinger & Swineford (1939)

Example purpose ： item analysis and test development 生活品質 QOL 生理 PHY 心理 PSY 社會 SOC Factor 1Factor 2Factor 3 X : : X X11 : :::: X20 X21 : :::: X30

Geometric Representation F2 F1 θ 21 angle θ  correlation Vector length --  communality

1.Communalities ( 估計共通值, h 2 ) meaning ： how much variance of each variable can be explained by each factor 每個變項之變異量能為各因素解釋之部分 Influence by the # of factors methods ： Iterative method estimates: Thurstone ’ s: largest |r| Squared multiple correlation (SMC) ： R 2

2.Number of Factors ( 決定因素的數目 ) 1 on diagonal: eigenvalue > 1 h 2 on diagonal: eigenvalue > 0 Scree test ML significant test Tucker-Lewis index Theory, integration, interpretability Parallel analysis Minimum average partial Bartlett ’ s test

3.Estimation/Extraction ( 估計因素負荷量 ) Principal component, PC Principal factor, PF Iterative principal factor, IPF [SAS: prinit = SPSS: principal axis factoring] Maximal likelihood, ML Image factor analysis Alpha factor analysis Minimum residual (Minres) ULS GLS Schmid-Leiman hierarchical factor analysis

4.Rotations ( 對因素做轉軸 ) Subjective vs. Objective rationale ： factor indeterminacy Goal ： simple structure Orthogonal rotations ( 正交轉軸 ) ： varimax, quartimax, tanderm criteria, orthomax Oblique rotation ( 斜交轉軸 ) ： oblimin, orthoblique, procrustes, promax

5.Interpretation ( 解釋結果 ) Naming ： creativity, ingenuity, familiarity with the data Factor loadings ≧ 0.3 or 0.4 Most salient loadings Relative importance of variables Literature and theory

6.Reports ( 報告結果 ) Background info. ： Sample size,composition, subject selection method Type of variables, descriptive statistics, correlation method methods ： communalities, number of factors, estimation on factor loadings, rotations results ： descriptive statistics of vars, correlation matrix, factor loadings, final communality estimates, inter-factor correlations

Good EFA Design Plan before data collection Hypothesize a factor structure based on theory An item should measure only one factor Aim at primary factors: do not analyze variables which belongs to different hierarchical levels Select suitable variables: normal,linear, reliable Choose appropriate sample Use proper correlations Use appropriate method on: communalities, # of factors, factor loadings, rotation Interpret the results, modify the hypothesis, cumulate study results Use other approaches Cross-validation, CFA

Problems of past studies (1) Wang, C.N., & Weng,L.J. (2002) ： Evaluation on the applications of EFA in Taiwan ： 1993~1999 。 Chinese Journal of Psychology, 44(2), 。 Investigate five journals on the areas of psychology, education, business management. 82/497(16%) papers use EFA ， 154 times results ： Item level (90%) Report on item distribution (0%) 5-point scale (34%) [No 14% ； 4-point 18% ； 7-point 19%] eigenvalue > 1 (29%) [No 53%] Principal component method (40%) [No 32% ； PF 22%] Varimax (26%) [No 33% ； orthogonal rot. 32% ； oblique rot. 31%]

Problems of past studies (2) Discussion ： # of point of response scale and distribution Decision on # of factors Estimation on factor loadings Rotation methods Conclusion ：