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Factor analysis
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- 2. Purpose To researchfor and define the fundamental constructs or dimensions assumed to underlie the original variables Data summarization Define a small numbers of factors that maximize the explanation of the entire variable set Data reduction Identifying representative variables from a much larger set of variables Exploratory and confirmatory
- 3. Kinds of researchquestions Defining the underlying structure Number of factor Nature of factor Importance of solutions and factors Testing theory in factor analysis Estimating scores on factors
- 4. Limitations to factoranalysis Theoretical issues Factors Selection of observed variable Practical issues Sample size and missing data Normality Linearity Outlier Multicollinearity and singularity Factorability of R
- 5. What is factor? Linear combination of the original variables Factor Principle component Common factor Total variance Shared variance
- 6. Variance Case V1 V2V3 V4 1 5 5 1 2 2 3 2 5 4 3 1 1 2 1 4 5 4 3 2 5 2 3 2 1 6 4 5 1 2 7 2 1 5 4 8 3 4 2 3 9 2 1 4 5 Variance 25 22.89 19.56 16 Total variance = 25+22.89+19.56+16=83.45
- 7. PCA and (C)FA Principle component analysis (PCA) Common factor analysis (FA) Total variance Shard (common) variance Unique variance Specific and error Variance extracted Variance excluded
- 8. Factor analysis 1 1 N P N*P 1 1 N Q N*Q 1 1 N P N*P F Q=P F<P PCA FA 5 53 1 3 2 5 5 4 3 5 4 3 2 5 3
- 9. Factor analysis V1 V2V3 V4 V5 V6 V7 V8 V9 F1 F2 F3
- 10. Fundamental equation forfactor analysis General form Matrix X: observed variables μ: means Λ: factors loadings F: factors E: errors matrix R: correlations D: unique variance (R − D) = AA′ X = 𝜇 + ΛF + E X = 𝜇 + ΛF R = AA′ PCAE=0 D=0
- 11. Fundamental equation forfactor analysis Eigen formulation λ: eigenvalue K: eigenvector R − λI K = 0
- 12. Eigenvalue and communalities Eigen formulation R − λI K = 0 m 5 5 3 1 3 2 5 5 4 3 5 4 3 2 5 3 1 1 P F = λ1 λ2 λ3 λ 𝑚 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒 = 𝑖=1 𝑝 𝑓𝑖𝑗 2 = 𝜆𝑗 𝑐11 𝑐22 𝑐33 𝑐 𝑝𝑝 𝑐𝑜𝑚𝑚𝑢𝑛𝑎𝑙𝑖𝑡𝑦 = 𝑖=1 𝑚 𝑓𝑖𝑗 2 = 𝐶𝑖𝑖 𝑓𝑖𝑗 facto rvariable
- 13. Axis rotations Orthogonalrotation Non-correlations among factors are assumed Method Quartimax Simplify the rows of a factor matrix (largest variance on factor) Varimax Simplify the columns of a factor matrix (largest variance on variable) Equimax Compromise quartimax and varimax Oblique rotation Correlations among factors are assumed
- 14. Orthogonal rotation Unrotated F2 UnrotatedF1 Rotated F1 Rotated F2 -1.0 -.50 0 +.50 +1.0 -.50 -1.0 +1.0 +.50 V1 V2 V3 V4 V5
- 15. Oblique rotation Unrotated F2 UnrotatedF1 Orthogonal Rotated F1 Orthogonal rotated F2 -1.0 -.50 0 +.50 +1.0 -.50 -1.0 +1.0 +.50 V1 V2 V3 V4 V5 Oblique-rotated F2 Oblique-rotated F1
- 16. Data summarization Numberof factor Kaiser rule Eigenvalue > 1 (Kaiser, 1970) Scree test Factors before inflection point (Carttelli, 1966)
- 17. Data reduction Factorloadings Correlation between item and factor Orthogonal rotation Factor matrix Oblique rotation Factor pattern matrix Unique contribution of item to factor (partial correlation) Factor structure matrix Unique variance and correlation among factors (correlation)
- 18. Data reduction Significantfactor loadings based on sample size Factor loadings Required sample size .30 350 .35 250 .40 100 .45 150 .50 120 .55 100 .60 85 .65 70 .70 60 .75 50
- 19. Data reduction Preservation Variable with significant factor loading Variable with communality > .50 Deletion Cross loading
- 20. Practice issue Metricvariables Sample size 200 (Comrey, 1973) 10 observations per variable (Kline, 2005) Some degree of multicollinearity Bartlett test of sphericity Measure of sampling adequacy (MSA) Kaiser-Meyer-Olkin measurement of sampling adequacy (KMO test)
- 21. Sampling adequacy test Bartlett test of sphericity (p < .05) Are correlations among observations big enough? MSA (> .05) and KMO (> .07) test Are partial correlations among observations small enough? V1 V3 V4
- 22. Comparison PCA FA Purpose Maximize explanation oftotal variance Maximize explanation of shared variance Term Principle component Common factor Meaning Linear combination Construct Relationship among factor 0 R Diagonal matrix 1 Communality
- 23. Comparison PCA andFA reveal almost identical results High ratio: variable/factor High common variance Different results (Widaman, 1993) Low ratio: variable/factor (3:1) Low common variance (<.4) Which one is preferred? Psychological concept Satisfaction on product/consumer behavior
- 24. Latent variable Psychologicalconcept Behavioral pattern/sampling behavior/item V1 V2 V3 V4 V5 V6 V7 V8 V9 F1 F2 F3 u u u u u u u u u
- 25. Categorization Product/object/… V1 V2V3 V4 V5 V6 V7 V8 V9 F1 F2 F3
- 26. Procedure 1. Research question 2.Select the type of factor analysis 3. Check the assumptions 4. Derive factor analysis and assess overall fit 5. Interpret the factors 6. Validation of factor analysis 7. Additional use of factor analysis results
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- 28. After EFA Itemanalysis Appropriateness of item Item discrimination Homogeneity of item Indices Critical ratio (CR) t > 3 Item total correlation >.3 Cronbach’s α ↑ if item deleted
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- 30. 運動員沮喪反應量表的效化檢驗 根據相關學理，沮喪這項心理特性會反應在三種向 度上，分別是： 1.認知性 2.身體性 3.人際關係 於是研究生迪蒲睿宣依各向度分別設計了8、9、5共 22題的問卷，他不太確定這份問卷是否能反應出三 種沮喪的向度，請幫他進行探索性因素分析。