Factor Analysis and Structural equation modelling


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1 Factor Analysis and Structural equation modelling Herman Adèr Previously: Department Clinical Epidemiology and Biostatistics, VU University medical center, Amsterdam Stavanger July 4 13, 2006 Herman Adèr 1
2 Overview 1 E C 2 EFA 3 CFA 4 SEM Exploratory Confirmatory methods Exploratory Factor Analysis Confirmatory Factor Analysis Structural Equation Modelling Herman Adèr 2
3 Overview 1 E C 2 EFA 3 CFA 4 SEM Exploratory Confirmatory methods Exploratory Factor Analysis Confirmatory Factor Analysis Structural Equation Modelling Herman Adèr 2
4 Overview 1 E C 2 EFA 3 CFA 4 SEM Exploratory Confirmatory methods Exploratory Factor Analysis Confirmatory Factor Analysis Structural Equation Modelling Herman Adèr 2
5 Overview 1 E C 2 EFA 3 CFA 4 SEM Exploratory Confirmatory methods Exploratory Factor Analysis Confirmatory Factor Analysis Structural Equation Modelling Herman Adèr 2
6 Overview 1 E C 2 EFA 3 CFA 4 SEM Exploratory Confirmatory methods Exploratory Factor Analysis Confirmatory Factor Analysis Structural Equation Modelling Herman Adèr 2
7 Part VI Factor analysis and Herman Adèr Factor Analysis and SEM 3
8 Detective Judge Herman Adèr Factor Analysis and SEM 5
9 Characterization of Tukey (1977) Exploratory data analysis is detective in character. Confirmatory data analysis is judicial or quasijudicial in character... Unless the detective finds the clues, judge or jury has nothing to consider. Unless exploratory data analysis uncovers indications, usually quantitative ones, there is likely to be nothing for confirmatory data analysis to consider. On the other hand: Exploratory data analysis can never be the whole story, but nothing else can serve as the foundation stone the first step. Herman Adèr Factor Analysis and SEM 7
10 Characterization of Tukey (1977) Exploratory data analysis is detective in character. Confirmatory data analysis is judicial or quasijudicial in character... Unless the detective finds the clues, judge or jury has nothing to consider. Unless exploratory data analysis uncovers indications, usually quantitative ones, there is likely to be nothing for confirmatory data analysis to consider. On the other hand: Exploratory data analysis can never be the whole story, but nothing else can serve as the foundation stone the first step. Herman Adèr Factor Analysis and SEM 7
11 Questions and answers Three steps Determining the meaningful factors Rotation Interpretation of the factor structure F = M U E, of which M indicate the meaningful factors, U socalled unique factors (factors on which only one item loads) and E error factors. Questions we try to settle using EFA 1 How many meaningful dimensions are present? 2 What is the structure of those dimensions? Herman Adèr Factor Analysis and SEM 8
12 Questions and answers Three steps Determining the meaningful factors Rotation Interpretation of the factor structure F = M U E, of which M indicate the meaningful factors, U socalled unique factors (factors on which only one item loads) and E error factors. Questions we try to settle using EFA 1 How many meaningful dimensions are present? 2 What is the structure of those dimensions? Herman Adèr Factor Analysis and SEM 8
13 Questions and answers Three steps Determining the meaningful factors Rotation Interpretation of the factor structure Data set: { } D = Subject Items Exploratory Factor Analysis factor structure determine meaningful factors unrotated Three steps 1 Determine the meaningful factors rotate factor structure k meaningful factors 2 Rotate 3 Interpret the factor structure determine factor meaning final factor solution Herman Adèr Factor Analysis and SEM 9
14 Questions and answers Three steps Determining the meaningful factors Rotation Interpretation of the factor structure determine meaningful factors: scree plot eigenvalues larger than 1 GoF change significant between solutions total variance explained more than 50% select factors with intelligible loading patterns and name them communalities reliabilities doubtful Herman Adèr Factor Analysis and SEM 10
15 Questions and answers Three steps Determining the meaningful factors Rotation Interpretation of the factor structure CESD: A multidimensional scale to assess Depression FACTOR /VARIABLES cesd1 cesd2 cesd3 cesd4 cesd5 cesd6 cesd7 cesd8 cesd9 cesd10 cesd11 cesd12 cesd13 cesd14 cesd15 cesd16 cesd17 cesd18 cesd19 cesd20 /MISSING LISTWISE /ANALYSIS cesd1 cesd2 cesd3 cesd4 cesd5 cesd6 cesd7 cesd8 cesd9 cesd10 cesd11 cesd12 cesd13 cesd14 cesd15 cesd16 cesd17 cesd18 cesd19 cesd20 /PRINT all /FORMAT BLANK(.30) /PLOT EIGEN /CRITERIA MINEIGEN(0) ITERATE(100) factors(3) /EXTRACTION pc /ROTATION norotate. Herman Adèr Factor Analysis and SEM 12
16 Questions and answers Three steps Determining the meaningful factors Rotation Interpretation of the factor structure CESD: A multidimensional scale to assess Depression FACTOR /VARIABLES cesd1 cesd2 cesd3 cesd4 cesd5 cesd6 cesd7 cesd8 cesd9 cesd10 cesd11 cesd12 cesd13 cesd14 cesd15 cesd16 cesd17 cesd18 cesd19 cesd20 /MISSING LISTWISE /ANALYSIS cesd1 cesd2 cesd3 cesd4 cesd5 cesd6 cesd7 cesd8 cesd9 cesd10 cesd11 cesd12 cesd13 cesd14 cesd15 cesd16 cesd17 cesd18 cesd19 cesd20 /PRINT all /FORMAT BLANK(.30) /PLOT EIGEN /CRITERIA MINEIGEN(0) ITERATE(100) factors(3) /EXTRACTION pc /ROTATION norotate. Herman Adèr Factor Analysis and SEM 12
17 Questions and answers Three steps Determining the meaningful factors Rotation Interpretation of the factor structure 10 Scree Plot Eigenvalue Component Number Herman Adèr Factor Analysis and SEM 13
18 Questions and answers Three steps Determining the meaningful factors Rotation Interpretation of the factor structure Component Initial Eigenvalues Total % of Variance Cumulative % Extraction Method: Principal Component Analysis. Herman Adèr Factor Analysis and SEM 14
19 Questions and answers Three steps Determining the meaningful factors Rotation Interpretation of the factor structure Interpretation of the unrotated factor structure Component CESD CESD CESD CESD CESD CESD6 CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD Extraction Method: Principal Component Analysis. Herman Adèr Factor Analysis and SEM 15
20 Questions and answers Three steps Determining the meaningful factors Rotation Interpretation of the factor structure Communalities: four factors and all factors Communalities Initial Extraction CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD Extraction Method: Principal Component Analysis. Communalities Initial Extraction CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD Extraction Method: Principal Component Analysis. Herman Adèr Factor Analysis and SEM 16
21 Questions and answers Three steps Determining the meaningful factors Rotation Interpretation of the factor structure Communalities: two and three factors Communalities Initial Extraction CESD CESD E02 CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD Extraction Method: Principal Component Analysis. Communalities Initial Extraction CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD CESD Extraction Method: Principal Component Analysis. Herman Adèr Factor Analysis and SEM 17
22 Questions and answers Three steps Determining the meaningful factors Rotation Interpretation of the factor structure Goodnessoffit criterium # Facts GoF χ 2 Df χ 2 change Df χ 2 α= As one can see by comparing the 0.05 threshold χ 2 in the last column Herman Adèr Factor Analysis and SEM 18
23 Questions and answers Three steps Determining the meaningful factors Rotation Interpretation of the factor structure Unrotated Rotated Factor Factor Item I II III I II III CESD CESD CESD CESD04 + CESD CESD CESD CESD08 + CESD CESD CESD CESD12 + CESD CESD CESD CESD16 + CESD CESD CESD CESD Herman Adèr Factor Analysis and SEM 19
24 Questions and answers Three steps Determining the meaningful factors Rotation Interpretation of the factor structure Interpretation of table I Unrotated Rotated Factor Factor Item I II III I II III CESD CESD CESD CESD04 + CESD CESD CESD CESD08 + CESD CESD CESD CESD12 + CESD CESD CESD CESD16 + CESD CESD CESD CESD All items with high positive loadings on the unrotated Factor I still load on this factor after rotation, but some of them (5, 6, 7, 9, 17, 18 and 20) also load on factor II. This makes that Factor I is now separated into two sublists, of which the above could be called the pure depression scale while (1, 3, 10, 11, 13, 14, 15, 19) includes a sublist indicating less severe symptoms of depression. Herman Adèr Factor Analysis and SEM 20
25 Questions and answers Three steps Determining the meaningful factors Rotation Interpretation of the factor structure Interpretation of table II Unrotated Rotated Factor Factor Item I II III I II III CESD CESD CESD CESD04 + CESD CESD CESD CESD08 + CESD CESD CESD CESD12 + CESD CESD CESD CESD16 + CESD CESD CESD CESD Factor II now also contains all the items that loaded negatively on Factor I before (4, 8, 12 and 16). We called this a factor that measures: general attitude towards life. Herman Adèr Factor Analysis and SEM 21
26 Questions and answers Three steps Determining the meaningful factors Rotation Interpretation of the factor structure Interpretation of table III Unrotated Rotated Factor Factor Item I II III I II III CESD CESD CESD CESD04 + CESD CESD CESD CESD08 + CESD CESD CESD CESD12 + CESD CESD CESD CESD16 + CESD CESD CESD CESD After rotation, Factor III now contains only item 2 ( I did not feel like eating: my appetite was poor ) and item 11 ( My sleep was restless ), suggesting that it is a factor that has to do with physical aspects of depression. Herman Adèr Factor Analysis and SEM 22
27 Example of EQS input Example of EQS input Conclusions of EFA and CFA combined Example of EQS input /TITLE Cesd; /SPECIFICATIONS VARIABLES=6; CASES= 200; DATAFILE= cesd.ess ; MATRIX=RAW; ME = ML; /EQUATIONS V1 = 2 * F1 + E1; V2 = 2 * F1 + E2; V3 = 2 * F2 + E3; V4 = 2 * F2 + E4; V5 = 2 * F3 + E5; V6 = 2 * F3 + E6; F1 = 2 * F2 + 2*F3+ D1; /VARIANCES E1 to E10 = 0.2*; D1, D2, D3 = 0.2*; /COVARIANCES F2,F3 =.5*; /END Herman Adèr Factor Analysis and SEM 23
28 Example of EQS input Example of EQS input Conclusions of EFA and CFA combined CESD: Conclusions of the CFA The varimax solution is not fully confirmed: item 5, 6, 7, 18 and 20 have negligible coefficients on the second factor. The assumption of a orthogonal factor structure is unfounded: The correlations between the factors are (F1, F2) = 0.241; (F1, F3) = and (F2, F3) = Herman Adèr Factor Analysis and SEM 24
29 Example of EQS input Example of EQS input Conclusions of EFA and CFA combined CESD: Conclusions of the CFA The varimax solution is not fully confirmed: item 5, 6, 7, 18 and 20 have negligible coefficients on the second factor. The assumption of a orthogonal factor structure is unfounded: The correlations between the factors are (F1, F2) = 0.241; (F1, F3) = and (F2, F3) = Herman Adèr Factor Analysis and SEM 24
30 Example of EQS input Example of EQS input Conclusions of EFA and CFA combined Conclusions of EFA and CFA combined 1 The first factor is a general depression factor (item 1, 3, 5, 6, 7, 10, 13, 14, 15, 18, 19 and 20) 2 The second factor contains the positively formulated items (4, 8, 9, 12, 16, 17). It represents a general attitude towards life. 3 The third factor contains items that have to do with physical aspects of depression. Herman Adèr Factor Analysis and SEM 25
31 Example: Posttraumatic stress disorder Example of a postulated structure Draw pictures! Herman Adèr Factor Analysis and SEM 26
32 Example: Posttraumatic stress disorder Example of a postulated structure Draw pictures! Herman Adèr Factor Analysis and SEM 26
33 Example: Posttraumatic stress disorder Example of a postulated structure Draw pictures! Herman Adèr Factor Analysis and SEM 26
34 Postulated structure Structure found Herman Adèr Factor Analysis and SEM 27
35 Recommended literature Factor analysis: Friendly (1995) ( Principal component analysis: Jackson (1991) : Bollen (1989), Kaplan (2000) Herman Adèr Factor Analysis and SEM 28
36 Summary Herman Adèr Factor Analysis and SEM 29
37 Summary In factor analysis the contrast between exploratory and confirmatory approaches is quite clear: the available techniques are different although related. can well be performed with principal component analysis, possibly combined with Maximum likelihood factor analysis. PCA is first used to determine the meaningful factors, MLFA for the final factor structure. Structural Equation Modelling makes it possible to analyze a research problem that has been represented as a diagram. Herman Adèr Factor Analysis and SEM 30
38 Bollen, K. A. (1989). Structural equations with latent variables. New York: John Wiley and Sons. Friendly, M. (1995). Annotated Factor Analysis Bibliography. Jackson, J. E. (1991). A user s guide to principal components. New York: Wiley. Kaplan, D. (2000). Structural Equation Modeling. Foundations and Extensions. Thousand Oaks London New Delhi: Sage Publications. Tukey, J. W. (1977). Exploratory data analysis. Reading, MA: Addison Wesley. Herman Adèr Factor Analysis and SEM 31
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