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Factor Analysis, Part 2

Factor Analysis, Part 2. BMTRY 726 4/5/14. Factor Rotation. Recall, can conduct orthogonal transformations of the factors and still reconstruct the covariance of X Means can use orthogonal transformations of the factor loading matrix to “simplify” the interpretation of the factors

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Factor Analysis, Part 2

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  1. Factor Analysis, Part 2 BMTRY 726 4/5/14

  2. Factor Rotation Recall, can conduct orthogonal transformations of the factors and still reconstruct the covariance of X Means can use orthogonal transformations of the factor loading matrix to “simplify” the interpretation of the factors We refer to any orthogonal transformation of the factor loading as factor rotation

  3. Factor Rotation Ideally, rotated factor loadings result in each variable having high loading on only one factor and moderate or small loadings on all other factors In 2-dimensional case (i.e. two factors) we may be able to visualize a rotation that makes sense….

  4. Factor Rotation Not always possible to find orthogonal rotation that yields high loads for each variable on only one factor Also difficult to visualize rotations for > 2 This lead to development of several analytical methods for choosing T to improve factor simplicity -Varimax rotation -Quartimaxrotation

  5. Factor Rotation Varimax Criterion: Define as the “scaled” loading of the ith variable on the jth factor after rotation The varimax procedure selects T to maximize Variables with smaller communalities more influential After rotation, each of pvariables should have high loading on only one factor

  6. Varimax Rotation Can do this rotation for any Factor Analysis method BUT… just like methods didn’t yield same factors, factor rotations will not be exactly the same! The Stock market example -Using PC solution: -Using ML solution:

  7. Varimax Rotation The Stock market example (PC factors)

  8. Factor Rotation Quartimax Criterion: Scaled loadings: Select an orthogonal transformation, T, of the loadings to maximize Difference from Varimax: 1. Each of the pvariables should have fairly high loading on the same factor 2. Each variable should have high loading on at most one other factor and near zero loadings on remaining factors

  9. Quartimax Rotation Can also do for any Factor Analysis method BUT… just like methods didn’t yield same factors, factor rotations will not be exactly the same! The Stock market example -Using PC solution: -Using ML solution:

  10. Quartimax Rotation The Stock market example (PC factors)

  11. Factor Rotation Varimax is most popular orthogonal rotation Orthogonal rotation may not be most “realistic”… -Believable to have 2 factors describing anxiety and depression -Less believable that these are independent! Oblique Rotations: -Non-orthogonal rotations -Accomplished with non-rigid rotation of axes -Ideally express each variable with minimum number of factors -Most commonly used oblique transformation is promax method

  12. Oblique Factor Rotation Promax Method: (1) Do varimax rotation and obtain loadings (2) Construct another p x m matrix Q where Note: k > 1 chosen by trial and error, usually < 4 (3) Find matrixU such that each column of L*U close to corresponding column of Q. Choose jth column of U to minimize This yields (4) Rescale U so transformed factors have unit variance

  13. Promax Rotation Promax Method yields factors with loadings: Also is the correlation matrix for the new factors

  14. Estimation of Factor Scores Factor scores: estimated values of common factors for j observations Used for (1) diagnostics and (2) inputs in subsequent analysis Recall the original factor model: -if this model is correct:

  15. Estimation of Factor Scores One method of estimation is the weighted least squares method Weighted least squares estimate of Fj is: If PC method used to estimate factors, often use unweighted OLS -in this case specific variances generally near equal

  16. Estimation of Factor Scores Alternatively we can use the regression method Consider joint distribution (xj–m) and Fj is: Obtain conditional mean

  17. Estimation of Factor Scores Use estimated conditional mean vector to estimate factor scores To reduce effect of poor choice of m replace with S

  18. Example of Factor Scores Consider a factor analysis that yields the following loading matrix and specific variance matrix

  19. Example of Factor Scores Consider a factor analysis that yields the following loading matrix and specific variance matrix

  20. General Strategy Factor analysis involves many decisions (each somewhat arbitrary) Most important is choice of m -proportion of variance explained -investigator knowledge of subject -how reasonable is result? Must also choose solution and rotation methods though these are less critical

  21. General Strategy One general approach: (1) Conduct PC factor analysis -check for unusual factor scores using methods we’ve discusses previously -Do varimax rotation (2) Conduct ML factor analysis (w/ varimax rotation) (3) Compare solutions -Are loadings similar (4) Repeat steps 1-3 with different m For large datasets, consider splitting the data and comparing analysis from the two sets

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