Factor Analysis Continued Psy 524 Ainsworth Equations Extraction Principal Components Analysis Skiers S1 S2 S3 S4 S5 Cost 32 61 59 36 62 Variables Lift

Depth 64 65 37 62 40 45 62 34 46 43 Powder 67 65 43 35 40 Equations Extraction Correlation matrix w/ 1s in the diag Cost

Lift Depth Powder Cost Lift Depth Powder 1 -0.952990 -0.055276 -0.129999 -0.952990 1 -0.091107 -0.036248 -0.055276 -0.091107 1 0.990174 -0.129999 -0.036248 0.990174 1 Large correlation between Cost and Lift and another between Depth and Powder Looks like two possible factors Equations Extraction

L= V= Reconfigure the variance of the correlation matrix into eigenvalues and eigenvectors 2.016356722 0.000000000 0.000000000 0.000000000 0.000000000 1.941495414 0.000000000 0.000000000 0.000000000 0.000000000 0.037784214 0.000000000 0.000000000

0.000000000 0.000000000 0.004363650 0.3525740484 -0.2513116190 -0.6273115310 -0.6473131033 0.6142350748 -0.6636944396 0.3224031907 0.2797876809 0.6627227864 0.6760812698 0.2748380988 -0.1678590017 0.2433214370 0.1981572044 -0.6534527107 0.6888598954

Equations Extraction L=VRV Where L is the eigenvalue matrix and V is the eigenvector matrix. This diagonalized the R matrix and reorganized the variance into eigenvalues A 4 x 4 matrix can be summarized by 4 numbers instead of 16. Equations Extraction R=VLV This exactly reproduces the R matrix if all eigenvalues are used SPSS matrix output factor_extraction.sps Gets pretty close even when you use only the eigenvalues larger than 1. More SPSS matrix output Equations Extraction

Since R=VLV R V L LV ' R (V L )( LV ') V L A, LV ' A ' R AA ', where A is the loading matrix and A' is the transpose of the loading matrix. See SPSS output from matrix syntax. Equations Extraction Cost Lift Depth Powder Factor 1 Factor 2 -0.401

0.907 0.251 -0.954 0.933 0.351 0.957 0.288 Here we see that factor 1 is mostly Depth and Powder (Snow Condition Factor) Factor 2 is mostly Cost and Lift, which is a resort factor Both factors have complex loadings Equations Orthogonal Rotation Factor extraction is usually followed by rotation in order to maximize large correlation and minimize small

correlations Rotation usually increases simple structure and interpretability. The most commonly used is the Varimax variance maximizing procedure which maximizes factor loading variance Equations Orthogonal Rotation The unrotated loading matrix is multiplied by a transformation matrix which is based on angle of rotation Aunrotated Arotated cos sin , where is the angle of rotation sin cos .946 .326 if = 19 then

.326 .946 See SPSS matrix syntax. Equations Other Stuff Communalities are found from the factor solution by the sum of the squared loadings 97% of cost is accounted for by Factors 1 and 2 Equations Other Stuff Proportion of variance in a variable set accounted for by a factor is the SSLs for a factor divided by the number of variables For factor 1 1.994/4 is .50 Equations Other Stuff

The proportion of covariance in a variable set accounted for by a factor is the SSLs divided by the total communality (or total SSLs across factors) 1.994/3.915 = .51 Equations Other Stuff The residual correlation matrix is found by subtracting the reproduced correlation matrix from the original correlation matrix. See matrix syntax output For a good factor solution these should be pretty small. The average should be below .05 or so. Equations Other Stuff Factor weight matrix is found by dividing

the loading matrix by the correlation matrix See matrix output 1 B R A Equations Other Stuff Factors scores are found by multiplying the standardized scores for each individual by the factor weight matrix and adding them up. F ZB Equations Other Stuff You can also estimate what each subject would score on the (standardized) variables

Z FA ' Equations Oblique Rotation In oblique rotation the steps for extraction are taken The variables are assessed for the unique relationship between each factor and the variables (removing relationships that are shared by multiple factors) The matrix of unique relationships is called the pattern matrix. The pattern matrix is treated like the loading matrix in orthogonal rotation. Equations Oblique Rotation The Factor weight matrix and factor scores are found in the same way

The factor scores are used to find correlations between the variables. .079 pattern A .981 .078 .978 .994 .033 .977 .033 Equations Oblique Rotation .104 .584 .081 .421 1 R A B .159 .020 .856 .034 1.12 -1.18

1.01 .88 F ZB .46 .68 1.07 .98 .59 .59 Equations Oblique Rotation Once you have the factor scores you can calculate the correlations between the factors (phi matrix; ) 1 F ' F N 1 Equations Oblique Rotation

1.12 -1.18 1.01 .88 1.00 .01 1 1.12 1.01 -0.46 -1.07 -0.59 * .46 .68 .01 1.00 4 -1.18 0.88 0.68 -0.98 0.59 1.07 .98 .59 .59 Equations Oblique Rotation

The structure matrix is the (zero-order) correlations between the variables and the factors. C A .079 .981 .069 .982 .078 .978 1.00 .01 .088 .977 C * .994 .033 .01 1.00 .994 .023 .977 .033

.997 .043 Equations Oblique Rotation With oblique rotation the reproduced factor matrix is found be multiplying the structure matrix by the pattern matrix. Rrep CA ' What else? How many factors do you extract? One convention is to extract all factors with eigenvalues greater than 1 (e.g. PCA) Another is to extract all factors with nonnegative eigenvalues Yet another is to look at the scree plot Number based on theory Try multiple numbers and see what gives best interpretation. Eigenvalues greater than 1

Total Variance Explained Initial Eigenvalues Factor 1 Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings Total 3.513 % of Variance 29.276 Cumulative % 29.276 Total 3.296 % of Variance 27.467

Cumulative % 27.467 Total 3.251 % of Variance 27.094 Cumulative % 27.094 2 3.141 26.171 55.447 2.681 22.338

49.805 1.509 12.573 39.666 3 1.321 11.008 66.455 .843 7.023 56.828 1.495

12.455 52.121 4 .801 6.676 73.132 .329 2.745 59.573 .894 7.452 59.573

5 .675 5.623 78.755 6 .645 5.375 84.131 7 .527 4.391 88.522

8 .471 3.921 92.443 9 .342 2.851 95.294 10 .232 1.936 97.231

11 .221 1.841 99.072 12 .111 .928 100.000 Extraction Method: Principal Axis Factoring. Scree Plot Scree Plot 4 3

Eigenvalue 2 1 0 1 2 3 Factor Number 4 5 6 7 8

9 10 11 12 What else? How do you know when the factor structure is good? When it makes sense and has a simple (relatively) structure. How do you interpret factors? Good question, that is where the true art of this comes in.