Factor Analysis ContinuedEquations – Extraction Principal Components AnalysisEquations – ExtractionSlide 4Slide 5Slide 6Slide 7Slide 8Equations – Orthogonal RotationSlide 10Equations – Other StuffSlide 12Slide 13Slide 14Slide 15Slide 16Slide 17Equations – Oblique RotationSlide 19Slide 20Slide 21Slide 22Slide 23Slide 24What else?Eigenvalues greater than 1Scree PlotSlide 28Factor Analysis ContinuedPsy 524AinsworthEquations – ExtractionPrincipal Components AnalysisEquations – Extraction•Correlation matrix w/ 1s in the diag•Large correlation between Cost and Lift and another between Depth and Powder•Looks like two possible factorsEquations – Extraction•Reconfigure the variance of the correlation matrix into eigenvalues and eigenvectorsEquations – Extraction•L=V’RV•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 outputEquations – Extraction•Since R=VLV’'R V L LV=( )( ')R V L LV=, ' 'V L A LV A= =', where A is the loading matrixand A' is the transpose of the loading matrix.See SPSS output from matrix syntax.R AA=Equations – Extraction•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 loadingsEquations – 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 varianceEquations – Orthogonal Rotation•The unrotated loading matrix is multiplied by a transformation matrix which is based on angle of rotationcos sin, where is the angle of rotationsin cos.946 .326if = 19 then .326 .946See SPSS matrix syntax.unrotated rotatedA AL =Y - Y� �L = Y� �Y Y� �-� �Y L =� �� �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 2Equations – 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 .50Equations – 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 = .51Equations – 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 output1B 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 .981.078 .978.994 .033.977 .033patt ern A-- -= =-Equations – Oblique Rotation1.104 .584.081 .421.159 .020.856 .0341.12 -1.181.01 .88.46 .681.07 .98.59 .59R A BF ZB--= =-= =-- --Equations – Oblique Rotation•Once you have the factor scores you can calculate the correlations between the factors (phi matrix; Φ)1'1F FN� �F =� �-� �Equations – Oblique Rotation1.12 -1.181.01 .881.12 1.01 -0.46 -1.07 -0.59 1.00 .011*.46 .68-1.18 0.88 0.68 -0.98 0.59 .01 1.0041.07 .98.59 .59� �� �� �-� � � �� �F = =-� � � �-� �� � � �- -� �� �-� �Equations – Oblique Rotation•The structure matrix is the (zero-order) correlations between the variables and the factors..079 .981 .069 .982.078 .978 1.00 .01 .088 .977*.994 .033 .01 1.00 .994 .023.977 .033 .997 .043C AC= F- -� � � �� � � �- - - - -� �� � � �= =� �� � � �-� �� � � �- -� � � �Equations – Oblique Rotation•With oblique rotation the reproduced factor matrix is found be multiplying the structure matrix by the pattern matrix.'repR 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 non-negative eigenvalues•Yet another is to look at the scree plot•Number based on theory•Try multiple numbers and see what gives best interpretation.Total Variance Explained3.513 29.276 29.276 3.296 27.467 27.467 3.251 27.094 27.0943.141 26.171 55.447 2.681 22.338 49.805 1.509 12.573 39.6661.321 11.008 66.455 .843 7.023 56.828 1.495 12.455 52.121.801 6.676 73.132 .329 2.745 59.573 .894 7.452 59.573.675 5.623 78.755.645 5.375 84.131.527 4.391 88.522.471 3.921 92.443.342 2.851 95.294.232 1.936 97.231.221 1.841 99.072.111 .928 100.000Fac tor123456789101112Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Varianc e Cumulative %Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared LoadingsExtrac tion Method: Principal Axis Factoring.Eigenvalues greater than 1Scree PlotScree PlotFactor Number121110987654321Eigenvalue43210What else?•How do you know when the factor structure is good?•When it makes sense and has a simple (relatively) structure.•How do you