STAT Factor Analysis in SAS

Transcription

1 STAT 5600 Factor Analysis in SAS The data for this example come from the decathlon results in the 1988 Olympics. The decathlon is a two-day competition, with the 100 m race, long jump, shot put, high jump, and 400 m on day 1, and 110 m hurdles, discus, pole vault, javelin, and 1500 m on day 2. Participants and their scores are shown in the table below: Athlete 100 m Long Jump Shot High Jump 400 m 110 m Hurdles Discus Pole Vault Javelin 1500 m TOTAL Schenk Voss Steen Thompson Blondel Plaziat Bright De wti Johnson Tarnovetsky Keskitalo Gaehwiler Szabo Smith Shirley Poelman Olander Freimuth Warming Hraban Werthner Gugler Penalver Kruger Lee Fu-An Mellado moser Valenta O'Connell Richards Gong Miller Kwang-Ik Kunwar

2 With the SAS commands below, we want to carry out some crude exploratory analyses, and then apply a factor analysis to these data along with an orthogonal and oblique rotation. Some fundamental questions we would like to answer: Are there any outliers in the dataset? What patterns do we observe in examining correlations between the events (variables)? Based on a factor analysis, how many latent factors appear to reasonably represent the 10 variables? How do the results compare between different factor rotations? Applying a factor rotation, can we identify any substantive patterns among the determined factors? SAS CODE: options ls=79 nodate; data decathlon; infile 'c:\decathlon88.txt' expandtabs; input run100 Ljump shot Hjump run400 hurdle discus polevlt javelin run1500 score; ***** EXPLORATORY ANALYSIS -- POTENTIAL OUTLIERS?; proc univariate data=decathlon plots; var score; run; ***** TRANSFORM RACING SCORES SO THAT DIRECTION IS CONSISTENT WITH OTHER EVENTS; data decathlon; set decathlon; if score > 6000; run100=run100*-1; run400=run400*-1; hurdle=hurdle*-1; run1500=run1500*-1; run; ***** COMPUTE CORRELATION MATRIX; proc corr data=decathlon; var run100--run1500; run; title "FACTOR ANALYSIS WITH NO ROTATION"; proc factor data=decathlon method=ml mineigen=1 rotate=none; var run100--run1500; run; title "FACTOR ANALYSIS WITH OBLIQUE ROTATION"; proc factor data=decathlon method=ml mineigen=1 rotate=obvarimax; var run100--run1500; run; title "FACTOR ANALYSES WITH ORTHOGONAL ROTATIONS"; proc factor data=decathlon method=ml mineigen=1 rotate=orthomax; var run100--run1500; run; proc factor data=decathlon method=ml mineigen=1 rotate=factorparsimax; var run100--run1500; run;

3 SAS OUTPUT: The UNIVARIATE Procedure Variable: score Moments N 34 Sum Weights 34 Mean Sum Observations Std Deviation Variance Skewness Kurtosis Uncorrected SS Corrected SS Coeff Variation Std Error Mean Basic Statistical Measures Location Variability Mean Std Deviation Median Variance Mode. Range 3149 Interquartile Range Tests for Location: Mu0=0 Test -Statistic p Value Student's t t Pr > t <.0001 Sign M 17 Pr >= M <.0001 Signed Rank S Pr >= S <.0001 Quantiles (Definition 5) Quantile Estimate 100% Max % % % % Q % Median % Q % % % % Min

4 The UNIVARIATE Procedure Variable: score Extreme Observations ----Lowest Highest--- Value Obs Value Obs Stem Leaf # Boxplot *-----* * Multiply Stem.Leaf by 10**+2 Normal Probability Plot * +**** * * ******** **+++ ****++ ****++ ***+++ * * *

5 The CORR Procedure 10 Variables: run100 Ljump shot Hjump run400 hurdle discus polevlt javelin run1500 Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum run Ljump shot Hjump run hurdle discus polevlt javelin run Pearson Correlation Coefficients, N = 33 Prob > r under H0: Rho=0 run100 Ljump shot Hjump run400 run Ljump shot hurdle discus polevlt javelin run1500 run < Ljump shot < run100 Ljump shot Hjump run400 Hjump run hurdle < discus < polevlt javelin run

6 hurdle discus polevlt javelin run1500 Hjump run hurdle discus polevlt javelin hurdle discus polevlt javelin run1500 run

7 FACTOR ANALYSIS WITH NO ROTATION Initial Factor Method: Maximum Likelihood Prior Communality Estimates: SMC run100 Ljump shot Hjump run hurdle discus polevlt javelin run Preliminary Eigenvalues: Total = Average = Eigenvalue Difference Proportion Cumulative factors will be retained by the MINEIGEN criterion. Iteration Criterion Ridge Change Communalities Convergence criterion satisfied. Significance Tests Based on 33 Observations Pr > Test DF Chi-Square ChiSq H0: No common factors <.0001 HA: At least one common factor H0: 2 Factors are sufficient HA: More factors are needed

8 Chi-Square without Bartlett's Correction Akaike's Information Criterion Schwarz's Bayesian Criterion Tucker and Lewis's Reliability Coefficient Squared Canonical Correlations Eigenvalues of the Weighted Reduced Correlation Matrix: Total = Average = Eigenvalue Difference Proportion Cumulative Initial Factor Method: Maximum Likelihood Factor Pattern run Ljump shot Hjump run hurdle discus polevlt javelin run Variance Explained by Each Factor Factor Weighted Unweighted Final Communality Estimates and Variable Weights Total Communality: Weighted = Unweighted = Variable Communality Weight run Ljump shot Hjump run hurdle discus polevlt javelin run

9 FACTOR ANALYSIS WITH OBLIQUE ROTATION Initial Factor Method: Maximum Likelihood Prior Communality Estimates: SMC run100 Ljump shot Hjump run hurdle discus polevlt javelin run Preliminary Eigenvalues: Total = Average = Eigenvalue Difference Proportion Cumulative factors will be retained by the MINEIGEN criterion. Pr > Test DF Chi-Square ChiSq H0: No common factors <.0001 HA: At least one common factor H0: 2 Factors are sufficient HA: More factors are needed Chi-Square without Bartlett's Correction Akaike's Information Criterion Schwarz's Bayesian Criterion Tucker and Lewis's Reliability Coefficient Squared Canonical Correlations Eigenvalues of the Weighted Reduced Correlation Matrix: Total = Average = Eigenvalue Difference Proportion Cumulative

10 Initial Factor Method: Maximum Likelihood Factor Pattern run Ljump shot Hjump run hurdle discus polevlt javelin run Variance Explained by Each Factor Factor Weighted Unweighted Final Communality Estimates and Variable Weights Total Communality: Weighted = Unweighted = Variable Communality Weight run Ljump shot Hjump run hurdle discus polevlt javelin run Rotation Method: Oblique Varimax Oblique Transformation Matrix Inter-Factor Correlations Rotated Factor Pattern (Standardized Regression Coefficients) run Ljump shot Hjump run hurdle discus polevlt javelin run

11 Reference Axis Correlations Reference Structure (Semipartial Correlations) run Ljump shot Hjump run hurdle discus polevlt javelin run Variance Explained by Each Factor Eliminating Other Factors Factor Weighted Unweighted Factor Structure (Correlations) run Ljump shot Hjump run hurdle discus polevlt javelin run Variance Explained by Each Factor Ignoring Other Factors Factor Weighted Unweighted Final Communality Estimates and Variable Weights Total Communality: Weighted = Unweighted = Variable Communality Weight run Ljump shot Hjump run hurdle discus polevlt javelin run

12 FACTOR ANALYSES WITH ORTHOGONAL ROTATIONS Initial Factor Method: Maximum Likelihood Prior Communality Estimates: SMC run100 Ljump shot Hjump run hurdle discus polevlt javelin run Preliminary Eigenvalues: Total = Average = Eigenvalue Difference Proportion Cumulative factors will be retained by the MINEIGEN criterion. Pr > Test DF Chi-Square ChiSq H0: No common factors <.0001 HA: At least one common factor H0: 2 Factors are sufficient HA: More factors are needed Chi-Square without Bartlett's Correction Akaike's Information Criterion Schwarz's Bayesian Criterion Tucker and Lewis's Reliability Coefficient Squared Canonical Correlations Eigenvalues of the Weighted Reduced Correlation Matrix: Total = Average = Eigenvalue Difference Proportion Cumulative

13 Initial Factor Method: Maximum Likelihood Factor Pattern run Ljump shot Hjump run hurdle discus polevlt javelin run Variance Explained by Each Factor Factor Weighted Unweighted Final Communality Estimates and Variable Weights Total Communality: Weighted = Unweighted = Variable Communality Weight run Ljump shot Hjump run hurdle discus polevlt javelin run Orthogonal Transformation Matrix Rotated Factor Pattern run Ljump shot Hjump run hurdle discus polevlt javelin run Variance Explained by Each Factor Factor Weighted Unweighted

14 Final Communality Estimates and Variable Weights Total Communality: Weighted = Unweighted = Variable Communality Weight run Ljump shot Hjump run hurdle discus polevlt javelin run

15 FACTOR ANALYSES WITH ORTHOGONAL ROTATIONS Initial Factor Method: Maximum Likelihood Prior Communality Estimates: SMC run100 Ljump shot Hjump run hurdle discus polevlt javelin run Preliminary Eigenvalues: Total = Average = Eigenvalue Difference Proportion Cumulative factors will be retained by the MINEIGEN criterion. Convergence criterion satisfied. Significance Tests Based on 33 Observations Pr > Test DF Chi-Square ChiSq H0: No common factors <.0001 HA: At least one common factor H0: 2 Factors are sufficient HA: More factors are needed Chi-Square without Bartlett's Correction Akaike's Information Criterion Schwarz's Bayesian Criterion Tucker and Lewis's Reliability Coefficient Squared Canonical Correlations Eigenvalues of the Weighted Reduced Correlation Matrix: Total = Average = Eigenvalue Difference Proportion Cumulative

16 Initial Factor Method: Maximum Likelihood Factor Pattern run Ljump shot Hjump run hurdle discus polevlt javelin run Variance Explained by Each Factor Factor Weighted Unweighted Final Communality Estimates and Variable Weights Total Communality: Weighted = Unweighted = Variable Communality Weight run Ljump shot Hjump run hurdle discus polevlt javelin run Rotation Method: Factor Parsimax Orthogonal Transformation Matrix Rotated Factor Pattern run Ljump shot Hjump run hurdle discus polevlt javelin run

17 Variance Explained by Each Factor Factor Weighted Unweighted Final Communality Estimates and Variable Weights Total Communality: Weighted = Unweighted = Variable Communality Weight run Ljump shot Hjump run hurdle discus polevlt javelin run