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 11.25 7.43 15.48 2.27 48.9 15.13 49.28 4.7 61.32 268.95 8488 Voss 10.87 7.45 14.97 1.97 47.71 14.46 44.36 5.1 61.76 273.02 8399 Steen 11.18 7.44 14.2 1.97 48.29 14.81 43.66 5.2 64.16 263.2 8328 Thompson 10.62 7.38 15.02 2.03 49.06 14.72 44.8 4.9 64.04 285.11 8306 Blondel 11.02 7.43 12.92 1.97 47.44 14.4 41.2 5.2 57.46 256.64 8286 Plaziat 10.83 7.72 13.58 2.12 48.34 14.18 43.06 4.9 52.18 274.07 8272 Bright 11.18 7.05 14.12 2.06 49.34 14.39 41.68 5.7 61.6 291.2 8216 De wti 11.05 6.95 15.34 2.00 48.21 14.36 41.32 4.8 63 265.86 8189 Johnson 11.15 7.12 14.52 2.03 49.15 14.66 42.36 4.9 66.46 269.62 8180 Tarnovetsky 11.23 7.28 15.25 1.97 48.6 14.76 48.02 5.2 59.48 292.24 8167 Keskitalo 10.94 7.45 15.34 1.97 49.94 14.25 41.86 4.8 66.64 295.89 8143 Gaehwiler 11.18 7.34 14.48 1.94 49.02 15.11 42.76 4.7 65.84 256.74 8114 Szabo 11.02 7.29 12.92 2.06 48.23 14.94 39.54 5 56.8 257.85 8093 Smith 10.99 7.37 13.61 1.97 47.83 14.7 43.88 4.3 66.54 268.97 8083 Shirley 11.03 7.45 14.2 1.97 48.94 15.44 41.66 4.7 64 267.48 8036 Poelman 11.09 7.08 14.51 2.03 49.89 14.78 43.2 4.9 57.18 268.54 8021 Olander 11.46 6.75 16.07 2.00 51.28 16.06 50.66 4.8 72.6 302.42 7869 Freimuth 11.57 7 16.6 1.94 49.84 15 46.66 4.9 60.2 286.04 7860 Warming 11.07 7.04 13.41 1.94 47.97 14.96 40.38 4.5 51.5 262.41 7859 Hraban 10.89 7.07 15.84 1.79 49.68 15.38 45.32 4.9 60.48 277.84 7781 Werthner 11.52 7.36 13.93 1.94 49.99 15.64 38.82 4.6 67.04 266.42 7753 Gugler 11.49 7.02 13.8 2.03 50.6 15.22 39.08 4.7 60.92 262.93 7745 Penalver 11.38 7.08 14.31 2.00 50.24 14.97 46.34 4.4 55.68 272.68 7743 Kruger 11.3 6.97 13.23 2.15 49.98 15.38 38.72 4.6 54.34 277.84 7623 Lee Fu-An 11 7.23 13.15 2.03 49.73 14.96 38.06 4.5 52.82 285.57 7579 Mellado 11.33 6.83 11.63 2.06 48.37 15.39 37.52 4.6 55.42 270.07 7517 moser 11.1 6.98 12.69 1.82 48.63 15.13 38.04 4.7 49.52 261.9 7505 Valenta 11.51 7.01 14.17 1.94 51.16 15.18 45.84 4.6 56.28 303.17 7422 O'Connell 11.26 6.9 12.41 1.88 48.24 15.61 38.02 4.4 52.68 272.06 7310 Richards 11.5 7.09 12.94 1.82 49.27 15.56 42.32 4.5 53.5 293.85 7237 Gong 11.43 6.22 13.98 1.91 51.25 15.88 46.18 4.6 57.84 294.99 7231 Miller 11.47 6.43 12.33 1.94 50.3 15 38.72 4 57.26 293.72 7016 Kwang-Ik 11.57 7.19 10.27 1.91 50.71 16.2 34.36 4.1 54.94 269.98 6907 Kunwar 12.12 5.83 9.71 1.7 52.32 17.05 27.1 2.6 39.1 281.24 5339
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;
SAS OUTPUT: The UNIVARIATE Procedure Variable: score Moments N 34 Sum Weights 34 Mean 7782.85294 Sum Observations 264617 Std Deviation 594.582723 Variance 353528.614 Skewness -2.2488675 Kurtosis 7.67309194 Uncorrected SS 2071141641 Corrected SS 11666444.3 Coeff Variation 7.63964997 Std Error Mean 101.970096 Basic Statistical Measures Location Variability Mean 7782.853 Std Deviation 594.58272 Median 7864.500 Variance 353529 Mode. Range 3149 Interquartile Range 663.00000 Tests for Location: Mu0=0 Test -Statistic- -----p Value------ Student's t t 76.32486 Pr > t <.0001 Sign M 17 Pr >= M <.0001 Signed Rank S 297.5 Pr >= S <.0001 Quantiles (Definition 5) Quantile Estimate 100% Max 8488.0 99% 8488.0 95% 8399.0 90% 8306.0 75% Q3 8180.0 50% Median 7864.5 25% Q1 7517.0 10% 7231.0 5% 6907.0 1% 5339.0 0% Min 5339.0
The UNIVARIATE Procedure Variable: score Extreme Observations ----Lowest---- ----Highest--- Value Obs Value Obs 5339 34 8286 5 6907 33 8306 4 7016 32 8328 3 7231 31 8399 2 7237 30 8488 1 Stem Leaf # Boxplot 84 09 2 82 27913 5 80 248914789 9 +-----+ 78 667 3 *-----* 76 24458 5 + 74 2028 4 +-----+ 72 341 3 70 2 1 68 1 1 66 64 62 60 58 56 54 52 4 1 * ----+----+----+----+ Multiply Stem.Leaf by 10**+2 Normal Probability Plot 8500+ +++ * +**** * * ******** **+++ ****++ ****++ ***+++ *++++ 6900+ *+++ +++ ++++ ++ 5300+ * +----+----+----+----+----+----+----+----+----+----+ -2-1 0 +1 +2
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 run100 33-11.19636 0.24332-369.48000-11.57000-10.62000 Ljump 33 7.13333 0.30434 235.40000 6.22000 7.72000 shot 33 13.97636 1.33199 461.22000 10.27000 16.60000 Hjump 33 1.98273 0.09398 65.43000 1.79000 2.27000 run400 33-49.27667 1.06966-1626 -51.28000-47.44000 hurdle 33-15.04879 0.50677-496.61000-16.20000-14.18000 discus 33 42.35394 3.71913 1398 34.36000 50.66000 polevlt 33 4.73939 0.33442 156.40000 4.00000 5.70000 javelin 33 59.43879 5.49600 1961 49.52000 72.60000 run1500 33-276.03848 13.65710-9109 -303.17000-256.64000 Pearson Correlation Coefficients, N = 33 Prob > r under H0: Rho=0 run100 Ljump shot Hjump run400 run100 1.00000 0.53957 0.20797 0.14591 0.60590 0.0012 0.2455 0.4178 0.0002 Ljump 0.53957 1.00000 0.14190 0.27314 0.51533 0.0012 0.4309 0.1241 0.0021 shot 0.20797 0.14190 1.00000 0.12208-0.09458 0.2455 0.4309 0.4986 0.6006 hurdle discus polevlt javelin run1500 run100 0.63836 0.04722 0.38914 0.06471 0.26103 <.0001 0.7941 0.0252 0.7205 0.1423 Ljump 0.47800 0.04192 0.34993 0.18167 0.39559 0.0049 0.8168 0.0459 0.3116 0.0227 shot 0.29572 0.80635 0.47998 0.59767-0.26883 0.0947 <.0001 0.0047 0.0002 0.1303 run100 Ljump shot Hjump run400 Hjump 0.14591 0.27314 0.12208 1.00000 0.08750 0.4178 0.1241 0.4986 0.6282 run400 0.60590 0.51533-0.09458 0.08750 1.00000 0.0002 0.0021 0.6006 0.6282 hurdle 0.63836 0.47800 0.29572 0.30674 0.54603 <.0001 0.0049 0.0947 0.0825 0.0010 discus 0.04722 0.04192 0.80635 0.14742-0.14219 0.7941 0.8168 <.0001 0.4130 0.4299 polevlt 0.38914 0.34993 0.47998 0.21323 0.31866 0.0252 0.0459 0.0047 0.2335 0.0707 javelin 0.06471 0.18167 0.59767 0.11594-0.12035 0.7205 0.3116 0.0002 0.5206 0.5047 run1500 0.26103 0.39559-0.26883 0.11409 0.58728 0.1423 0.0227 0.1303 0.5273 0.0003
hurdle discus polevlt javelin run1500 Hjump 0.30674 0.14742 0.21323 0.11594 0.11409 0.0825 0.4130 0.2335 0.5206 0.5273 run400 0.54603-0.14219 0.31866-0.12035 0.58728 0.0010 0.4299 0.0707 0.5047 0.0003 hurdle 1.00000 0.11050 0.52155 0.06282 0.14330 0.5404 0.0019 0.7284 0.4263 discus 0.11050 1.00000 0.34397 0.44291-0.40232 0.5404 0.0500 0.0098 0.0203 polevlt 0.52155 0.34397 1.00000 0.27424 0.03150 0.0019 0.0500 0.1225 0.8619 javelin 0.06282 0.44291 0.27424 1.00000-0.09638 0.7284 0.0098 0.1225 0.5937 hurdle discus polevlt javelin run1500 run1500 0.14330-0.40232 0.03150-0.09638 1.00000 0.4263 0.0203 0.8619 0.5937
FACTOR ANALYSIS WITH NO ROTATION Initial Factor Method: Maximum Likelihood Prior Communality Estimates: SMC run100 Ljump shot Hjump run400 0.55457740 0.45256490 0.80445733 0.21341936 0.68782612 hurdle discus polevlt javelin run1500 0.62561769 0.73943287 0.42243571 0.41879238 0.55199588 Preliminary Eigenvalues: Total = 15.8533372 Average = 1.58533372 Eigenvalue Difference Proportion Cumulative 1 8.89838207 2.21951018 0.5613 0.5613 2 6.67887188 5.79654533 0.4213 0.9826 3 0.88232656 0.43776469 0.0557 1.0382 4 0.44456186 0.19762584 0.0280 1.0663 5 0.24693603 0.18365534 0.0156 1.0819 6 0.06328069 0.18838261 0.0040 1.0859 7 -.12510193 0.14994679-0.0079 1.0780 8 -.27504872 0.04051625-0.0173 1.0606 9 -.31556497 0.32974126-0.0199 1.0407 10 -.64530624-0.0407 1.0000 2 factors will be retained by the MINEIGEN criterion. Iteration Criterion Ridge Change Communalities 1 0.7285725 0.0000 0.1488 0.59384 0.46599 0.94737 0.08387 0.72801 0.56093 0.68895 0.44242 0.32332 0.40324 2 0.7221374 0.0000 0.0335 0.59884 0.45475 0.96928 0.06850 0.71726 0.57614 0.68533 0.42789 0.35677 0.38623 3 0.7216597 0.0000 0.0120 0.60212 0.45394 0.97536 0.06753 0.71508 0.57833 0.68328 0.42613 0.36047 0.37424 4 0.7216141 0.0000 0.0031 0.60328 0.45358 0.97709 0.06716 0.71355 0.57973 0.68252 0.42554 0.36149 0.37114 5 0.7216093 0.0000 0.0012 0.60368 0.45352 0.97750 0.06713 0.71308 0.58021 0.68238 0.42541 0.36180 0.36991 6 0.7216088 0.0000 0.0004 0.60382 0.45350 0.97766 0.06714 0.71288 0.58038 0.68231 0.42539 0.36186 0.36956 Convergence criterion satisfied. Significance Tests Based on 33 Observations Pr > Test DF Chi-Square ChiSq H0: No common factors 45 137.1386 <.0001 HA: At least one common factor H0: 2 Factors are sufficient 26 19.1226 0.8312 HA: More factors are needed
Chi-Square without Bartlett's Correction 23.091482 Akaike's Information Criterion -28.908518 Schwarz's Bayesian Criterion -67.817715 Tucker and Lewis's Reliability Coefficient 1.129187 Squared Canonical Correlations 0.97932390 0.87049551 Eigenvalues of the Weighted Reduced Correlation Matrix: Total = 54.0867551 Average = 5.40867551 Eigenvalue Difference Proportion Cumulative 1 47.3650155 40.6432756 0.8757 0.8757 2 6.7217399 6.0853763 0.1243 1.0000 3 0.6363636 0.2644105 0.0118 1.0118 4 0.3719530 0.1504848 0.0069 1.0186 5 0.2214682 0.1425255 0.0041 1.0227 6 0.0789427 0.1557214 0.0015 1.0242 7-0.0767787 0.1917958-0.0014 1.0228 8-0.2685744 0.0537115-0.0050 1.0178 9-0.3222859 0.3188028-0.0060 1.0119 10-0.6410887-0.0119 1.0000 Initial Factor Method: Maximum Likelihood Factor Pattern run100 0.21972 0.74536 Ljump 0.15527 0.65528 shot 0.98867-0.01359 Hjump 0.13490 0.22123 run400-0.08346 0.84018 hurdle 0.30854 0.69658 discus 0.81265-0.14808 polevlt 0.49326 0.42671 javelin 0.59918-0.05364 run1500-0.27318 0.54297 Variance Explained by Each Factor Factor Weighted Unweighted 47.3650155 2.50756391 6.7217399 2.72691426 Final Communality Estimates and Variable Weights Total Communality: Weighted = 54.086755 Unweighted = 5.234478 Variable Communality Weight run100 0.60384600 2.5241161 Ljump 0.45349671 1.8298317 shot 0.97765555 44.7534978 Hjump 0.06714079 1.0719691 run400 0.71286418 3.4829107 hurdle 0.58041971 2.3831226 discus 0.68232597 3.1477626 polevlt 0.42538373 1.7402965 javelin 0.36189692 1.5670528 run1500 0.36944861 1.5861952
FACTOR ANALYSIS WITH OBLIQUE ROTATION Initial Factor Method: Maximum Likelihood Prior Communality Estimates: SMC run100 Ljump shot Hjump run400 0.55457740 0.45256490 0.80445733 0.21341936 0.68782612 hurdle discus polevlt javelin run1500 0.62561769 0.73943287 0.42243571 0.41879238 0.55199588 Preliminary Eigenvalues: Total = 15.8533372 Average = 1.58533372 Eigenvalue Difference Proportion Cumulative 1 8.89838207 2.21951018 0.5613 0.5613 2 6.67887188 5.79654533 0.4213 0.9826 3 0.88232656 0.43776469 0.0557 1.0382 4 0.44456186 0.19762584 0.0280 1.0663 5 0.24693603 0.18365534 0.0156 1.0819 6 0.06328069 0.18838261 0.0040 1.0859 7 -.12510193 0.14994679-0.0079 1.0780 8 -.27504872 0.04051625-0.0173 1.0606 9 -.31556497 0.32974126-0.0199 1.0407 10 -.64530624-0.0407 1.0000 2 factors will be retained by the MINEIGEN criterion. Pr > Test DF Chi-Square ChiSq H0: No common factors 45 137.1386 <.0001 HA: At least one common factor H0: 2 Factors are sufficient 26 19.1226 0.8312 HA: More factors are needed Chi-Square without Bartlett's Correction 23.091482 Akaike's Information Criterion -28.908518 Schwarz's Bayesian Criterion -67.817715 Tucker and Lewis's Reliability Coefficient 1.129187 Squared Canonical Correlations 0.97932390 0.87049551 Eigenvalues of the Weighted Reduced Correlation Matrix: Total = 54.0867551 Average = 5.40867551 Eigenvalue Difference Proportion Cumulative 1 47.3650155 40.6432756 0.8757 0.8757 2 6.7217399 6.0853763 0.1243 1.0000 3 0.6363636 0.2644105 0.0118 1.0118 4 0.3719530 0.1504848 0.0069 1.0186 5 0.2214682 0.1425255 0.0041 1.0227 6 0.0789427 0.1557214 0.0015 1.0242 7-0.0767787 0.1917958-0.0014 1.0228 8-0.2685744 0.0537115-0.0050 1.0178 9-0.3222859 0.3188028-0.0060 1.0119 10-0.6410887-0.0119 1.0000
Initial Factor Method: Maximum Likelihood Factor Pattern run100 0.21972 0.74536 Ljump 0.15527 0.65528 shot 0.98867-0.01359 Hjump 0.13490 0.22123 run400-0.08346 0.84018 hurdle 0.30854 0.69658 discus 0.81265-0.14808 polevlt 0.49326 0.42671 javelin 0.59918-0.05364 run1500-0.27318 0.54297 Variance Explained by Each Factor Factor Weighted Unweighted 47.3650155 2.50756391 6.7217399 2.72691426 Final Communality Estimates and Variable Weights Total Communality: Weighted = 54.086755 Unweighted = 5.234478 Variable Communality Weight run100 0.60384600 2.5241161 Ljump 0.45349671 1.8298317 shot 0.97765555 44.7534978 Hjump 0.06714079 1.0719691 run400 0.71286418 3.4829107 hurdle 0.58041971 2.3831226 discus 0.68232597 3.1477626 polevlt 0.42538373 1.7402965 javelin 0.36189692 1.5670528 run1500 0.36944861 1.5861952 Rotation Method: Oblique Varimax Oblique Transformation Matrix 1 2 1 0.08424332 0.98862745 2 1.0008292-0.1771208 Inter-Factor Correlations 1.00000 0.09317 0.09317 1.00000 Rotated Factor Pattern (Standardized Regression Coefficients) run100 0.76449 0.08520 Ljump 0.66890 0.03744 shot 0.06969 0.97983 Hjump 0.23278 0.09418 run400 0.83384-0.23133 hurdle 0.72315 0.18165 discus -0.07975 0.82964 polevlt 0.46862 0.41207 javelin -0.00321 0.60187 run1500 0.52041-0.36625
Reference Axis Correlations 1.00000-0.09317-0.09317 1.00000 Reference Structure (Semipartial Correlations) run100 0.76117 0.08483 Ljump 0.66599 0.03728 shot 0.06938 0.97557 Hjump 0.23177 0.09377 run400 0.83022-0.23032 hurdle 0.72000 0.18086 discus -0.07940 0.82603 polevlt 0.46658 0.41028 javelin -0.00320 0.59925 run1500 0.51815-0.36465 Variance Explained by Each Factor Eliminating Other Factors Factor Weighted Unweighted 7.0076789 2.78160269 46.1010308 2.39760046 Factor Structure (Correlations) run100 0.77243 0.15643 Ljump 0.67239 0.09976 shot 0.16097 0.98633 Hjump 0.24155 0.11586 run400 0.81229-0.15364 hurdle 0.74007 0.24902 discus -0.00245 0.82221 polevlt 0.50701 0.45573 javelin 0.05286 0.60157 run1500 0.48629-0.31776 Variance Explained by Each Factor Ignoring Other Factors Factor Weighted Unweighted 7.9857246 2.83687771 47.0790765 2.45287548 Final Communality Estimates and Variable Weights Total Communality: Weighted = 54.086755 Unweighted = 5.234478 Variable Communality Weight run100 0.60384600 2.5241161 Ljump 0.45349671 1.8298317 shot 0.97765555 44.7534978 Hjump 0.06714079 1.0719691 run400 0.71286418 3.4829107 hurdle 0.58041971 2.3831226 discus 0.68232597 3.1477626 polevlt 0.42538373 1.7402965 javelin 0.36189692 1.5670528 run1500 0.36944861 1.5861952
FACTOR ANALYSES WITH ORTHOGONAL ROTATIONS Initial Factor Method: Maximum Likelihood Prior Communality Estimates: SMC run100 Ljump shot Hjump run400 0.55457740 0.45256490 0.80445733 0.21341936 0.68782612 hurdle discus polevlt javelin run1500 0.62561769 0.73943287 0.42243571 0.41879238 0.55199588 Preliminary Eigenvalues: Total = 15.8533372 Average = 1.58533372 Eigenvalue Difference Proportion Cumulative 1 8.89838207 2.21951018 0.5613 0.5613 2 6.67887188 5.79654533 0.4213 0.9826 3 0.88232656 0.43776469 0.0557 1.0382 4 0.44456186 0.19762584 0.0280 1.0663 5 0.24693603 0.18365534 0.0156 1.0819 6 0.06328069 0.18838261 0.0040 1.0859 7 -.12510193 0.14994679-0.0079 1.0780 8 -.27504872 0.04051625-0.0173 1.0606 9 -.31556497 0.32974126-0.0199 1.0407 10 -.64530624-0.0407 1.0000 2 factors will be retained by the MINEIGEN criterion. Pr > Test DF Chi-Square ChiSq H0: No common factors 45 137.1386 <.0001 HA: At least one common factor H0: 2 Factors are sufficient 26 19.1226 0.8312 HA: More factors are needed Chi-Square without Bartlett's Correction 23.091482 Akaike's Information Criterion -28.908518 Schwarz's Bayesian Criterion -67.817715 Tucker and Lewis's Reliability Coefficient 1.129187 Squared Canonical Correlations 0.97932390 0.87049551 Eigenvalues of the Weighted Reduced Correlation Matrix: Total = 54.0867551 Average = 5.40867551 Eigenvalue Difference Proportion Cumulative 1 47.3650155 40.6432756 0.8757 0.8757 2 6.7217399 6.0853763 0.1243 1.0000 3 0.6363636 0.2644105 0.0118 1.0118 4 0.3719530 0.1504848 0.0069 1.0186 5 0.2214682 0.1425255 0.0041 1.0227 6 0.0789427 0.1557214 0.0015 1.0242 7-0.0767787 0.1917958-0.0014 1.0228 8-0.2685744 0.0537115-0.0050 1.0178 9-0.3222859 0.3188028-0.0060 1.0119 10-0.6410887-0.0119 1.0000
Initial Factor Method: Maximum Likelihood Factor Pattern run100 0.21972 0.74536 Ljump 0.15527 0.65528 shot 0.98867-0.01359 Hjump 0.13490 0.22123 run400-0.08346 0.84018 hurdle 0.30854 0.69658 discus 0.81265-0.14808 polevlt 0.49326 0.42671 javelin 0.59918-0.05364 run1500-0.27318 0.54297 Variance Explained by Each Factor Factor Weighted Unweighted 47.3650155 2.50756391 6.7217399 2.72691426 Final Communality Estimates and Variable Weights Total Communality: Weighted = 54.086755 Unweighted = 5.234478 Variable Communality Weight run100 0.60384600 2.5241161 Ljump 0.45349671 1.8298317 shot 0.97765555 44.7534978 Hjump 0.06714079 1.0719691 run400 0.71286418 3.4829107 hurdle 0.58041971 2.3831226 discus 0.68232597 3.1477626 polevlt 0.42538373 1.7402965 javelin 0.36189692 1.5670528 run1500 0.36944861 1.5861952 Orthogonal Transformation Matrix 1 2 1 0.17070 0.98532 2 0.98532-0.17070 Rotated Factor Pattern run100 0.77193 0.08927 Ljump 0.67216 0.04114 shot 0.15537 0.97648 Hjump 0.24101 0.09515 run400 0.81360-0.22565 hurdle 0.73902 0.18511 discus -0.00719 0.82600 polevlt 0.50464 0.41318 javelin 0.04942 0.59955 run1500 0.48837-0.36186 Variance Explained by Each Factor Factor Weighted Unweighted 7.9059780 2.83367841 46.1807773 2.40079975
Final Communality Estimates and Variable Weights Total Communality: Weighted = 54.086755 Unweighted = 5.234478 Variable Communality Weight run100 0.60384600 2.5241161 Ljump 0.45349671 1.8298317 shot 0.97765555 44.7534978 Hjump 0.06714079 1.0719691 run400 0.71286418 3.4829107 hurdle 0.58041971 2.3831226 discus 0.68232597 3.1477626 polevlt 0.42538373 1.7402965 javelin 0.36189692 1.5670528 run1500 0.36944861 1.5861952
FACTOR ANALYSES WITH ORTHOGONAL ROTATIONS Initial Factor Method: Maximum Likelihood Prior Communality Estimates: SMC run100 Ljump shot Hjump run400 0.55457740 0.45256490 0.80445733 0.21341936 0.68782612 hurdle discus polevlt javelin run1500 0.62561769 0.73943287 0.42243571 0.41879238 0.55199588 Preliminary Eigenvalues: Total = 15.8533372 Average = 1.58533372 Eigenvalue Difference Proportion Cumulative 1 8.89838207 2.21951018 0.5613 0.5613 2 6.67887188 5.79654533 0.4213 0.9826 3 0.88232656 0.43776469 0.0557 1.0382 4 0.44456186 0.19762584 0.0280 1.0663 5 0.24693603 0.18365534 0.0156 1.0819 6 0.06328069 0.18838261 0.0040 1.0859 7 -.12510193 0.14994679-0.0079 1.0780 8 -.27504872 0.04051625-0.0173 1.0606 9 -.31556497 0.32974126-0.0199 1.0407 10 -.64530624-0.0407 1.0000 2 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 45 137.1386 <.0001 HA: At least one common factor H0: 2 Factors are sufficient 26 19.1226 0.8312 HA: More factors are needed Chi-Square without Bartlett's Correction 23.091482 Akaike's Information Criterion -28.908518 Schwarz's Bayesian Criterion -67.817715 Tucker and Lewis's Reliability Coefficient 1.129187 Squared Canonical Correlations 0.97932390 0.87049551 Eigenvalues of the Weighted Reduced Correlation Matrix: Total = 54.0867551 Average = 5.40867551 Eigenvalue Difference Proportion Cumulative 1 47.3650155 40.6432756 0.8757 0.8757 2 6.7217399 6.0853763 0.1243 1.0000 3 0.6363636 0.2644105 0.0118 1.0118 4 0.3719530 0.1504848 0.0069 1.0186 5 0.2214682 0.1425255 0.0041 1.0227 6 0.0789427 0.1557214 0.0015 1.0242 7-0.0767787 0.1917958-0.0014 1.0228 8-0.2685744 0.0537115-0.0050 1.0178 9-0.3222859 0.3188028-0.0060 1.0119 10-0.6410887-0.0119 1.0000
Initial Factor Method: Maximum Likelihood Factor Pattern run100 0.21972 0.74536 Ljump 0.15527 0.65528 shot 0.98867-0.01359 Hjump 0.13490 0.22123 run400-0.08346 0.84018 hurdle 0.30854 0.69658 discus 0.81265-0.14808 polevlt 0.49326 0.42671 javelin 0.59918-0.05364 run1500-0.27318 0.54297 Variance Explained by Each Factor Factor Weighted Unweighted 47.3650155 2.50756391 6.7217399 2.72691426 Final Communality Estimates and Variable Weights Total Communality: Weighted = 54.086755 Unweighted = 5.234478 Variable Communality Weight run100 0.60384600 2.5241161 Ljump 0.45349671 1.8298317 shot 0.97765555 44.7534978 Hjump 0.06714079 1.0719691 run400 0.71286418 3.4829107 hurdle 0.58041971 2.3831226 discus 0.68232597 3.1477626 polevlt 0.42538373 1.7402965 javelin 0.36189692 1.5670528 run1500 0.36944861 1.5861952 Rotation Method: Factor Parsimax Orthogonal Transformation Matrix 1 2 1-0.12296 0.99241 2 0.99241 0.12296 Rotated Factor Pattern run100 0.71269 0.30971 Ljump 0.63121 0.23467 shot -0.13506 0.97950 Hjump 0.20297 0.16108 run400 0.84406 0.02048 hurdle 0.65335 0.39185 discus -0.24689 0.78827 polevlt 0.36282 0.54198 javelin -0.12691 0.58804 run1500 0.57245-0.20434
Variance Explained by Each Factor Factor Weighted Unweighted 7.3362764 2.64149774 46.7504789 2.59298042 Final Communality Estimates and Variable Weights Total Communality: Weighted = 54.086755 Unweighted = 5.234478 Variable Communality Weight run100 0.60384600 2.5241161 Ljump 0.45349671 1.8298317 shot 0.97765555 44.7534978 Hjump 0.06714079 1.0719691 run400 0.71286418 3.4829107 hurdle 0.58041971 2.3831226 discus 0.68232597 3.1477626 polevlt 0.42538373 1.7402965 javelin 0.36189692 1.5670528 run1500 0.36944861 1.5861952