OPTIONS PAGENO=1 PAGESIZE=56 NOLABEL; * Kim #13, Table 3, Table 4, Figure 5. DATA FACTOR(TYPE=CORR); _TYPE_='CORR'; INPUT _TYPE_ $ 1-4 _NAME_ $ 6-10 X1 11-15 X2 17-21 X3 23-27 X4 29-33 X5 35-49; CARDS; N 200 200 200 200 200 CORR X1 1.000.... CORR X2.560 1.000... CORR X3.480.420 1.000.. CORR X4.000.000.480 1.000. CORR X5.000.000.360.480 1.000 PROC REG; MODEL X1 = X2 X3 X4 X5 / STB; PROC REG; MODEL X2 = X1 X3 X4 X5 / STB; PROC REG; MODEL X3 = X1 X2 X4 X5 / STB; PROC REG; MODEL X4 = X1 X2 X3 X5 / STB; PROC REG; MODEL X5 = X1 X2 X3 X4 / STB; PROC FACTOR METHOD=ML MIN=0 ROTATE=VARIMAX; VAR X1-X5; RUN;
Monday, September 23, 2013 2:13 PM 1 The REG Procedure Model: MODEL1 Dependent Variable: X1 Analysis of Variance Source DF Sum of Squares Mean Square F Value Pr > F Model 4 83.27793 20.81948 35.08 <.0001 Error 195 115.72207 0.59345 Corrected Total 199 199.00000 Root MSE 0.77035 R-Square 0.4185 Dependent Mean 0 Adj R-Sq 0.4066 Coeff Var. Parameter s Variable DF Parameter Standard Error t Value Pr > t Standardized Intercept 1 0 0.05447 0.00 1.0000 0 X2 1 0.38006 0.06247 6.08 <.0001 0.38006 X3 1 0.42844 0.07224 5.93 <.0001 0.42844 X4 1-0.17102 0.06860-2.49 0.0135-0.17102 X5 1-0.07215 0.06343-1.14 0.2567-0.07215
Monday, September 23, 2013 2:13 PM 2 The REG Procedure Model: MODEL1 Dependent Variable: X2 Analysis of Variance Source DF Sum of Squares Mean Square F Value Pr > F Model 4 71.19968 17.79992 27.16 <.0001 Error 195 127.80032 0.65539 Corrected Total 199 199.00000 Root MSE 0.80956 R-Square 0.3578 Dependent Mean 0 Adj R-Sq 0.3446 Coeff Var. Parameter s Variable DF Parameter Standard Error t Value Pr > t Standardized Intercept 1 0 0.05724 0.00 1.0000 0 X1 1 0.41972 0.06899 6.08 <.0001 0.41972 X3 1 0.29224 0.07978 3.66 0.0003 0.29224 X4 1-0.11665 0.07276-1.60 0.1105-0.11665 X5 1-0.04921 0.06678-0.74 0.4621-0.04921
Monday, September 23, 2013 2:13 PM 3 The REG Procedure Model: MODEL1 Dependent Variable: X3 Analysis of Variance Source DF Sum of Squares Mean Square F Value Pr > F Model 4 102.67023 25.66756 51.96 <.0001 Error 195 96.32977 0.49400 Corrected Total 199 199.00000 Root MSE 0.70285 R-Square 0.5159 Dependent Mean 0 Adj R-Sq 0.5060 Coeff Var. Parameter s Variable DF Parameter Standard Error t Value Pr > t Standardized Intercept 1 0 0.04970 0.00 1.0000 0 X1 1 0.35664 0.06014 5.93 <.0001 0.35664 X2 1 0.22028 0.06014 3.66 0.0003 0.22028 X4 1 0.39917 0.05679 7.03 <.0001 0.39917 X5 1 0.16840 0.05679 2.97 0.0034 0.16840
Monday, September 23, 2013 2:13 PM 4 The REG Procedure Model: MODEL1 Dependent Variable: X4 Analysis of Variance Source DF Sum of Squares Mean Square F Value Pr > F Model 4 76.80426 19.20106 30.64 <.0001 Error 195 122.19574 0.62664 Corrected Total 199 199.00000 Root MSE 0.79161 R-Square 0.3860 Dependent Mean 0 Adj R-Sq 0.3734 Coeff Var. Parameter s Variable DF Parameter Standard Error t Value Pr > t Standardized Intercept 1 0 0.05598 0.00 1.0000 0 X1 1-0.18059 0.07244-2.49 0.0135-0.18059 X2 1-0.11154 0.06957-1.60 0.1105-0.11154 X3 1 0.50635 0.07204 7.03 <.0001 0.50635 X5 1 0.29771 0.06182 4.82 <.0001 0.29771
Monday, September 23, 2013 2:13 PM 5 The REG Procedure Model: MODEL1 Dependent Variable: X5 Analysis of Variance Source DF Sum of Squares Mean Square F Value Pr > F Model 4 52.45659 13.11415 17.45 <.0001 Error 195 146.54341 0.75150 Corrected Total 199 199.00000 Root MSE 0.86689 R-Square 0.2636 Dependent Mean 0 Adj R-Sq 0.2485 Coeff Var. Parameter s Variable DF Parameter Standard Error t Value Pr > t Standardized Intercept 1 0 0.06130 0.00 1.0000 0 X1 1-0.09137 0.08032-1.14 0.2567-0.09137 X2 1-0.05643 0.07658-0.74 0.4621-0.05643 X3 1 0.25618 0.08640 2.97 0.0034 0.25618 X4 1 0.35703 0.07414 4.82 <.0001 0.35703
Monday, September 23, 2013 2:13 PM 6 The FACTOR Procedure Input Data Type Correlations N Set/Assumed in Data Set 200 N for Significance Tests 200
Monday, September 23, 2013 2:13 PM 7 The FACTOR Procedure Initial Factor Method: Maximum Likelihood Prior Communality s: SMC X1 X2 X3 X4 X5 0.41848207 0.35778733 0.51593083 0.38595104 0.26360097 Preliminary Eigenvalues: Total = 3.32906852 Average = 0.6658137 Eigenvalue Difference Proportion Cumulative 1 2.84255362 1.47391287 0.8539 0.8539 2 1.36864075 1.56814272 0.4111 1.2650 3 -.19950197 0.07452993-0.0599 1.2050 4 -.27403189 0.13456010-0.0823 1.1227 5 -.40859200-0.1227 1.0000 2 factors will be retained by the MINEIGEN criterion. Iteration Criterion Ridge Change Communalities 1 0.0000618 0.0000 0.2518 0.63911 0.48752 0.72086 0.63774 0.35268 2 0.0000000 0.0000 0.0073 0.64000 0.49000 0.72000 0.64000 0.36000 3 0.0000000 0.0000 0.0000 0.64000 0.49000 0.72000 0.64000 0.36000 Convergence criterion satisfied. Significance Tests Based on 200 Observations Test DF Chi-Square Pr > ChiSq H0: No common factors 10 267.9683 <.0001 HA: At least one common factor H0: 2 Factors are sufficient 1 0.0000 1.0000 HA: More factors are needed Chi-Square without Bartlett's Correction 0.0000000 Akaike's Information Criterion -2.0000000 Schwarz's Bayesian Criterion -5.2983174 Tucker and Lewis's Reliability Coefficient 1.0387644 Squared Canonical Correlations Factor1 Factor2 0.83676613 0.71623872 Eigenvalues of the Weighted Reduced Correlation Matrix: Total = 7.65026844 Average = 1.53005369 Eigenvalue Difference Proportion 1 5.12617950 2.60209056 0.6701 2 2.52408894 2.52408894 0.3299
Monday, September 23, 2013 2:13 PM 8 The FACTOR Procedure Initial Factor Method: Maximum Likelihood 2 factors will be retained by the MINEIGEN criterion. Eigenvalues of the Weighted Reduced Correlation Matrix: Total = 7.65026844 Average = 1.53005369 Cumulative 0.6701 1.0000 Eigenvalues of the Weighted Reduced Correlation Matrix: Total = 7.65026844 Average = 1.53005369 Eigenvalue Difference Proportion 3 -.00000000 0.00000000-0.0000 4 -.00000000 0.00000000-0.0000 5 -.00000000-0.0000 Eigenvalues of the Weighted Reduced Correlation Matrix: Total = 7.65026844 Average = 1.53005369 Cumulative 1.0000 1.0000 1.0000 Factor Pattern Factor1 Factor2 X1 0.60744-0.52060 X2 0.53151-0.45552 X3 0.84602 0.06513 X4 0.52060 0.60744 X5 0.39045 0.45558 Variance Explained by Each Factor Factor Weighted Unweighted Factor1 5.12617950 1.79070701 Factor2 2.52408894 1.05929299
Monday, September 23, 2013 2:13 PM 9 The FACTOR Procedure Initial Factor Method: Maximum Likelihood 2 factors will be retained by the MINEIGEN criterion. Final Communality s and Variable Weights Total Communality: Weighted = 7.650268 Unweighted = 2.850000 Variable Communality Weight X1 0.64000000 2.77777778 X2 0.49000000 1.96078431 X3 0.72000000 3.57142857 X4 0.64000000 2.77777778 X5 0.36000000 1.56250000
Monday, September 23, 2013 2:13 PM 10 The FACTOR Procedure Rotation Method: Varimax Orthogonal Transformation Matrix 1 2 1 0.75930 0.65074 2-0.65074 0.75930 Rotated Factor Pattern Factor1 Factor2 X1 0.80000 0.00000 X2 0.70000 0.00000 X3 0.60000 0.60000 X4 0.00000 0.80000 X5-0.00000 0.60000 Variance Explained by Each Factor Factor Weighted Unweighted Factor1 4.02427638 1.49000000 Factor2 3.62599206 1.36000000 Final Communality s and Variable Weights Total Communality: Weighted = 7.650268 Unweighted = 2.850000 Variable Communality Weight X1 0.64000000 2.77777778 X2 0.49000000 1.96078431 X3 0.72000000 3.57142857 X4 0.64000000 2.77777778 X5 0.36000000 1.56250000
* * CALCULATE WEIGHTED INPUT MATRIX FOR MAXIMUM LIKELIHOOD SOLUTION * TO DATA IN KIM #13, TABLE 4. * * USE CALL EIGEN PROCEDURE TO PRINT EIGENVALUES FOR COMPARISION OF RESULTS. TITLE1 'Results'; OPTIONS NOCENTER; PROC IML; START MAIN; * * DEFINE CORRELATION MATRIX. R = {1.000 0.560 0.480 0.000 0.000, 0.560 1.000 0.420 0.000 0.000, 0.480 0.420 1.000 0.480 0.360, 0.000 0.000 0.480 1.000 0.480, 0.000 0.000 0.360 0.480 1.000}; * * DEFINE COMMUNALITIES H2 = {0.64 0.00 0.00 0.00 0.00, 0.00 0.49 0.00 0.00 0.00, 0.00 0.00 0.72 0.00 0.00, 0.00 0.00 0.00 0.64 0.00, 0.00 0.00 0.00 0.00 0.36}; ID5 = I(5); U2 = ID5-H2; U = SQRT(U2); IU = INV(U); RR = IU*(R-U2)*IU; M = EIGVAL(RR); CALL EIGEN(M,E,RR); PRINT, RR; PRINT, M; FINISH MAIN; RUN; Results RR 1.7777778 1.3069281 1.5118579 0 0 1.3069281 0.9607843 1.1114379 0 0 1.5118579 1.1114379 2.5714286 1.5118579 0.8504201 0 0 1.5118579 1.7777778 1 0 0 0.8504201 1 0.5625 M (Eigenvalues) 5.1261795 2.5240889 2.704E-16 1.176E-16-2.36E-16