Nonlinear Regression Examples

3 /**************************************************************/ 4 /*** EXST7034 Example -- Growth Models ***/ 5 /**************************************************************/ 6 OPTIONS NOCENTER PS=61 LS=78 nodate NOCENTER NONUMBER; 7 8 DATA ONE; INFILE CARDS MISSOVER; 9 TITLE1 'Various Growth Curves fitted to Vermillion Lake Cisco data'; 10 INPUT AGE N LENGTH; 11 LABEL AGE='Age in years'; 12 LABEL LENGTH='Mean length in mm'; 13 LABEL N='Sample size'; 14 CARDS; NOTE: The data set WORK.ONE has 10 observations and 3 variables. NOTE: DATA statement used: real time 0.32 seconds 14 ! RUN; 25 ; 26 27 PROC PRINT DATA=ONE; 27 ! RUN; NOTE: There were 10 observations read from the data set WORK.ONE. NOTE: The PROCEDURE PRINT printed page 1. NOTE: PROCEDURE PRINT used: real time 0.27 seconds Various Growth Curves fitted to Vermillion Lake Cisco data Obs AGE N LENGTH 1 2 101 172 2 3 14 210 3 4 136 241 4 5 52 265 5 6 67 280 6 7 81 289 7 8 54 294 8 9 20 302 9 10 6 299 10 11 2 306 28 29 PROC NLIN DATA=ONE CONVERGE=10E-12 MAXITER=200; _WEIGHT_ = N; 30 TITLE2 'Traditional von Bertlanffy growth model used in fisheries 30 ! research'; 31 PARAMETERS LINF = 300.00 K = 0.3000 T0 = 0; 32 MODEL Length = LINF * (1 - EXP(-K*(AGE-T0))); 33 DER.LINF = 1 - EXP(-K * (AGE - T0)); 34 DER.T0 = -K * LINF * EXP(-K * (AGE - T0)); 35 DER.K = LINF * EXP(-K * (AGE - T0)) * (AGE - T0); 35 ! RUN; NOTE: PROC NLIN grid search time was 0: 0: 0. NOTE: Convergence criterion met. NOTE: The PROCEDURE NLIN printed pages 2-3. NOTE: PROCEDURE NLIN used: real time 0.10 seconds Various Growth Curves fitted to Vermillion Lake Cisco data Traditional von Bertlanffy growth model used in fisheries research The NLIN Procedure Dependent Variable LENGTH Method: Gauss-Newton Iterative Phase Weighted Iter LINF K T0 SS 0 300.0 0.3000 0 484898 1 312.7 0.3394 -0.3522 2015.5 2 314.0 0.3433 -0.3003 1282.3 3 314.1 0.3431 -0.3021 1282.0 4 314.1 0.3431 -0.3021 1282.0 5 314.1 0.3431 -0.3021 1282.0 6 314.1 0.3431 -0.3021 1282.0 7 314.1 0.3431 -0.3021 1282.0 8 314.1 0.3431 -0.3021 1282.0 NOTE: Convergence criterion met. Estimation Summary Method Gauss-Newton Iterations 8 R 8.16E-12 PPC 6.37E-12 RPC 1.82E-10 Object 1.05E-14 Objective 1282.043 Observations Read 10 Observations Used 10 Observations Missing 0 Sum of Mean Approx Source DF Squares Square F Value Pr > F Regression 3 34388121 11462707 2787.62 <.0001 Residual 7 1282.0 183.1 Uncorrected Total 10 34389403 Corrected Total 9 1022381 Approx Parameter Estimate Std Error Approximate 95% Confidence Limits LINF 314.1 2.8780 307.3 320.9 K 0.3431 0.0161 0.3050 0.3811 T0 -0.3021 0.0945 -0.5256 -0.0785 Approximate Correlation Matrix LINF K T0 LINF 1.0000000 -0.9328561 -0.7932938 K -0.9328561 1.0000000 0.9426013 T0 -0.7932938 0.9426013 1.0000000 37 PROC NLIN DATA=ONE; _WEIGHT_ = N; 38 TITLE2 'vBert with Y-axis intercept'; 39 PARAMETERS LINF=300 K=.3 L0=0; E = EXP(-K*AGE); 40 MODEL Length = LINF - (LINF - L0) * E; NOTE: PROC NLIN grid search time was 0: 0: 0. NOTE: Convergence criterion met. NOTE: The PROCEDURE NLIN printed page 4. NOTE: PROCEDURE NLIN used: real time 0.00 seconds Various Growth Curves fitted to Vermillion Lake Cisco data vBert with Y-axis intercept The NLIN Procedure Dependent Variable LENGTH Method: Gauss-Newton Iterative Phase Weighted Iter LINF K L0 SS 0 300.0 0.3000 0 484898 1 312.7 0.3394 31.7012 2908.4 2 314.0 0.3432 30.8685 1282.1 3 314.1 0.3431 30.9198 1282.0 4 314.1 0.3431 30.9174 1282.0 NOTE: Convergence criterion met. Estimation Summary Method Gauss-Newton Iterations 4 R 4.031E-6 PPC(L0) 2.602E-6 RPC(L0) 0.000075 Object 1.283E-8 Objective 1282.043 Observations Read 10 Observations Used 10 Observations Missing 0 Sum of Mean Approx Source DF Squares Square F Value Pr > F Regression 3 34388121 11462707 2787.62 <.0001 Residual 7 1282.0 183.1 Uncorrected Total 10 34389403 Corrected Total 9 1022381 Approx Parameter Estimate Std Error Approximate 95% Confidence Limits LINF 314.1 2.8780 307.3 320.9 K 0.3431 0.0161 0.3050 0.3811 L0 30.9174 8.1185 11.7201 50.1147 Approximate Correlation Matrix LINF K L0 LINF 1.0000000 -0.9328560 0.7742829 K -0.9328560 1.0000000 -0.9294754 L0 0.7742829 -0.9294754 1.0000000 45 PROC NLIN DATA=ONE MAXITER=200; _WEIGHT_ = N; 46 TITLE2 'Gompertz growth model'; 47 PARAMETERS LINF=1.000 K=.3000 L0=100.000; 48 E = EXP(LINF *(1 - EXP(-K*AGE))); 49 MODEL Length = L0 * E; 52 ! RUN; NOTE: PROC NLIN grid search time was 0: 0: 0. NOTE: Convergence criterion met. NOTE: The PROCEDURE NLIN printed pages 5-6. NOTE: PROCEDURE NLIN used: real time 0.04 seconds Various Growth Curves fitted to Vermillion Lake Cisco data Gompertz growth model The NLIN Procedure Dependent Variable LENGTH Method: Gauss-Newton Iterative Phase Weighted Iter LINF K L0 SS 0 1.0000 0.3000 100.0 895786 1 1.3543 0.4768 74.0223 71793.1 2 1.3989 0.4310 76.2124 1082.1 3 1.4104 0.4398 75.2141 653.2 4 1.4104 0.4397 75.2337 652.3 5 1.4104 0.4397 75.2332 652.3 NOTE: Convergence criterion met. Estimation Summary Method Gauss-Newton Iterations 5 R 1.458E-6 PPC(L0) 1.249E-7 RPC(L0) 6.642E-6 Object 5.795E-9 Objective 652.31 Observations Read 10 Observations Used 10 Observations Missing 0 NOTE: An intercept was not specified for this model. Sum of Mean Approx Source DF Squares Square F Value Pr > F Regression 3 34388751 11462917 123010 <.0001 Residual 7 652.3 93.1871 Uncorrected Total 10 34389403 Corrected Total 9 1022381 Approx Parameter Estimate Std Error Approximate 95% Confidence Limits LINF 1.4104 0.0328 1.3328 1.4880 K 0.4397 0.0120 0.4112 0.4682 L0 75.2332 2.7359 68.7638 81.7025 Various Growth Curves fitted to Vermillion Lake Cisco data Gompertz growth model The NLIN Procedure Approximate Correlation Matrix LINF K L0 LINF 1.0000000 0.8879184 -0.9937163 K 0.8879184 1.0000000 -0.9294274 L0 -0.9937163 -0.9294274 1.0000000 53 54 PROC NLIN DATA=ONE MAXITER=200; _WEIGHT_ = N; 55 TITLE2 'Logistic Growth Curve'; 56 PARAMETERS LINF=300 K=.3000 L0=100.000; 57 R = (LINF - L0) / L0; E = EXP(-K * AGE); 58 MODEL Length = LINF / (1 + R*E); 59 DER.LINF = (1/(1+R*E)) - (LINF/(1+R*E)**2)*(E/L0); 60 DER.L0 = (-LINF/(1+R*E)**2) * ((-LINF/L0**2)*E); 61 DER.K = (-LINF/(1+R*E)**2) * R*E*(-AGE); 61 ! RUN; NOTE: PROC NLIN grid search time was 0: 0: 0. NOTE: Convergence criterion met. NOTE: The PROCEDURE NLIN printed pages 7-8. NOTE: PROCEDURE NLIN used: real time 0.00 seconds Various Growth Curves fitted to Vermillion Lake Cisco data Logistic Growth Curve The NLIN Procedure Dependent Variable LENGTH Method: Gauss-Newton Iterative Phase Weighted Iter LINF K L0 SS 0 300.0 0.3000 100.0 1184765 1 258.1 0.5194 101.7 537894 2 304.6 0.5334 91.7871 2168.5 3 304.2 0.5410 93.0269 412.5 4 304.2 0.5413 92.9954 411.7 5 304.2 0.5413 92.9954 411.7 NOTE: Convergence criterion met. Estimation Summary Method Gauss-Newton Iterations 5 R 7.841E-9 PPC 3.92E-10 RPC(L0) 1.307E-7 Object 2.84E-10 Objective 411.6518 Observations Read 10 Observations Used 10 Observations Missing 0 NOTE: An intercept was not specified for this model. Sum of Mean Approx Source DF Squares Square F Value Pr > F Regression 3 34388991 11462997 194924 <.0001 Residual 7 411.7 58.8074 Uncorrected Total 10 34389403 Corrected Total 9 1022381 Approx Parameter Estimate Std Error Approximate 95% Confidence Limits LINF 304.2 1.0754 301.6 306.7 K 0.5413 0.0102 0.5171 0.5655 L0 92.9954 1.6233 89.1570 96.8338 Approximate Correlation Matrix LINF K L0 LINF 1.0000000 -0.8329083 0.6571073 K -0.8329083 1.0000000 -0.9315397 L0 0.6571073 -0.9315397 1.0000000 63 PROC NLIN DATA=ONE MAXITER=200;_WEIGHT_ = N; 64 TITLE2 'Richards 4 parameter model'; 65 PARAMETERS LINF=300 K=0.300 L0=-1.000 P=-1.000; 66 E = EXP(-K*AGE); BASE = 1 - L0*E; 67 MODEL Length = LINF * (1 - L0*E)**P; 72 RUN; NOTE: PROC NLIN grid search time was 0: 0: 0. NOTE: Convergence criterion met. NOTE: The PROCEDURE NLIN printed pages 9-10. NOTE: PROCEDURE NLIN used: real time 0.00 seconds Various Growth Curves fitted to Vermillion Lake Cisco data Richards 4 parameter model The NLIN Procedure Dependent Variable LENGTH Method: Gauss-Newton Iterative Phase Weighted Iter LINF K L0 P SS 0 300.0 0.3000 -1.0000 -1.0000 184826 1 299.7 0.3147 -0.5927 -1.3836 163358 2 298.1 0.4184 -0.6394 -1.7821 63210.3 3 298.3 0.4276 -0.7396 -1.5920 61824.5 4 298.5 0.4365 -0.8429 -1.4490 59241.0 5 298.8 0.4452 -0.9488 -1.3385 55741.5 6 299.3 0.4620 -1.1647 -1.1645 53257.9 7 300.2 0.4863 -1.5105 -0.9922 49647.1 8 302.2 0.5359 -2.3350 -0.7872 48039.2 9 303.5 0.5582 -2.9291 -0.8149 2204.9 10 303.7 0.5566 -2.8005 -0.8694 419.3 11 303.7 0.5566 -2.7991 -0.8720 407.8 12 303.7 0.5566 -2.7987 -0.8721 407.8 NOTE: Convergence criterion met. Estimation Summary Method Gauss-Newton Iterations 12 Subiterations 13 Average Subiterations 1.083333 R 2.594E-6 PPC(L0) 4.2E-6 RPC(L0) 0.000136 Object 3.883E-8 Objective 407.7838 Observations Read 10 Observations Used 10 Observations Missing 0 NOTE: An intercept was not specified for this model. Sum of Mean Approx Source DF Squares Square F Value Pr > F Regression 4 34388995 8597249 126497 <.0001 Residual 6 407.8 67.9640 Uncorrected Total 10 34389403 Corrected Total 9 1022381 Approx Parameter Estimate Std Error Approximate 95% Confidence Limits LINF 303.7 2.2942 298.1 309.3 K 0.5566 0.0639 0.4002 0.7130 L0 -2.7987 2.3081 -8.4464 2.8491 P -0.8721 0.4592 -1.9956 0.2514 Approximate Correlation Matrix LINF K L0 P LINF 1.0000000 -0.9276161 0.8772081 -0.8702797 K -0.9276161 1.0000000 -0.9887872 0.9847621 L0 0.8772081 -0.9887872 1.0000000 -0.9995605 P -0.8702797 0.9847621 -0.9995605 1.0000000

Ciscoes data set from Ricker
						Likelihood Ratio Test
Model	df	SS	MS	R²	Ratio	*-ln(ratio)n**	df	P>Chi2
von Bertalanffy	7	1282	128.2043	99.87460222	0.318	11.45	1	0.00071
Gompertz	7	652	65.2310	99.93619695	0.625	4.70	1	0.03020
Logistic	7	412	41.1652	99.95973595	0.991	0.09	1	0.75865
Richards	6	408	40.7784	99.96011429
C Total	10	1022381

		0.05	0.01
ChiSq 1 df =		3.84146	6.63490
ChiSq 2 df =		5.99147	9.21034

1 *************************************************************; 2 *** EXST7034 Homework Example 1 ***; 3 *** Problem from Neter, Wasserman & Kuttner 1989, #11.16 ***; 4 *************************************************************; 5 OPTIONS LS=82 PS=61 NOCENTER NODATE NONUMBER; 6 7 DATA ONE; INFILE CARDS MISSOVER; 8 TITLE1 'EXST7034 - Homework Example NWK 11.16 (based on # 1.27) : Muscle mass'; 9 * LABEL X1 = 'Age (years)'; 10 * LABEL X2 = '(X1-60)*Indicator of Age'; 11 * LABEL X3 = 'Indicator of Age > 60'; 12 * LABEL Y = 'Muscle mass'; 13 INPUT Y X1; 14 X3 = 0; IF X1 GE 60 THEN X3 = 1; 15 X2 = (X1-60)*X3; 16 X1X3 = X1*X3; 17 CARDS; NOTE: The data set WORK.ONE has 16 observations and 5 variables. NOTE: DATA statement used: real time 0.11 seconds 17 ! RUN; 34 ; 35 PROC SORT DATA=ONE; BY X1; RUN; NOTE: There were 16 observations read from the data set WORK.ONE. NOTE: The data set WORK.ONE has 16 observations and 5 variables. NOTE: PROCEDURE SORT used: real time 0.04 seconds 36 PROC PRINT DATA=ONE; RUN; NOTE: There were 16 observations read from the data set WORK.ONE. NOTE: The PROCEDURE PRINT printed page 1. NOTE: PROCEDURE PRINT used: real time 0.05 seconds EXST7034 - Homework Example NWK 11.16 (based on # 1.27) : Muscle mass Obs Y X1 X3 X2 X1X3 1 100 43 0 0 0 2 116 45 0 0 0 3 97 45 0 0 0 4 105 49 0 0 0 5 100 53 0 0 0 6 87 56 0 0 0 7 80 56 0 0 0 8 76 58 0 0 0 9 91 64 1 4 64 10 84 65 1 5 65 11 68 67 1 7 67 12 78 68 1 8 68 13 82 71 1 11 71 14 73 73 1 13 73 15 65 76 1 16 76 16 77 78 1 18 78 37 proc plot data=one; plot Y*X1; run; NOTE: There were 16 observations read from the data set WORK.ONE. NOTE: The PROCEDURE PLOT printed page 2. NOTE: PROCEDURE PLOT used: real time 0.00 seconds EXST7034 - Homework Example NWK 11.16 (based on # 1.27) : Muscle mass Plot of Y*X1. Legend: A = 1 obs, B = 2 obs, etc. Y | | 120 + | | | A 115 + | | | 110 + | | | 105 + A | | | 100 + A A | | A | 95 + | | | A 90 + | | A | 85 + | A | A | 80 + A | | A A | A 75 + | | A | 70 + | | A | 65 + A | ---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+-- 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 X1 38 PROC REG; TITLE2 'Piecewise regression with SAS REG procedure'; 39 TITLE3 'Level adjustment included'; 40 MODEL Y = X1 X2 X3; RUN; NOTE: 16 observations read. NOTE: 16 observations used in computations. NOTE: The PROCEDURE REG printed page 3. NOTE: PROCEDURE REG used: real time 0.16 seconds EXST7034 - Homework Example NWK 11.16 (based on # 1.27) : Muscle mass Piecewise regression with SAS REG procedure Level adjustment included The REG Procedure Model: MODEL1 Dependent Variable: Y Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 3 2226.68272 742.22757 11.03 0.0009 Error 12 807.75478 67.31290 Corrected Total 15 3034.43750 Root MSE 8.20444 R-Square 0.7338 Dependent Mean 86.18750 Adj R-Sq 0.6673 Coeff Var 9.51930 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 186.30233 26.86367 6.94 <.0001 X1 1 -1.80103 0.52754 -3.41 0.0051 X2 1 0.85553 0.80320 1.07 0.3078 X3 1 8.70111 8.93463 0.97 0.3493 41 PROC REG; TITLE2 'Piecewise regression with SAS REG procedure'; 42 TITLE3 'Level adjustment NOT included'; 43 MODEL Y = X1 X2; RUN; NOTE: 16 observations read. NOTE: 16 observations used in computations. NOTE: The PROCEDURE REG printed page 4. NOTE: PROCEDURE REG used: real time 0.00 seconds EXST7034 - Homework Example NWK 11.16 (based on # 1.27) : Muscle mass Piecewise regression with SAS REG procedure Level adjustment NOT included The REG Procedure Model: MODEL1 Dependent Variable: Y Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 2 2162.84260 1081.42130 16.13 0.0003 Error 13 871.59490 67.04576 Corrected Total 15 3034.43750 Root MSE 8.18815 R-Square 0.7128 Dependent Mean 86.18750 Adj R-Sq 0.6686 Coeff Var 9.50039 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 172.82270 22.97757 7.52 <.0001 X1 1 -1.51665 0.43847 -3.46 0.0042 X2 1 0.98098 0.79122 1.24 0.2370 45 PROC NLIN DATA=ONE MAXITER=200; 46 TITLE2 'Piece wise regression where join-point is unknown'; 47 PARAMETERS B0 = 172.8 b1=-1.52 b2=0.981 xvalue = 60; 48 x2 = 0; if x1 gt xvalue then x2=1; 49 MODEL Y = b0 + b1*x1 + b2*(x1-xvalue)*x2; 50 RUN; NOTE: DER.B0 not initialized or missing. It will be computed automatically. NOTE: DER.b1 not initialized or missing. It will be computed automatically. NOTE: DER.b2 not initialized or missing. It will be computed automatically. NOTE: DER.xvalue not initialized or missing. It will be computed automatically. NOTE: PROC NLIN grid search time was 0: 0: 0. WARNING: PROC NLIN failed to converge. NOTE: The PROCEDURE NLIN printed pages 5-7. NOTE: PROCEDURE NLIN used: real time 0.04 seconds EXST7034 - Homework Example NWK 11.16 (based on # 1.27) : Muscle mass Piece wise regression where join-point is unknown The NLIN Procedure Dependent Variable Y Method: Gauss-Newton Iterative Phase Sum of Iter B0 b1 b2 xvalue Squares 0 172.8 -1.5200 0.9810 60.0000 872.4 1 179.6 -1.6605 0.9183 55.5652 869.7 2 172.9 -1.5135 0.7965 57.5730 856.6 3 173.0 -1.5160 0.8384 58.2299 854.3 . . . 51 174.3 -1.5449 0.8792 58.0000 849.7 52 174.3 -1.5449 0.8792 58.0000 849.7 WARNING: Step size shows no improvement. WARNING: PROC NLIN failed to converge. Estimation Summary (Not Converged) Method Gauss-Newton Iterations 52 Subiterations 163 Average Subiterations 3.134615 R 0.106711 PPC(b2) 0.16296 RPC . Object 1.43E-13 Objective 849.6886 Observations Read 16 Observations Used 16 Observations Missing 0 Sum of Mean Approx Source DF Squares Square F Value Pr > F Regression 4 121037 30259.3 10.28 0.0012 Residual 12 849.7 70.8074 Uncorrected Total 16 121887 Corrected Total 15 3034.4 Approx Parameter Estimate Std Error Approximate 95% Confidence Limits B0 174.3 31.2778 106.2 242.5 b1 -1.5449 0.6277 -2.9125 -0.1773 b2 0.8792 0.7858 -0.8329 2.5913 xvalue 58.0000 9.6824 36.9038 79.0962 Approximate Correlation Matrix B0 b1 b2 xvalue B0 1.0000000 -0.9948167 0.7946792 -0.5802775 b1 -0.9948167 1.0000000 -0.7988197 0.6214894 b2 0.7946792 -0.7988197 1.0000000 -0.1327208 xvalue -0.5802775 0.6214894 -0.1327208 1.0000000 51 PROC NLIN DATA=ONE MAXITER=200 METHOD=MARQUARDT; * Methods: GAUSS | MARQUARDT | NEWTON | GRADIENT | DUD ; 52 TITLE2 'Piece wise regression where join-point is unknown'; 53 PARAMETERS B0 = 172.8 b1=-1.52 b2=0.981 xvalue = 60; 54 x2 = 0; if x1 gt xvalue then x2=1; 55 MODEL Y = b0 + b1*x1 + b2*(x1-xvalue)*x2; 56 RUN; NOTE: DER.B0 not initialized or missing. It will be computed automatically. NOTE: DER.b1 not initialized or missing. It will be computed automatically. NOTE: DER.b2 not initialized or missing. It will be computed automatically. NOTE: DER.xvalue not initialized or missing. It will be computed automatically. NOTE: PROC NLIN grid search time was 0: 0: 0. WARNING: PROC NLIN failed to converge. NOTE: The PROCEDURE NLIN printed pages 8-9. NOTE: PROCEDURE NLIN used: real time 0.00 seconds EXST7034 - Homework Example NWK 11.16 (based on # 1.27) : Muscle mass Piece wise regression where join-point is unknown The NLIN Procedure Dependent Variable Y Method: Marquardt Iterative Phase Sum of Iter B0 b1 b2 xvalue Squares 0 172.8 -1.5200 0.9810 60.0000 872.4 1 172.8 -1.5248 0.8858 57.8779 850.2 2 172.8 -1.5247 0.8863 57.9153 849.9 3 172.8 -1.5246 0.8870 57.9602 849.6 4 172.8 -1.5244 0.8880 58.0139 849.5 . . . 27 172.8 -1.5244 0.8881 58.0000 849.3 28 172.8 -1.5244 0.8881 58.0000 849.3 29 172.8 -1.5244 0.8881 58.0000 849.3 WARNING: Step size shows no improvement. WARNING: PROC NLIN failed to converge. Estimation Summary (Not Converged) Method Marquardt Iterations 29 Subiterations 165 Average Subiterations 5.689655 R 0.104617 PPC(b2) 0.151255 RPC . Object 8.03E-16 Objective 849.3083 Observations Read 16 Observations Used 16 Observations Missing 0 EXST7034 - Homework Example NWK 11.16 (based on # 1.27) : Muscle mass Piece wise regression where join-point is unknown The NLIN Procedure Sum of Mean Approx Source DF Squares Square F Value Pr > F Regression 4 121038 30259.4 10.29 0.0012 Residual 12 849.3 70.7757 Uncorrected Total 16 121887 Corrected Total 15 3034.4 Approx Parameter Estimate Std Error Approximate 95% Confidence Limits B0 172.8 31.2708 104.7 240.9 b1 -1.5244 0.6276 -2.8917 -0.1571 b2 0.8881 0.7856 -0.8235 2.5998 xvalue 58.0000 9.5828 37.1208 78.8792 Approximate Correlation Matrix B0 b1 b2 xvalue B0 1.0000000 -0.9948167 0.7946792 -0.5802775 b1 -0.9948167 1.0000000 -0.7988197 0.6214894 b2 0.7946792 -0.7988197 1.0000000 -0.1327208 xvalue -0.5802775 0.6214894 -0.1327208 1.0000000 57 PROC NLIN DATA=ONE MAXITER=200 METHOD=NEWTON; 58 TITLE2 'Piece wise regression where join-point is unknown'; 59 PARAMETERS B0 = 172.8 b1=-1.52 b2=0.981 xvalue = 60; 60 x2 = 0; if x1 gt xvalue then x2=1; 61 MODEL Y = b0 + b1*x1 + b2*(x1-xvalue)*x2; 62 RUN; NOTE: DER.B0 not initialized or missing. It will be computed automatically. NOTE: DER.b1 not initialized or missing. It will be computed automatically. NOTE: DER.b2 not initialized or missing. It will be computed automatically. NOTE: DER.xvalue not initialized or missing. It will be computed automatically. NOTE: PROC NLIN grid search time was 0: 0: 0. WARNING: PROC NLIN failed to converge. NOTE: The PROCEDURE NLIN printed pages 10-11. NOTE: PROCEDURE NLIN used: real time 0.00 seconds EXST7034 - Homework Example NWK 11.16 (based on # 1.27) : Muscle mass Piece wise regression where join-point is unknown The NLIN Procedure Dependent Variable Y Method: Newton Iterative Phase Sum of Iter B0 b1 b2 xvalue Squares 0 172.8 -1.5200 0.9810 60.0000 872.4 1 186.1 -1.7908 1.2431 56.4388 866.0 2 184.8 -1.7583 1.3009 58.1663 855.8 3 185.3 -1.7715 1.2363 57.2498 854.8 . . . 26 181.2 -1.6865 1.1413 58.0000 846.7 27 181.2 -1.6865 1.1413 58.0000 846.7 WARNING: Step size shows no improvement. WARNING: PROC NLIN failed to converge. EXST7034 - Homework Example NWK 11.16 (based on # 1.27) : Muscle mass Piece wise regression where join-point is unknown The NLIN Procedure Estimation Summary (Not Converged) Method Newton Iterations 27 Subiterations 104 Average Subiterations 3.851852 R 0.191664 PPC(b1) 0.114274 RPC . Object 5.37E-16 Objective 846.7078 Observations Read 16 Observations Used 16 Observations Missing 0 Sum of Mean Approx Source DF Squares Square F Value Pr > F Regression 4 121040 30260.1 10.34 0.0012 Residual 12 846.7 70.5590 Uncorrected Total 16 121887 Corrected Total 15 3034.4 Approx Parameter Estimate Std Error Approximate 95% Confidence Limits B0 181.2 31.0976 113.5 249.0 b1 -1.6865 0.6166 -3.0299 -0.3431 b2 1.1413 0.7778 -0.5533 2.8358 xvalue 58.0000 7.9145 40.7557 75.2443 Approximate Correlation Matrix B0 b1 b2 xvalue B0 1.0000000 -0.9954024 0.7424834 -0.5756454 b1 -0.9954024 1.0000000 -0.7507783 0.6050888 b2 0.7424834 -0.7507783 1.0000000 -0.0273950 xvalue -0.5756454 0.6050888 -0.0273950 1.0000000 63 PROC NLIN DATA=ONE MAXITER=32000 METHOD=GRADIENT; * Methods: GAUSS | MARQUARDT | NEWTON | GRADIENT | DUD ; 64 TITLE2 'Piece wise regression where join-point is unknown'; 65 PARAMETERS B0 = 172.8 b1=-1.52 b2=0.981 xvalue = 60; 66 x2 = 0; if x1 gt xvalue then x2=1; 67 MODEL Y = b0 + b1*x1 + b2*(x1-xvalue)*x2; 68 RUN; NOTE: DER.B0 not initialized or missing. It will be computed automatically. NOTE: DER.b1 not initialized or missing. It will be computed automatically. NOTE: DER.b2 not initialized or missing. It will be computed automatically. NOTE: DER.xvalue not initialized or missing. It will be computed automatically. NOTE: PROC NLIN grid search time was 0: 0: 0. WARNING: PROC NLIN failed to converge. NOTE: The PROCEDURE NLIN printed pages 12-206. NOTE: PROCEDURE NLIN used: real time 1.31 seconds EXST7034 - Homework Example NWK 11.16 (based on # 1.27) : Muscle mass Piece wise regression where join-point is unknown The NLIN Procedure Dependent Variable Y Method: Gradient Iterative Phase Sum of Iter B0 b1 b2 xvalue Squares 0 172.8 -1.5200 0.9810 60.0000 872.4 1 172.8 -1.5132 0.9817 59.9997 872.2 2 172.8 -1.5168 0.9813 59.9996 871.6 3 172.8 -1.5161 0.9814 59.9994 871.6 4 172.8 -1.5165 0.9813 59.9992 871.6 . . . 9480 172.9 -1.5269 0.8972 58.0000 849.1 9481 172.9 -1.5269 0.8972 58.0000 849.1 9482 172.9 -1.5269 0.8972 58.0000 849.1 WARNING: Step size shows no improvement. WARNING: PROC NLIN failed to converge. Estimation Summary (Not Converged) Method Gradient Iterations 9482 Subiterations 22062 Average Subiterations 2.326724 R 0.146898 PPC(b2) 0.185486 RPC . Object 1.61E-15 Objective 849.0904 Observations Read 16 Observations Used 16 Observations Missing 0 Sum of Mean Approx Source DF Squares Square F Value Pr > F Regression 4 121038 30259.5 10.30 0.0012 Residual 12 849.1 70.7575 Uncorrected Total 16 121887 Corrected Total 15 3034.4 Approx Parameter Estimate Std Error Approximate 95% Confidence Limits B0 172.9 27.5425 112.9 232.9 b1 -1.5269 0.5409 -2.7054 -0.3485 b2 0.8972 0.8235 -0.8971 2.6914 xvalue 58.0000 10.6597 34.7744 81.2256 Approximate Correlation Matrix B0 b1 b2 xvalue B0 1.0000000 -0.9941532 0.6529582 -0.3810699 b1 -0.9941532 1.0000000 -0.6567984 0.4170864 b2 0.6529582 -0.6567984 1.0000000 0.3258316 xvalue -0.3810699 0.4170864 0.3258316 1.0000000 69 PROC NLIN DATA=ONE MAXITER=200 METHOD=DUD; 69 * Methods: GAUSS | MARQUARDT | NEWTON | GRADIENT | DUD ; 70 TITLE2 'Piece wise regression where join-point is unknown'; 71 PARAMETERS B0 = 172.8 b1=-1.52 b2=0.981 xvalue = 60; 72 x2 = 0; if x1 gt xvalue then x2=1; 73 MODEL Y = b0 + b1*x1 + b2*(x1-xvalue)*x2; 74 RUN; NOTE: PROC NLIN grid search time was 0: 0: 0. NOTE: Convergence criterion met. NOTE: The PROCEDURE NLIN printed pages 207-208. NOTE: PROCEDURE NLIN used: real time 0.00 seconds DUD Initialization Sum of DUD B0 b1 b2 xvalue Squares -5 172.8 -1.5200 0.9810 60.0000 872.4 -4 190.1 -1.5200 0.9810 60.0000 5525.7 -3 172.8 -1.6720 0.9810 60.0000 2336.2 -2 172.8 -1.5200 1.0791 60.0000 878.0 -1 172.8 -1.5200 0.9810 66.0000 1175.0 Iterative Phase Sum of Iter B0 b1 b2 xvalue Squares 0 172.8 -1.5200 0.9810 60.0000 872.4 1 175.7 -1.5801 0.9143 57.5053 850.5 . . . 13 174.4 -1.5536 0.9100 58.0000 848.0 14 174.4 -1.5536 0.9100 58.0000 848.0 NOTE: Convergence criterion met. Estimation Summary Method DUD Iterations 14 Object 2.821E-9 Objective 848.0422 Observations Read 16 Observations Used 16 Observations Missing 0 Sum of Mean Approx Source DF Squares Square F Value Pr > F Regression 4 121039 30259.7 10.31 0.0012 Residual 12 848.0 70.6702 Uncorrected Total 16 121887 Corrected Total 15 3034.4 Approx Parameter Estimate Std Error Approximate 95% Confidence Limits B0 174.4 31.2477 106.3 242.5 b1 -1.5536 0.6271 -2.9199 -0.1873 b2 0.9100 0.7850 -0.8004 2.6204 xvalue 58.0000 9.3450 37.6390 78.3610 Approximate Correlation Matrix B0 b1 b2 xvalue B0 1.0000000 -0.9948168 0.7946828 -0.5801406 b1 -0.9948168 1.0000000 -0.7988232 0.6213573 b2 0.7946828 -0.7988232 1.0000000 -0.1325735 xvalue -0.5801406 0.6213573 -0.1325735 1.0000000

1 *****************************************************************; 2 *** Chapter 10 ***; 3 *** Problem from Neter, Kutner, Nachtsheim, & Wasserman 1996 ***; 4 ****************************************************************; 5 OPTIONS LS=99 PS=256 NOCENTER NODATE NONUMBER; 6 7 DATA ONE; INFILE CARDS MISSOVER; 8 TITLE1 'EXST7034 - NKNW Table 10.11'; 9 INPUT state $ 1-25 mathprof parents homelib reading tvwatch absences; 10 weight = 1; 11 * label mathprof = 'proficiency in math'; 12 * label parents = '% 8th grade with both parents at home'; 13 * label homelib = ''% 8th grade with 3 or more types of reference materials'; 14 * label reading = ''% 8th grade who read more than 10 pages/day'; 15 * label TVWatch = ''% 8th grade who watch TV 6 hrs or more daily'; 16 * label Absences = ''% 8th grade absent 3 or more days last month'; 17 18 *---+----1----+----2----+----3----+----4----+----5----+----6; 19 CARDS; NOTE: The data set WORK.ONE has 40 observations and 8 variables. NOTE: DATA statement used: real time 0.04 seconds 19 ! RUN; 60 ; 61 Title2 'Robust regression'; 62 OPTIONS LS=99 PS=56; 63 PROC REG DATA=ONE lineprinter; weight weight; id state; 64 MODEL mathprof = homelib / influence; 65 output out=next1a r=e p=yhat; 66 plot residual.*homelib; 67 run; NOTE: 40 observations read. NOTE: 40 observations used in computations. NOTE: Some ID variables have been truncated to 16 characters. NOTE: The data set WORK.NEXT1A has 40 observations and 10 variables. NOTE: The PROCEDURE REG printed pages 1-5. NOTE: PROCEDURE REG used: real time 0.17 seconds 68 data next1a; set next1a; abse = abs(e); run; . . . 127 proc print data=next4c; run; NOTE: There were 41 observations read from the data set WORK.NEXT4C. NOTE: The PROCEDURE PRINT printed page 12. NOTE: PROCEDURE PRINT used: real time 0.00 seconds 128 data next4c; set next4c; if mathprof eq . then delete; 129 drop mad mu yhat e abse; run; NOTE: There were 41 observations read from the data set WORK.NEXT4C. NOTE: The data set WORK.NEXT4C has 40 observations and 9 variables. NOTE: DATA statement used: real time 0.00 seconds 130 131 OPTIONS LS=99 PS=56; 132 PROC REG DATA=next4c lineprinter; weight weight; 133 MODEL mathprof = homelib; 134 plot residual.*homelib; 135 run; NOTE: 40 observations read. NOTE: 40 observations used in computations. Earlier in the semester robust regression was demonstrated by mechanically recalculating weights and refitting the regression. Final results of that exercise are given below. EXST7034 - NKNW Table 10.11 Robust regression m a a p h r t b m t a o e v s w e s h r m a w e e d t p e e d a n i y a O i a r n l i t c g h b m b a t o t i n c e h a s a m s n e f s b g h s t t e e d u 1 4.46039 . . . . . . 1.00000 . . . 6.61288 . 2 . Alabama 252 75 78 34 18 18 1.00000 258.350 -6.3505 6.3505 6.61288 -0.96032 3 . Arizona 259 75 73 41 12 26 1.00000 250.950 8.0505 8.0505 6.61288 1.21739 4 . Arkansas 256 77 77 28 20 23 1.00000 256.870 -0.8703 0.8703 6.61288 -0.13161 5 . California 256 78 68 42 11 28 0.71432 243.549 12.4514 12.4514 6.61288 1.88291 6 . Colorado 267 78 85 38 9 25 1.00000 268.712 -1.7118 1.7118 6.61288 -0.25886 7 . Connecticut 270 79 86 43 12 22 1.00000 270.192 -0.1920 0.1920 6.61288 -0.02904 8 . Delaware 261 75 83 32 18 28 1.00000 265.751 -4.7514 4.7514 6.61288 -0.71851 9 . Distric of Columbia 231 47 76 24 33 37 0.36467 255.390 -24.3901 24.3901 6.61288 -3.68827 10 . Florida 255 75 73 31 19 27 1.00000 250.950 4.0505 4.0505 6.61288 0.61251 11 . Georgia 258 73 80 36 17 22 1.00000 261.311 -3.3109 3.3109 6.61288 -0.50067 12 . Guam 231 81 64 32 20 28 1.00000 237.628 -6.6278 6.6278 6.61288 -1.00225 13 . Hawaii 251 78 69 36 23 26 1.00000 245.029 5.9713 5.9713 6.61288 0.90297 14 . Idaho 272 84 84 48 7 21 1.00000 267.232 4.7684 4.7684 6.61288 0.72107 15 . Illinois 260 78 82 43 14 21 1.00000 264.271 -4.2713 4.2713 6.61288 -0.64590 16 . Indiana 267 81 84 37 11 23 1.00000 267.232 -0.2316 0.2316 6.61288 -0.03503 17 . Iowa 278 83 88 43 8 20 1.00000 273.152 4.8476 4.8476 6.61288 0.73305 18 . Kentucky 256 79 78 36 14 23 1.00000 258.350 -2.3505 2.3505 6.61288 -0.35544 19 . Louisiana 246 73 76 36 19 27 0.94720 255.390 -9.3901 9.3901 6.61288 -1.41997 20 . Maryland 260 75 83 34 19 27 1.00000 265.751 -5.7514 5.7514 6.61288 -0.86973 21 . Michigan 264 77 84 31 14 25 1.00000 267.232 -3.2316 3.2316 6.61288 -0.48869 22 . Minnesota 276 83 88 36 7 20 1.00000 273.152 2.8476 2.8476 6.61288 0.43061 23 . Montana 280 83 88 44 6 21 1.00000 273.152 6.8476 6.8476 6.61288 1.03549 24 . Nebraska 276 85 88 42 9 19 1.00000 273.152 2.8476 2.8476 6.61288 0.43061 25 . New Hampshire 273 83 88 40 7 22 1.00000 273.152 -0.1524 0.1524 6.61288 -0.02305 26 . New Jersey 269 79 84 41 13 23 1.00000 267.232 1.7684 1.7684 6.61288 0.26741 27 . New Mexico 256 77 72 40 11 27 1.00000 249.469 6.5307 6.5307 6.61288 0.98757 28 . New York 261 76 79 35 17 29 1.00000 259.831 1.1693 1.1693 6.61288 0.17683 29 . North Carolina 250 74 78 37 21 25 1.00000 258.350 -8.3505 8.3505 6.61288 -1.26276 30 . North Dakota 281 85 90 41 6 14 1.00000 276.113 4.8872 4.8872 6.61288 0.73904 31 . Ohio 264 79 84 36 11 22 1.00000 267.232 -3.2316 3.2316 6.61288 -0.48869 32 . Oklahoma 263 78 78 37 14 22 1.00000 258.350 4.6495 4.6495 6.61288 0.70310 33 . Oregon 271 81 82 41 9 31 1.00000 264.271 6.7287 6.7287 6.61288 1.01752 34 . Pennsylvania 266 80 86 34 10 24 1.00000 270.192 -4.1920 4.1920 6.61288 -0.63392 35 . Rhode Island 260 78 80 38 12 28 1.00000 261.311 -1.3109 1.3109 6.61288 -0.19823 36 . Texas 258 77 70 34 15 18 0.77402 246.509 11.4911 11.4911 6.61288 1.73768 37 . Virgin Islands 218 63 76 23 27 22 0.23788 255.390 -37.3901 37.3901 6.61288 -5.65413 38 . Virginia 264 78 82 33 16 24 1.00000 264.271 -0.2713 0.2713 6.61288 -0.04102 39 . West Virginia 256 82 80 36 16 25 1.00000 261.311 -5.3109 5.3109 6.61288 -0.80311 40 . Wisconsin 274 81 86 38 8 21 1.00000 270.192 3.8080 3.8080 6.61288 0.57584 41 . Wyoming 272 85 86 43 7 23 1.00000 270.192 1.8080 1.8080 6.61288 0.27340 EXST7034 - NKNW Table 10.11 Robust regression The REG Procedure Model: MODEL1 Dependent Variable: mathprof Weight: weight Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 3170.39031 3170.39031 78.00 <.0001 Error 38 1544.50010 40.64474 Corrected Total 39 4714.89041 Root MSE 6.37532 R-Square 0.6724 Dependent Mean 262.38623 Adj R-Sq 0.6638 Coeff Var 2.42975 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 142.99058 13.55814 10.55 <.0001 homelib 1 1.47908 0.16747 8.83 <.0001 139 title 'Beaton/Tukey Biweight Robust Regression using IRLS'; NOTE: The PROCEDURE REG printed pages 13-14. NOTE: PROCEDURE REG used: real time 0.05 seconds 140 proc nlin data=one nohalve; 141 parms b0=142 b1=1.48; 142 model mathprof = b0 + b1*homelib; 143 resid=mathprof-model.mathprof; 144 sigma=2; 145 b=4.685; 146 r=abs(resid/sigma); 147 if r<=b then _weight_=(1-(r/b)**2)**2; 148 else _weight_=0; 149 output out=c r=rbi; 150 run; NOTE: DER.b0 not initialized or missing. It will be computed automatically. NOTE: DER.b1 not initialized or missing. It will be computed automatically. NOTE: PROC NLIN grid search time was 0: 0: 0. WARNING: PROC NLIN failed to converge. NOTE: The data set WORK.C has 40 observations and 9 variables. NOTE: The PROCEDURE NLIN printed pages 15-16. NOTE: PROCEDURE NLIN used: real time 0.11 seconds 151 152 data c; 153 set c; 154 sigma=2; 155 b=4.685; 156 r=abs(rbi/sigma); 157 if r<=b then _weight_=(1-(r/b)**2)**2; 158 else _weight_=0; NOTE: There were 40 observations read from the data set WORK.C. NOTE: The data set WORK.C has 40 observations and 13 variables. NOTE: DATA statement used: real time 0.00 seconds 159 proc print; 160 run; NOTE: There were 40 observations read from the data set WORK.C. NOTE: The PROCEDURE PRINT printed page 17. NOTE: PROCEDURE PRINT used: real time 0.00 seconds Beaton/Tukey Biweight Robust Regression using IRLS

The NLIN Procedure Dependent Variable mathprof Method: Gauss-Newton Iterative Phase Weighted Iter b0 b1 SS 0 142.0 1.4800 249.9 1 138.7 1.5205 246.9 2 138.3 1.5262 246.3 3 138.0 1.5290 246.0 . . . 21 137.9 1.5309 245.8 22 137.9 1.5309 245.8 23 137.9 1.5309 245.8 WARNING: Step size shows no improvement. WARNING: PROC NLIN failed to converge. Estimation Summary (Not Converged) Method Gauss-Newton Iterations 23 Subiterations 49 Average Subiterations 2.130435 R 0.274148 PPC(b1) 0.120528 RPC . Object 1 Objective 245.784 Observations Read 40 Observations Used 36 Observations Missing 4

Beaton/Tukey Biweight Robust Regression using IRLS The NLIN Procedure Sum of Mean Approx Source DF Squares Square F Value Pr > F Regression 2 1565584 782792 213.13 <.0001 Residual 34 245.8 7.2289 Uncorrected Total 36 1565830 Corrected Total 35 1786.5 Approx Parameter Estimate Std Error Approximate 95% Confidence Limits b0 137.9 9.6465 118.3 157.5 b1 1.5309 0.1168 1.2935 1.7682 Approximate Correlation Matrix b0 b1 b0 1.0000000 -0.9982646 b1 -0.9982646 1.0000000 Beaton/Tukey Biweight Robust Regression using IRLS m a _ a p h r t b w t a o e v s w e s h r m a w e e s i t p e e d a n i i g O a r n l i t c g R g h b t o t i n c e h B m t s e f s b g h s t I a b r _ 1 Alabama 252 75 78 34 18 18 1 -5.2843 2 4.685 2.6421 0.46506 2 Arizona 259 75 73 41 12 26 1 9.3700 2 4.685 4.6850 0.00000 3 Arkansas 256 77 77 28 20 23 1 0.2466 2 4.685 0.1233 0.99862 4 California 256 78 68 42 11 28 1 14.0243 2 4.685 7.0121 0.00000 5 Colorado 267 78 85 38 9 25 1 -1.0003 2 4.685 0.5001 0.97734 6 Connecticut 270 79 86 43 12 22 1 0.4689 2 4.685 0.2344 0.99500 7 Delaware 261 75 83 32 18 28 1 -3.9386 2 4.685 1.9693 0.67785 8 Distric of Columbia 231 47 76 24 33 37 1 -23.2226 2 4.685 11.6113 0.00000 9 Florida 255 75 73 31 19 27 1 5.3700 2 4.685 2.6850 0.45098 10 Georgia 258 73 80 36 17 22 1 -2.3460 2 4.685 1.1730 0.87856 11 Guam 231 81 64 32 20 28 1 -4.8523 2 4.685 2.4262 0.53557 12 Hawaii 251 78 69 36 23 26 1 7.4934 2 4.685 3.7467 0.12992 13 Idaho 272 84 84 48 7 21 1 5.5306 2 4.685 2.7653 0.42460 14 Illinois 260 78 82 43 14 21 1 -3.4077 2 4.685 1.7038 0.75297 15 Indiana 267 81 84 37 11 23 1 0.5306 2 4.685 0.2653 0.99360 16 Iowa 278 83 88 43 8 20 1 5.4072 2 4.685 2.7036 0.44487 17 Kentucky 256 79 78 36 14 23 1 -1.2843 2 4.685 0.6421 0.96278 18 Louisiana 246 73 76 36 19 27 1 -8.2226 2 4.685 4.1113 0.05286 19 Maryland 260 75 83 34 19 27 1 -4.9386 2 4.685 2.4693 0.52158 20 Michigan 264 77 84 31 14 25 1 -2.4694 2 4.685 1.2347 0.86591 21 Minnesota 276 83 88 36 7 20 1 3.4072 2 4.685 1.7036 0.75304 22 Montana 280 83 88 44 6 21 1 7.4072 2 4.685 3.7036 0.14068 23 Nebraska 276 85 88 42 9 19 1 3.4072 2 4.685 1.7036 0.75304 24 New Hampshire 273 83 88 40 7 22 1 0.4072 2 4.685 0.2036 0.99623 25 New Jersey 269 79 84 41 13 23 1 2.5306 2 4.685 1.2653 0.85944 26 New Mexico 256 77 72 40 11 27 1 7.9009 2 4.685 3.9504 0.08352 27 New York 261 76 79 35 17 29 1 2.1849 2 4.685 1.0924 0.89421 28 North Carolina 250 74 78 37 21 25 1 -7.2843 2 4.685 3.6421 0.15653 29 North Dakota 281 85 90 41 6 14 1 5.3455 2 4.685 2.6727 0.45501 30 Ohio 264 79 84 36 11 22 1 -2.4694 2 4.685 1.2347 0.86591 31 Oklahoma 263 78 78 37 14 22 1 5.7157 2 4.685 2.8579 0.39425 32 Oregon 271 81 82 41 9 31 1 7.5923 2 4.685 3.7962 0.11796 33 Pennsylvania 266 80 86 34 10 24 1 -3.5311 2 4.685 1.7656 0.73613 34 Rhode Island 260 78 80 38 12 28 1 -0.3460 2 4.685 0.1730 0.99727 35 Texas 258 77 70 34 15 18 1 12.9626 2 4.685 6.4813 0.00000 36 Virgin Islands 218 63 76 23 27 22 1 -36.2226 2 4.685 18.1113 0.00000 37 Virginia 264 78 82 33 16 24 1 0.5923 2 4.685 0.2962 0.99202 38 West Virginia 256 82 80 36 16 25 1 -4.3460 2 4.685 2.1730 0.61602 39 Wisconsin 274 81 86 38 8 21 1 4.4689 2 4.685 2.2344 0.59681 40 Wyoming 272 85 86 43 7 23 1 2.4689 2 4.685 1.2344 0.86597