1 ************************************************************************;
2 *** EXST7034 Example 1 using PC-SAS - Toluca Company Example ***;
3 *** Problem from Neter, Kutner, Nachtsheim & Wasserman 1996, #1.21 ***;
4 ************************************************************************;
5 ODS HTML style=minimal rs=none
6 body='C:\Geaghan\EXST\EXST7034New\Fall2002\SAS\03b-Toluca-Nknw1-Ex.html' ;
NOTE: Writing HTML Body file: C:\Geaghan\EXST\EXST7034New\Fall2002\SAS\03b-Toluca-Nknw1-Ex.html
9 OPTIONS LS=155 PS=256 NOCENTER NODATE NONUMBER;
10 DATA ONE; INFILE CARDS MISSOVER;
11 TITLE1 'EXST7034 - Chapter 3 examples : Toluca example';
12 * LABEL X = 'Lot size';
13 * LABEL Y = 'work hours';
14 INPUT X Y;
15 group = 'Upper'; If X lt 80 then group = 'Lower';
16 anotherX = X;
17 CARDS;
NOTE: The data set WORK.ONE has 25 observations and 4 variables.
NOTE: DATA statement used:
real time 0.10 seconds
43 ;
44 PROC SORT; BY group X Y; run;
NOTE: There were 25 observations read from the data set WORK.ONE.
NOTE: The data set WORK.ONE has 25 observations and 4 variables.
NOTE: PROCEDURE SORT used: real time 0.05 seconds
45 PROC REG DATA=ONE lineprinter; id x;
46 TITLE2 'Regression Models done with SAS REG procedure';
47 MODEL Y = X / XPX I P CLM CLI R CLB alpha=0.01;
48 TEST X = 5;
49 OUTPUT OUT=Next2 PREDICTED=YHat RESIDUAL=E; RUN;
NOTE: 25 observations read.
NOTE: 25 observations used in computations.
50 OPTIONS PS=35 ls=80; PLOT Y*X='O' PREDICTED.*X='P'/ OVERLAY;
51 PLOT RESIDUAL.*X='e'; RUN;
NOTE: The data set WORK.NEXT2 has 25 observations and 6 variables.
NOTE: The PROCEDURE REG printed pages 1-6.
NOTE: PROCEDURE REG used: real time 0.17 seconds
EXST7034 - Chapter 3 examples : Toluca example
Regression Models done with SAS REG procedure
The REG Procedure
Model: MODEL1
Model Crossproducts X'X X'Y Y'Y
Variable Intercept X Y
Intercept 25 1750 7807
X 1750 142300 617180
Y 7807 617180 2745173
X'X Inverse, Parameter Estimates, and SSE
Variable Intercept X Y
Intercept 0.2874747475 -0.003535354 62.365858586
X -0.003535354 0.0000505051 3.5702020202
Y 62.365858586 3.5702020202 54825.459192
Analysis of Variance Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 252378 252378 105.88 <.0001
Error 23 54825 2383.71562
Corrected Total 24 307203
Root MSE 48.82331 R-Square 0.8215
Dependent Mean 312.28000 Adj R-Sq 0.8138
Coeff Var 15.63447
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t| 99% Confidence Limits
Intercept 1 62.36586 26.17743 2.38 0.0259 -11.12299 135.85470
X 1 3.57020 0.34697 10.29 <.0001 2.59613 4.54427EXST7034 - Chapter 3 examples : Toluca example
Regression Models done with SAS REG procedure
The REG Procedure
Model: MODEL1
Dependent Variable: Y
Output Statistics
Dep Var Predicted Std Error Std Error Student Cook's
Obs X Y Value Mean Predict 99% CL Mean 99% CL Predict Residual Residual Residual -2-1 0 1 2 D
1 20 113.0000 133.7699 19.9079 77.8819 189.6579 -14.2499 281.7897 -20.7699 44.580 -0.466 | | | 0.022
2 30 121.0000 169.4719 16.9697 121.8322 217.1117 24.3653 314.5785 -48.4719 45.779 -1.059 | **| | 0.077
3 30 212.0000 169.4719 16.9697 121.8322 217.1117 24.3653 314.5785 42.5281 45.779 0.929 | |* | 0.059
4 30 273.0000 169.4719 16.9697 121.8322 217.1117 24.3653 314.5785 103.5281 45.779 2.261 | |**** | 0.351
5 40 160.0000 205.1739 14.2723 165.1067 245.2412 62.3742 347.9737 -45.1739 46.691 -0.968 | *| | 0.044
6 40 244.0000 205.1739 14.2723 165.1067 245.2412 62.3742 347.9737 38.8261 46.691 0.832 | |* | 0.032
7 50 157.0000 240.8760 11.9793 207.2459 274.5060 99.7471 382.0048 -83.8760 47.331 -1.772 | ***| | 0.101
8 50 221.0000 240.8760 11.9793 207.2459 274.5060 99.7471 382.0048 -19.8760 47.331 -0.420 | | | 0.006
9 50 268.0000 240.8760 11.9793 207.2459 274.5060 99.7471 382.0048 27.1240 47.331 0.573 | |* | 0.011
10 60 224.0000 276.5780 10.3628 247.4861 305.6698 136.4612 416.6948 -52.5780 47.711 -1.102 | **| | 0.029
11 70 252.0000 312.2800 9.7647 284.8673 339.6927 172.5022 452.0578 -60.2800 47.837 -1.260 | **| | 0.033
12 70 323.0000 312.2800 9.7647 284.8673 339.6927 172.5022 452.0578 10.7200 47.837 0.224 | | | 0.001
13 70 361.0000 312.2800 9.7647 284.8673 339.6927 172.5022 452.0578 48.7200 47.837 1.018 | |** | 0.022
14 80 342.0000 347.9820 10.3628 318.8902 377.0739 207.8652 488.0988 -5.9820 47.711 -0.125 | | | 0.000
15 80 352.0000 347.9820 10.3628 318.8902 377.0739 207.8652 488.0988 4.0180 47.711 0.0842 | | | 0.000
16 80 399.0000 347.9820 10.3628 318.8902 377.0739 207.8652 488.0988 51.0180 47.711 1.069 | |** | 0.027
17 90 376.0000 383.6840 11.9793 350.0540 417.3141 242.5552 524.8129 -7.6840 47.331 -0.162 | | | 0.001
18 90 377.0000 383.6840 11.9793 350.0540 417.3141 242.5552 524.8129 -6.6840 47.331 -0.141 | | | 0.001
19 90 389.0000 383.6840 11.9793 350.0540 417.3141 242.5552 524.8129 5.3160 47.331 0.112 | | | 0.000
20 90 468.0000 383.6840 11.9793 350.0540 417.3141 242.5552 524.8129 84.3160 47.331 1.781 | |*** | 0.102
21 100 353.0000 419.3861 14.2723 379.3188 459.4533 276.5863 562.1858 -66.3861 46.691 -1.422 | **| | 0.094
22 100 420.0000 419.3861 14.2723 379.3188 459.4533 276.5863 562.1858 0.6139 46.691 0.0131 | | | 0.000
23 110 421.0000 455.0881 16.9697 407.4483 502.7278 309.9815 600.1947 -34.0881 45.779 -0.745 | *| | 0.038
24 110 435.0000 455.0881 16.9697 407.4483 502.7278 309.9815 600.1947 -20.0881 45.779 -0.439 | | | 0.013
25 120 546.0000 490.7901 19.9079 434.9021 546.6781 342.7703 638.8099 55.2099 44.580 1.238 | |** | 0.153
Sum of Residuals 0
Sum of Squared Residuals 54825
Predicted Residual SS (PRESS) 65818
Test 1 Results for Dependent Variable Y
Mean
Source DF Square F Value Pr > F
Numerator 1 40478 16.98 0.0004
Denominator 23 2383.71562
EXST7034 - Chapter 3 examples : Toluca example
Regression Models done with SAS REG procedure
The REG Procedure
Model: MODEL1
Dependent Variable: Y
--+------+------+------+------+------+------+------+------+------+------+---
Y | |
600 + +
| |
| O |
| P |
| O P |
| ? O |
400 + O ? +
| O O O O |
| O ? |
| P |
| O O P O |
| O ? O |
200 + O P +
| P O O |
| P O |
| O |
| |
| |
0 + +
| |
--+------+------+------+------+------+------+------+------+------+------+---
20 30 40 50 60 70 80 90 100 110 120
X
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+----
100 + e +
| |
| e |
| |
| e |
50 + e e +
R | e e |
e RESIDUAL | e |
s | |
i | e e |
d 0 + e e +
u | e e |
a | e e e |
l | e |
| |
-50 + e e e +
| e |
| e |
| e |
| |
-100 + +
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+----
20 30 40 50 60 70 80 90 100 110 120
X
52 PROC PLOT DATA=Next2; PLOT E*X='x' / VREF=0; RUN;
53 OPTIONS PS=256 ls=88;
NOTE: There were 25 observations read from the data set WORK.NEXT2.
NOTE: The PROCEDURE PLOT printed page 7.
NOTE: PROCEDURE PLOT used:
real time 0.00 seconds
EXST7034 - Chapter 3 examples : Toluca example
Regression Models done with SAS REG procedure
Plot of E*X. Symbol used is 'x'.
|
|
100 + x
|
| x
|
| x
50 + x x
R | x x
e | x
s |
i | x x
d 0 +------------------------------------------x-------------x---------------
u | x x
a |x x x
l | x
|
-50 + x x x
| x
| x
| x
|
-100 +
|
-+------+------+------+------+------+------+------+------+------+------+-
20 30 40 50 60 70 80 90 100 110 120
X
NOTE: 1 obs hidden.
55 PROC UNIVARIATE DATA=Next2 PLOT NORMAL; VAR E; RUN;
NOTE: The PROCEDURE UNIVARIATE printed page 8.
NOTE: PROCEDURE UNIVARIATE used:
real time 0.00 seconds
EXST7034 - Chapter 3 examples : Toluca example
Regression Models done with SAS REG procedure
The UNIVARIATE Procedure
Variable: E (Residual)
Moments
N 25 Sum Weights 25
Mean 0 Sum Observations 0
Std Deviation 47.7953359 Variance 2284.39413
Skewness 0.31691262 Kurtosis -0.3941493
Uncorrected SS 54825.4592 Corrected SS 54825.4592
Coeff Variation . Std Error Mean 9.55906718
Basic Statistical Measures
Location Variability
Mean 0.00000 Std Deviation 47.79534
Median -5.98202 Variance 2284
Mode . Range 187.40404
Interquartile Range 72.91414
Tests for Location: Mu0=0
Test -Statistic- -----p Value------
Student's t t 0 Pr > |t| 1.0000
Sign M -0.5 Pr >= |M| 1.0000
Signed Rank S -7.5 Pr >= |S| 0.8448
Tests for Normality
Test --Statistic--- -----p Value------
Shapiro-Wilk W 0.978904 Pr < W 0.8626
Kolmogorov-Smirnov D 0.09572 Pr > D >0.1500
Cramer-von Mises W-Sq 0.033263 Pr > W-Sq >0.2500
Anderson-Darling A-Sq 0.207142 Pr > A-Sq >0.2500
Quantiles (Definition 5)
Quantile Estimate
100% Max 103.52808
99% 103.52808
95% 84.31596
90% 55.20990
75% Q3 38.82606
50% Median -5.98202
25% Q1 -34.08808
10% -60.28000
5% -66.38606
1% -83.87596
0% Min -83.87596
Extreme Observations
------Lowest----- ------Highest-----
Value Obs Value Obs
-83.8760 7 48.7200 13
-66.3861 21 51.0180 16
-60.2800 11 55.2099 25
-52.5780 10 84.3160 20
-48.4719 2 103.5281 4
Stem Leaf # Boxplot
10 4 1 |
8 4 1 |
6 |
4 3915 4 |
2 79 2 +-----+
0 1451 4 | + |
-0 876 3 *-----*
-2 4100 4 +-----+
-4 385 3 |
-6 60 2 |
-8 4 1 |
----+----+----+----+
Multiply Stem.Leaf by 10**+1
Normal Probability Plot
110+ *+++++
| * ++++
| ++++
| **+*+*
| **++
10+ +****
| +****
| ++***
| +*+**
| +*+*
-90+ *+++
+----+----+----+----+----+----+----+----+----+----+
-2 -1 0 +1 +2
57 PROC SORT; BY group X Y; run;
NOTE: There were 25 observations read from the data set WORK.NEXT2.
NOTE: The data set WORK.NEXT2 has 25 observations and 6 variables.
NOTE: PROCEDURE SORT used:
real time 0.04 seconds
58 PROC means data=next2 noprint; BY group; var e;
59 OUTPUT OUT=NEXT3 median=med;
60 run;
NOTE: There were 25 observations read from the data set WORK.NEXT2.
NOTE: The data set WORK.NEXT3 has 2 observations and 4 variables.
NOTE: PROCEDURE MEANS used:
real time 0.06 seconds
61 data next2; merge next2 next3; by group;
62 absE = abs(e);
63 esquared = e*e;
64 logesq = log(esquared);
65 logX = Log(X);
66 leveneTest = abs(e - med);
67 drop _TYPE_ _FREQ_;
68 run;
NOTE: There were 25 observations read from the data set WORK.NEXT2.
NOTE: There were 2 observations read from the data set WORK.NEXT3.
NOTE: The data set WORK.NEXT2 has 25 observations and 12 variables.
NOTE: DATA statement used:
real time 0.00 seconds
69 proc print data=next2; run;
NOTE: There were 25 observations read from the data set WORK.NEXT2.
NOTE: The PROCEDURE PRINT printed page 9.
NOTE: PROCEDURE PRINT used:
real time 0.04 seconds
EXST7034 - Chapter 3 examples : Toluca example
Regression Models done with SAS REG procedure
l
e
a e v
n s e
o q l n
g t u o e
r h Y a a g l T
O o e H m b r e o e
b u r a e s e s g s
s X Y p X t E d E d q X t
1 20 113 Lower 20 133.770 -20.770 -19.8760 20.770 431.39 6.06701 2.99573 0.894
2 30 121 Lower 30 169.472 -48.472 -19.8760 48.472 2349.53 7.76197 3.40120 28.596
3 30 212 Lower 30 169.472 42.528 -19.8760 42.528 1808.64 7.50033 3.40120 62.404
4 30 273 Lower 30 169.472 103.528 -19.8760 103.528 10718.06 9.27969 3.40120 123.404
5 40 160 Lower 40 205.174 -45.174 -19.8760 45.174 2040.68 7.62104 3.68888 25.298
6 40 244 Lower 40 205.174 38.826 -19.8760 38.826 1507.46 7.31818 3.68888 58.702
7 50 157 Lower 50 240.876 -83.876 -19.8760 83.876 7035.18 8.85868 3.91202 64.000
8 50 221 Lower 50 240.876 -19.876 -19.8760 19.876 395.05 5.97902 3.91202 0.000
9 50 268 Lower 50 240.876 27.124 -19.8760 27.124 735.71 6.60084 3.91202 47.000
10 60 224 Lower 60 276.578 -52.578 -19.8760 52.578 2764.44 7.92459 4.09434 32.702
11 70 252 Lower 70 312.280 -60.280 -19.8760 60.280 3633.68 8.19800 4.24850 40.404
12 70 323 Lower 70 312.280 10.720 -19.8760 10.720 114.92 4.74422 4.24850 30.596
13 70 361 Lower 70 312.280 48.720 -19.8760 48.720 2373.64 7.77218 4.24850 68.596
14 80 342 Upper 80 347.982 -5.982 -2.6840 5.982 35.78 3.57752 4.38203 3.298
15 80 352 Upper 80 347.982 4.018 -2.6840 4.018 16.14 2.78156 4.38203 6.702
16 80 399 Upper 80 347.982 51.018 -2.6840 51.018 2602.83 7.86436 4.38203 53.702
17 90 376 Upper 90 383.684 -7.684 -2.6840 7.684 59.04 4.07829 4.49981 5.000
18 90 377 Upper 90 383.684 -6.684 -2.6840 6.684 44.68 3.79945 4.49981 4.000
19 90 389 Upper 90 383.684 5.316 -2.6840 5.316 28.26 3.34143 4.49981 8.000
20 90 468 Upper 90 383.684 84.316 -2.6840 84.316 7109.18 8.86914 4.49981 87.000
21 100 353 Upper 100 419.386 -66.386 -2.6840 66.386 4407.11 8.39097 4.60517 63.702
22 100 420 Upper 100 419.386 0.614 -2.6840 0.614 0.38 -0.97572 4.60517 3.298
23 110 421 Upper 110 455.088 -34.088 -2.6840 34.088 1162.00 7.05790 4.70048 31.404
24 110 435 Upper 110 455.088 -20.088 -2.6840 20.088 403.53 6.00025 4.70048 17.404
25 120 546 Upper 120 490.790 55.210 -2.6840 55.210 3048.13 8.02228 4.78749 57.894
70 proc ttest data=next2;
71 TITLE2 'TTest for the Modified Levene test';
72 class group; var leveneTest;
73 run;
NOTE: There were 25 observations read from the data set WORK.NEXT2.
NOTE: The PROCEDURE TTEST printed page 10.
NOTE: PROCEDURE TTEST used:
real time 0.00 seconds
EXST7034 - Chapter 3 examples : Toluca example
TTest for the Modified Levene test
The TTEST Procedure
Statistics
Lower CL Upper CL Lower CL Upper CL
Variable group N Mean Mean Mean Std Dev Std Dev Std Dev
leveneTest Lower 13 25.26 44.815 64.37 23.205 32.361 53.419
leveneTest Upper 12 9.6702 28.45 47.23 20.939 29.558 50.186
leveneTest Diff (1-2) -9.35 16.365 42.08 24.134 31.052 43.558
Statistics
Variable group Std Err Minimum Maximum
leveneTest Lower 8.9753 0 123.4
leveneTest Upper 8.5326 3.298 87
leveneTest Diff (1-2) 12.431
T-Tests
Variable Method Variances DF t Value Pr > |t|
leveneTest Pooled Equal 23 1.32 0.2010
leveneTest Satterthwaite Unequal 23 1.32 0.1993
Equality of Variances
Variable Method Num DF Den DF F Value Pr > F
leveneTest Folded F 12 11 1.20 0.7710
75 proc npar1way data=next2;
76 TITLE2 'Nonparametric analysis of abs(residuals)';
77 TITLE3 'Other tests of residuals for homogeneity';
78 class group; var abse;
79 run;
NOTE: There were 25 observations read from the data set WORK.NEXT2.
NOTE: The PROCEDURE NPAR1WAY printed pages 11-16.
NOTE: PROCEDURE NPAR1WAY used:
real time 0.00 seconds
80 proc npar1way data=next2;
81 TITLE2 'Nonparametric analysis of abs(residuals)';
82 TITLE3 'Other tests of residuals for homogeneity';
83 class group; var e;
84 run;
NOTE: There were 25 observations read from the data set WORK.NEXT2.
NOTE: The PROCEDURE NPAR1WAY printed pages 17-22.
NOTE: PROCEDURE NPAR1WAY used:
real time 0.04 seconds
EXST7034 - Chapter 3 examples : Toluca example
Nonparametric analysis of abs(residuals)
Other tests of residuals for homogeneity
The NPAR1WAY Procedure
Analysis of Variance for Variable absE
Classified by Variable group
group N Mean
Lower 13 46.343994
Upper 12 28.450337
Source DF Sum of Squares Mean Square F Value Pr > F
Among 1 1997.941694 1997.941694 2.6730 0.1157
Within 23 17191.444414 747.454105
Wilcoxon Scores (Rank Sums) for Variable absE
Classified by Variable group
Sum of Expected Std Dev Mean
group N Scores Under H0 Under H0 Score
Lower 13 199.0 169.0 18.384776 15.307692
Upper 12 126.0 156.0 18.384776 10.500000
Wilcoxon Two-Sample Test
Statistic 126.0000
Normal Approximation
Z -1.6046
One-Sided Pr < Z 0.0543
Two-Sided Pr > |Z| 0.1086
t Approximation
One-Sided Pr < Z 0.0608
Two-Sided Pr > |Z| 0.1217
Z includes a continuity correction of 0.5.
Kruskal-Wallis Test
Chi-Square 2.6627
DF 1
Pr > Chi-Square 0.1027
Median Scores (Number of Points Above Median) for Variable absE
Classified by Variable group
Sum of Expected Std Dev Mean
group N Scores Under H0 Under H0 Score
Lower 13 8.0 6.240 1.273735 0.615385
Upper 12 4.0 5.760 1.273735 0.333333
Median Two-Sample Test
Statistic 4.0000
Z -1.3818
One-Sided Pr < Z 0.0835
Two-Sided Pr > |Z| 0.1670
Median One-Way Analysis
Chi-Square 1.9093
DF 1
Pr > Chi-Square 0.1670
Van der Waerden Scores (Normal) for Variable absE
Classified by Variable group
Sum of Expected Std Dev Mean
group N Scores Under H0 Under H0 Score
Lower 13 3.826510 0.0 2.264058 0.294347
Upper 12 -3.826510 0.0 2.264058 -0.318876
Van der Waerden Two-Sample Test
Statistic -3.8265
Z -1.6901
One-Sided Pr < Z 0.0455
Two-Sided Pr > |Z| 0.0910
Van der Waerden One-Way Analysis
Chi-Square 2.8565
DF 1
Pr > Chi-Square 0.0910
Savage Scores (Exponential) for Variable absE
Classified by Variable group
Sum of Expected Std Dev Mean
group N Scores Under H0 Under H0 Score
Lower 13 2.470789 0.0 2.346881 0.190061
Upper 12 -2.470789 0.0 2.346881 -0.205899
Savage Two-Sample Test
Statistic -2.4708
Z -1.0528
One-Sided Pr < Z 0.1462
Two-Sided Pr > |Z| 0.2924
Savage One-Way Analysis
Chi-Square 1.1084
DF 1
Pr > Chi-Square 0.2924
Kolmogorov-Smirnov Test for Variable absE
Classified by Variable group
EDF at Deviation from Mean
group N Maximum at Maximum
Lower 13 0.000 -0.865332
Upper 12 0.500 0.900666
Total 25 0.240
Maximum Deviation Occurred at Observation 17
Value of absE at Maximum = 7.684040
Kolmogorov-Smirnov Two-Sample Test (Asymptotic)
KS 0.249800 D 0.500000
KSa 1.249000 Pr > KSa 0.0883
Cramer-von Mises Test for Variable absE
Classified by Variable group
Summed Deviation
group N from Mean
Lower 13 0.192246
Upper 12 0.208267
Cramer-von Mises Statistics (Asymptotic)
CM 0.016021 CMa 0.400513
Kuiper Test for Variable absE
Classified by Variable group
Deviation
group N from Mean
Lower 13 0.025641
Upper 12 0.500000
Kuiper Two-Sample Test (Asymptotic)
K 0.525641 Ka 1.313051 Pr > Ka 0.3751
86 TITLE2 'Other tests for homogeneity of residuals';
87 proc reg data=next2; TITLE3 'SLR'; model Y = x; run;
NOTE: 25 observations read.
NOTE: 25 observations used in computations.
NOTE: The PROCEDURE REG printed page 23.
NOTE: PROCEDURE REG used: real time 0.05 seconds
88 proc reg data=next2; TITLE3 'e*e on X'; model esquared = x; run;
NOTE: 25 observations read.
NOTE: 25 observations used in computations.
NOTE: The PROCEDURE REG printed page 24.
NOTE: PROCEDURE REG used: real time 0.00 seconds
EXST7034 - Chapter 3 examples : Toluca example
Other tests for homogeneity of residuals
SLR
The REG Procedure
Model: MODEL1
Dependent Variable: Y
Analysis of Variance Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 252378 252378 105.88 <.0001
Error 23 54825 2383.71562
Corrected Total 24 307203
Root MSE 48.82331 R-Square 0.8215
Dependent Mean 312.28000 Adj R-Sq 0.8138
Coeff Var 15.63447
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 62.36586 26.17743 2.38 0.0259
X 1 3.57020 0.34697 10.29 <.0001
Other tests for homogeneity of residuals
e*e on X
Dependent Variable: esquared
Analysis of Variance Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 7896142 7896142 1.09 0.3070
Error 23 166395896 7234604
Corrected Total 24 174292038
Root MSE 2689.72195 R-Square 0.0453
Dependent Mean 2193.01837 Adj R-Sq 0.0038
Coeff Var 122.64931
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 3590.90811 1442.13939 2.49 0.0204
X 1 -19.96985 19.11502 -1.04 0.3070
Breusch-Pagan Test -
P(>Chi square with 1 d.f.) = 0.364907535493054
89 proc reg data=next2; TITLE3 'log(e*e) on X'; model logesq = x; run;
NOTE: 25 observations read.
NOTE: 25 observations used in computations.
NOTE: The PROCEDURE REG printed page 25.
NOTE: PROCEDURE REG used:
real time 0.05 seconds
90 proc reg data=next2; TITLE3 'Log(e*e) on log(X)'; model logesq = logx;
NOTE: 25 observations read.
NOTE: 25 observations used in computations.
91 options ps=55;
NOTE: The PROCEDURE REG printed page 26.
NOTE: PROCEDURE REG used:
real time 0.05 seconds
EXST7034 - Chapter 3 examples : Toluca example
Other tests for homogeneity of residuals
log(e*e) on X
The REG Procedure
Model: MODEL1
Dependent Variable: logesq
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 15.35148 15.35148 2.78 0.1093
Error 23 127.19606 5.53026
Corrected Total 24 142.54754
Root MSE 2.35165 R-Square 0.1077
Dependent Mean 6.33733 Adj R-Sq 0.0689
Coeff Var 37.10793
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 8.28646 1.26088 6.57 <.0001
X 1 -0.02784 0.01671 -1.67 0.1093
Log(e*e) on log(X)
Dependent Variable: logesq
Analysis of Variance Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 15.55203 15.55203 2.82 0.1068
Error 23 126.99551 5.52154
Corrected Total 24 142.54754
Root MSE 2.34980 R-Square 0.1091
Dependent Mean 6.33733 Adj R-Sq 0.0704
Coeff Var 37.07867
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 13.17710 4.10248 3.21 0.0039
logX 1 -1.64898 0.98254 -1.68 0.1068
91 PROC PLOT DATA=next2; PLOT e*x / href=75 vref=0; RUN;
NOTE: There were 25 observations read from the data set WORK.NEXT2.
NOTE: The PROCEDURE PLOT printed page 27.
NOTE: PROCEDURE PLOT used: real time 0.00 seconds
92 PROC PLOT DATA=next2; PLOT logesq*logx / href=4.32; RUN;
NOTE: There were 25 observations read from the data set WORK.NEXT2.
NOTE: The PROCEDURE PLOT printed page 28.
NOTE: PROCEDURE PLOT used: real time 0.00 seconds
EXST7034 - Chapter 3 examples : Toluca example
Other tests for homogeneity of residuals
Log(e*e) on log(X)
Plot of E*X. Legend: A = 1 obs, B = 2 obs, etc.
| |
125 + |
| |
| |
| A |
100 + |
| |
| |
| | A
75 + |
| |
| |
| | A
50 + A | A
| A |
R | A |
e | |
s 25 + A |
i | |
d | A |
u | | A A
a 0 +-----------------------------------------+----------------A----------------
l | | A B
| |
| A A | A
-25 + |
| | A
| |
| A |
-50 + A A |
| |
| A |
| | A
-75 + |
| A |
| |
| |
-100 + |
---+------+------+------+------+------+------+------+------+------+------+--
20 30 40 50 60 70 80 90 100 110 120
X
Log(e*e) on log(X)
Plot of logesq*logX. Legend: A = 1 obs, B = 2 obs, etc.
10 + |
| |
| |
| A |
| A | A
| |
| A | A
8 + A | A A
| A A A |
| A A |
| | A
| |
| A |
| |
6 + A A | A
| |
| |
logesq | |
| A |
| |
| |
4 + | A
| | A A
| | A
| |
| | A
| |
| |
2 + |
| |
| |
| |
| |
| |
| |
0 + |
| |
| |
| | A
| |
---+---------------+---------------+---------------+---------------+--
3.0 3.5 4.0 4.5 5.0
logX
94 proc freq data=one; table x / NOROW NOCOL NOPERCENT; run;
NOTE: There were 25 observations read from the data set WORK.ONE.
NOTE: The PROCEDURE FREQ printed page 29.
NOTE: PROCEDURE FREQ used:
real time 0.04 seconds
EXST7034 - Chapter 3 examples : Toluca example
Other tests for homogeneity of residuals
Log(e*e) on log(X)
The FREQ Procedure
Cumulative
X Frequency Frequency
------------------------------
20 1 1
30 3 4
40 2 6
50 3 9
60 1 10
70 3 13
80 3 16
90 4 20
100 2 22
110 2 24
120 1 25
95 proc mixed DATA=ONE; CLASSES AnotherX;
96 title2 'Analysis of Lack of Fit using PROC MIXED - Full Model';
97 model Y = AnotherX / htype=1 3 DDFM=Satterthwaite solution;
98 run;
NOTE: The PROCEDURE MIXED printed pages 30-31.
NOTE: PROCEDURE MIXED used:
real time 0.05 seconds
EXST7034 - Chapter 3 examples : Toluca example
Analysis of Lack of Fit using PROC MIXED - Full Model
The Mixed Procedure
Model Information
Data Set WORK.ONE
Dependent Variable Y
Covariance Structure Diagonal
Estimation Method REML
Residual Variance Method Profile
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Residual
Class Level Information
Class Levels Values
anotherX 11 20 30 40 50 60 70 80 90 100 110 120
Dimensions
Covariance Parameters 1
Columns in X 12
Columns in Z 0
Subjects 1
Max Obs Per Subject 25
Observations Used 25
Observations Not Used 0
Total Observations 25
EXST7034 - Chapter 3 examples : Toluca example
Analysis of Lack of Fit using PROC MIXED - Full Model
The Mixed Procedure
Covariance Parameter Estimates
Cov Parm Estimate
Residual 2684.35
Fit Statistics
-2 Res Log Likelihood 158.1
AIC (smaller is better) 160.1
AICC (smaller is better) 160.5
BIC (smaller is better) 160.8
Solution for Fixed Effects
another Standard
Effect X Estimate Error DF t Value Pr > |t|
Intercept 546.00 51.8107 14 10.54 <.0001
anotherX 20 -433.00 73.2713 14 -5.91 <.0001
anotherX 30 -344.00 59.8258 14 -5.75 <.0001
anotherX 40 -344.00 63.4548 14 -5.42 <.0001
anotherX 50 -330.67 59.8258 14 -5.53 <.0001
anotherX 60 -322.00 73.2713 14 -4.39 0.0006
anotherX 70 -234.00 59.8258 14 -3.91 0.0016
anotherX 80 -181.67 59.8258 14 -3.04 0.0089
anotherX 90 -143.50 57.9261 14 -2.48 0.0266
anotherX 100 -159.50 63.4548 14 -2.51 0.0248
anotherX 110 -118.00 63.4548 14 -1.86 0.0841
anotherX 120 0 . . . .
Type 1 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
anotherX 10 14 10.04 <.0001
Type 3 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
anotherX 10 14 10.04 <.0001
99 proc mixed DATA=ONE; CLASSES AnotherX;
100 title2 'Analysis of Lack of Fit using PROC MIXED';
101 model Y = X AnotherX/ htype=1 3 DDFM=Satterthwaite solution;
102 run;
NOTE: The PROCEDURE MIXED printed pages 32-33.
NOTE: PROCEDURE MIXED used:
real time 0.00 seconds
EXST7034 - Chapter 3 examples : Toluca example
Analysis of Lack of Fit using PROC MIXED
The Mixed Procedure
Model Information
Data Set WORK.ONE
Dependent Variable Y
Covariance Structure Diagonal
Estimation Method REML
Residual Variance Method Profile
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Residual
Class Level Information
Class Levels Values
anotherX 11 20 30 40 50 60 70 80 90 100
110 120
Dimensions
Covariance Parameters 1
Columns in X 13
Columns in Z 0
Subjects 1
Max Obs Per Subject 25
Observations Used 25
Observations Not Used 0
Total Observations 25
Covariance Parameter Estimates
Cov Parm Estimate
Residual 2684.35
Fit Statistics
-2 Res Log Likelihood 162.7
AIC (smaller is better) 164.7
AICC (smaller is better) 165.1
BIC (smaller is better) 165.4
Solution for Fixed Effects
another Standard
Effect X Estimate Error DF t Value Pr > |t|
Intercept -870.00 719.78 14 -1.21 0.2468
X 11.8000 6.3455 14 1.86 0.0841
anotherX 20 747.00 595.26 14 1.25 0.2301
anotherX 30 718.00 530.48 14 1.35 0.1973
anotherX 40 600.00 467.73 14 1.28 0.2204
anotherX 50 495.33 404.10 14 1.23 0.2405
anotherX 60 386.00 343.67 14 1.12 0.2803
anotherX 70 356.00 278.21 14 1.28 0.2215
anotherX 80 290.33 215.71 14 1.35 0.1997
anotherX 90 210.50 153.26 14 1.37 0.1912
anotherX 100 76.5000 96.9289 14 0.79 0.4431
anotherX 110 0 . . . .
anotherX 120 0 . . . .
Type 1 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
X 1 14 94.02 <.0001
anotherX 9 14 0.71 0.6893
Type 3 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
X 0 . . .
anotherX 9 14 0.71 0.6893
103 proc GLM DATA=ONE; CLASSES AnotherX;
104 title2 'Analysis of Lack of Fit using PROC glm';
105 model Y = X AnotherX / solution;
106 run;
NOTE: Unable to find the "Note" style element. Default style attributes will be used.
NOTE: The PROCEDURE GLM printed pages 34-35.
NOTE: PROCEDURE GLM used:
real time 0.04 seconds
EXST7034 - Chapter 3 examples : Toluca example
Analysis of Lack of Fit using PROC glm
The GLM Procedure
Class Level Information
Class Levels Values
anotherX 11 20 30 40 50 60 70 80 90 100 110 120
Number of observations 25
Dependent Variable: Y
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 10 269622.2067 26962.2207 10.04 <.0001
Error 14 37580.8333 2684.3452
Corrected Total 24 307203.0400
R-Square Coeff Var Root MSE Y Mean
0.877668 16.59109 51.81067 312.2800
Source DF Type I SS Mean Square F Value Pr > F
X 1 252377.5808 252377.5808 94.02 <.0001
anotherX 9 17244.6259 1916.0695 0.71 0.6893
Source DF Type III SS Mean Square F Value Pr > F
X 0 0.00000 . . .
anotherX 9 17244.62586 1916.06954 0.71 0.6893
Standard
Parameter Estimate Error t Value Pr > |t|
Intercept -870.0000000 B 719.7767925 -1.21 0.2468
X 11.8000000 B 6.3454849 1.86 0.0841
anotherX 20 747.0000000 B 595.2592472 1.25 0.2301
anotherX 30 718.0000000 B 530.4798386 1.35 0.1973
anotherX 40 600.0000000 B 467.7329761 1.28 0.2204
anotherX 50 495.3333333 B 404.1010624 1.23 0.2405
anotherX 60 386.0000000 B 343.6730866 1.12 0.2803
anotherX 70 356.0000000 B 278.2060766 1.28 0.2215
anotherX 80 290.3333333 B 215.7050087 1.35 0.1997
anotherX 90 210.5000000 B 153.2580205 1.37 0.1912
anotherX 100 76.5000000 B 96.9288829 0.79 0.4431
anotherX 110 0.0000000 B . . .
anotherX 120 0.0000000 B . . .
NOTE: The X'X matrix has been found to be singular, and a generalized inverse was
used to solve the normal equations. Terms whose estimates are followed by the
letter 'B' are not uniquely estimable.
108 proc transreg data=one;
109 MODEL BOXCOX(Y) = identity(X);
110 run;
NOTE: Algorithm converged.
NOTE: There were 25 observations read from the data set WORK.ONE.
NOTE: The PROCEDURE TRANSREG printed pages 36-37.
NOTE: PROCEDURE TRANSREG used: real time 0.05 seconds
EXST7034 - Chapter 3 examples : Toluca example
Box-Cox transformation with PROC TRANSREG
The TRANSREG Procedure
Transformation Information for BoxCox(Y)
Lambda R-Square Log Like
-3.00 0.42 -143.735
-2.75 0.45 -139.597
-2.50 0.47 -135.541
-2.25 0.50 -131.573
-2.00 0.53 -127.700
-1.75 0.56 -123.933
-1.50 0.59 -120.282
-1.25 0.62 -116.762
-1.00 0.66 -113.394
-0.75 0.69 -110.205
-0.50 0.72 -107.231
-0.25 0.75 -104.519
0.00 0.77 -102.129
0.25 0.79 -100.133
0.50 0.81 -98.605 *
0.75 0.82 -97.614 *
1.00 + 0.82 -97.205 <
1.25 0.82 -97.388 *
1.50 0.82 -98.132 *
1.75 0.81 -99.376
2.00 0.79 -101.039
2.25 0.78 -103.039
2.50 0.76 -105.301
2.75 0.74 -107.765
3.00 0.71 -110.382
< - Best Lambda
* - Confidence Interval
+ - Convenient Lambda
EXST7034 - Chapter 3 examples : Toluca example
Analysis of Lack of Fit using PROC glm
The TRANSREG Procedure
TRANSREG Univariate Algorithm Iteration History for BoxCox(Y)
Iteration Average Maximum Criterion
Number Change Change R-Square Change Note
-------------------------------------------------------------------------
1 0.00000 0.00000 0.82153 Converged
Algorithm converged.
138 PROC loess DATA=ONE; TITLE2 'Loess Models';
139 MODEL Y = X / CLM R smooth=0.1 to 0.8 by 0.1;
140 ods output OutputStatistics=Results;
141 run;
NOTE: The DFMETHOD=EXACT option is implicitly selected when the STD CLM or T option is
used.
WARNING: At smoothing parameter 0.1, the local SSCP matrix for 11 fit point(s) is
numerically singular. The fitted value and standard errors at those points are
not uniquely defined.
WARNING: At smoothing parameter 0.2, the local SSCP matrix for 9 fit point(s) is
numerically singular. The fitted value and standard errors at those points are
not uniquely defined.
WARNING: At smoothing parameter 0.3, the local SSCP matrix for 6 fit point(s) is
numerically singular. The fitted value and standard errors at those points are
not uniquely defined.
WARNING: At smoothing parameter 0.4, the local SSCP matrix for 1 fit point(s) is
numerically singular. The fitted value and standard errors at those points are
not uniquely defined.
NOTE: The data set WORK.RESULTS has 200 observations and 8 variables.
NOTE: The PROCEDURE LOESS printed pages 48-56.
NOTE: PROCEDURE LOESS used:
real time 0.10 seconds
143 proc sort data=Results; by SmoothingParameter; run;
NOTE: There were 200 observations read from the data set WORK.RESULTS.
NOTE: The data set WORK.RESULTS has 200 observations and 8 variables.
NOTE: PROCEDURE SORT used:
real time 0.00 seconds
144
145 options ps=65 ls=123;
146 proc plot data=results; by SmoothingParameter; plot depvar*x='o' pred*x='p' / overlay; run;
NOTE: There were 200 observations read from the data set WORK.RESULTS.
NOTE: The PROCEDURE PLOT printed pages 57-64.
NOTE: PROCEDURE PLOT used:
real time 0.05 seconds
149 data results; set results; format x 3.0 depvar 3.0; run;
NOTE: There were 200 observations read from the data set WORK.RESULTS.
NOTE: The data set WORK.RESULTS has 200 observations and 8 variables.
NOTE: DATA statement used:
real time 0.05 seconds
150 proc print data=Results; run;
NOTE: There were 200 observations read from the data set WORK.RESULTS.
NOTE: The PROCEDURE PRINT printed pages 65-68.
NOTE: PROCEDURE PRINT used:
real time 0.04 seconds
152 GOPTIONS DEVICE=CGMflwa GSFMODE=REPLACE GSFNAME=OUT1 NOPROMPT noROTATE
153 ftext='TimesRoman' ftitle='TimesRoman' htext=1 htitle=1 ctitle=black ctext=black;
154 FILENAME OUT1 'C:\Geaghan\EXST\EXST7034New\Fall2002\SAS\Lowess for Toluca.cgm';
155 PROC GPLOT DATA=results; by SmoothingParameter; TITLE1 H=1 'Lowess line';
156 PLOT depvar*x=1 depvar*x=2 pred*x=3 / HAXIS=AXIS1 VAXIS=AXIS2 overlay;
157 AXIS1 LABEL=(H=1 'Lot size') WIDTH=1 MINOR=(N=1)
158 VALUE=(H=1) ORDER=20 TO 120 BY 10;
159 AXIS2 LABEL=(ANGLE=90 H=1 'Hours') WIDTH=1
160 VALUE=(H=1) MINOR=(N=4) ORDER=100 TO 600 BY 50;
161 SYMBOL1 C=red V=J I=none W=1 H=1 F=SPECIAL MODE=INCLUDE;
162 SYMBOL2 C=blue V=none I=RLclm95 L=1 W=1 H=1 MODE=INCLUDE;
163 SYMBOL3 C=orange V=none I=join L=1 W=1 H=1 MODE=INCLUDE;
164 RUN;
EXST7034 - Chapter 3 examples : Toluca example
Loess Models
The LOESS Procedure
Independent Variable Scaling
Scaling applied: None
Statistic X
Minimum Value 20.00000
Maximum Value 120.00000
Smoothing Parameter: 0.1
Dependent Variable: Y
Fit Summary
Fit Method kd Tree
Blending Linear
Number of Observations 25
Number of Fitting Points 11
kd Tree Bucket Size 1
Degree of Local Polynomials 1
Smoothing Parameter 0.10000
Points in Local Neighborhood 2
Residual Sum of Squares 48793
Trace[L] 11.00000
GCV 248.94133
AICC 10.57646
AICC1 257.06446
Delta1 17.00000
Delta2 22.00000
Equivalent Number of Parameters 14.00000
Lookup Degrees of Freedom 13.13636
Residual Standard Error 53.57375
Smoothing Parameter: 0.2
Dependent Variable: Y
Fit Summary
Fit Method kd Tree
Blending Linear
Number of Observations 25
Number of Fitting Points 11
kd Tree Bucket Size 1
Degree of Local Polynomials 1
Smoothing Parameter 0.20000
Points in Local Neighborhood 5
Residual Sum of Squares 37581
Trace[L] 11.00000
GCV 191.73895
AICC 10.31537
AICC1 257.88434
Delta1 14.00000
Delta2 14.00000
Equivalent Number of Parameters 11.00000
Lookup Degrees of Freedom 14.00000
Residual Standard Error 51.81067
Smoothing Parameter: 0.3
Dependent Variable: Y
Fit Summary
Fit Method kd Tree
Blending Linear
Number of Observations 25
Number of Fitting Points 11
kd Tree Bucket Size 1
Degree of Local Polynomials 1
Smoothing Parameter 0.30000
Points in Local Neighborhood 7
Residual Sum of Squares 39843
Trace[L] 9.31323
GCV 161.91504
AICC 9.88087
AICC1 247.61182
Delta1 15.32538
Delta2 14.62872
Equivalent Number of Parameters 8.95184
Lookup Degrees of Freedom 16.05522
Residual Standard Error 50.98841
Smoothing Parameter: 0.4
Dependent Variable: Y
Fit Summary
Fit Method kd Tree
Blending Linear
Number of Observations 25
Number of Fitting Points 11
kd Tree Bucket Size 2
Degree of Local Polynomials 1
Smoothing Parameter 0.40000
Points in Local Neighborhood 10
Residual Sum of Squares 45775
Trace[L] 5.81868
GCV 124.41338
AICC 9.30634
AICC1 233.26857
Delta1 18.56795
Delta2 18.14650
Equivalent Number of Parameters 5.20530
Lookup Degrees of Freedom 18.99919
Residual Standard Error 49.65125
Smoothing Parameter: 0.5
Dependent Variable: Y
Fit Summary
Fit Method kd Tree
Blending Linear
Number of Observations 25
Number of Fitting Points 11
kd Tree Bucket Size 2
Degree of Local Polynomials 1
Smoothing Parameter 0.50000
Points in Local Neighborhood 12
Residual Sum of Squares 46341
Trace[L] 5.03022
GCV 116.20295
AICC 9.19605
AICC1 230.41014
Delta1 19.35613
Delta2 19.02074
Equivalent Number of Parameters 4.41657
Lookup Degrees of Freedom 19.69744
Residual Standard Error 48.92971
Smoothing Parameter: 0.6
Dependent Variable: Y
Fit Summary
Fit Method kd Tree
Blending Linear
Number of Observations 25
Number of Fitting Points 11
kd Tree Bucket Size 2
Degree of Local Polynomials 1
Smoothing Parameter 0.60000
Points in Local Neighborhood 15
Residual Sum of Squares 50029
Trace[L] 3.87019
GCV 112.05442
AICC 9.11065
AICC1 228.03777
Delta1 20.66097
Delta2 20.41581
Equivalent Number of Parameters 3.40134
Lookup Degrees of Freedom 20.90907
Residual Standard Error 49.20790
Smoothing Parameter: 0.7
Dependent Variable: Y
Fit Summary
Fit Method kd Tree
Blending Linear
Number of Observations 25
Number of Fitting Points 9
kd Tree Bucket Size 3
Degree of Local Polynomials 1
Smoothing Parameter 0.70000
Points in Local Neighborhood 17
Residual Sum of Squares 52150
Trace[L] 3.38284
GCV 111.59776
AICC 9.08984
AICC1 227.42141
Delta1 21.23026
Delta2 20.95352
Equivalent Number of Parameters 2.99594
Lookup Degrees of Freedom 21.51066
Residual Standard Error 49.56198
Smoothing Parameter: 0.8
Dependent Variable: Y
Fit Summary
Fit Method kd Tree
Blending Linear
Number of Observations 25
Number of Fitting Points 9
kd Tree Bucket Size 3
Degree of Local Polynomials 1
Smoothing Parameter 0.80000
Points in Local Neighborhood 20
Residual Sum of Squares 53326
Trace[L] 3.07846
GCV 110.96674
AICC 9.07475
AICC1 227.00497
Delta1 21.59115
Delta2 21.37400
Equivalent Number of Parameters 2.74806
Lookup Degrees of Freedom 21.81050
Residual Standard Error 49.69694
EXST7034 - Chapter 3 examples : Toluca example
Loess Models
Smoothing Dep
Obs Parameter Obs X Var Pred Residual LowerCL UpperCL
1 0.1 1 80 399 375.50000 23.50000 293.74640 457.25360
2 0.1 2 30 121 166.50000 -45.50000 84.74640 248.25360
3 0.1 3 50 221 244.50000 -23.50000 162.74640 326.25360
. . .
24 0.1 24 80 342 375.50000 -33.50000 293.74640 457.25360
25 0.1 25 70 323 287.50000 35.50000 205.74640 369.25360
26 0.2 1 80 399 364.33333 34.66667 300.17654 428.49013
27 0.2 2 30 121 202.00000 -81.00000 137.84320 266.15680
28 0.2 3 50 221 215.33333 5.66667 151.17654 279.49013
. . .
49 0.2 24 80 342 364.33333 -22.33333 300.17654 428.49013
50 0.2 25 70 323 312.00000 11.00000 247.84320 376.15680
51 0.3 1 80 399 364.33333 34.66667 301.94464 426.72203
52 0.3 2 30 121 185.39119 -64.39119 139.14129 231.64108
53 0.3 3 50 221 214.46246 6.53754 168.21256 260.71236
. . .
74 0.3 24 80 342 364.33333 -22.33333 301.94464 426.72203
75 0.3 25 70 323 312.00000 11.00000 249.61130 374.38870
76 0.4 1 80 399 364.33333 34.66667 304.33420 424.33247
77 0.4 2 30 121 181.31970 -60.31970 137.83003 224.80938
78 0.4 3 50 221 214.46246 6.53754 169.98399 258.94093
. . .
99 0.4 24 80 342 364.33333 -22.33333 304.33420 424.33247
100 0.4 25 70 323 304.84581 18.15419 261.59917 348.09245
101 0.5 1 80 399 360.04835 38.95165 326.88093 393.21577
102 0.5 2 30 121 180.61465 -59.61465 138.09368 223.13562
103 0.5 3 50 221 214.46246 6.53754 170.73529 258.18963
104 0.5 4 90 376 390.43038 -14.43038 355.30511 425.55565
105 0.5 5 70 361 304.84581 56.15419 262.32966 347.36196
106 0.5 6 60 224 255.76420 -31.76420 216.65128 294.87712
107 0.5 7 120 546 503.21276 42.78724 427.82299 578.60254
108 0.5 8 80 352 360.04835 -8.04835 326.88093 393.21577
109 0.5 9 100 353 404.93224 -51.93224 366.50549 443.35899
110 0.5 10 50 157 214.46246 -57.46246 170.73529 258.18963
111 0.5 11 40 160 203.53581 -43.53581 169.91203 237.15958
112 0.5 12 70 252 304.84581 -52.84581 262.32966 347.36196
113 0.5 13 90 389 390.43038 -1.43038 355.30511 425.55565
114 0.5 14 20 113 156.52948 -43.52948 89.98035 223.07861
115 0.5 15 110 435 452.24519 -17.24519 405.20963 499.28076
116 0.5 16 100 420 404.93224 15.06776 366.50549 443.35899
117 0.5 17 30 212 180.61465 31.38535 138.09368 223.13562
118 0.5 18 50 268 214.46246 53.53754 170.73529 258.18963
119 0.5 19 90 377 390.43038 -13.43038 355.30511 425.55565
120 0.5 20 110 421 452.24519 -31.24519 405.20963 499.28076
121 0.5 21 30 273 180.61465 92.38535 138.09368 223.13562
122 0.5 22 90 468 390.43038 77.56962 355.30511 425.55565
123 0.5 23 40 244 203.53581 40.46419 169.91203 237.15958
124 0.5 24 80 342 360.04835 -18.04835 326.88093 393.21577
125 0.5 25 70 323 304.84581 18.15419 262.32966 347.36196
126 0.6 1 80 399 354.47845 44.52155 324.28913 384.66778
127 0.6 2 30 121 178.90856 -57.90856 137.26480 220.55231
128 0.6 3 50 221 227.31067 -6.31067 194.66589 259.95546
. . .
149 0.6 24 80 342 354.47845 -12.47845 324.28913 384.66778
150 0.6 25 70 323 307.47386 15.52614 276.75071 338.19701
151 0.7 1 80 399 354.47845 44.52155 324.12383 384.83308
152 0.7 2 30 121 175.84907 -54.84907 135.04365 216.65450
153 0.7 3 50 221 235.10419 -14.10419 206.79752 263.41086
. . .
174 0.7 24 80 342 354.47845 -12.47845 324.12383 384.83308
175 0.7 25 70 323 307.47386 15.52614 276.58248 338.36524
176 0.8 1 80 399 348.82099 50.17901 321.86329 375.77869
177 0.8 2 30 121 175.84907 -54.84907 134.96590 216.73225
178 0.8 3 50 221 235.10419 -14.10419 206.74359 263.46480
. . .
199 0.8 24 80 342 348.82099 -6.82099 321.86329 375.77869
200 0.8 25 70 323 308.22776 14.77224 281.50283 334.95268