data lntrans;
input case x1 x2 x3 x4 y logy;
cards;
01 6.7 62 81 2.59 200 2.3010
02 5.1 59 66 1.70 101 2.0043
03 7.4 57 83 2.16 204 2.3096
04 6.5 73 41 2.01 101 2.0043
05 7.8 65 115 4.30 509 2.7067
06 5.8 38 72 1.42 80 1.9031
07 5.7 46 63 1.91 80 1.9031
08 3.7 68 81 2.57 127 2.1038
09 6.0 67 93 2.50 202 2.3054
10 3.7 76 94 2.40 203 2.3075
11 6.3 84 83 4.13 329 2.5172
12 6.7 51 43 1.86 65 1.8129
13 5.8 96 114 3.95 830 2.9191
14 5.8 83 88 3.95 330 2.5185
15 7.7 62 67 3.40 168 2.2253
16 7.4 74 68 2.40 217 2.3365
17 6.0 85 28 2.98 87 1.9395
18 3.7 51 41 1.55 34 1.5315
19 7.3 68 74 3.56 215 2.3324
20 5.6 57 87 3.02 172 2.2355
21 5.2 52 76 2.85 109 2.0374
22 3.4 83 53 1.12 136 2.1335
23 6.7 26 68 2.10 70 1.8451
24 5.8 67 86 3.40 220 2.3424
25 6.3 59 100 2.95 276 2.4409
26 5.8 61 73 3.50 144 2.1584
27 5.2 52 86 2.45 181 2.2577
28 11.2 76 90 5.59 574 2.7589
29 5.2 54 56 2.71 72 1.8573
30 5.8 76 59 2.58 178 2.2504
31 3.2 64 65 0.74 71 1.8513
32 8.7 45 23 2.52 58 1.7634
33 5.0 59 73 3.50 116 2.0645
34 5.8 72 93 3.30 295 2.4698
35 5.4 58 70 2.64 115 2.0607
36 5.3 51 99 2.60 184 2.2648
37 2.6 74 86 2.05 118 2.0719
38 4.3 8 119 2.85 120 2.0792
39 4.8 61 76 2.45 151 2.1790
40 5.4 52 88 1.81 148 2.1703
41 5.2 49 72 1.84 95 1.9777
42 3.6 28 99 1.30 75 1.8751
43 8.8 86 88 6.40 483 2.6840
44 6.5 56 77 2.85 153 2.1847
45 3.4 77 93 1.48 191 2.2810
46 6.5 40 84 3.00 123 2.0899
47 4.5 73 106 3.05 311 2.4928
48 4.8 86 101 4.10 398 2.5999
49 5.1 67 77 2.86 158 2.1987
50 3.9 82 103 4.55 310 2.4914
51 6.6 77 46 1.95 124 2.0934
52 6.4 85 40 1.21 125 2.0969
53 6.4 59 85 2.33 198 2.2967
54 8.8 78 72 3.20 313 2.4955
;
proc chart;
hbar x1 x2 x3 x4 y logy;
proc reg simple corr;
model y=x1 x2 x3 x4 /adjrsq corrb dw stb tol vif r pcorr2 scorr2 sse mse;
output out=resid student=sresid;
plot (y)*(x1 x2 x3 x4) (residual.)*(x1 x2 x3 x4)
residual.*predicted. / hplots=2 vplots=2;
proc chart data=resid;
vbar sresid;
proc univariate plot;
var sresid;
proc reg;
model y=x1 x2 x3 x4 /adjrsq stb tol selection=forward sle=.05;
model y=x1 x2 x3 x4 / adjrsq stb tol selection=backward sle=.1;
model y=x1 x2 x3 x4 / adjrsq stb tol selection=stepwise sle=.05;
proc reg simple corr;
model logy=x1 x2 x3 x4 /adjrsq corrb dw stb tol vif r pcorr2 scorr2 sse
mse;
output out=resid2 student=sresid2;
plot (logy)*(x1 x2 x3 x4) (residual.)*(x1 x2 x3 x4)
residual.*predicted. / hplots=2 vplots=2;
proc chart data=resid2;
vbar sresid2;
proc univariate plot;
var sresid2;
proc reg;
model logy=x1 x2 x3 x4 / adjrsq stb tol selection=forward sle=.05;
model logy=x1 x2 x3 x4 / adjrsq stb tol selection=backward sle=.1;
model logy=x1 x2 x3 x4 / adjrsq stb tol selection=stepwise sle=.05;
proc reg;
model x1 = x2 x3 x4 /adjrsq stb;
model x2 = x1 x3 x4 /adjrsq stb;
model x3 = x1 x2 x4 /adjrsq stb;
model x4 = x1 x2 x3 /adjrsq stb;
run;
The SAS System 14:43 Thursday, December 23, 1999 1
X1 Cum. Cum.
Midpoint Freq Freq Percent Percent
|
3.0 |******** 8 8 14.81 14.81
|
4.5 |************ 12 20 22.22 37.04
|
6.0 |************************* 25 45 46.30 83.33
|
7.5 |***** 5 50 9.26 92.59
|
9.0 |*** 3 53 5.56 98.15
|
10.5 |* 1 54 1.85 100.00
|
-----+----+----+----+----+
5 10 15 20 25
Frequency
The SAS System 14:43 Thursday, December 23, 1999 2
X2 Cum. Cum.
Midpoint Freq Freq Percent Percent
|
15 |** 1 1 1.85 1.85
|
30 |**** 2 3 3.70 5.56
|
45 |********************** 11 14 20.37 25.93
|
60 |************************************ 18 32 33.33 59.26
|
75 |**************************** 14 46 25.93 85.19
|
90 |**************** 8 54 14.81 100.00
|
----+---+---+---+---+---+---+---+---+
2 4 6 8 10 12 14 16 18
Frequency
The SAS System 14:43 Thursday, December 23, 1999 3
X3 Cum. Cum.
Midpoint Freq Freq Percent Percent
|
22.5 |**** 2 2 3.70 3.70
|
37.5 |******** 4 6 7.41 11.11
|
52.5 |******** 4 10 7.41 18.52
|
67.5 |************************** 13 23 24.07 42.59
|
82.5 |********************************** 17 40 31.48 74.07
|
97.5 |******************** 10 50 18.52 92.59
|
112.5 |******** 4 54 7.41 100.00
|
----+---+---+---+---+---+---+---+--
2 4 6 8 10 12 14 16
Frequency
The SAS System 14:43 Thursday, December 23, 1999 4
X4 Cum. Cum.
Midpoint Freq Freq Percent Percent
|
1 |************ 6 6 11.11 11.11
|
2 |******************************** 16 22 29.63 40.74
|
3 |****************************************** 21 43 38.89 79.63
|
4 |**************** 8 51 14.81 94.44
|
5 |** 1 52 1.85 96.30
|
6 |**** 2 54 3.70 100.00
|
----+---+---+---+---+---+---+---+---+---+--
2 4 6 8 10 12 14 16 18 20
Frequency
The SAS System 14:43 Thursday, December 23, 1999 5
Y Cum. Cum.
Midpoint Freq Freq Percent Percent
|
75 |************************* 25 25 46.30 46.30
|
225 |******************* 19 44 35.19 81.48
|
375 |****** 6 50 11.11 92.59
|
525 |*** 3 53 5.56 98.15
|
675 | 0 53 0.00 98.15
|
825 |* 1 54 1.85 100.00
|
-----+----+----+----+----+
5 10 15 20 25
Frequency
The SAS System 14:43 Thursday, December 23, 1999 6
LOGY Cum. Cum.
Midpoint Freq Freq Percent Percent
|
1.625 |** 1 1 1.85 1.85
|
1.875 |******************** 10 11 18.52 20.37
|
2.125 |************************************** 19 30 35.19 55.56
|
2.375 |********************************** 17 47 31.48 87.04
|
2.625 |********** 5 52 9.26 96.30
|
2.875 |**** 2 54 3.70 100.00
|
----+---+---+---+---+---+---+---+---+--
2 4 6 8 10 12 14 16 18
Frequency
The SAS System 14:43 Thursday, December 23, 1999 7
Descriptive Statistics
Variables Sum Mean Uncorrected SS
INTERCEP 54 1 54
X1 312.3 5.7833333333 1942.33
X2 3415 63.240740741 231109
X3 4164 77.111111111 345032
X4 148.19 2.7442592593 467.3919
Y 10647 197.16666667 3218165
Variables Variance Std Deviation
INTERCEP 0 0
X1 2.5697169811 1.6030336806
X2 285.69566737 16.902534347
X3 451.72327044 21.253782497
X4 1.1456626485 1.0703563185
Y 21111.915094 145.29939812
Correlation
CORR X1 X2 X3 X4 Y
X1 1.0000 0.0901 -0.1496 0.5024 0.3725
X2 0.0901 1.0000 -0.0236 0.3690 0.5540
X3 -0.1496 -0.0236 1.0000 0.4164 0.5802
X4 0.5024 0.3690 0.4164 1.0000 0.7223
Y 0.3725 0.5540 0.5802 0.7223 1.0000
The SAS System 14:43 Thursday, December 23, 1999 8
Model: MODEL1
Dependent Variable: Y
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 4 936264.53767 234066.13442 62.788 0.0001
Error 49 182666.96233 3727.89719
C Total 53 1118931.5
Root MSE 61.05651 R-square 0.8367
Dep Mean 197.16667 Adj R-sq 0.8234
C.V. 30.96695
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP 1 -621.597550 64.80042601 -9.592 0.0001
X1 1 33.163828 7.01727463 4.726 0.0001
X2 1 4.271860 0.56338454 7.582 0.0001
X3 1 4.125738 0.51116093 8.071 0.0001
X4 1 14.091563 12.52532754 1.125 0.2661
Squared Squared
Standardized Semi-partial Partial
Variable DF Estimate Corr Type II Corr Type II Tolerance
INTERCEP 1 0.00000000 . . .
X1 1 0.36588406 0.07441368 0.31310333 0.55586050
X2 1 0.49694120 0.19155106 0.53988110 0.77566563
X3 1 0.60349558 0.21704434 0.57072538 0.59593691
X4 1 0.10380630 0.00421697 0.02518072 0.39133899
Variance
Variable DF Inflation
INTERCEP 1 0.00000000
X1 1 1.79901252
X2 1 1.28921530
X3 1 1.67802997
X4 1 2.55532933
Correlation of Estimates
CORRB INTERCEP X1 X2 X3 X4
INTERCEP 1.0000 -0.7167 -0.6382 -0.7516 0.5155
X1 -0.7167 1.0000 0.2464 0.4958 -0.6535
X2 -0.6382 0.2464 1.0000 0.2987 -0.4668
X3 -0.7516 0.4958 0.2987 1.0000 -0.6248
X4 0.5155 -0.6535 -0.4668 -0.6248 1.0000
The SAS System 14:43 Thursday, December 23, 1999 9
Durbin-Watson D 1.720
(For Number of Obs.) 54
1st Order Autocorrelation 0.134
The SAS System 14:43 Thursday, December 23, 1999 10
Dep Var Predict Std Err Std Err Student
Obs Y Value Predict Residual Residual Residual
1 200.0 236.1 12.166 -36.1374 59.832 -0.604
2 101.0 95.8321 11.742 5.1679 59.917 0.086
3 204.0 240.2 19.814 -36.1848 57.752 -0.627
4 101.0 103.3 17.502 -2.2924 58.494 -0.039
5 509.0 449.8 21.570 59.1952 57.120 1.036
6 80.0000 50.1465 17.392 29.8535 58.527 0.510
7 80.0000 50.7782 13.299 29.2218 59.591 0.490
8 127.0 162.0 14.930 -34.9952 59.203 -0.591
9 202.0 282.5 14.780 -80.5226 59.241 -1.359
10 203.0 247.4 16.756 -44.4091 58.712 -0.756
11 329.0 346.8 15.652 -17.8052 59.016 -0.302
12 65.0000 22.0820 17.433 42.9180 58.515 0.733
13 830.0 506.8 24.210 323.2 56.052 5.765
14 330.0 344.0 15.035 -14.0437 59.177 -0.237
15 168.0 223.0 13.683 -54.9550 59.504 -0.924
16 217.0 254.3 17.396 -37.3024 58.526 -0.637
17 87.0000 98.0070 26.476 -11.0070 55.018 -0.200
18 34.0000 -90.0293 23.686 124.0 56.275 2.204
19 215.0 266.5 11.990 -51.4555 59.868 -0.859
20 172.0 209.1 10.051 -37.1116 60.224 -0.616
21 109.0 126.7 12.691 -17.7081 59.723 -0.297
22 136.0 80.1705 22.860 55.8295 56.615 0.986
23 70.0000 21.8109 21.503 48.1891 57.145 0.843
24 220.0 259.7 10.194 -39.6921 60.200 -0.659
25 276.0 293.5 14.435 -17.5182 59.326 -0.295
26 144.0 181.8 14.160 -37.8355 59.392 -0.637
27 181.0 162.3 10.802 18.6711 60.093 0.311
28 574.0 524.6 31.422 49.4130 52.350 0.944
29 72.0000 50.7643 17.528 21.2357 58.486 0.363
30 178.0 175.2 12.671 2.8112 59.727 0.047
31 71.0000 36.5265 19.833 34.4735 57.746 0.597
32 58.0000 -10.4358 28.677 68.4358 53.903 1.270
33 116.0 146.8 18.272 -30.7607 58.258 -0.528
34 295.0 308.5 11.367 -13.5224 59.989 -0.225
35 115.0 131.3 10.286 -16.2584 60.184 -0.270
36 184.0 217.1 13.783 -33.1217 59.481 -0.557
37 118.0 164.4 19.870 -46.4472 57.733 -0.805
38 120.0 86.3056 34.172 33.6944 50.598 0.666
39 151.0 146.3 10.028 4.7473 60.227 0.079
40 148.0 168.2 15.094 -20.1945 59.161 -0.341
41 95.0000 83.1571 11.924 11.8429 59.881 0.198
42 75.0000 44.1714 23.515 30.8286 56.347 0.547
43 483.0 490.9 31.129 -7.8751 52.525 -0.150
44 153.0 191.0 9.945 -38.0343 60.241 -0.631
45 191.0 224.6 22.221 -33.6418 56.869 -0.592
46 123.0 153.7 15.811 -30.6784 58.974 -0.520
47 311.0 319.8 16.127 -8.7930 58.888 -0.149
48 398.0 379.4 19.382 18.5563 57.898 0.320
49 158.0 191.7 10.007 -33.7363 60.231 -0.560
50 310.0 347.1 26.016 -37.1015 55.236 -0.672
51 124.0 143.5 17.903 -19.4794 58.373 -0.334
52 125.0 135.8 25.418 -10.8394 55.514 -0.195
53 198.0 226.2 13.676 -28.2118 59.505 -0.474
54 313.0 345.6 21.471 -32.5954 57.157 -0.570
The SAS System 14:43 Thursday, December 23, 1999 11
Cook's
Obs -2-1-0 1 2 D
1 | *| | 0.003
2 | | | 0.000
3 | *| | 0.009
4 | | | 0.000
5 | |** | 0.031
6 | |* | 0.005
7 | | | 0.002
8 | *| | 0.004
9 | **| | 0.023
10 | *| | 0.009
11 | | | 0.001
12 | |* | 0.010
13 | |******| 1.240
14 | | | 0.001
15 | *| | 0.009
16 | *| | 0.007
17 | | | 0.002
18 | |**** | 0.172
19 | *| | 0.006
20 | *| | 0.002
21 | | | 0.001
22 | |* | 0.032
23 | |* | 0.020
24 | *| | 0.002
25 | | | 0.001
26 | *| | 0.005
27 | | | 0.001
28 | |* | 0.064
29 | | | 0.002
30 | | | 0.000
31 | |* | 0.008
32 | |** | 0.091
33 | *| | 0.005
34 | | | 0.000
35 | | | 0.000
36 | *| | 0.003
37 | *| | 0.015
38 | |* | 0.040
39 | | | 0.000
40 | | | 0.002
41 | | | 0.000
42 | |* | 0.010
43 | | | 0.002
44 | *| | 0.002
45 | *| | 0.011
46 | *| | 0.004
47 | | | 0.000
48 | | | 0.002
49 | *| | 0.002
50 | *| | 0.020
51 | | | 0.002
52 | | | 0.002
53 | | | 0.002
54 | *| | 0.009
The SAS System 14:43 Thursday, December 23, 1999 12
Sum of Residuals 0
Sum of Squared Residuals 182666.9623
Predicted Resid SS (Press) 247880.7920
The SAS System 14:43 Thursday, December 23, 1999 13
---+----+----+----+----+----+--- ---+----+----+----+----+----+---
| | | |
1000 + + 1000 + +
| | | |
| | | |
| 1 | | 1 |
800 + + 800 + +
| | | |
Y | | Y | |
| | | |
600 + + 600 + +
| 1 | | 1 |
| 1 1 | | 1 1 |
| | | |
400 + 1 + 400 + 1 +
| 11 | | 2 |
| 11 11 1 | | 1 2 11 |
| | | |
200 + 2 2 31112 + 200 + 21113 4 +
| 2 11212 1 | | 12212 2 |
| 1 1 1 42221 | | 1 1 2 21 3 12 1 |
| 11 1 2 1 | | 1 1 21 1 |
0 + + 0 + +
| | | |
---+----+----+----+----+----+--- ---+----+----+----+----+----+---
2 4 6 8 10 12 0 20 40 60 80 100
X1 X2
---+----+----+----+----+----+--- -+------+------+------+------+--
| | | |
1000 + + 1000 + +
| | | |
| | | |
| 1 | | 1 |
800 + + 800 + +
| | | |
Y | | Y | |
| | | |
600 + + 600 + +
| 1 | | 1 |
| 1 1 | | 1 1 |
| | | |
400 + 1 + 400 + 1 +
| 11 | | 2 |
| 1 1 111 | | 121 1 |
| | | |
200 + 1 1 1122211 + 200 + 1 45 2 +
| 1 1 1131 2 | | 2 1 2212 |
| 1 1 1 1141 11 1 1 | | 224 1311 |
| 1 11 1 11 | | 1 1 2 2 |
0 + + 0 + +
| | | |
---+----+----+----+----+----+--- -+------+------+------+------+--
20 40 60 80 100 120 0 2 4 6 8
X3 X4
The SAS System 14:43 Thursday, December 23, 1999 14
---+----+----+----+----+----+--- ---+----+----+----+----+----+---
| | | |
400 + + 400 + +
| | | |
| | | |
| 1 | | 1 |
300 + + 300 + +
| | | |
R | | R | |
E | | E | |
S 200 + + S 200 + +
I | | I | |
D | | D | |
U | 1 | U | 1 |
A 100 + + A 100 + +
L | 1 | L | 1 |
| 1 2 1 1 | | 1 1 1 1 1 |
| 11 11211 | | 1 1 1 111 1 1 |
0 + 112 22 1 + 0 + 1 2 21 21 +
| 21 43242 2 1 | | 1 324121313 |
| 1 1 21 11 | | 1112 2 |
| 1 | | 1 |
-100 + + -100 + +
| | | |
---+----+----+----+----+----+--- ---+----+----+----+----+----+---
2 4 6 8 10 12 0 20 40 60 80 100
X1 X2
---+----+----+----+----+----+--- -+------+------+------+------+--
| | | |
400 + + 400 + +
| | | |
| | | |
| 1 | | 1 |
300 + + 300 + +
| | | |
R | | R | |
E | | E | |
S 200 + + S 200 + +
I | | I | |
D | | D | |
U | 1 | U | 1 |
A 100 + + A 100 + +
L | 1 | L | 1 |
| 1 1 1 1 1 | | 1 2 1 1 |
| 1 2 1 1 2 1 | | 1 2 1 21 1 |
0 + 1 2 1 111 1 1 + 0 + 1 21 211 1 +
| 1 1322432 21 | | 11134332 2 1 |
| 112 2 1 | | 11 1 4 |
| 1 | | 1 |
-100 + + -100 + +
| | | |
---+----+----+----+----+----+--- -+------+------+------+------+--
20 40 60 80 100 120 0 2 4 6 8
X3 X4
The SAS System 14:43 Thursday, December 23, 1999 15
-+------+------+------+------+--
| |
400 + +
| |
| |
| 1 |
300 + +
| |
R | |
E | |
S 200 + +
I | |
D | |
U | 1 |
A 100 + +
L | 1 |
| 2 1 1 1 |
| 141 1 1 |
0 + 3121 1 1 +
| 142251114 |
| 2113 |
| 1 |
-100 + +
| |
-+------+------+------+------+--
-200 0 200 400 600
PRED
The SAS System 14:43 Thursday, December 23, 1999 16
Frequency
| *****
| *****
30 + *****
| *****
| *****
| *****
| *****
25 + *****
| *****
| *****
| *****
| *****
20 + *****
| *****
| *****
| *****
| *****
15 + *****
| *****
| ***** *****
| ***** *****
| ***** *****
10 + ***** *****
| ***** *****
| ***** *****
| ***** ***** *****
| ***** ***** *****
5 + ***** ***** *****
| ***** ***** *****
| ***** ***** *****
| ***** ***** *****
| ***** ***** ***** ***** *****
----------------------------------------------------------------------------
-1.2 0.0 1.2 2.4 3.6 4.8 6.0
Studentized Residual
The SAS System 14:43 Thursday, December 23, 1999 17
Univariate Procedure
Variable=SRESID Studentized Residual
Moments
N 54 Sum Wgts 54
Mean 0.01449 Sum 0.782471
Std Dev 1.036889 Variance 1.07514
Skewness 3.486046 Kurtosis 17.50163
USS 56.99373 CSS 56.9824
CV 7155.8 Std Mean 0.141103
T:Mean=0 0.102692 Pr>|T| 0.9186
Num ^= 0 54 Num > 0 20
M(Sign) -7 Pr>=|M| 0.0759
Sgn Rank -137.5 Pr>=|S| 0.2400
Quantiles(Def=5)
100% Max 5.765284 99% 5.765284
75% Q3 0.363088 95% 1.269614
50% Med -0.25373 90% 0.943893
25% Q1 -0.59156 10% -0.67169
0% Min -1.35925 5% -0.85949
1% -1.35925
Range 7.12453
Q3-Q1 0.954651
Mode -1.35925
Extremes
Lowest Obs Highest Obs
-1.35925( 9) 0.986117( 22)
-0.92356( 15) 1.036339( 5)
-0.85949( 19) 1.269614( 32)
-0.80452( 37) 2.203981( 18)
-0.75638( 10) 5.765284( 13)
The SAS System 14:43 Thursday, December 23, 1999 18
Univariate Procedure
Variable=SRESID Studentized Residual
Stem Leaf # Boxplot
5 8 1 *
5
4
4
3
3
2
2 2 1 0
1
1 003 3 |
0 55567789 8 |
0 0112334 7 +--+--+
-0 3333332222110 13 *-----*
-0 99887766666666666555 20 +-----+
-1 4 1 |
----+----+----+----+
Normal Probability Plot
5.75+ *
|
|
|
|
|
| ++
2.25+ *++++
| +++++
| +++++* *
| +++*******
| ++++****
| +*******
| * * ************
-1.25+ * +++++
+----+----+----+----+----+----+----+----+----+----+
-2 -1 0 +1 +2
The SAS System 14:43 Thursday, December 23, 1999 19
Forward Selection Procedure for Dependent Variable Y
Step 1 Variable X4 Entered R-square = 0.52175569 C(p) = 93.54543578
DF Sum of Squares Mean Square F Prob>F
Regression 1 583808.87323857 583808.87323857 56.73 0.0001
Error 52 535122.62676143 10290.81974541
Total 53 1118931.5000000
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP -71.92123497 38.30031842 36287.70264864 3.53 0.0660
X4 98.05483965 13.01843184 583808.87323857 56.73 0.0001
Bounds on condition number: 1, 1
--------------------------------------------------------------------------------
Step 2 Variable X2 Entered R-square = 0.61738863 C(p) = 66.84112703
DF Sum of Squares Mean Square F Prob>F
Regression 2 690815.58520223 345407.79260112 41.15 0.0001
Error 51 428115.91479777 8394.42970192
Total 53 1118931.5000000
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP -207.06423185 51.27700590 136885.06002868 16.31 0.0002
X2 2.86025069 0.80111375 107006.71196366 12.75 0.0008
X4 81.38681690 12.65078976 347427.38408526 41.39 0.0001
Bounds on condition number: 1.157648, 4.630594
--------------------------------------------------------------------------------
Step 3 Variable X3 Entered R-square = 0.76233507 C(p) = 25.33532953
DF Sum of Squares Mean Square F Prob>F
Regression 3 853000.72546101 284333.57515367 53.46 0.0001
Error 50 265930.77453899 5318.61549078
Total 53 1118931.5000000
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP -402.09953137 53.97534792 295171.77519430 55.50 0.0001
X2 3.61581816 0.65218694 163481.10413865 30.74 0.0001
X3 2.92792762 0.53021756 162185.14025877 30.49 0.0001
X4 52.77323058 11.32476709 115496.12099960 21.72 0.0001
Bounds on condition number: 1.464174, 11.82183
--------------------------------------------------------------------------------
The SAS System 14:43 Thursday, December 23, 1999 20
Step 4 Variable X1 Entered R-square = 0.83674875 C(p) = 5.00000000
DF Sum of Squares Mean Square F Prob>F
Regression 4 936264.53766549 234066.13441637 62.79 0.0001
Error 49 182666.96233451 3727.89719050
Total 53 1118931.5000000
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP -621.59755029 64.80042601 343025.80578100 92.02 0.0001
X1 33.16382813 7.01727463 83263.81220448 22.34 0.0001
X2 4.27185982 0.56338454 214332.51418236 57.49 0.0001
X3 4.12573829 0.51116093 242857.75291129 65.15 0.0001
X4 14.09156259 12.52532754 4718.50099971 1.27 0.2661
Bounds on condition number: 2.555329, 29.28635
--------------------------------------------------------------------------------
All variables have been entered into the model.
Summary of Forward Selection Procedure for Dependent Variable Y
Variable Number Partial Model
Step Entered In R**2 R**2 C(p) F Prob>F
1 X4 1 0.5218 0.5218 93.5454 56.7310 0.0001
2 X2 2 0.0956 0.6174 66.8411 12.7473 0.0008
3 X3 3 0.1449 0.7623 25.3353 30.4939 0.0001
4 X1 4 0.0744 0.8367 5.0000 22.3353 0.0001
The SAS System 14:43 Thursday, December 23, 1999 21
Model: MODEL1
Dependent Variable: Y
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 4 936264.53767 234066.13442 62.788 0.0001
Error 49 182666.96233 3727.89719
C Total 53 1118931.5
Root MSE 61.05651 R-square 0.8367
Dep Mean 197.16667 Adj R-sq 0.8234
C.V. 30.96695
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP 1 -621.597550 64.80042601 -9.592 0.0001
X1 1 33.163828 7.01727463 4.726 0.0001
X2 1 4.271860 0.56338454 7.582 0.0001
X3 1 4.125738 0.51116093 8.071 0.0001
X4 1 14.091563 12.52532754 1.125 0.2661
Standardized
Variable DF Estimate Tolerance
INTERCEP 1 0.00000000 .
X1 1 0.36588406 0.55586050
X2 1 0.49694120 0.77566563
X3 1 0.60349558 0.59593691
X4 1 0.10380630 0.39133899
The SAS System 14:43 Thursday, December 23, 1999 22
Backward Elimination Procedure for Dependent Variable Y
Step 0 All Variables Entered R-square = 0.83674875 C(p) = 5.00000000
DF Sum of Squares Mean Square F Prob>F
Regression 4 936264.53766549 234066.13441637 62.79 0.0001
Error 49 182666.96233451 3727.89719050
Total 53 1118931.5000000
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP -621.59755029 64.80042601 343025.80578100 92.02 0.0001
X1 33.16382813 7.01727463 83263.81220448 22.34 0.0001
X2 4.27185982 0.56338454 214332.51418236 57.49 0.0001
X3 4.12573829 0.51116093 242857.75291129 65.15 0.0001
X4 14.09156259 12.52532754 4718.50099971 1.27 0.2661
Bounds on condition number: 2.555329, 29.28635
--------------------------------------------------------------------------------
Step 1 Variable X4 Removed R-square = 0.83253178 C(p) = 4.26572723
DF Sum of Squares Mean Square F Prob>F
Regression 3 931546.03666578 310515.34555526 82.85 0.0001
Error 50 187385.46333422 3747.70926668
Total 53 1118931.5000000
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP -659.17941990 55.67408110 525372.16182084 140.18 0.0001
X1 38.32274412 5.32589144 194041.43220438 51.78 0.0001
X2 4.56773202 0.49955913 313323.76018378 83.60 0.0001
X3 4.48503622 0.40017410 470760.31016036 125.61 0.0001
Bounds on condition number: 1.030813, 9.186356
--------------------------------------------------------------------------------
All variables left in the model are significant at the 0.1000 level.
Summary of Backward Elimination Procedure for Dependent Variable Y
Variable Number Partial Model
Step Removed In R**2 R**2 C(p) F Prob>F
1 X4 3 0.0042 0.8325 4.2657 1.2657 0.2661
The SAS System 14:43 Thursday, December 23, 1999 23
Model: MODEL2
Dependent Variable: Y
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 3 931546.03667 310515.34556 82.855 0.0001
Error 50 187385.46333 3747.70927
C Total 53 1118931.5
Root MSE 61.21854 R-square 0.8325
Dep Mean 197.16667 Adj R-sq 0.8225
C.V. 31.04913
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP 1 -659.179420 55.67408110 -11.840 0.0001
X1 1 38.322744 5.32589144 7.196 0.0001
X2 1 4.567732 0.49955913 9.144 0.0001
X3 1 4.485036 0.40017410 11.208 0.0001
Standardized
Variable DF Estimate Tolerance
INTERCEP 1 0.00000000 .
X1 1 0.42280044 0.97010805
X2 1 0.53135972 0.99177367
X3 1 0.65605216 0.97750637
The SAS System 14:43 Thursday, December 23, 1999 24
Stepwise Procedure for Dependent Variable Y
Step 1 Variable X4 Entered R-square = 0.52175569 C(p) = 93.54543578
DF Sum of Squares Mean Square F Prob>F
Regression 1 583808.87323857 583808.87323857 56.73 0.0001
Error 52 535122.62676143 10290.81974541
Total 53 1118931.5000000
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP -71.92123497 38.30031842 36287.70264864 3.53 0.0660
X4 98.05483965 13.01843184 583808.87323857 56.73 0.0001
Bounds on condition number: 1, 1
--------------------------------------------------------------------------------
Step 2 Variable X2 Entered R-square = 0.61738863 C(p) = 66.84112703
DF Sum of Squares Mean Square F Prob>F
Regression 2 690815.58520223 345407.79260112 41.15 0.0001
Error 51 428115.91479777 8394.42970192
Total 53 1118931.5000000
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP -207.06423185 51.27700590 136885.06002868 16.31 0.0002
X2 2.86025069 0.80111375 107006.71196366 12.75 0.0008
X4 81.38681690 12.65078976 347427.38408526 41.39 0.0001
Bounds on condition number: 1.157648, 4.630594
--------------------------------------------------------------------------------
Step 3 Variable X3 Entered R-square = 0.76233507 C(p) = 25.33532953
DF Sum of Squares Mean Square F Prob>F
Regression 3 853000.72546101 284333.57515367 53.46 0.0001
Error 50 265930.77453899 5318.61549078
Total 53 1118931.5000000
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP -402.09953137 53.97534792 295171.77519430 55.50 0.0001
X2 3.61581816 0.65218694 163481.10413865 30.74 0.0001
X3 2.92792762 0.53021756 162185.14025877 30.49 0.0001
X4 52.77323058 11.32476709 115496.12099960 21.72 0.0001
Bounds on condition number: 1.464174, 11.82183
--------------------------------------------------------------------------------
The SAS System 14:43 Thursday, December 23, 1999 25
Step 4 Variable X1 Entered R-square = 0.83674875 C(p) = 5.00000000
DF Sum of Squares Mean Square F Prob>F
Regression 4 936264.53766549 234066.13441637 62.79 0.0001
Error 49 182666.96233451 3727.89719050
Total 53 1118931.5000000
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP -621.59755029 64.80042601 343025.80578100 92.02 0.0001
X1 33.16382813 7.01727463 83263.81220448 22.34 0.0001
X2 4.27185982 0.56338454 214332.51418236 57.49 0.0001
X3 4.12573829 0.51116093 242857.75291129 65.15 0.0001
X4 14.09156259 12.52532754 4718.50099971 1.27 0.2661
Bounds on condition number: 2.555329, 29.28635
--------------------------------------------------------------------------------
Step 5 Variable X4 Removed R-square = 0.83253178 C(p) = 4.26572723
DF Sum of Squares Mean Square F Prob>F
Regression 3 931546.03666578 310515.34555526 82.85 0.0001
Error 50 187385.46333422 3747.70926668
Total 53 1118931.5000000
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP -659.17941990 55.67408110 525372.16182084 140.18 0.0001
X1 38.32274412 5.32589144 194041.43220438 51.78 0.0001
X2 4.56773202 0.49955913 313323.76018378 83.60 0.0001
X3 4.48503622 0.40017410 470760.31016036 125.61 0.0001
Bounds on condition number: 1.030813, 9.186356
--------------------------------------------------------------------------------
All variables left in the model are significant at the 0.1500 level.
No other variable met the 0.0500 significance level for entry into the model.
Summary of Stepwise Procedure for Dependent Variable Y
Variable Number Partial Model
Step Entered Removed In R**2 R**2 C(p) F Prob>F
1 X4 1 0.5218 0.5218 93.5454 56.7310 0.0001
2 X2 2 0.0956 0.6174 66.8411 12.7473 0.0008
3 X3 3 0.1449 0.7623 25.3353 30.4939 0.0001
4 X1 4 0.0744 0.8367 5.0000 22.3353 0.0001
5 X4 3 0.0042 0.8325 4.2657 1.2657 0.2661
The SAS System 14:43 Thursday, December 23, 1999 26
Model: MODEL3
Dependent Variable: Y
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 3 931546.03667 310515.34556 82.855 0.0001
Error 50 187385.46333 3747.70927
C Total 53 1118931.5
Root MSE 61.21854 R-square 0.8325
Dep Mean 197.16667 Adj R-sq 0.8225
C.V. 31.04913
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP 1 -659.179420 55.67408110 -11.840 0.0001
X1 1 38.322744 5.32589144 7.196 0.0001
X2 1 4.567732 0.49955913 9.144 0.0001
X3 1 4.485036 0.40017410 11.208 0.0001
Standardized
Variable DF Estimate Tolerance
INTERCEP 1 0.00000000 .
X1 1 0.42280044 0.97010805
X2 1 0.53135972 0.99177367
X3 1 0.65605216 0.97750637
The SAS System 14:43 Thursday, December 23, 1999 27
Descriptive Statistics
Variables Sum Mean Uncorrected SS
INTERCEP 54 1 54
X1 312.3 5.7833333333 1942.33
X2 3415 63.240740741 231109
X3 4164 77.111111111 345032
X4 148.19 2.7442592593 467.3919
LOGY 119.1318 2.2061444444 266.79473098
Variables Variance Std Deviation
INTERCEP 0 0
X1 2.5697169811 1.6030336806
X2 285.69566737 16.902534347
X3 451.72327044 21.253782497
X4 1.1456626485 1.0703563185
LOGY 0.074957967 0.2737845267
Correlation
CORR X1 X2 X3 X4 LOGY
X1 1.0000 0.0901 -0.1496 0.5024 0.3464
X2 0.0901 1.0000 -0.0236 0.3690 0.5929
X3 -0.1496 -0.0236 1.0000 0.4164 0.6651
X4 0.5024 0.3690 0.4164 1.0000 0.7262
LOGY 0.3464 0.5929 0.6651 0.7262 1.0000
The SAS System 14:43 Thursday, December 23, 1999 28
Model: MODEL1
Dependent Variable: LOGY
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 4 3.86300 0.96575 431.097 0.0001
Error 49 0.10977 0.00224
C Total 53 3.97277
Root MSE 0.04733 R-square 0.9724
Dep Mean 2.20614 Adj R-sq 0.9701
C.V. 2.14541
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP 1 0.488756 0.05023319 9.730 0.0001
X1 1 0.068520 0.00543978 12.596 0.0001
X2 1 0.009254 0.00043673 21.189 0.0001
X3 1 0.009475 0.00039625 23.910 0.0001
X4 1 0.001925 0.00970961 0.198 0.8436
Squared Squared
Standardized Semi-partial Partial
Variable DF Estimate Corr Type II Corr Type II Tolerance
INTERCEP 1 0.00000000 . . .
X1 1 0.40119198 0.08946853 0.76404020 0.55586050
X2 1 0.57131764 0.25318025 0.90160386 0.77566563
X3 1 0.73550521 0.32238275 0.92105814 0.59593691
X4 1 0.00752741 0.00002217 0.00080187 0.39133899
Variance
Variable DF Inflation
INTERCEP 1 0.00000000
X1 1 1.79901252
X2 1 1.28921530
X3 1 1.67802997
X4 1 2.55532933
Correlation of Estimates
CORRB INTERCEP X1 X2 X3 X4
INTERCEP 1.0000 -0.7167 -0.6382 -0.7516 0.5155
X1 -0.7167 1.0000 0.2464 0.4958 -0.6535
X2 -0.6382 0.2464 1.0000 0.2987 -0.4668
X3 -0.7516 0.4958 0.2987 1.0000 -0.6248
X4 0.5155 -0.6535 -0.4668 -0.6248 1.0000
The SAS System 14:43 Thursday, December 23, 1999 29
Durbin-Watson D 2.146
(For Number of Obs.) 54
1st Order Autocorrelation -0.073
The SAS System 14:43 Thursday, December 23, 1999 30
Dep Var Predict Std Err Std Err Student
Obs LOGY Value Predict Residual Residual Residual
1 2.3010 2.2940 0.009 0.00698 0.046 0.150
2 2.0043 2.0128 0.009 -0.00849 0.046 -0.183
3 2.3096 2.3138 0.015 -0.00424 0.045 -0.095
4 2.0043 2.0020 0.014 0.00229 0.045 0.050
5 2.7067 2.7226 0.017 -0.0159 0.044 -0.359
6 1.9031 1.9227 0.013 -0.0196 0.045 -0.433
7 1.9031 1.9056 0.010 -0.00248 0.046 -0.054
8 2.1038 2.1439 0.012 -0.0401 0.046 -0.875
9 2.3054 2.4058 0.011 -0.1004 0.046 -2.187
10 2.3075 2.3408 0.013 -0.0333 0.046 -0.732
11 2.5172 2.4921 0.012 0.0251 0.046 0.548
12 1.8129 1.8308 0.014 -0.0179 0.045 -0.394
13 2.9191 2.8623 0.019 0.0568 0.043 1.308
14 2.5185 2.4956 0.012 0.0229 0.046 0.499
15 2.2253 2.2315 0.011 -0.00616 0.046 -0.133
16 2.3365 2.3295 0.013 0.00700 0.045 0.154
17 1.9395 1.9575 0.021 -0.0180 0.043 -0.422
18 1.5315 1.6057 0.018 -0.0742 0.044 -1.700
19 2.3324 2.3262 0.009 0.00620 0.046 0.134
20 2.2355 2.2301 0.008 0.00545 0.047 0.117
21 2.0374 2.0518 0.010 -0.0144 0.046 -0.312
22 2.1335 1.9941 0.018 0.1394 0.044 3.176
23 1.8451 1.8368 0.017 0.00834 0.044 0.188
24 2.3424 2.3276 0.008 0.0148 0.047 0.318
25 2.4409 2.4196 0.011 0.0213 0.046 0.464
26 2.1584 2.1491 0.011 0.00935 0.046 0.203
27 2.2577 2.1458 0.008 0.1119 0.047 2.402
28 2.7589 2.8230 0.024 -0.0641 0.041 -1.579
29 1.8573 1.8806 0.014 -0.0233 0.045 -0.513
30 2.2504 2.1535 0.010 0.0969 0.046 2.094
31 1.8513 1.9176 0.015 -0.0663 0.045 -1.480
32 1.7634 1.7241 0.022 0.0393 0.042 0.941
33 2.0645 2.0757 0.014 -0.0112 0.045 -0.249
34 2.4698 2.4400 0.009 0.0298 0.047 0.642
35 2.0607 2.0638 0.008 -0.00310 0.047 -0.067
36 2.2648 2.2669 0.011 -0.00206 0.046 -0.045
37 2.0719 2.1705 0.015 -0.0986 0.045 -2.202
38 2.0792 1.9904 0.026 0.0888 0.039 2.264
39 2.1790 2.1069 0.008 0.0721 0.047 1.544
40 2.1703 2.1772 0.012 -0.00692 0.046 -0.151
41 1.9777 1.9842 0.009 -0.00652 0.046 -0.141
42 1.8751 1.9350 0.018 -0.0599 0.044 -1.372
43 2.6840 2.7337 0.024 -0.0497 0.041 -1.220
44 2.1847 2.1874 0.008 -0.00269 0.047 -0.058
45 2.2810 2.3183 0.017 -0.0373 0.044 -0.845
46 2.0899 2.1059 0.012 -0.0160 0.046 -0.351
47 2.4928 2.4828 0.013 0.00998 0.046 0.219
48 2.5999 2.5783 0.015 0.0216 0.045 0.481
49 2.1987 2.1933 0.008 0.00542 0.047 0.116
50 2.4914 2.4995 0.020 -0.00806 0.043 -0.188
51 2.0934 2.0931 0.014 0.000261 0.045 0.006
52 2.0969 2.0952 0.020 0.00170 0.043 0.040
53 2.2967 2.2831 0.011 0.0136 0.046 0.295
54 2.4955 2.5019 0.017 -0.00638 0.044 -0.144
The SAS System 14:43 Thursday, December 23, 1999 31
Cook's
Obs -2-1-0 1 2 D
1 | | | 0.000
2 | | | 0.000
3 | | | 0.000
4 | | | 0.000
5 | | | 0.004
6 | | | 0.003
7 | | | 0.000
8 | *| | 0.010
9 | ****| | 0.060
10 | *| | 0.009
11 | |* | 0.004
12 | | | 0.003
13 | |** | 0.064
14 | | | 0.003
15 | | | 0.000
16 | | | 0.000
17 | | | 0.008
18 | ***| | 0.102
19 | | | 0.000
20 | | | 0.000
21 | | | 0.001
22 | |******| 0.329
23 | | | 0.001
24 | | | 0.001
25 | | | 0.003
26 | | | 0.000
27 | |**** | 0.037
28 | ***| | 0.180
29 | *| | 0.005
30 | |**** | 0.039
31 | **| | 0.052
32 | |* | 0.050
33 | | | 0.001
34 | |* | 0.003
35 | | | 0.000
36 | | | 0.000
37 | ****| | 0.115
38 | |**** | 0.468
39 | |*** | 0.013
40 | | | 0.000
41 | | | 0.000
42 | **| | 0.066
43 | **| | 0.105
44 | | | 0.000
45 | *| | 0.022
46 | | | 0.002
47 | | | 0.001
48 | | | 0.005
49 | | | 0.000
50 | | | 0.002
51 | | | 0.000
52 | | | 0.000
53 | | | 0.001
54 | | | 0.001
The SAS System 14:43 Thursday, December 23, 1999 32
Sum of Residuals 0
Sum of Squared Residuals 0.1098
Predicted Resid SS (Press) 0.1456
The SAS System 14:43 Thursday, December 23, 1999 33
-+-----+-----+-----+-----+-----+- -+-----+-----+-----+-----+-----+-
| | | |
3.0 + + 3.0 + +
| 1 | | 1 |
| | | |
| 1 | | 1 |
| 1 1 | | 1 1 |
| | | |
L | 1 | L | 1 |
O 2.5 + 1 1 2 1 1 + O 2.5 + 21 3 +
G | 1 | G | 1 |
Y | 1 2 | Y | 2 1 |
| 11 21111 1 | | 111111 3 |
| 11 1 1 1 | | 2111 |
| 1 11 | | 1 1 1 |
| 1 1 1 12 12 | | 1 1 111 1 11 1 |
2.0 + 11 1 + 2.0 + 1 1 1 +
| 21 | | 1 1 1 |
| 11 1 1 | | 2 1 1 |
| 1 1 | | 11 |
| | | |
| | | |
| | | |
1.5 + 1 + 1.5 + 1 +
-+-----+-----+-----+-----+-----+- -+-----+-----+-----+-----+-----+-
2 4 6 8 10 12 0 20 40 60 80 100
X1 X2
-+-----+-----+-----+-----+-----+- --+---+---+---+---+---+---+---+--
| | | |
3.0 + + 3.0 + +
| 1 | | 1 |
| | | |
| 1 | | 1 |
| 1 1 | | 1 1 |
| | | |
L | 1 | L | 1 |
O 2.5 + 1 11 1 11 + O 2.5 + 12 111 +
G | 1 | G | 1 |
Y | 1 1 1 | Y | 1 2 |
| 1 112 3 1 | | 1 26 |
| 1 3 1 | | 121 1 |
| 1 1 1 | | 1 1 1 |
| 1 1 111111 1| | 1 2 131 1 |
2.0 + 1 1 1 + 2.0 + 21 +
| 1 1 1 | | 1 1 1 |
| 1 2 1 | | 1 1 1 1 |
| 1 1 | | 1 1 |
| | | |
| | | |
| | | |
1.5 + 1 + 1.5 + 1 +
-+-----+-----+-----+-----+-----+- --+---+---+---+---+---+---+---+--
20 40 60 80 100 120 0 1 2 3 4 5 6 7
X3 X4
The SAS System 14:43 Thursday, December 23, 1999 34
---+----+----+----+----+----+--- ---+----+----+----+----+----+---
| | | |
0.2 + + 0.2 + +
| | | |
| | | |
| | | |
| 1 | | 1 |
| | | |
R | 1 | R | 1 |
E 0.1 + 1 + E 0.1 + 1 +
S | 1 | S | 1 |
I | 1 | I | 1 |
D | 1 | D | 1 |
U | 1 | U | 1 |
A | 1 21 | A | 1 21 |
L | 1 221 | L | 1 3 11 |
0.0 + 34 3213 1 + 0.0 + 2231221211 +
| 1 3 211 1 | | 2 2 21 2 |
| 1 1 | | 1 1 |
| 2 1 | | 1 1 1 |
| 1 1 | | 1 1 |
| 11 | | 1 1 |
| | | |
-0.1 + 1 1 + -0.1 + 1 1 +
---+----+----+----+----+----+--- ---+----+----+----+----+----+---
2 4 6 8 10 12 0 20 40 60 80 100
X1 X2
---+----+----+----+----+----+--- -+------+------+------+------+--
| | | |
0.2 + + 0.2 + +
| | | |
| | | |
| | | |
| 1 | | 1 |
| | | |
R | 1 | R | 1 |
E 0.1 + 1 + E 0.1 + 1 +
S | 1 | S | 1 |
I | 1 | I | 1 |
D | 1 | D | 1 |
U | 1 | U | 1 |
A | 111 1 | A | 1 3 |
L | 11 11 1 1 | L | 11 112 |
0.0 + 2 1 1233112 1 + 0.0 + 1 2323222 +
| 1 1 121 1 1 1 | | 111 211 11 |
| 1 1 | | 11 |
| 1 11 | | 1 1 1 |
| 1 1 | | 1 1 |
| 1 1 | | 1 1 |
| | | |
-0.1 + 11 + -0.1 + 1 1 +
---+----+----+----+----+----+--- -+------+------+------+------+--
20 40 60 80 100 120 0 2 4 6 8
X3 X4
The SAS System 14:43 Thursday, December 23, 1999 35
-----+------+------+------+-----
| |
0.2 + +
| |
| |
| |
| 1 |
| |
R | 1 |
E 0.1 + 1 +
S | 1 |
I | 1 |
D | 1 |
U | 1 |
A | 121 |
L | 1 1 1111 |
0.0 + 1231432 1 +
| 1213 1 1 |
| 1 1 |
| 1 1 1 |
| 1 1 |
| 1 1 |
| |
-0.1 + 1 1 +
-----+------+------+------+-----
1.5 2.0 2.5 3.0
PRED
The SAS System 14:43 Thursday, December 23, 1999 36
Frequency
| *****
| *****
| *****
| *****
30 + *****
| *****
| *****
| *****
| *****
25 + *****
| *****
| *****
| *****
| *****
20 + *****
| *****
| *****
| *****
| *****
15 + *****
| *****
| *****
| *****
| *****
10 + *****
| *****
| *****
| ***** *****
| ***** *****
5 + ***** *****
| ***** ***** ***** ***** *****
| ***** ***** ***** ***** *****
| ***** ***** ***** ***** *****
| ***** ***** ***** ***** ***** *****
-------------------------------------------------------------------------
-2 -1 0 1 2 3
Studentized Residual
The SAS System 14:43 Thursday, December 23, 1999 37
Univariate Procedure
Variable=SRESID2 Studentized Residual
Moments
N 54 Sum Wgts 54
Mean -0.00244 Sum -0.13196
Std Dev 1.029039 Variance 1.058921
Skewness 0.627116 Kurtosis 1.794059
USS 56.12312 CSS 56.1228
CV -42110.8 Std Mean 0.140034
T:Mean=0 -0.01745 Pr>|T| 0.9861
Num ^= 0 54 Num > 0 25
M(Sign) -2 Pr>=|M| 0.6835
Sgn Rank -49.5 Pr>=|S| 0.6740
Quantiles(Def=5)
100% Max 3.175723 99% 3.175723
75% Q3 0.294823 95% 2.26434
50% Med -0.05574 90% 1.307878
25% Q1 -0.39434 10% -1.37195
0% Min -2.20248 5% -1.70045
1% -2.20248
Range 5.378208
Q3-Q1 0.689165
Mode -2.20248
Extremes
Lowest Obs Highest Obs
-2.20248( 37) 1.54351( 39)
-2.18732( 9) 2.093907( 30)
-1.70045( 18) 2.26434( 38)
-1.5787( 28) 2.402034( 27)
-1.48006( 31) 3.175723( 22)
Stem Leaf # Boxplot
3 2 1 *
2
2 134 3 0
1 5 1 0
1 3 1 |
0 555569 6 |
0 0011112222233 13 +-----+
-0 444443222211111110 18 *--+--*
-0 9875 4 |
-1 42 2 |
-1 765 3 0
-2 22 2 0
----+----+----+----+
The SAS System 14:43 Thursday, December 23, 1999 38
Univariate Procedure
Variable=SRESID2 Studentized Residual
Normal Probability Plot
3.25+ *
| +
| * * *+++++
| *+++++
| +++*+
| +++++***
| *********
| **********
| ***++++
| ***++
| ++*+*
-2.25+ +*+++*
+----+----+----+----+----+----+----+----+----+----+
-2 -1 0 +1 +2
The SAS System 14:43 Thursday, December 23, 1999 39
Forward Selection Procedure for Dependent Variable LOGY
Step 1 Variable X4 Entered R-square = 0.52737486 C(p) =788.14813627
DF Sum of Squares Mean Square F Prob>F
Regression 1 2.09514023 2.09514023 58.02 0.0001
Error 52 1.87763202 0.03610831
Total 53 3.97277225
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP 1.69638489 0.07174324 20.18803240 559.10 0.0001
X4 0.18575488 0.02438581 2.09514023 58.02 0.0001
Bounds on condition number: 1, 1
--------------------------------------------------------------------------------
Step 2 Variable X3 Entered R-square = 0.68653440 C(p) =507.89639740
DF Sum of Squares Mean Square F Prob>F
Regression 2 2.72744481 1.36372240 55.85 0.0001
Error 51 1.24532745 0.02441819
Total 53 3.97277225
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP 1.38877757 0.08446782 6.60079464 270.32 0.0001
X3 0.00565254 0.00111080 0.63230458 25.89 0.0001
X4 0.13901497 0.02205693 0.96994304 39.72 0.0001
Bounds on condition number: 1.209789, 4.839155
--------------------------------------------------------------------------------
Step 3 Variable X2 Entered R-square = 0.88290076 C(p) =161.66248928
DF Sum of Squares Mean Square F Prob>F
Regression 3 3.50756363 1.16918788 125.66 0.0001
Error 50 0.46520862 0.00930417
Total 53 3.97277225
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP 0.94226338 0.07138961 1.62088545 174.21 0.0001
X2 0.00789866 0.00086260 0.78011882 83.85 0.0001
X3 0.00699974 0.00070128 0.92694429 99.63 0.0001
X4 0.08184608 0.01497852 0.27780226 29.86 0.0001
Bounds on condition number: 1.464174, 11.82183
--------------------------------------------------------------------------------
The SAS System 14:43 Thursday, December 23, 1999 40
Step 4 Variable X1 Entered R-square = 0.97236929 C(p) = 5.00000000
DF Sum of Squares Mean Square F Prob>F
Regression 4 3.86300172 0.96575043 431.10 0.0001
Error 49 0.10977053 0.00224021
Total 53 3.97277225
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP 0.48875592 0.05023319 0.21207636 94.67 0.0001
X1 0.06852018 0.00543978 0.35543809 158.66 0.0001
X2 0.00925411 0.00043673 1.00582746 448.99 0.0001
X3 0.00947455 0.00039625 1.28075323 571.71 0.0001
X4 0.00192542 0.00970961 0.00008809 0.04 0.8436
Bounds on condition number: 2.555329, 29.28635
--------------------------------------------------------------------------------
All variables have been entered into the model.
Summary of Forward Selection Procedure for Dependent Variable LOGY
Variable Number Partial Model
Step Entered In R**2 R**2 C(p) F Prob>F
1 X4 1 0.5274 0.5274 788.1481 58.0238 0.0001
2 X3 2 0.1592 0.6865 507.8964 25.8948 0.0001
3 X2 3 0.1964 0.8829 161.6625 83.8461 0.0001
4 X1 4 0.0895 0.9724 5.0000 158.6625 0.0001
The SAS System 14:43 Thursday, December 23, 1999 41
Model: MODEL1
Dependent Variable: LOGY
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 4 3.86300 0.96575 431.097 0.0001
Error 49 0.10977 0.00224
C Total 53 3.97277
Root MSE 0.04733 R-square 0.9724
Dep Mean 2.20614 Adj R-sq 0.9701
C.V. 2.14541
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP 1 0.488756 0.05023319 9.730 0.0001
X1 1 0.068520 0.00543978 12.596 0.0001
X2 1 0.009254 0.00043673 21.189 0.0001
X3 1 0.009475 0.00039625 23.910 0.0001
X4 1 0.001925 0.00970961 0.198 0.8436
Standardized
Variable DF Estimate Tolerance
INTERCEP 1 0.00000000 .
X1 1 0.40119198 0.55586050
X2 1 0.57131764 0.77566563
X3 1 0.73550521 0.59593691
X4 1 0.00752741 0.39133899
The SAS System 14:43 Thursday, December 23, 1999 42
Backward Elimination Procedure for Dependent Variable LOGY
Step 0 All Variables Entered R-square = 0.97236929 C(p) = 5.00000000
DF Sum of Squares Mean Square F Prob>F
Regression 4 3.86300172 0.96575043 431.10 0.0001
Error 49 0.10977053 0.00224021
Total 53 3.97277225
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP 0.48875592 0.05023319 0.21207636 94.67 0.0001
X1 0.06852018 0.00543978 0.35543809 158.66 0.0001
X2 0.00925411 0.00043673 1.00582746 448.99 0.0001
X3 0.00947455 0.00039625 1.28075323 571.71 0.0001
X4 0.00192542 0.00970961 0.00008809 0.04 0.8436
Bounds on condition number: 2.555329, 29.28635
--------------------------------------------------------------------------------
Step 1 Variable X4 Removed R-square = 0.97234711 C(p) = 3.03932316
DF Sum of Squares Mean Square F Prob>F
Regression 3 3.86291363 1.28763788 586.04 0.0001
Error 50 0.10985863 0.00219717
Total 53 3.97277225
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP 0.48362086 0.04262871 0.28279376 128.71 0.0001
X1 0.06922508 0.00407795 0.63315225 288.17 0.0001
X2 0.00929454 0.00038250 1.29732180 590.45 0.0001
X3 0.00952364 0.00030641 2.12262768 966.07 0.0001
Bounds on condition number: 1.030813, 9.186356
--------------------------------------------------------------------------------
All variables left in the model are significant at the 0.1000 level.
Summary of Backward Elimination Procedure for Dependent Variable LOGY
Variable Number Partial Model
Step Removed In R**2 R**2 C(p) F Prob>F
1 X4 3 0.0000 0.9723 3.0393 0.0393 0.8436
The SAS System 14:43 Thursday, December 23, 1999 43
Model: MODEL2
Dependent Variable: LOGY
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 3 3.86291 1.28764 586.043 0.0001
Error 50 0.10986 0.00220
C Total 53 3.97277
Root MSE 0.04687 R-square 0.9723
Dep Mean 2.20614 Adj R-sq 0.9707
C.V. 2.12470
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP 1 0.483621 0.04262871 11.345 0.0001
X1 1 0.069225 0.00407795 16.975 0.0001
X2 1 0.009295 0.00038250 24.299 0.0001
X3 1 0.009524 0.00030641 31.082 0.0001
Standardized
Variable DF Estimate Tolerance
INTERCEP 1 0.00000000 .
X1 1 0.40531921 0.97010805
X2 1 0.57381347 0.99177367
X3 1 0.73931630 0.97750637
The SAS System 14:43 Thursday, December 23, 1999 44
Stepwise Procedure for Dependent Variable LOGY
Step 1 Variable X4 Entered R-square = 0.52737486 C(p) =788.14813627
DF Sum of Squares Mean Square F Prob>F
Regression 1 2.09514023 2.09514023 58.02 0.0001
Error 52 1.87763202 0.03610831
Total 53 3.97277225
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP 1.69638489 0.07174324 20.18803240 559.10 0.0001
X4 0.18575488 0.02438581 2.09514023 58.02 0.0001
Bounds on condition number: 1, 1
--------------------------------------------------------------------------------
Step 2 Variable X3 Entered R-square = 0.68653440 C(p) =507.89639740
DF Sum of Squares Mean Square F Prob>F
Regression 2 2.72744481 1.36372240 55.85 0.0001
Error 51 1.24532745 0.02441819
Total 53 3.97277225
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP 1.38877757 0.08446782 6.60079464 270.32 0.0001
X3 0.00565254 0.00111080 0.63230458 25.89 0.0001
X4 0.13901497 0.02205693 0.96994304 39.72 0.0001
Bounds on condition number: 1.209789, 4.839155
--------------------------------------------------------------------------------
Step 3 Variable X2 Entered R-square = 0.88290076 C(p) =161.66248928
DF Sum of Squares Mean Square F Prob>F
Regression 3 3.50756363 1.16918788 125.66 0.0001
Error 50 0.46520862 0.00930417
Total 53 3.97277225
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP 0.94226338 0.07138961 1.62088545 174.21 0.0001
X2 0.00789866 0.00086260 0.78011882 83.85 0.0001
X3 0.00699974 0.00070128 0.92694429 99.63 0.0001
X4 0.08184608 0.01497852 0.27780226 29.86 0.0001
Bounds on condition number: 1.464174, 11.82183
--------------------------------------------------------------------------------
The SAS System 14:43 Thursday, December 23, 1999 45
Step 4 Variable X1 Entered R-square = 0.97236929 C(p) = 5.00000000
DF Sum of Squares Mean Square F Prob>F
Regression 4 3.86300172 0.96575043 431.10 0.0001
Error 49 0.10977053 0.00224021
Total 53 3.97277225
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP 0.48875592 0.05023319 0.21207636 94.67 0.0001
X1 0.06852018 0.00543978 0.35543809 158.66 0.0001
X2 0.00925411 0.00043673 1.00582746 448.99 0.0001
X3 0.00947455 0.00039625 1.28075323 571.71 0.0001
X4 0.00192542 0.00970961 0.00008809 0.04 0.8436
Bounds on condition number: 2.555329, 29.28635
--------------------------------------------------------------------------------
Step 5 Variable X4 Removed R-square = 0.97234711 C(p) = 3.03932316
DF Sum of Squares Mean Square F Prob>F
Regression 3 3.86291363 1.28763788 586.04 0.0001
Error 50 0.10985863 0.00219717
Total 53 3.97277225
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP 0.48362086 0.04262871 0.28279376 128.71 0.0001
X1 0.06922508 0.00407795 0.63315225 288.17 0.0001
X2 0.00929454 0.00038250 1.29732180 590.45 0.0001
X3 0.00952364 0.00030641 2.12262768 966.07 0.0001
Bounds on condition number: 1.030813, 9.186356
--------------------------------------------------------------------------------
All variables left in the model are significant at the 0.1500 level.
No other variable met the 0.0500 significance level for entry into the model.
Summary of Stepwise Procedure for Dependent Variable LOGY
Variable Number Partial Model
Step Entered Removed In R**2 R**2 C(p) F Prob>F
1 X4 1 0.5274 0.5274 788.1481 58.0238 0.0001
2 X3 2 0.1592 0.6865 507.8964 25.8948 0.0001
3 X2 3 0.1964 0.8829 161.6625 83.8461 0.0001
4 X1 4 0.0895 0.9724 5.0000 158.6625 0.0001
5 X4 3 0.0000 0.9723 3.0393 0.0393 0.8436
The SAS System 14:43 Thursday, December 23, 1999 46
Model: MODEL3
Dependent Variable: LOGY
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 3 3.86291 1.28764 586.043 0.0001
Error 50 0.10986 0.00220
C Total 53 3.97277
Root MSE 0.04687 R-square 0.9723
Dep Mean 2.20614 Adj R-sq 0.9707
C.V. 2.12470
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP 1 0.483621 0.04262871 11.345 0.0001
X1 1 0.069225 0.00407795 16.975 0.0001
X2 1 0.009295 0.00038250 24.299 0.0001
X3 1 0.009524 0.00030641 31.082 0.0001
Standardized
Variable DF Estimate Tolerance
INTERCEP 1 0.00000000 .
X1 1 0.40531921 0.97010805
X2 1 0.57381347 0.99177367
X3 1 0.73931630 0.97750637
The SAS System 14:43 Thursday, December 23, 1999 47
Model: MODEL1
Dependent Variable: X1
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 3 60.48958 20.16319 13.317 0.0001
Error 50 75.70542 1.51411
C Total 53 136.19500
Root MSE 1.23049 R-square 0.4441
Dep Mean 5.78333 Adj R-sq 0.4108
C.V. 21.27650
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP 1 6.618597 0.91069854 7.268 0.0001
X2 1 -0.019782 0.01100402 -1.798 0.0783
X3 1 -0.036118 0.00894609 -4.037 0.0002
X4 1 1.166381 0.19107702 6.104 0.0001
Standardized
Variable DF Estimate
INTERCEP 1 0.00000000
X2 1 -0.20858157
X3 1 -0.47886942
X4 1 0.77880058
The SAS System 14:43 Thursday, December 23, 1999 48
Model: MODEL2
Dependent Variable: X2
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 3 3396.84194 1132.28065 4.820 0.0050
Error 50 11745.02843 234.90057
C Total 53 15141.87037
Root MSE 15.32647 R-square 0.2243
Dep Mean 63.24074 Adj R-sq 0.1778
C.V. 24.23512
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP 1 73.410773 12.52232622 5.862 0.0001
X1 1 -3.068979 1.70717633 -1.798 0.0783
X3 1 -0.271051 0.12245259 -2.214 0.0315
X4 1 10.377992 2.78054642 3.732 0.0005
Standardized
Variable DF Estimate
INTERCEP 1 0.00000000
X1 1 -0.29106144
X3 1 -0.34082752
X4 1 0.65718839
The SAS System 14:43 Thursday, December 23, 1999 49
Model: MODEL3
Dependent Variable: X3
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 3 9673.80903 3224.60301 11.300 0.0001
Error 50 14267.52430 285.35049
C Total 53 23941.33333
Root MSE 16.89232 R-square 0.4041
Dep Mean 77.11111 Adj R-sq 0.3683
C.V. 21.90647
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP 1 95.287238 11.82476250 8.058 0.0001
X1 1 -6.806834 1.68598977 -4.037 0.0002
X2 1 -0.329264 0.14875190 -2.214 0.0315
X4 1 15.309410 2.70574448 5.658 0.0001
Standardized
Variable DF Estimate
INTERCEP 1 0.00000000
X1 1 -0.51339493
X2 1 -0.26185471
X4 1 0.77099327
The SAS System 14:43 Thursday, December 23, 1999 50
Model: MODEL4
Dependent Variable: X4
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 3 36.95797 12.31932 25.922 0.0001
Error 50 23.76215 0.47524
C Total 53 60.72012
Root MSE 0.68938 R-square 0.6087
Dep Mean 2.74426 Adj R-sq 0.5852
C.V. 25.12076
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP 1 -2.666977 0.62694289 -4.254 0.0001
X1 1 0.366100 0.05997458 6.104 0.0001
X2 1 0.020996 0.00562551 3.732 0.0005
X3 1 0.025497 0.00450634 5.658 0.0001
Standardized
Variable DF Estimate
INTERCEP 1 0.00000000
X1 1 0.54829410
X2 1 0.33156483
X3 1 0.50629475