以下三種方法：1.向前選擇法(Forward selection) 2. 向後消去法(Backward elimination) 3. 逐步 選擇法(Stepwise regression)。
此解釋變數之F2*|4 = 719. >Fin，統計上顯著(P-value = 0.0033 < 0.05)，由 ANOVA Table 4.1.4
此方式為Forward 和 Backward 合併使用，發現選變數之過程與 Forward selection 相同，
選入的變數亦相同，在此不做贅述，其結果呈現於Table 4.1.17－4.1.25。
Table 4.1.1 Parameter estimates (Forward Selection: Step 1)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 16.3030 1.4128 2043.8377 133.15 <.0001
2
x1 1 0.4812 0.0717 690.9604 45.01 <.0001
Table 4.1.2 Analysis of Variance (Forward Selection: Step 1) Source DF Sum of
Squares
Mean
Square F Value P-value
Model 1 690.9604 690.9604 45.01 <.0001
Error 42 644.6849 15.3496
Total 43 1335.6453
R-Square 0.5173 C(p) 26.7792
Table 4.1.3 Parameter estimates (Forward Selection: Step 2)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 9.0606 2.6560 147.9408 11.64 0.0015 x2 1 0.5259 0.1688 123.4553 9.71 0.0033
2
x1 1 0.3084 0.0857 164.7976 12.96 0.0008
Table 4.1.4 Analysis of Variance (Forward Selection: Step 2) Source DF Sum of
Squares
Mean
Square F Value P-value
Model 2 814.4157 407.2079 32.03 <.0001
Error 41 521.2296 12.7129
Total 43 1335.6453
R-Square 0.6098 C(p) 15.9912
Table 4.1.5 Parameter estimates (Forward Selection: Step 3)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 3.0397 3.4053 8.8920 0.80 0.3774
x2 1 0.4864 0.1588 104.6439 9.38 0.0039 x 3 1 2.7053 1.0445 74.8636 6.71 0.0133
2
x1 1 0.2657 0.0819 117.4015 10.52 0.0024
Table 4.1.6 Analysis of Variance (Forward Selection: Step 3) Source DF Sum of
R-Square 0.6658 C(p) 10.2365
Table 4.1.7 Parameter estimates (Forward Selection: Step 4)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 23.5358 8.7641 71.0460 7.21 0.0106
x1 1 -11.1275 4.4297 62.1645 6.31 0.0163 x2 1 0.5443 0.1510 127.9864 12.99 0.0009 x 3 1 3.3897 1.0185 109.1184 11.08 0.0019
2
x1 1 1.5126 0.5023 89.3353 9.07 0.0045
Table 4.1.8 Analysis of Variance (Forward Selection: Step 4) Source DF Sum of
Table 4.1.9 Summary of Forward Selection
Table 4.1.10 Parameter estimates (Backward Elimination: Step 0)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 6.0356 13.9553 1.8058 0.19 0.6679
x1 1 -13.4205 4.5977 82.2551 8.52 0.0059
Table 4.1.11 Analysis of Variance (Backward Elimination: Step 0) Source DF Sum of
Table 4.1.12 Parameter estimates (Backward Elimination: Step 1)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 12.0304 11.5921 10.2909 1.08 0.3059
x1 1 -13.0405 4.5483 78.5437 8.22 0.0067 x2 1 0.6062 0.1544 147.2173 15.41 0.0004 x 3 1 14.3242 7.4225 35.5844 3.72 0.0611
2
x1 1 1.6693 0.5058 104.0735 10.89 0.0021
Table 4.1.13 Analysis of Variance (Backward Elimination: Step 1)
R-Square 0.7282 C(p) 5.6094
Table 4.1.14 Parameter estimates (Backward Elimination: Step 2)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 23.5358 8.7641 71.0460 7.21 0.0106 x1 1 -11.1275 4.4297 62.1645 6.31 0.0163 x2 1 0.5443 0.1510 127.9864 12.99 0.0009 x 3 1 3.3897 1.0185 109.1184 11.08 0.0019
2
x1 1 1.5126 0.5023 89.3353 9.07 0.0045
Table 4.1.15 Analysis of Variance (Backward Elimination: Step 2) Source DF Sum of
R-Square 0.7123 C(p) 5.7972
Table 4.1.16 Summary of Backward Elimination Step Variable
Table 4.1.17 Parameter estimates (Stepwise Selection: Step 1)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 16.3030 1.4128 2043.8377 133.15 <.0001
2
x1 1 0.4812 0.0717 690.9604 45.01 <.0001
Table 4.1.18 Analysis of Variance (Stepwise Selection: Step 1) Source DF Sum of
Table 4.1.19 Parameter estimates (Stepwise Selection: Step 2)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 9.0606 2.6560 147.9408 11.64 0.0015
x2 1 0.5259 0.1688 123.4553 9.71 0.0033
2
x1 1 0.3084 0.0857 164.7976 12.96 0.0008 Table 4.1.20 Analysis of Variance (Stepwise Selection: Step 2)
Source DF Sum of
R-Square 0.6098 C(p) 15.9912
Table 4.1.21 Parameter estimates (Stepwise Selection: Step 3)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 3.0397 3.4053 8.8920 0.80 0.3774
x2 1 0.4864 0.1588 104.6439 9.38 0.0039 x 3 1 2.7054 1.0445 74.8636 6.71 0.0133
2
Table 4.1.22 Analysis of Variance (Stepwise Selection: Step 3)
R-Square 0.6658 C(p) 10.2365
Table 4.1.23 Parameter estimates (Stepwise Selection: Step 4)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 23.5358 8.7641 71.0460 7.21 0.0106
x1 1 -11.1275 4.4297 62.1645 6.31 0.0163 x2 1 0.5443 0.1510 127.9864 12.99 0.0009 x 3 1 3.3897 1.0185 109.1184 11.08 0.0019
2
x1 1 1.5126 0.5023 89.3353 9.07 0.0045
Table 4.1.24 Analysis of Variance (Stepwise Selection: Step 4) Source DF Sum of
R-Square 0.7123 C(p) 5.7972
Table 4.1.25 Summary of Stepwise Selection Step Variable
Table 4.1.26.1 Summary of All Possible Regressions 1 x1 0.4951 0.4831 29.8561 124.1032 16.0569 1 x2 0.4864 0.4741 31.0616 124.8560 16.3340 1 x 3 0.2375 0.2194 65.4866 142.2376 24.2468
Table 4.1.26.2 Summary of All Possible Regressions Number of
3 x2 x12 x 32 0.6610 0.6356 10.9037 110.5762 11.3202 3 x3 x12 x 22 0.6602 0.6348 11.0070 110.6729 11.3451 3 x12 x22 x 32 0.6552 0.6293 11.7092 111.3254 11.5146 3 x1 x2 x 3 0.6455 0.6189 13.0510 112.5458 11.8384 3 x1 x3 x 22 0.6407 0.6138 13.7039 113.1276 11.9960 3 x1 x2 x 32 0.6403 0.6133 13.7682 113.1845 12.0115 3 x1 x22 x 32 0.6353 0.6080 14.4537 113.7864 12.1770 3 x1 x2 x 12 0.6307 0.6029 15.1002 114.3466 12.3330 3 x1 x12 x 22 0.6273 0.5994 15.5592 114.7401 12.4438 3 x2 x3 x 12 0.6165 0.5878 17.0546 115.9981 12.8047 3 x3 x22 x 32 0.6134 0.5844 17.4851 116.3537 12.9086 3 x2 x3 x 32 0.6118 0.5827 17.7015 116.5314 12.9608
3 x1 x12 x 32 0.6116 0.5825 17.7321 116.5564 12.9682 3 x2 x12 x 22 0.6098 0.5805 17.9911 116.7679 13.0307 3 x1 x2 x 22 0.5920 0.5614 20.4421 118.7214 13.6223 3 x3 x12 x 32 0.5875 0.5565 21.0736 119.2109 13.7747 3 x2 x3 x 22 0.5782 0.5466 22.3578 120.1900 14.0846 3 x2 x22 x 32 0.5603 0.5274 24.8285 122.0145 14.6809 3 x1 x3 x 32 0.5585 0.5254 25.0838 122.1987 14.7425 4 x1 x2 x3 x 12 0.7123 0.6828 5.7972 105.3470 9.8513 4 x1 x3 x12 x 22 0.7029 0.6724 7.1050 106.7696 10.1750
2 2
4 x1 x12 x22 x 32 0.6922 0.6606 8.5848 108.3258 10.5413 4 x2 x3 x12 x 32 0.6694 0.6354 11.7453 111.4758 11.3237 4 x2 x3 x12 x 22 0.6689 0.6349 11.8096 111.5376 11.3396 4 x2 x12 x22 x 32 0.6648 0.6304 12.3730 112.0755 11.4791 4 x3 x12 x22 x 32 0.6648 0.6304 12.3731 112.0756 11.4791 4 x1 x2 x3 x 32 0.6502 0.6144 14.3898 113.9486 11.9783 4 x1 x2 x3 x 22 0.6469 0.6107 14.8464 114.3618 12.0913 4 x1 x3 x22 x 32 0.6469 0.6106 14.8589 114.3731 12.0944 4 x1 x2 x22 x 32 0.6422 0.6055 15.4997 114.9465 12.2531 4 x1 x2 x12 x 22 0.6309 0.5930 17.0721 116.3224 12.6423
Table 4.1.26.3 Summary of All Possible Regressions Number of
Regressors in Model
Regressors in Model
2
Rp R2Adj,p C p AIC MSRes(p)
4 x1 x3 x12 x 32 0.6179 0.5788 18.8588 117.8354 13.0846 4 x2 x3 x22 x 32 0.6137 0.5741 19.4408 118.3172 13.2286 5 x1 x2 x3 x12 x 32 0.7282 0.6924 5.6094 104.8591 9.5547 5 x1 x2 x3 x12 x 22 0.7214 0.6848 6.5392 105.9336 9.7910 5 x1 x3 x12 x22 x 32 0.7202 0.6834 6.7094 106.1275 9.8342 5 x1 x2 x12 x22 x 32 0.7113 0.6733 7.9412 107.5059 10.1471 5 x2 x3 x12 x22 x 32 0.6710 0.6277 13.5203 113.2589 11.5645 5 x x x x2 x 2 0.6506 0.6047 16.3350 115.8988 12.2796
6 x1 x2 x3 x12 x22 x 32 0.7326 0.6892 7.0000 106.1403 9.6540
Figure 4.1.1 Plot R2p versus p
Figure 4.1.3 Plot MSRes(p versus p ) 第二節
本節為考慮反應變數經轉換後(y(−0.22))與解釋變數膀胱癌(x )、肺癌(1 x )、腎癌(2 x )與其解3 釋變數之平方項，利用逐步回歸分析(Stepwise regression methods)來選擇較佳模型之組合。而 逐步回歸包含以下三種方法：1.向前選擇法(Forward selection) 2. 向後消去法(Backward elimination) 3. 逐步選擇法(Stepwise regression)。
假設考慮之解釋變數為Xk ={x1,x2,x3,x12,x22,x32}，k = 1, …, 6；以下為說明逐步回歸選取法 之結果：(α =0.05,Fin =Fout ≈4)
1. 向前選擇法(Forward selection)
此法一開始假設模式中沒有任何解釋變數，而後第一步驟選入之變數為x2(k = 2)，由 Table 4.2.1 得知此解釋變數之F2* =44.65>Fin，統計上顯著(P-value < 0.05)，由 ANOVA Table4.2.2 得知，R2 = 0.5153，Cp = 34.0582；第二步驟選入之變數為x (k = 3)，由 Table 4.2.33 得知此解釋變數之F3*|2 =14.26>Fin，統計上顯著(P-value = 0.0005 < 0.05)，由 ANOVA Table 4.2.4 得知，R2 = 0.6404，C = 16.9496；第三步驟選入之變數為x (k = 4)，由 Table 4.2.5 得知2
此解釋變數之F4*|2,3 = 027. >Fin，統計上顯著(P-value = 0.0115 < 0.05)，由 Table 4.2.6 得知，R2
此方式為Forward 和 Backward 合併使用，發現選變數之過程與 Forward selection 相同，
選入的變數亦相同，在此不做贅述，其結果呈現於Table 4.2.13－4.2.19 中。
型4.2.3 為我們所選取之最佳回歸模型。
Table 4.2.1 Parameter estimates (Forward Selection: Step 1)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 0.5749 0.0121 0.6293 2268.34 <0.0001 x2 1 -0.0040 0.0006 0.0124 44.65 <0.0001
Table 4.2.2 Analysis of Variance (Forward Selection: Step 1) Source DF Sum of
R-Square 0.5153 C(p) 34.0582
Table 4.2.3 Parameter estimates (Forward Selection: Step 2)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 0.6104 0.0141 0.3948 1872.02 <0.0001 x2 1 -0.0034 0.0005 0.0083 39.49 <0.0001 x 3 1 -0.0168 0.0045 0.0030 14.26 0.0005
Table 4.2.4 Analysis of Variance (Forward Selection: Step 2) Source DF Sum of
R-Square 0.6404 C(p) 16.9496
Table 4.2.5 Parameter estimates (Forward Selection: Step 3)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 0.5993 0.0138 0.3456 1879.59 <0.0001 x2 1 -0.0024 0.0006 0.0025 13.69 0.0006 x 1 -0.0145 0.0042 0.0022 11.75 0.0014
Table 4.2.6 Analysis of Variance (Forward Selection: Step 3)
R-Square 0.6940 C(p) 10.7462
Table 4.2.7 Summary of Forward Selection Step Variable
Table 4.2.8 Parameter estimates (Backward Elimination: Step 0)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 0.6648 0.0563 0.0219 139.25 <0.0001 x1 1 0.0424 0.0186 0.0008 5.22 0.0282
Table 4.2.9 Analysis of Variance (Backward Elimination: Step 0) Source DF Sum of
Table 4.2.10 Parameter estimates (Backward Elimination: Step 1)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 0.6209 0.0477 0.0274 169.71 <0.0001 x1 1 0.0396 0.0187 0.0007 4.49 0.0408
Table 4.2.11 Analysis of Variance (Backward Elimination: Step 1) Source DF Sum of
R-Square 0.7447 C(p) 7.0072
Table 4.2.12 Summary of Backward Elimination Step Variable
Table 4.2.13 Parameter estimates (Stepwise Selection: Step 1)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 0.5749 0.0121 0.6293 2268.34 <0.0001 x2 1 -0.0040 0.0006 0.0124 44.65 <0.0001
Table 4.2.14 Analysis of Variance (Stepwise Selection: Step 1) Source DF Sum of
Table 4.2.15 Parameter estimates (Stepwise Selection: Step 2)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 0.6104 0.0141 0.3948 1872.02 <0.0001 x2 1 -0.0034 0.0005 0.0083 39.49 <0.0001 x 3 1 -0.0168 0.0045 0.0030 14.26 0.0005
Table 4.2.16 Analysis of Variance (Stepwise Selection: Step 2) Source DF Sum of
R-Square 0.6404 C(p) 16.9496
Table 4.2.17 Parameter estimates (Stepwise Selection: Step 3)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 0.5992 0.0138 0.3456 1879.59 <0.0001
x2 1 -0.0024 0.0006 0.0025 13.69 0.0006 x 3 1 -0.0145 0.0042 0.0022 11.75 0.0014
2
x1 1 -0.0009 0.0003 0.0013 7.02 0.0115 Table 4.2.18 Analysis of Variance (Stepwise Selection: Step 3)
Source DF Sum of
R-Square 0.6940 C(p) 10.7462
Table 4.2.19 Summary of Stepwise Selection Step Variable
Table 4.2.20.1 Summary of All Possible Regressions 1 x2 0.5153 0.5038 34.0582 -358.4039 0.0002774 1 x 22 0.5135 0.5019 34.3368 -358.2386 0.0002785 1 x 12 0.4788 0.4664 39.6311 -355.2115 0.0002983 1 x1 0.4755 0.4630 40.1397 -354.9314 0.0003002 1 x 3 0.2940 0.2771 67.8764 -341.8539 0.0004041
Table 4.2.20.2 Summary of All Possible Regressions Number of
3 x1 x2 x 0.6714 32 0.6468 14.2008 -371.5120 0.000197
Table 4.2.20.3 Summary of All Possible Regressions Number of
5 x1 x2 x12 x22 x 0.7181 0.6810 32 11.0791 -374.2437 0.000178 5 x1 x2 x3 x22 x 0.7131 0.6754 32 11.8302 -373.4831 0.000181 6 x1 x2 x3 x12 x22 x 0.7578 0.7186 7.0000 32 -378.9369 0.000157
Figure 4.2.1 Plot R2p versus p
Figure 4.2.2 The Cp plot
第三節
本節為考慮去除影響點之資料，反應變數後y 與解釋變數膀胱癌(* x )、肺癌(1* x )、腎癌(*2 x )*3 與解釋變數膀胱癌(x )之平方項，利用逐步回歸分析(Stepwise regression methods)來選擇較佳1* 模型之組合。而逐步回歸包含以下三種方法：1.向前選擇法(Forward selection) 2. 向後消去法 (Backward elimination) 3. 逐步選擇法(Stepwise regression)。
假設考慮之解釋變數為Xk = { , , ,( ) }x x x1* *2 3* x1* 2 ， k = 1, …, 4；以下先說明逐步回歸選取法之結
此方式為Forward 和 Backward 合併使用，發現選變數之過程與 Forward selection 相同，
選入的變數亦相同，在此不做贅述，其結果呈現於Table 4.3.13－4.3.17。
而後利用所有回歸式的比較選取法中之選模指標(Table 4.3.18)：R2p, R2Adj,p, C , AIC, p
Table 4.3.1 Parameter estimates (Forward Selection: Step 1)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 9.6329 2.2932 164.5969 17.65 0.0001
*
x 2 1 0.7475 0.1157 389.1066 41.71 <0.0001
Table 4.3.2 Analysis of Variance (Forward Selection: Step 1) Source DF Sum of
Squares
Mean
Square F Value P-value
Model 1 389.1066 389.1066 41.71 <.0001
Error 40 373.1277 9.3282
Total 41 762.2344
R-Square 0.5105 C(p) 19.2627
Table 4.3.3 Parameter estimates (Forward Selection: Step 2)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 2.3987 2.5108 5.8338 0.91 0.3453
*
x 2 1 0.6308 0.0994 257.3525 40.26 <0.0001
*
x 3 1 3.4100 0.7746 123.8486 19.38 <0.0001
Source DF Sum of Squares
Mean
Square F Value P-value
Model 2 512.9552 256.4776 40.13 <.0001
Error 39 249.2791 6.3918
Total 41 762.2344
R-Square 0.6730 C(p) 2.2561
Table 4.3.5 Summary of Forward Selection Step Variable
Table 4.3.6 Parameter estimates (Backward Elimination: Step 0)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 -3.2198 9.7293 0.7136 0.11 0.7426
*
Table 4.3.7 Analysis of Variance (Backward Elimination: Step 0) Source DF Sum of
Squares
Mean
Square F Value P-value
Model 4 521.1399 130.2850 19.99 <.0001
Error 37 241.0945 6.5161
Total 41 762.2344
R-Square 0.6837 C(p) 5.0000
Table 4.3.8 Parameter estimates (Backward Elimination: Step 1)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 2.0274 2.5405 4.0748 0.64 0.4298
*
Table 4.3.9 Analysis of Variance (Backward Elimination: Step 1) Source DF Sum of
Squares
Mean
Square F Value P-value
Model 3 519.1032 173.0344 27.04 <.0001
Error 38 243.1312 6.3982
Total 41 762.2344
R-Square 0.6810 C(p) 3.3126
Table 4.3.10 Parameter estimates (Backward Elimination: Step 2)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 2.3987 2.5108 5.8338 0.91 0.3453
*
x 2 1 0.6307 0.0994 257.3525 40.26 <0.0001
*
x 3 1 3.4100 0.7747 123.8486 19.38 <0.0001
Table 4.3.11 Analysis of Variance (Backward Elimination: Step2) Source DF Sum of
Squares
Mean
Square F Value P-value
Model 2 512.9552 256.4776 40.13 <.0001
Error 39 249.2791 6.3918
R-Square 0.6730 C(p) 2.2561
Table 4.3.12 Summary of Backward Elimination Step Variable
Table 4.3.13 Parameter estimates (Stepwise Selection: Step 1)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 9.6329 2.2932 164.5969 17.65 0.0001
*
x 2 1 0.7475 0.1157 389.1066 41.71 <0.0001
Table 4.3.14 Analysis of Variance (Stepwise Selection: Step 1) Source DF Sum of
Squares
Mean
Square F Value P-value
Model 1 389.1066 389.1066 41.71 <.0001
Error 40 373.1277 9.3282
Total 41 762.2344
R-Square 0.5105 C(p) 19.2627
Table 4.3.15 Parameter estimates (Stepwise Selection: Step 2)
Variable DF Parameter Estimate Standard Error Type II SS F Value P-value Intercept 1 2.3987 2.5108 5.8338 0.91 0.3453
*
x 2 1 0.6308 0.0994 257.3525 40.26 <0.0001
Table 4.3.16 Analysis of Variance (Stepwise Selection: Step 2) Source DF Sum of
Squares
Mean
Square F Value P-value
Model 2 512.9552 256.4776 40.13 <.0001
Error 39 249.2791 6.3918
Total 41 762.2344
R-Square 0.6730 C(p) 2.2561
Table 4.3.17 Summary of Stepwise Selection Step Variable
Table 4.3.18 Summary of All Possible Regressions Number of
Figure 4.3.1 Plot R2p versus p
Figure 4.3.3 Plot MSRes(p versus p )
23.5358-11.1275x 0.5443x 3.3897x 1.5126x
= + + +