DATA SET "lcavol" "lweight" "age" "lbph" "svi" "lcp" "gleason" "pgg45" "lpsa" -0.5798185 2.7695 50 -1.386294 0 -1.38629 6 0 -0.43078 -0.9942523 3.3196 58 -1.386294 0 -1.38629 6 0 -0.16252 -0.5108256 2.6912 74 -1.386294 0 -1.38629 7 20 -0.16252 -1.2039728 3.2828 58 -1.386294 0 -1.38629 6 0 -0.16252 0.7514161 3.4324 62 -1.386294 0 -1.38629 6 0 0.37156 -1.0498221 3.2288 50 -1.386294 0 -1.38629 6 0 0.76547 0.7371641 3.4735 64 0.615186 0 -1.38629 6 0 0.76547 0.6931472 3.5395 58 1.536867 0 -1.38629 6 0 0.85442 -0.7765288 3.5395 47 -1.386294 0 -1.38629 6 0 1.04732 0.2231436 3.2445 63 -1.386294 0 -1.38629 6 0 1.04732 0.2546422 3.6041 65 -1.386294 0 -1.38629 6 0 1.26695 -1.3470736 3.5987 63 1.266948 0 -1.38629 6 0 1.26695 1.6134299 3.0229 63 -1.386294 0 -0.59784 7 30 1.26695 1.4770487 2.9982 67 -1.386294 0 -1.38629 7 5 1.34807 1.2059708 3.442 57 -1.386294 0 -0.43078 7 5 1.39872 1.5411591 3.0611 66 -1.386294 0 -1.38629 6 0 1.44692 -0.4155154 3.516 70 1.244155 0 -0.59784 7 30 1.47018 2.2884862 3.6494 66 -1.386294 0 0.37156 6 0 1.4929 -0.5621189 3.2677 41 -1.386294 0 -1.38629 6 0 1.55814 0.1823216 3.8254 70 1.658228 0 -1.38629 6 0 1.59939 1.1474025 3.4194 59 -1.386294 0 -1.38629 6 0 1.639 2.0592388 3.501 60 1.474763 0 1.34807 7 20 1.65823 -0.5447272 3.3759 59 -0.798508 0 -1.38629 6 0 1.69562 1.7817091 3.4516 63 0.438255 0 1.17865 7 60 1.7138 0.3852624 3.6674 69 1.599388 0 -1.38629 6 0 1.73166 1.446919 3.1246 68 0.300105 0 -1.38629 6 0 1.76644 0.5128236 3.7197 65 -1.386294 0 -0.79851 7 70 1.80006 -0.4004776 3.866 67 1.816452 0 -1.38629 7 20 1.81645 1.0402767 3.129 67 0.223144 0 0.04879 7 80 1.84845 2.4096442 3.3759 65 -1.386294 0 1.61939 6 0 1.89462 0.2851789 4.0902 65 1.962908 0 -0.79851 6 0 1.92425 0.1823216 6.1076 65 1.704748 0 -1.38629 6 0 2.00821 1.2753628 3.0374 71 1.266948 0 -1.38629 6 0 2.00821 0.0099503 3.2677 54 -1.386294 0 -1.38629 6 0 2.02155 -0.0100503 3.2169 63 -1.386294 0 -0.79851 6 0 2.04769 1.3083328 4.1198 64 2.171337 0 -1.38629 7 5 2.08567 1.4231083 3.6571 73 -0.579818 0 1.65823 8 15 2.15756 0.4574248 2.3749 64 -1.386294 0 -1.38629 7 15 2.19165 2.6609586 4.0851 68 1.373716 1 1.83258 7 35 2.21375 0.7975072 3.0131 56 0.936093 0 -0.16252 7 5 2.27727 0.6205765 3.142 60 -1.386294 0 -1.38629 9 80 2.29757 1.442202 3.6826 68 -1.386294 0 -1.38629 7 10 2.30757 0.5822156 3.866 62 1.713798 0 -0.43078 6 0 2.32728 1.7715568 3.8969 61 -1.386294 0 0.81093 7 6 2.37491 1.4861397 3.4095 66 1.7492 0 -0.43078 7 20 2.52172 1.6639261 3.3928 61 0.615186 0 -1.38629 7 15 2.55334 2.7278528 3.9954 79 1.879465 1 2.65676 9 100 2.56879 1.1631508 4.0351 68 1.713798 0 -0.43078 7 40 2.56879 1.7457155 3.498 43 -1.386294 0 -1.38629 6 0 2.59152 1.2208299 3.5681 70 1.373716 0 -0.79851 6 0 2.59152 1.0919233 3.9936 68 -1.386294 0 -1.38629 7 50 2.65676 1.660131 4.2348 64 2.073172 0 -1.38629 6 0 2.67759 0.5128236 3.6336 64 1.492904 0 0.04879 7 70 2.68444 2.1270405 4.1215 68 1.766442 0 1.44692 7 40 2.69124 3.1535904 3.516 59 -1.386294 0 -1.38629 7 5 2.70471 1.2669476 4.2801 66 2.122262 0 -1.38629 7 15 2.718 0.9745596 2.8651 47 -1.386294 0 0.50078 7 4 2.78809 0.463734 3.7647 49 1.423108 0 -1.38629 6 0 2.79423 0.5423243 4.1782 70 0.438255 0 -1.38629 7 20 2.80639 1.0612565 3.8512 61 1.294727 0 -1.38629 7 40 2.81241 0.4574248 4.5245 73 2.326302 0 -1.38629 6 0 2.842 1.9974177 3.7197 63 1.619388 1 1.90954 7 40 2.85359 2.7757088 3.5249 72 -1.386294 0 1.55814 9 95 2.85359 2.0347056 3.917 66 2.008214 1 2.11021 7 60 2.882 2.0731719 3.623 64 -1.386294 0 -1.38629 6 0 2.882 1.458615 3.8362 61 1.321756 0 -0.43078 7 20 2.88759 2.0228712 3.8785 68 1.783391 0 1.32176 7 70 2.92047 2.1983351 4.0509 72 2.307573 0 -0.43078 7 10 2.96269 -0.4462871 4.4085 69 -1.386294 0 -1.38629 6 0 2.96269 1.1939225 4.7804 72 2.326302 0 -0.79851 7 5 2.97298 1.8640801 3.5932 60 -1.386294 1 1.32176 7 60 3.01308 1.1600209 3.3411 77 1.7492 0 -1.38629 7 25 3.03735 1.2149127 3.8254 69 -1.386294 1 0.22314 7 20 3.05636 1.8389611 3.2367 60 0.438255 1 1.17865 9 90 3.07501 2.9992262 3.8491 69 -1.386294 1 1.90954 7 20 3.27526 3.1411305 3.2638 68 -0.051293 1 2.42037 7 50 3.33755 2.010895 4.4338 72 2.122262 0 0.50078 7 60 3.39283 2.5376572 4.3548 78 2.326302 0 -1.38629 7 10 3.4356 2.6483002 3.5821 69 -1.386294 1 2.584 7 70 3.45789 2.7794402 3.8232 63 -1.386294 0 0.37156 7 50 3.51304 1.4678743 3.0704 66 0.559616 0 0.22314 7 40 3.51601 2.5136561 3.4735 57 0.438255 0 2.32728 7 60 3.53076 2.6130067 3.8888 77 -0.527633 1 0.55962 7 30 3.5653 2.677591 3.8384 65 1.115142 0 1.7492 9 70 3.57094 1.5623463 3.7099 60 1.695616 0 0.81093 7 30 3.58768 3.3028493 3.519 64 -1.386294 1 2.32728 7 60 3.63099 2.0241931 3.7317 58 1.638997 0 -1.38629 6 0 3.68009 1.7316555 3.369 62 -1.386294 1 0.3001 7 30 3.71235 2.8075938 4.7181 65 -1.386294 1 2.46385 7 60 3.98434 1.5623463 3.6951 76 0.936093 1 0.81093 7 75 3.9936 3.246491 4.1018 68 -1.386294 0 -1.38629 6 0 4.02981 2.5329028 3.6776 61 1.348073 1 -1.38629 7 15 4.12955 2.8302678 3.8764 68 -1.386294 1 1.32176 7 60 4.38515 3.8210036 3.8969 44 -1.386294 1 2.16905 7 40 4.68444 2.9074474 3.3962 52 -1.386294 1 2.46385 7 10 5.14312 2.8825636 3.7739 68 1.558145 1 1.55814 7 80 5.47751 3.4719665 3.975 68 0.438255 1 2.90417 7 20 5.58293 

We consider the data (prostate.txt) from the study of Stamey (1989). It was a study on 97 men with prostate cancer who were about to receive a radical prostatectomy (an operation). The relationship between the level of prostate-specific antigen and a number of clinical measures were studied.

Variable Meaning

X1 lcavol log(cancer volume)

X2 lweight log(prostate weight)

X3 age age

X4 lbph log(benign prostatic hyperplasia amount)

X5 svi seminal vesicle invasion

X6 lcp log(capsular penetration)

X7 gleason Gleason score

X8 pgg45 percentage Gleason scores 4 or 5

Y lpsa log(prostate specific antigen)

(a) [20 marks] Consider the full model

Y = 0 + 1X1 + . . . + 8X8 + Error .

Here, the error terms are assumed to be independent and identically distributed random variables N(0, 2 ). Suppose that the value of X of a new patient is given as follows

X1 = 1.1474025, X2 = 3.4194, X3 = 59, X4 = -1.386294, X5 = 0, X6 = -1.38629, X7 = 6, X8 = 0

You are interested in predicting Y , the logarithm of amount of prostate specific antigen. Give the predicted value of Y and express the variance of prediction error in terms of 2 .

(b) [20 marks] Now, you do not want a model with eight predictors as the prediction error is not satisfactory. You prefer a model with only five predictors. Select a model with BACKWARD selection approach. Report the t statistic of each remained variables in each step.

(c) [20 marks] Now, you prefer a model with only four predictors. Select a model with FORWARD selection approach. Report the t statistic of each unselected variables in each step.

(d) [20 marks] Consider the reduced model obtained in part (c). You are interested in predicting Y , the logarithm of amount of prostate specific antigen. The value of X for a new patient is given in (a). Give the predicted value of Y and express the variance of prediction error in terms of 2 . Comparing (a) and (d), which model tends to give smaller estimation error?

Please answer all questions and show your steps clearly. Round off your numerical answers to 4 significant figures. You may use SAS and/or R to do calculations. Please include the program codes in your answer script.