STA2005Nov08 - UNIVERSITY OF CAPE TOWN DEPARTMENT OF STATISTICAL SCIENCES STA2005S NOVEMBER 2008 EXAMINATION INTERNAL EXAMINERS Mr A Clark Dr B Erni

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Unformatted text preview: UNIVERSITY OF CAPE TOWN DEPARTMENT OF STATISTICAL SCIENCES STA2005S NOVEMBER 2008 EXAMINATION INTERNAL EXAMINERS: Mr A Clark, Dr B Erni INTERNAL ASSESSOR: Dr F Little EXTERNAL EXAMINER: Dr C Lombard AVAILABLE MARKS: 100 MAXIMUM MARKS: 100 TIME ALLOWED: 3 hours PAGES: 16 + 6 TABLES PLEASE ANSWER EACH SECTION IN A SEPARATE BOOK SECTION A: REGRESSION ANALYSIS [45 Marks] 1. The general linear regression model is deﬁned as Y(n×1) = X(n×k) β(k×1) + e(n×1) where e ∼ N (0, σ 2 I ). Assume that X is full rank and that (X X )−1 exists. ˆ ˆ (a) Show that β and cov (β ) are functions of the eigenvalues of X X . (4) ˆ (b) Use the previous answer and explain when β will be unstable. (1) ˆ (c) Now suggest what could be done when β is unstable. (1) [6] ˆ 2. Deﬁne Yave as the average ﬁtted value when undertaking linear regression. Now prove that the following identities are true: ˆ ¯ (a) Yave = Y n (b) j =1 ¯ Yj − Y (2) ˆ ¯ Yj − Yave = n j =1 ˆ ¯ Yj2 − nY 2 (5) [7] 1 ˆ 3. Let X be a n × k design matrix and β be the least squares estimate of β from the general linear model. Now deﬁne X(i) as the design matrix with the ith row deleted. ˆ Deﬁne Y(i) similarly and β(i) is the least squares estimate of β if the ith row of X and Y are deleted. Denote xi as the ith row of X , yi as the ith element of Y and ei ˆ th ˆ. Further as the i estimated residual when all observations are used to estimate β deﬁne H = X (X X )−1 X with diagonal elements hii . Now use the following results (X(i) X(i) )−1 = (X X )−1 + ˆ ˆ β(i) = β − (X X )−1 xi xi (X X )−1 1 − xi (X X )−1 xi ei ˆ (X X )−1 xi 1 − hii to show that the studentised residuals, ti , is equal to ei ˆ ti = s2i) (1 − hii ) ( [8] 4. In a diabetes study, 442 diabetes patients were measured on 10 explanatory variables. The aim of the study was to identify a predictive model of the response variable (Y), a measure of disease progression. The following explanatory variables were used: AGE - the age of the patient in years SEX - SEX=1 if the patient is a male and 2 if the patient is a female BMI - the patients body mass index and is calculated as a patients weight (kg ) divided by their height squared (m2 ) BP - a patients blood pressure Serum measurements, S1, S2, S3, S4, S5 and S6 (continuous variables) Use the attached outputs to answer the following questions: (a) Explain what the AIC of a model represents. (1) (b) Brieﬂy explain what is meant by stepwise regression with speciﬁc reference to the R function, stepAIC and to the output in Appendix 2. In your answer explain what happens when the computer moves from STEP 4 to STEP 5 in Appendix 2. (4) (c) Use the model suggested by the stepwise regression procedure and interpret the sex.2 coeﬃcient. Now test the hypothesis that H0 : βsex.2 = 0. The alternative hypothesis is H1 : βsex.2 = 0 (4) 2 (d) Use MODEL 1 to construct a point prediction and a prediction interval of a future observation if a male patient has a body mass index of 32, a blood pressure of 101, S1=93, S2=100 and S5 =5. (See Appendix 3.) (3) (e) Use MODEL 1 to test the following hypothesis at the 5% signiﬁcance level, H0 : βBM I + βBP βS 1 + βS 2 + βS 5 = 8 72 Explain how the required test statistic is constructed and identify the distribution of test statistic used when undertaking the hypothesis test. (6) (f) Use the plots as well as the estimation output provided (Appendix 1-7) and brieﬂy discuss whether MODEL 1 adequately captures the linear relationship between the response variable and the explanatory variables. Suggest what might be done to rectify potential problems with the model. (6) [24] 3 Appendix 1 : Correlation matrix (Diabetes Data set) ================================================================ #the correlation matrix round(cor(a),digits=3) AGE SEX BMI BP S1 S2 S3 S4 S5 S6 Y AGE 1.000 0.174 0.185 0.335 0.260 0.219 -0.075 0.204 0.271 0.302 0.188 SEX 0.174 1.000 0.088 0.241 0.035 0.143 -0.379 0.332 0.150 0.208 0.043 BMI 0.185 0.088 1.000 0.395 0.250 0.261 -0.367 0.414 0.446 0.389 0.586 BP 0.335 0.241 0.395 1.000 0.242 0.186 -0.179 0.258 0.393 0.390 0.441 S1 0.260 0.035 0.250 0.242 1.000 0.897 0.052 0.542 0.516 0.326 0.212 S2 0.219 0.143 0.261 0.186 0.897 1.000 -0.196 0.660 0.318 0.291 0.174 S3 -0.075 -0.379 -0.367 -0.179 0.052 -0.196 1.000 -0.738 -0.399 -0.274 -0.395 S4 0.204 0.332 0.414 0.258 0.542 0.660 -0.738 1.000 0.618 0.417 0.430 S5 0.271 0.150 0.446 0.393 0.516 0.318 -0.399 0.618 1.000 0.465 0.566 S6 0.302 0.208 0.389 0.390 0.326 0.291 -0.274 0.417 0.465 1.000 0.382 Y 0.188 0.043 0.586 0.441 0.212 0.174 -0.395 0.430 0.566 0.382 1.000 Appendix 2 : Stepwise Selection (Diabetes data set) ================================================================= step.<-stepAIC(lm(Y~AGE+sex.+BMI+BP+S1+S2+S3+S4+S5+S6), scope=list(upper=Y~AGE+sex.+BMI+BP+S1+S2+S3+S4+S5+S6,lower=~1), direction=c("both")) Start: AIC=3539.64 (STEP 1) Y ~ AGE + sex. + BMI + BP + S1 + S2 + S3 + S4 + S5 + S6 Df Sum of Sq RSS - AGE 1 82 1264068 - S3 1 663 1264649 - S6 1 3080 1267066 - S4 1 3526 1267512 <none> 1263986 - S2 1 5799 1269785 - S1 1 10600 1274586 - sex. 1 44999 1308984 - S5 1 56016 1320001 - BP 1 72100 1336086 - BMI 1 179033 1443019 AIC 3538 3538 3539 3539 3540 3540 3541 3553 3557 3562 3596 Step: AIC=3537.67 (STEP 2) Y ~ sex. + BMI + BP + S1 + S2 + S3 + S4 + S5 + S6 4 Df Sum of Sq RSS - S3 1 646 1264715 - S6 1 3001 1267069 - S4 1 3543 1267611 <none> 1264068 - S2 1 5751 1269820 - S1 1 10569 1274637 + AGE 1 82 1263986 - sex. 1 45830 1309898 - S5 1 55964 1320032 - BP 1 73847 1337915 - BMI 1 179084 1443152 AIC 3536 3537 3537 3538 3538 3539 3540 3551 3555 3561 3594 Step: AIC=3535.9 (STEP 3) Y ~ sex. + BMI + BP + S1 + S2 + S4 + S5 + S6 Df Sum of Sq RSS - S6 1 3093 1267808 - S4 1 3247 1267961 <none> 1264715 - S2 1 7505 1272219 + S3 1 646 1264068 + AGE 1 66 1264649 - S1 1 26839 1291554 - sex. 1 46381 1311096 - BP 1 73533 1338248 - S5 1 97508 1362223 - BMI 1 178542 1443256 AIC 3535 3535 3536 3537 3538 3538 3543 3550 3559 3567 3592 Step: AIC=3534.98 (STEP 4) Y ~ sex. + BMI + BP + S1 + S2 + S4 + S5 - S4 <none> - S2 + S6 + S3 + AGE - S1 - sex. - BP - S5 - BMI Df Sum of Sq RSS 1 3686 1271494 1267808 1 7472 1275280 1 3093 1264715 1 738 1267069 1 0.4385 1267807 1 26378 1294186 1 44684 1312492 1 82152 1349960 1 102520 1370328 1 189976 1457784 AIC 3534 3535 3536 3536 3537 3537 3542 3548 3561 3567 3595 Step: AIC=3534.26 (STEP 5) Y ~ sex. + BMI + BP + S1 + S2 + S5 5 Df Sum of Sq <none> + S4 + S6 + S3 + AGE - S2 - sex. - S1 - BP - BMI - S5 1 1 1 1 1 1 1 1 1 1 3686 3533 395 11 39377 41856 65236 79625 190592 294092 RSS 1271494 1267808 1267961 1271099 1271483 1310871 1313350 1336730 1351119 1462086 1565586 AIC 3534 3535 3535 3536 3536 3546 3547 3554 3559 3594 3624 Appendix 3 : MODEL 1 =============================================================== sex.=factor(SEX) step.m=lm(Y ~ sex. + BMI + BP + S1 + S2 + S5) step.m.s=summary(step.m) b=step.m\$coef Coefficients: Estimate Std. Error t value Pr(>|t|) -------------------------------------------------------------(Intercept) -335.3576 25.3234 -13.243 < 2e-16 *** sex.2 -21.5910 ? ? ? ? BMI 5.7111 0.7073 8.075 6.69e-15 *** BP 1.1266 0.2158 5.219 2.79e-07 *** S1 -1.0429 0.2208 -4.724 3.12e-06 *** S2 0.8433 0.2298 3.670 0.000272 *** S5 73.3065 7.3083 10.031 < 2e-16 *** -------------------------------------------------------------Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 -------------------------------------------------------------Residual standard error: 54.06 on ? degrees of freedom Multiple R-squared: 0.5149, Adjusted R-squared: 0.5082 F-statistic: 76.95 on ? and ? DF, p-value: < 2.2e-16 -------------------------------------------------------------sex.in=ifelse(SEX==1,0,1) X=cbind(1,sex.in,BMI,BP,S1,S2,S5) 6 C=solve(crossprod(X)) round(C,digits=5) sex.in 0.21939 0.00504 sex.in 0.00504 0.01114 BMI -0.00075 0.00018 BP -0.00062 -0.00009 S1 -0.00005 0.00015 S2 -0.00001 -0.00016 S5 -0.02805 -0.00348 BMI -0.00075 0.00018 0.00017 -0.00001 0.00001 -0.00002 -0.00070 BP S1 S2 S5 -0.00062 -0.00005 -0.00001 -0.02805 -0.00009 0.00015 -0.00016 -0.00348 -0.00001 0.00001 -0.00002 -0.00070 0.00002 0.00000 0.00000 -0.00007 0.00000 0.00002 -0.00002 -0.00033 0.00000 -0.00002 0.00002 0.00028 -0.00007 -0.00033 0.00028 0.01827 x.f=matrix(c(1,0,32,101,93,100,5),nrow=1) x.f%*%C%*%t(x.f) [,1] [1,] 0.1410783 A=matrix(0,nrow=2,ncol=7) A[1,3:4]=1 A[2,5:7]=1 A%*%b [,1] [1,] 6.837659 [2,] 73.106927 B=matrix(c(8,72),ncol=1) V=solve(A%*%C%*%t(A)) t(A%*%b-B)%*%V%*%(A%*%b-B)/(NROW(V)*step.m.s\$sigma^2) [,1] [1,] 1.713070 7 Appendix 4 : Model Checking (Diabetes data set) ==================================================================== shapiro.test(step.m\$res) Shapiro-Wilk normality test ---------------------------data: step.m\$res W = 0.9972, p-value = 0.6718 QQ normal of the Estimated Residuals −100 0 50 100 300 250 200 150 100 50 −3 step.m\$res 50 0 −150 0.000 −200 step.m\$fit −50 step.m\$res 0.004 0.002 Density 0.006 100 150 0.008 Empirical Dist of the Estimated Residuals −2 −1 0 1 2 3 norm quantiles Residuals vs fitted of the Diabetes data set 115 406 333 322 252 251 323 395263 257 368 391 168 216 140 367 146 255 33 383 250 24404 429139 39131 160 351 269 337 304 72 138 241 226 109249 118 363 206 148 413 162235136 328 417 173 123 178 45 218 291 314 209 326441 292301 142 37 377170433 208358217117 10 1 17 303 423 169151 57 324130114 325 98 311 97 122372317 185 405 254 409 187 156 422 274 277 400 361 290 369 50 438 271 286 342153 147 93 237 18616418 92 320204 239 191 321 126 198 77 398 116 35530 329 253 34766145294 3 212 200 236 370 203 331 81 27528 150 334312 37639979 205 360 315 381 21018112562401 318 143 240 86 338425183 16 14 341 396 101 175 343 231 196 180410 4 378 408 389415 416 10452 281 272 364 124319 31285119 411 379233 9 88232 25 155 29635432751 273 84 177 365 103 427 108 207 366 141189 184 288 113 38 344 105 276 19554 213340 412 121 223 424 7541 73 26 120 157 418 152 47 19 43 234 199 308 345386 222 220 135 67 163 40 55 260 100 19429 403 165 245 110 30642160 53266 268310374 211 283 380112 99 435 247 307 428 6874375 96 219 356 5349192 339 305 46 158 76 302 262 346 20 280 224 8 300 432 414313 385 82154 149176 256242 436 128 265 13 21 229 144 384 440 42670 65 6 23 279 106190 133 299 69362 61 95 278 316 48 431 37334228 332 258259 246 387430 394 102 244 1535015330 29835717911 64 27 59 439 193 230 348 90 214 197174393 201 419 129 2 12282 182293 309 388 172159 270 80 161335 36 352 22243402 24835 336 295227 390 289382 7 284 58 22591 107132 78 49 42032 297 89166 244 287 134 188 42238 202 167 94 56 71 83 353 63 407127 39785 137 371 434 261442 359 437221 264 392 171 87 111 267 −150 −50 0 50 100 150 step.m\$res Figure 1: Model Checking plots of the full model of the Diabetes Data set 8 Appendix 5 : Outlier plots (Diabetes data set) =============================================================== The Leverage of the i’th Observation 0 100 200 300 0.05 354 231 257 262 124 24 406 368 442 261 267 341 372 409 36 59 324 367 395 323 353 383 73 255 30 33 77 111 146 305 408 79 115 8 417 287 326 118 171 200 294 337 72 344 12 203 249 277 16 263288 333 377401 63 270 11 274 318 350 388 418 139 136 175 216 245268 108131 162 194 289 282 328 363 390 419 391 424 347 380404 434 142 96 126 382 412 439 426 210 44 93 344862 83102 138 164 192 218 247 279304 351376 415 397 221 260 150 195219 253 286 228252 98 100 240 87105 148 174199 225 251 290 322 355 381 423 92 431 17 39 10274258 86 109134 160183 209 236259283306329 362385 413 172 154 188 40 9 28 54 85 116140 167191215239264 299 325 369 405429 173 189 271 312 348 427 94 128153 186211 243 269 301 342 375 421 285 311 66 91 127151 180 206 234 265 292317340364 392416 156 187 220 4964 88 123 149 178201226250 278302 331 359 386410 437 169 295 338 135159182 217241 275 310335 371 403 163 293 321 352 37 5670 89 120145168 198222 248273296320345370 398 428 411 407 433 45 67 32 61 81 1 21385571 104 132 161184 223 258 291 332 358 389 420 2 22 4157 75 97 137 165190213237 4 264360 78 107 144 179202 235 309 343 3 1835506580101 141166 193 224 254 298 327 366 393 430 5 19 7 29 6 23 46 6984103 130 158 185208232256 284307330 374399 425 117 214238 122 197 233 244 114 314 346 378 121 440 308 339 365 402 432 212 242 441 316 349373 414438 25 51 76 95113 143 181205229 272 303 336361 153147 7490110133 177 204227 266 297 334357 396 422 119 112 147 82 106129 176 207230 246 276300 53 14 5268 99 125 155 196 319 394 435 400 436 280 313 356379 281 315 360384 387 157 13 20 152 170 0.03 Leverage 360 305 78 38 291 365388 205 233 142 284 281 113 153 191 223 10 30 33 405 283 114139 332 361 396 339363386 132 173 219 240 277 355 342 379 59 79 111 151 177 218 257280 431 130 165188211 239264 301 337 378 56 185 13 7 26 52 708498117 143 169 201 231255 288 331 362 399 429 242 118 345 73 116 37 5569 107 138 120 155 192 220245 274 366 411 263 27 47 341 229 256 4 20 4560 94 121 163 207 241 286 324 359 400 426 96 127 217 251 292 325 104 136 413 137 48 72 91 131 204 249273 299 330 376 403 140 309 224 267 308 349 391 417 393 423 404 44 318 346 392 422 313336 2539 6480 119 146 184 221246269294317 343 374398421 129 159 203 235 265 297 328353 382407 434 358 395 440 238 282 310 214 36 67 85 109134 167 410 442 178 416 176 515 49 87 228 271 304 24 11 29 63 90 128 171 199 225 270293 320 350375 401 437 289312335 371 402 439 372 406 441 102 145 174 216 414 279 162 189 227 261 287 323 351 397 430 166 193 42 74 95 61632 516683 108133156179 208232 259 285 316 348 35 7188 243268 303326 352 175 180 334 370 409 438 262 306 333 367390 419 12 34 627792 122147172197 226250 327 368 394 420 252 384 412 68 89 123 149 183 315 126 161 194 22 58 86 115 141 168 202 427 300 344 387 415 3 1731 506581101125 158182 215 248272295 322347 373 408433 99 135160 190214 247 278 357 18 54 23 154 198 236 921 41 314 354 389 425 110 150 196 230 258 432 436 97 428 1819 40 6175 100 144 170195 222 254 298321 356 385 424 106 105 152 187 234 266 302 340 369 164 209 244 276 319 181 213 296 338 53 82 112 46 76 124 435 186210 253 260 307 200 2843 380 377 418 311 148 329 364 212 275 383 237 157 381 290 206 93 57 0.01 2 1 0 −1 −2 −3 Studentised Residuals 3 Studentised Residuals 103 400 0 100 200 Index 300 400 Index Cooks Statistic Cooks* Statistic 305 305 0 124 100 200 300 0.020 0.010 57 257 381 290 388 206 142 200 377 418 329 59 78 103 170 210 237 277 79 380 153 148 111 139 363 253 354 275 10 38 191 219 255 283 231 260 291 337 364 405 28 186 212 240 360 288311 355 223 8 332 365 173 205 233 98118 151 195218 338 369 284 431 342 339 114 157 43 73 100 132 164188213 239264 296321 362386 424 105 341 301 340 435 56 76 113 150 181 209 247 281 361 396 429 344 46 75 107 144169 198 236 280 307331356 385 428 187 1717 37537084 106130154177201 234 263286 322345 378 408432 921 4054 72 96115138 165 192 222245 274298 324347 379 417 117 112136160185 211 242 426 26 425 244268 302325 366389 415 314 193652668297116140163 190214 241 267 299323 353 387411434 319 126 168 23 131 18355064 86104127 155180203226249272295318 352376399422 94 120143166 4143145617791110134158182207230254278303326349373397420 3132741556983101125149172196220243266289312335358382406430 141 178 22 4258 87 109 135159183208232256279304328351375400423 229 258 137 167 193 228251 293316 359 390413436 404 409 20 39 65 8599 121146 174197221 252276300 327350374398421 403427 412 71 92 123 162 189 216 262287310334 372395 440 25 49 67 89 122147 176 204227 259282306330 357 384407 433 235 265 309 336 616 344863 81102 128152175199224 250273297320 348371394419442 401 441 402 410 515324762 8095 119 145 171194217 246269292315 346370393416439 2122944607488108133156179202225248271294317 343367391414437 5168 90 129 161184 215238261285308333 368392 438 270 313 24 11 30 33 Cooks* Statistic 383 93 0.000 0.020 0.010 0.000 Cooks Statistic 383 400 93 57 257 381 290 388 206 200 377 418 329 59 78 103 170 210 237 277 79 380 153 148 111 139 363 405 253 354 364 10 38 191 219 260283 231 275 291 28 186 212 240 355 223 255 288311 337360 8 332 365 173 205 233 98118 151 195218 338 369 284 431 342 339 114 157 43 73 100 132 164188213 239264 296321 362386 424 341 113 76 105 150 185209 236 281 307331 361385 428 435 5670 107130154177201 234 263 301324347 379 344 389 429 466175 97115138 165 192 222245 274298322345 378 211 247 280 340 1717 3753 72 96116 144169 198 244 278302325 356 396 425 921 4054 82 106 136160 187 117 112 26 417 408432 314 415 411 19365266 86104127 155 181204229252276299323 353376399423 319 366 403427 262 23 131 168 18355065 8499120143166189214 241265 295318 350374398421 94 126 163 193 235258 286309 336359 390413438 4143145627791110134158182207230254279303326349373397420 3132741556983101125149172196220243266289312335358382406430 141 180 227251 22 4258 87 121146 174197221 249272 300 327351375400 426 92 123 162 190 216 242267 293316 352 387410 436 140 178 137 167 404 409 20 39 67 85 109 135159183208232256 282306330 357 384407 433 412 25 49 68 89 122147 176 203226 259 285308333 368392 422 616 34486481102 128152175199224 250273297320 348371394419442 401 440 402 441 5153247638095 119 145 171194217 246269292315 346370393416439 2122944607488108133156179202225248271294317 343367391414437 228 268 304328 51 71 90 129 161184 215238261 287310334 372395 434 270 313 24 11 30 33 0 124 142 100 0.01 0.02 0.03 0.04 0.05 2 1 0 −1 354 −2 57 257 124 381 30 290 388 33 206 142418 200 377 329 10378 59 170 237 153 210 277 79 148 380 111 38 260 364 283 363 275 10253 139 231 291 191405 337 255 186 219 311 360212 223 28 240 288 8 218 233 338173 355 157284332 188 98 151 73 365339386 239195 118 20543 296 431 424 369 114 342 164 264 105 362 341 113 132331 301 307 281 21356321 209 274 76298 201 150 396 165 429100245 344 324 435 187 361 75302 154 247 46 70 9 53 107 40 3791441211385 347 177399198160 192 136 28037842834054 426 263 115 181 389169 236 82 726 117 356110278 325322 130185 11284345 58 138 106 234 17 96 432 425 222 244 127 43621 37 242 9761116 286 143 120 314258 254 123 2766914135927 415390 72417 19 5522 266214 41194376131 319 366135 52 4145 427 1335750220215 251126 249 408 323 262 2334 168 121241 413 194 101104 196 433 183 1865 125 42016386 48 1524723060 89 25262 391 203 158182 248 81 292 353395 250 229 190 8892404 295 256137 268 373 161273 423 382 294 267 15530821766 140 44 333 326 77367 409 368 300272403226 221318 12 99 374 64 299 434 400122 317 207 224 91 20 204 184265 34412 387330 149189 397 175 39431202 134 87 309 80 315 179 348 442 327 156 146 36 261 38419723524342 68953931269259 419 42234671306 83 162 438 6 392 421 246343398172 334 80 25 352 349535407271 279 108401 129238 375 406 141472 410 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• Winter '13
• ernie
• Regression Analysis, Statistical hypothesis testing, Hebrew numerals, diabetes data

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