Use R Software to solve the following questions:

1. Cutaneous Melanoma DataThe data set contained in the file cutaneous.txt are part of an assay on cutaneousmelanoma, a type of malignant cancer, for the evaluation of postoperative treatment performance with a high dose of interferon alpha-2b as a drug to prevent recurrence(Ibrahim et al., 2001). Patients were allowed to enter the study from 1991 to 1995 andwere followed up until 1998. The data set contains information on 417 patients and present their survival times, representing the time in years (t) until the patients death or the censoring time. The other covariate information available for patients is the nodulecategory x, taking the values 1, 2, 3 or 4, which categorizes patient based on the severityof the disease, i.e., patients who were at the lowest risk of cancer were categorized intonodule category 1 and so on patients at the highest risk of cancer were categorized in to nodule category 4.Note: The entire data set is confidential and as such only a part of the data is provided1(a) Calculate the five number summary of the lifetime t using the technique shown inclass(b) Represent the results obtained in part (a) graphically and interpret(c) Use the QQ-plot to check if the lifetime t can be assumed to come from a normalpopulation. Can a similar conclusion be drawn from part (b)(d) As you might have noticed, the data set provided is in the form of a data frame.Extract the data frame for x = 1, i.e., for the first nodule category. So, in this case,you should have t values only for the first nodule category. Assign the name data1to this data frameHint: If data is the name of the initial data frame that is provided. Then,data1=data[data$x==1,]Calculate the number of patients belonging to nodule category 1 by using the command length(data1$t) or length(data1$x)(e) Repeat part (d) for nodule categories 2, 3 and 4. So, now you should be having4 different data frames: data1, data2, data3, and data4, where the data frame icontains the lifetime (t) information for the i-th nodule category (x) , i = 1, 2, 3, 4(f) The clinician conjectures that the average lifetime for patients belonging to nodulecategory 1 is significantly higher than those belonging to nodule category 4. Testthe clinicians claim at 5% level of significance and draw appropriate conclusion(g) The way the nodule category is introduced (see the data description), it is clear thatone would conjecture that the average lifetime of patients would decrease as thenodule category increases. This requires us to carry out 5 more tests of hypothesesbesides the one in part (f). So, carry out the following testing: Average lifetime of patients belonging to nodule category 1 is higher than thosebelonging to nodule category 2 Average lifetime of patients belonging to nodule category 1 is higher than thosebelonging to nodule category 3 Average lifetime of patients belonging to nodule category 2 is higher than thosebelonging to nodule category 3 Average lifetime of patients belonging to nodule category 2 is higher than thosebelonging to nodule category 4 Average lifetime of patients belonging to nodule category 3 is higher than thosebelonging to nodule category 4(h) Summarize the results obtained in parts (f) and (g) and comment on the lifetimesof patients belonging to different nodule categoriesNote: In all testing problems, specify the null and the alternative hypotheses22. Anti-Anginal Response against Disease HistoryTreadmill stress tests were applied to patients with angina pectoris before and 4 weeksafter once-daily dosing with an experimental anti-anginal medication. The main objectivefor carrying out this study is to check if the improvement in exercise duration is dependenton the patients disease history. The data file disease.txt represents the disease durationtime (in years) since initial diagnosis and percent-improvement in treadmill walking times.(a) Identify the response and the explanatory variables with proper reasoning(b) Check graphically if the dependency between improvement and disease duration timecan be considered to be linear(c) Now, use a formal test of hypothesis to check for the linear relationship in part (b) anddraw appropriate conclusion(d) How does the disease duration effect the percentage improvement in exercise duration(e) What assumption did you make to carry out the testing in part (c). Is there any other assumptionsrequired to formulate a linear model for improvement and disease durationtime(f) Check all the model assumptions made in part (d) graphically and comment on theviolation or non violation of the model assumptions with proper reasoning.

cutaneous melanoma data set:

Patient t x 1 0.72279 3 2 6.29979 1 3 6.62286 3 4 3.47159 4 5 7.01164 1 6 6.64750 1 7 6.97604 2 8 0.67077 4 9 6.88022 1 10 6.60917 1 11 1.06776 4 12 6.32991 2 13 6.32170 1 14 1.50308 3 15 0.77207 3 16 1.78234 4 17 6.65845 2 18 3.53730 1 19 5.74949 2 20 3.81383 1 21 5.97673 2 22 1.68925 1 23 5.73854 3 24 5.87543 1 25 1.57426 1 26 5.56879 3 27 1.57153 3 28 2.43669 1 29 5.78782 3 30 1.59890 2 31 0.60780 2 32 5.67556 1 33 0.24093 4 34 5.54689 3 35 6.04517 4 36 1.90007 2 37 2.13279 4 38 0.35592 2 39 2.43395 4 40 1.45380 3 41 5.24846 2 42 1.53046 4 43 5.32786 3 44 0.93361 4 45 2.22313 3 46 4.94456 1 47 5.07598 4 48 5.10609 1 49 5.05133 4 50 5.73032 2 51 1.24572 4 52 1.41273 3 53 0.94182 2 54 5.46475 2 55 1.62902 3 56 5.41547 2 57 5.63997 2 58 3.14853 1 59 5.63450 3 60 1.69199 4 61 5.13347 3 62 3.24162 3 63 5.24298 2 64 5.31143 4 65 2.27515 1 66 5.33060 1 67 2.56810 2 68 5.04312 3 69 4.95003 1 70 4.98015 3 71 1.54415 3 72 4.72005 4 73 5.15811 1 74 5.06776 1 75 1.71389 3 76 5.15264 3 77 1.92197 4 78 4.68720 3 79 4.95825 1 80 4.14784 2 81 1.44559 2 82 4.73922 2 83 5.10883 1 84 4.36961 2 85 3.33470 4 86 3.30459 4 87 3.61123 2 88 0.58590 3 89 1.48665 4 90 4.63792 1 91 4.32033 2 92 4.46270 3 93 4.90349 1 94 2.19576 1 95 1.18549 4 96 4.42710 2 97 1.53593 2 98 4.37782 2 99 4.33676 1 100 4.50103 3 101 4.46543 1 102 2.79808 1 103 4.71458 3 104 3.03080 3 105 3.33744 2 106 2.77892 1 107 4.13963 1 108 4.48734 1 109 0.76660 1 110 4.19439 2 111 4.57769 1 112 1.53320 2 113 1.48118 4 114 4.49555 3 115 4.46270 2 116 4.49008 1 117 4.07392 3 118 4.07666 2 119 3.39493 3 120 3.96167 1 121 3.93977 3 122 4.18617 1 123 2.00958 4 124 4.21903 2 125 4.02190 2 126 1.96851 3 127 2.08350 4 128 3.50719 1 129 3.76728 1 130 1.15264 3 131 0.81314 2 132 4.06845 2 133 2.98973 2 134 1.34155 2 135 3.84942 2 136 2.22040 1 137 3.74538 2 138 3.60301 4 139 0.68446 4 140 1.06229 2 141 3.55921 1 142 3.38672 2 143 3.46612 1 144 3.53730 2 145 3.68241 2 146 3.57563 4 147 0.96099 2 148 1.42368 2 149 2.15469 1 150 3.61670 2 151 3.23614 1 152 0.92539 1 153 0.95551 4 154 3.21150 2 155 3.20329 2 156 3.44422 2 157 2.64203 3 158 1.94114 3 159 0.98015 2 160 0.95551 4 161 3.10198 2 162 3.20055 1 163 2.96509 2 164 3.12389 2 165 2.37919 2 166 2.92676 4 167 3.06639 3 168 3.04723 3 169 2.58453 1 170 2.05886 4 171 0.79124 1 172 2.89665 1 173 3.12663 3 174 0.34771 4 175 5.99863 4 176 5.96851 4 177 6.29432 1 178 6.58179 3 179 6.67488 2 180 0.61328 2 181 0.87611 2 182 5.84531 2 183 6.10541 3 184 1.70294 3 185 6.01506 1 186 1.66461 2 187 2.11088 4 188 1.59617 2 189 0.77207 3 190 1.40726 2 191 1.68104 2 192 1.60438 4 193 1.81793 3 194 3.41136 2 195 5.19370 4 196 5.95483 4 197 5.71116 2 198 3.39220 1 199 1.88638 1 200 3.96167 1 201 1.00753 4 202 2.08624 1 203 1.62902 2 204 6.03970 3 205 1.20465 3 206 3.30732 1 207 5.50308 4 208 1.73580 1 209 3.72348 1 210 2.07529 3 211 0.82136 3 212 5.80698 1 213 2.87201 2 214 1.57974 2 215 4.98289 2 216 5.77139 4 217 0.96372 1 218 1.32512 2 219 5.22656 1 220 1.51403 1 221 0.67625 4 222 1.01574 3 223 5.04312 4 224 5.22382 2 225 4.88433 2 226 0.45996 3 227 5.01300 1 228 5.09240 1 229 5.40452 2 230 2.86379 2 231 4.98289 2 232 5.08966 4 233 0.80219 4 234 3.65777 2 235 4.80493 2 236 2.40383 1 237 4.17248 2 238 1.73854 2 239 0.60233 3 240 5.16359 2 241 1.29500 2 242 4.80767 4 243 5.14990 4 244 4.52293 1 245 4.75565 2 246 4.27105 1 247 0.52567 3 248 5.01574 1 249 2.51608 1 250 4.80219 1 251 0.85969 2 252 4.73648 2 253 4.92266 3 254 4.48460 3 255 0.45996 2 256 4.85147 1 257 3.30732 1 258 4.49281 1 259 0.47912 3 260 3.88227 1 261 4.68172 2 262 4.59959 3 263 3.50445 4 264 4.34497 3 265 1.67009 3 266 2.40110 4 267 4.34497 3 268 4.61328 2 269 3.44148 1 270 3.73717 2 271 3.22793 4 272 4.12868 4 273 0.54757 4 274 1.21287 2 275 4.14784 3 276 1.60712 4 277 4.54209 4 278 2.02053 1 279 0.16975 4 280 4.26557 2 281 4.20808 1 282 1.33881 4 283 3.51814 3 284 0.59685 4 285 3.29637 1 286 4.12047 1 287 1.76318 1 288 3.03901 3 289 2.25873 3 290 0.55578 2 291 1.24298 2 292 2.81725 1 293 4.05749 2 294 3.74538 1 295 2.86653 3 296 3.61396 1 297 4.03285 2 298 2.80082 3 299 1.47296 2 300 0.74196 4 301 4.06297 2 302 3.98905 2 303 3.57016 4 304 3.58385 1 305 3.59754 4 306 1.85900 2 307 0.85969 4 308 1.63723 1 309 1.69747 1 310 1.50856 3 311 3.86858 3 312 3.65777 1 313 1.99589 3 314 0.33128 3 315 3.63860 3 316 3.60301 2 317 2.54620 1 318 2.17385 1 319 3.52088 2 320 2.82272 2 321 2.63655 2 322 2.62560 1 323 2.80356 2 324 3.54552 2 325 0.99384 4 326 2.71869 1 327 0.91992 4 328 1.85079 2 329 0.91170 4 330 5.12526 3 331 5.79877 2 332 3.20055 2 333 4.43258 2 334 1.15811 1 335 0.26557 3 336 5.49760 4 337 1.24298 4 338 5.50308 2 339 5.65092 2 340 3.49350 2 341 0.14784 3 342 5.93018 2 343 0.78303 2 344 1.92471 2 345 3.11020 2 346 4.93361 3 347 1.70294 4 348 1.73580 4 349 1.43737 4 350 2.24778 2 351 5.40999 4 352 4.26283 1 353 4.73922 3 354 4.47639 1 355 0.53662 3 356 0.97741 1 357 5.07598 4 358 5.00205 2 359 3.05818 2 360 4.41342 1 361 1.91923 3 362 0.88980 4 363 1.31691 4 364 4.07392 2 365 2.68036 1 366 3.89049 2 367 4.04654 3 368 4.49281 3 369 1.54415 3 370 2.09172 4 371 3.80287 2 372 3.15674 2 373 1.13895 3 374 2.30801 2 375 4.64887 2 376 0.28474 2 377 1.26762 4 378 3.70431 1 379 3.25804 3 380 4.13689 3 381 0.42437 4 382 2.93771 2 383 1.83710 2 384 4.00274 3 385 3.08556 1 386 3.65229 3 387 3.78919 1 388 1.51129 2 389 3.37577 1 390 2.99521 3 391 1.31964 1 392 3.41136 2 393 0.29295 2 394 2.93224 1 395 2.89117 2 396 3.01985 1 397 3.17317 2 398 3.25257 1 399 2.73785 2 400 0.88433 4 401 3.36208 1 402 0.68994 4 403 2.98700 1 404 3.26899 4 405 3.22245 4 406 1.78234 2 407 1.80698 2 408 2.88296 3 409 2.71595 2 410 3.19233 4 411 2.94319 1 412 0.73374 4 413 2.77070 1 414 1.36893 1 415 2.67762 2 416 2.77618 2 417 2.97878 1

Disease Data Set:

Patient_number Disease_duration Improvement 1 1 40 2 1 90 3 3 30 4 2 30 5 1 80 6 5 60 7 1 10 8 4 -10 9 2 50 10 6 40 11 1 60 12 4 0 13 2 50 14 2 100 15 3 20 16 3 70 17 5 -30 18 3 20 19 1 40 20 6 0