5 (A5) MATH 627: Generalized Linear Models SPRING 2017 Due: Monday, March 27, 2017 Q1. [8] The following data presents the lymph node involvement of 53 patients for prostate cancer with the level of Acid Phosphates in blood and their Age. . Table 1. Brown (1980) Cancer data Index LNI A.P Age Index LNI A.P Age 1 0 48 66 28 0 50 61 2 0 56 68 29 0 50 64 3 0 50 66 30 0 40 63 4 0 52 56 31 0 55 52 5 0 50 58 32 0 59 66 6 0 49 60 33 1 48 58 7 0 46 65 34 1 51 57 8 0 62 60 35 1 49 65 9 1 56 50 36 0 48 65 10 0 55 49 37 0 63 59 11 0 62 61 38 0 102 61 12 0 71 58 39 0 76 53 13 0 65 51 40 0 95 67 14 1 67 67 41 0 66 53 15 0 47 67 42 1 84 65 16 0 49 51 43 1 81 50 17 0 50 56 44 1 76 60 18 0 78 60 45 1 70 45 19 0 83 52 46 1 78 56 20 0 98 56 47 1 70 46 21 0 52 67 48 1 67 67 22 0 75 63 49 1 82 63 23 1 99 59 50 1 67 57 24 0 187 64 51 1 72 51 25 1 136 61 52 1 89 64 26 1 82 56 53 1 126 68 27 0 40 64 (a) [1+2] Fit a binary logistic regression model to this data to determine whether AP and Age have any significant impact on LNI (use Wald statistics at the 10% level). (b) [2] Compute the deviance and test the goodness of fit of this model (use the 10% level). (c) [1] Is there any evidence of outlier in this data? (d) [2] Find the probability of a person developing LNI when his AP level is 90 and Age is 60 years. Q2. [10]. PRINCIPLE describes the application of rotenone in a medium of 0.5% saponin. Insects were examined and classified 1 day after spraying. Dose of Rotenone (mg/l) 10.2 7.7 5.1 3.8 2.6 1.2 Number of Insects 50 49 46 48 50 49 Number Killed 44 42 24 16 6 0 (a) [1] Plot the proportion of insects killed against the log dose of rotenone. Which type of model should be appropriate for this data? (b) [1+1] Fit the above data by a logit model and test the goodness of fit. Do you think that the dose of rotenone has a significant impact on the proportion of insects killed? (c) [1+1] Fit the above data by a probit model and test the goodness of fit. Do you think that the dose of rotenone has a significant impact on the proportion of insects killed? (d) [1+1] Fit the above data by a complementary log log model and test the goodness of fit. Do you think that the dose of rotenone has a significant impact on the proportion of insects killed? (e) [2] Which of the above model more adequately fits the data? Give reasons in favor of your answer. (f) [1] Compute the median lethal dose for each of the above three models. Q3. [7] The following table gives a bioassay problem concerning a standard preparation and a test preparation for different doses injected to laboratory rats. Dose 10 25 50 Test Preparation Cured Treated 53 105 102 144 81 104 Standard Preparation Cured Treated 48 160 47 100 54 83 (a) [3] Plot the recovery rates against log dose for test, standard and overall preparation. Comment on the shape of each of the graph. (b) [4] Use the deviance to compare the recovery rates between standard and test preparations