Q1: Below are some data on an arthritis treatment. The data were obtained from a web site http://www.psych.yorku.ca/friendly/lab/files/psy6140/examples/logistic We have an arthritis treatment indicator, a gender variable, age, and the response, 0 if no improvement was observred, 1 if improvement was observed. Caveats: I do not know if these data are real nor do I know what treatment was applied. (A) As you see I have run a main effects logitic model. Explain how you would get the probability of improvement for a 50 year old female in the control group. Include an explanation of how the logit is computed from the parameter estimates in the printout and how the probability is computed from the logit. This is to show understanding of the logistic model and how it is used - I've supplied the program. (B) Compute the odds that a 50 year old woman in the control group will feel better at the end of the trial and do the same for a 50 year old woman in the treatment group. Take the ratio of these. (and of course document these three calculations). Explain how this "odds ratio" is related to the printout. Explain how the odds ratio would change if (i) Instead of a 50 year old woman we used a 40 year old woman. (ii) Instead of a 50 year old woman we used a 50 year old man. (C) A company dealing with women's health products wants to market the treatment to women of whatever age group shows probability 0.75 or more of feeling better. (i) What logit goes with this probability? (ii) What age in women produces this logit? (iii) What age group in women should be the target market for the company? "Studies show 75% of women over age ____ feel better after using the new Arthaway Creme!" (D) The program produces a plot of the probabilities and one of the logits against age. On each plot there are 4 lines representing the 4 combinations of gender and treatment. The two male lines seem as far apart (vertically) as the two female lines. Is that coincidence or is it forced by the model? Exactly how far apart are they (vertically)? (E) Add an interaction between the treatment and gender. Is the term significant? In words what is that term testing? What would be the implication if it were significant? Q2: This summer I showed you the results of an analysis of the famous Framingham heart study and I have loaded that data on our AAEMDATA file. (A) Build a diagram that brings in the Framingham data, eliminates all of the systolic blood pressure measurements except sbp32, eliminates cholest2 and keeps cholest3 and declares firschd (First stage coronary heart disease) as the target variable. . Using the Explore feature from the previous module, check out the distributions of your inputs. Note anything unusual in your report (we'll not do any transforming etc. for now). (B) Pull in a regression node, connect and run. Write down the equation for the logit as a function of your inputs. Discuss statistical significance of the inputs. (C) I am a 63 year old non-smoker and I would like to have my probability of first stage coronary heart disease at most 0.10. Perhaps by diet and exercise I can control my cholesterol and blood pressure. Now in the cholesterol-blood pressure plane, there would be (with this main effects model) a linbe separating the (cholesterol, blood pressure) pairs that predict exactly a 10% probability and thus split off the desirable from the undesirable pairs. According to your model, what is the equation of that line? BP = _____ + ______Chol. Common sense tells us that points below that line (low BP's) are good. What amount of blood pressure lowering will keep me on that line if my cholesterol goes up by 5? (D) If we think of odds as measuring risk, what is the odds ratio for smokers versus nonsmokers (of the same age, BP, etc.). Of course do more than just give the number- interpret, for your mom or a future employer, what that means. ************ SAS Code for Arthritis data******************; title h=1.6 'Logistic Regression: Arthritis Treatment data'; axis1 label=(angle=90 font=triplex h=1.2); proc format; value st 1="Ctrl M" 2="Ctrl F" 3="Trtd M" 4="Trtd F" 5="Pred C M" 6="Pred C F" 7="Pred T M" 8="Pred T F"; data arthrit; length treat $7. sex $6. ; input id treat $ sex $ age improve @@ ; better = (improve > 0); /* Dichotomous response */ _treat_ = (treat ='Treated') ; /* Dummy var for treatment */ _female_ = (sex = 'Female'); /* and sex */ st =1 + 2*_treat_+_female_; format st st.; cards ; 57 Treated Male 27 1 9 Placebo Male 37 0 46 Treated Male 29 0 14 Placebo Male 44 0 77 Treated Male 30 0 73 Placebo Male 50 0 17 Treated Male 32 2 74 Placebo Male 51 0 36 Treated Male 46 2 25 Placebo Male 52 0 23 Treated Male 58 2 18 Placebo Male 53 0 75 Treated Male 59 0 21 Placebo Male 59 0 39 Treated Male 59 2 52 Placebo Male 59 0 33 Treated Male 63 0 45 Placebo Male 62 0 55 Treated Male 63 0 41 Placebo Male 62 0 30 Treated Male 64 0 8 Placebo Male 63 2 5 Treated Male 64 1 80 Placebo Female 23 0 63 Treated Male 69 0 12 Placebo Female 30 0 83 Treated Male 70 2 29 Placebo Female 30 0 66 Treated Female 23 0 50 Placebo Female 31 1 40 Treated Female 32 0 38 Placebo Female 32 0 6 Treated Female 37 1 35 Placebo Female 33 2 7 Treated Female 41 0 51 Placebo Female 37 0 72 Treated Female 41 2 54 Placebo Female 44 0 37 Treated Female 48 0 76 Placebo Female 45 0 82 Treated Female 48 2 16 Placebo Female 46 0 53 Treated Female 55 2 69 Placebo Female 48 0 79 Treated Female 55 2 31 Placebo Female 49 0 26 Treated Female 56 2 20 Placebo Female 51 0 28 Treated Female 57 2 68 Placebo Female 53 0 60 Treated Female 57 2 81 Placebo Female 54 0 22 Treated Female 57 2 4 Placebo Female 54 0 27 Treated Female 58 0 78 Placebo Female 54 2 2 Treated Female 59 2 70 Placebo Female 55 2 59 Treated Female 59 2 49 Placebo Female 57 0 62 Treated Female 60 2 10 Placebo Female 57 1 84 Treated Female 61 2 47 Placebo Female 58 1 64 Treated Female 62 1 44 Placebo Female 59 1 34 Treated Female 62 2 24 Placebo Female 59 2 58 Treated Female 66 2 48 Placebo Female 61 0 13 Treated Female 67 2 19 Placebo Female 63 1 61 Treated Female 68 1 3 Placebo Female 64 0 65 Treated Female 68 2 67 Placebo Female 65 2 11 Treated Female 69 0 32 Placebo Female 66 0 56 Treated Female 69 1 42 Placebo Female 66 0 43 Treated Female 70 1 15 Placebo Female 66 1 71 Placebo Female 68 1 1 Placebo Female 74 2 ; proc logistic nosimple data=arthrit descending; model better = age _female_ _treat_ ; output out = results pred=phat xbeta=Lhat; ** GRAPHICS **; proc sort data=results; by age; data results; set results; part=1; Y=better; output; st=st+4; part=2; Y=phat; output; proc print data=results; proc gplot; plot Y*age=st Lhat*age=st/vaxis=axis1; label Y = "Pr{Improvement}"; label Lhat="Logit"; symbol1 v="M" i=none c=blue; symbol2 v="F" i=none c=red; symbol3 v=circle c=blue i=none h=1.5; symbol4 v=circle c=red i=none h=1.5; symbol5 v=none i=smooth10 c=blue; symbol6 v=none i=smooth10 c=red; symbol7 v=none i=smooth10 c=blue w=2; symbol8 v=none i = smooth10 c=red w=2; run;