STATS 210: Problem Set Four—Due Thursday August 5th 1. The following is the 2x2 table that resulted from a case-control study of breast cancer and exposure to antibiotics: cancer control High antibiotic use 10 5 Little/no antibiotics 40 55 50 60 You can enter these data in SAS by using a count variable to group identical observations: data table; input IsCase IsControl freq; datalines; 1 1 10 0 1 5 1 0 40 0 0 55 ; run; To incorporate counts in PROC FREQ, use a “weight” statement, as follows: proc freq data=table; tables IsCase*IsControl; weight freq; run; (a) What proportion of cases had been exposed and what proportion of controls had been exposed? (b) Test the hypothesis that breast cancer and antibiotic use are independent (use α=.05 and any appropriate statistical test). (c) Build a 95% confidence interval for the odds ratio between breast cancer and antibiotic use. 12. A study was undertaken to examine the prevalence of abnormal hematologic (blood cell) profiles in elite cross-country skiers at the 2001 World Ski Championships. Abnormal hematologic profiles—measured as increased red blood cells or hemoglobin—may indicate blood doping. Sixty-eight percent of all skiers and 92% of those finishing in the top 10 places were tested. Hemoglobin levels in the athletes were compared against established reference data (hemoglobin concentration is normally distributed). Values of >2 SD above average were classified as “abnormal,” and values >3 SD above average were classified as “highly abnormal.” Results for the top 50 finishers in each of the 9 races of the Championships are represented in the figure below (assume each individual skier only competed once): Top ten finishers above this line Each of the 9 races is represented by a column. The hematologic result for an athlete is placed in the position of their race result (1st to 50th) in each column. A black oval indicates a “highly abnormal” (>3 SD) hematologic profile in the skier who obtained that race result. A speckled oval indicates an “abnormal” (>2 SD but <3 SD) hematologic profile in the skier who obtained that race result. A white oval indicates a “normal” hematologic profile in the skier who obtained that race result. A blank area indicates that a sample was not obtained from the athlete who achieved that race result. 22. Continued…These data are available on the class website, www.stanford.edu/~kcobb/stats210, as columns 1-4, respectively: place (1-50); moderately abnormal blood profile (1=yes, 0=no); highly abnormal blood profile (1=yes, 0=no); and frequency (e.g., the number of 3rd placers with highly abnormal blood profiles=5). To incorporate counts into SAS procedures, use a “weight” statement within PROC FREQ and a “freq” statement within PROC LOGISTIC; for example: proc freq data=d2; tables HiAb*decile; weight freq; run; proc logistic descending data=doping; model HiAb = decile; freq freq; run; (a) What is the probability of having a “highly abnormal” test result in each decile of finishing place (1-10, 11-20, 21-30, 31-40, 41-50)? What is the probability of having an “abnormal” test result in each decile of finishing place (1-10, 11-20, 21-30, 31-40, 41-50)? (b) Using logistic regression, calculate the odds ratio that represents the increase in the odds of a “highly abnormal” test result for every one-unit higher finishing place (e.g., going from 15th place to 16th place or from 40th to 41st place). Note: use “abnormal” (gray circles) and “normal” skiers combined as the reference group. (c) Using logistic regression, calculate the odds ratio that represents the increase in the odds of a “highly abnormal” test result for every ten-unit increase in finishing place (e.g., going from 15th place to 25th place or from 38th to 48th place). 3(d) Using logistic regression, calculate the odds ratio that represents the increase in the odds of a highly abnormal test result for every jump in decile of finishing place (e.g., going from a 11-20 finisher to a 21-30 finisher). (e) Compare the odds ratios in (d) and (e). Explain why they differ. (f) Calculate the odds ratio that represents the increase in the odds of having a “highly abnormal” test result for top-ten finishers (compared with all other finishers). (g) Calculate the odds ratio that represents the increase in the odds of being in the top ten given that you have an “abnormal” test result. (h) Calculate the odds ratio that represents the increase in the odds of being in the top ten given that you have a “highly abnormal” test result. (i) Briefly interpret these results.