MIT OpenCourseWare http://ocw.mit.edu 18.443 Statistics for Applications Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.18.443 Problem Set 7 1. Rice, §13.8, Problem 2, but omit week +2, leaving a 3 × 2 table. 2. A small preliminary study is done of a new medication intended to prevent further heart attacks in patients who recently survived a first one of a particular type. Suppose 9 such patients are given the new medication and another 9 are given a placebo (blank pills). After a year, 8 of the 18 patients have had another heart attack. Let j be the number of patients having a second attack who got the new medication. Different possible tests could be done of the hypothesis H0 that the new drug is equivalent to a placeb o. (a) Suppose the study was on the issue of safety, using a 1-sided test at level 0.1. For what values of j would one decide that the new drug appears to be unsafe and should not be tested further (unless some adjustment is made)? (b) For effectiveness, at level 0.05, for what values of j would one decide in a one-sided test that there is significant evidence of effectiveness (relative to placebo) and go on to test the new medication against other effective ones to see which may be more effective? 3. Rice, §13.8, problem 24, parts a and b only. (But you may consider part c a s providing a hint.) 4. For the same data as in the previous problem, if we focus on the boxing data, the bino-mial probability E(148, 268, 1/2), the probability of 1 48 or more wins in 268 independent trials with probability 1/2 of winning on each trial, is 0.04945. (a) If we had only the boxing data and were testing at the 0.05 level H0: πR = 1/2 against the composite alternative H1: πR > 1/2, what would we conclude? (b) What about the two-sided alternative πR 6= 1/2? (c) Suppose we were considering the 1-sided alternative πR > 1/2 but with the data for all four sports, where πR can be different for different sports but larger than 1/2 in each case. The p-values for t he other three sports, using the binomial distribution as in part (a ), are .all larger, for example E(45, 80, 1/2) = 0.157 for Tae Kwon Do, and still larger for the two forms of wrestling. What would be the conclusion(s)? 5. In a study of hospitalized Medicare beneficiaries, 150119 had alcohol-related disease (ARD) and of these, 6973 had hip fractures and 21467 had diab etes. Of 726218 patients who did not have ARD in the same study, 13647 had hip fractures and 159042 had diabetes. (a) Estimate the odds ratio Δ where D = hip fracture and X =ARD (Rice’s notation). (b) Is this odds ratio (statistically) significantly different from 1? Hint: You may not need to do a very precise or long calculatio n to decide this. (c) Estimate the odds ratio Δ where D = diabetes and X =ARD ( Rice’s notati on). (d) Is the latter odds ratio (statistically) significantly different from 1? In what direction? (e) In part (a) we would like to estimate the odds ratio in the general population. Could there be a problem with an estimate based on the hospitalized population in the study? Hint: consider parts (c)-(d). [Actually, the authors of the study didn’t just accept the estimate in (a), they made various adjustments for “confounding” to try to isolat e the direct effect of ARD on hip fracture.]