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# An Estimate of the Odds Ratio That Always Exists

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An Estimate of the Odds Ratio That Always Exists Michael PARZEN Stuart LIPSITZ Joseph IBRAHIM and Neil KLAR This article proposes an estimate of the odds ratio in a 2 2 table obtained from studies in which the row totals are xed by design such as a phase II clinical trial Our estimate based on the median unbiasedestimate of the probabilitiesof success in the 2 2 table will always be in the interval 0 1 Another estimate of the odds ratio which has such properties is obtained when adding 5 to each cell of the table Using simulations we compared our proposed estimate to that obtained by adding 5 to every cell and found that our estimate had smaller nite sample bias and larger mean square error We also propose the use of the bootstrap to form a con dence interval for the odds ratio based on our proposed estimate Instead of a Monte Carlo bootstrap one can easily calculate the exact bootstrap distribution of our estimate of the odds ratio and use this distribution to calculate con dence intervals Key Words Median unbiased estimator Phase II clinical trials Small samples 1 INTRODUCTION Phase II cancer clinical trials are designed to determine if a new treatment produces favorable results proportion of success when compared to a known standard treatment For a given subject the outcome of the phase II trial is success or failure If the new treatment produces favorable results then further testing will be done in a phase III study in which patients will be randomized to the new treatment or the industry standard Often instead of one new promising treatment there are two new promising treatments In an effort to reduce the time necessary to determine if either or both of the new treatments are effective a randomized phase II trial is often conducted In the randomized phase II trial patients are randomized to receive one of the two new therapies The data can be arranged in a 2 2 Michael Parzen is Associate Professor Graduate School of Business University of Chicago 101 East 58th

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