An Estimate of the Odds RatioThat Always ExistsMichael PARZEN , Stuart LIPSITZ , Joseph IBRAHIM, and Neil KLARThis article proposes an estimat e of the odds ratio in a (2 £ 2) table obtained fromstudi es in which the row totals are  xed by design, such as a phase II clinical trial. Ourestima te,based on the median unbiased estimateof the probabilitiesof success in the (2£2)table , will always be in the interval (0; 1): Another estimate of the odds ratio which hassuch properties is obtained wh en addin g .5 to each cell of the table. Using simulations,we compared our proposed estimate to that obtained by adding .5 to every cell, and foundthat our estimate had smaller  nite sample bias, and larger mean square error. We als opropo se the use of the bootstrap to form a con den ce interval for the odds ratio based onour 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 distribu tion tocalcu late con dence intervals.Key Words: Median unbiased estimator; Phase II clinical trials; Small samples.1. INTRODUCTIONPhase II cancer clinical trials are designed to determine if a new treatment producesfavorable results (proportion of success), when compared to a known, “standard treatment.”For a given subject, the outcome of the phase II trial is success o r fai lure. If the new treatmentproduces favorable results, then further testing will be done in a phase III study, in whichpatients will be randomized to the new treatment or the “industry standard.” Often, insteadof one new promising treatmen t, there are two new promising treatments. In an effort toreduce the time necessary to determine if either or both of the new treatments are effective,a randomi zed phase II trial is often conducted. In the randomized phase II trial, patients arerandomized 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 Street,Chicago, IL 60637. Stuart Lipsitz is Professor, Department of Biometry an d Epidemiology, Medical University ofSouth Carolina, 135 Rutledge Avenue, Charleston, SC 29425. Joseph Ibrahim is Associate Professor, Departmentof Biostatistics, Harvard School of Public Health and Dana-Farber Cancer Institute, 44 Binney Street, Boston, MA02115 (E-mail: [email protected]). Neil Klar is Assistant Professor, Division of Preventive Oncology,Cancer Care Ontario, 620 University Avenue, Toronto, Ontario, Canada M5G 2L7.c® 2002 American Statistical Association, Institute of Mathematical Statistics,and Interface Foundation of North AmericaJournal of Computational and Graphical Statistics, Volume 11, Number 2, Pages 420–436420ESTIMATE OF THE ODDS RATIO 421Table 1. Data for a Randomized Phas e II Clini cal TrialOutcomeTreatment Success Failure Total1 Y1n1¡ Y1n12 Y2n2¡ Y2n2Total Y1+ Y2(n1+ n2)¡ n1+ n2(Y1+ Y2)table as in Table 1, with the rows representing treatment, and the columns representingthe outcome (success or failure). In Table 1, there are n1subjects on treatment 1, with Y1successes, and n2subjects on treatment 2, with Y2successes.Because phase II trials are pilot studies, most have small samples, many with samplesizes of less than 20 subjects per treatment. Thus, the sample sizes in phase II randomizedstudies are not large enough to e stimate with precision the odds ratio for success betweenthe two treatments. Furthermore, the nature of the cancer is su ch that the probability ofsuccess is often very small, even for promising new t herapies. Nevertheless, even thoughthe sample sizes and the probability of successes on each treatment arm may be small, in a nal report on the trial, investigatorsusually report an estimate of this odds ratio between thetwo treatments. With such small sample sizes and probabilitiesof success, such an estimateof the odds ratio will have a large standard error, and thus the investigator will get only arough idea of the value of the true odds ratio. Furthermore, besides estimation of the oddsratio, investigators are also interested in obtaining a c on dence interval to get a range ofpossible values for the odds ratio.Our example is a ran domized phase II clinicalfor the evaluation of two new chemother-apy treatme nts in p atie nts with advanced large bowel cancer. The two treatments were Ho-moharringtonine and Caracemide. The clinical trial was developed by the United States’Eastern Cooperative Oncology Group (ECOG), and open for patient accrual from 1987through 1990. In this clinical trial, the investigators were interested in two binary outcomesfor each treatment. The  rst binary outcome of interest is th e “tumor shrinkage,” with the“successful” outcome de ned as the t umor shrinking at least 50%; a successful outcome inthis context is called a “response.” The second binary outcome of int erest is the toxicity, orside effects of the treatment, with the “successful” outcome de ned as “life-threatening”toxicity. In order to prese nt a complete picture of the effectiveness of a treatment, one mustlook at both the “tumor shrinkage” and “toxicity.” If, for example, the tumor shrinks in allsubjects, but all subjects also die as a result of toxicity, then the treatment will not warrantfurther study in a phase III trial. Thus, we are interested in estimating the odds ratio forresponse and well as the odds ratio for toxicity.Prior to the trial, the commonly accepted treatment was 5-Flurouracil, whic h gave aresponse rate of 15–25% on previous trials (David 1982; Heal and Schein 1977). Homohar-ringtonine is a cephalotoxine ester from the bark of an evergreen tree in China. It was usedmostly for its anti-l eukemia properties in China, and was also shown to have anti-leukemiaproperties in phase II trials. Caracemide is a water soluble c ompound which functions as a422 M. PARZEN, S. LIPSITZ, J. IBRAHIM, N. KLARnonspeci c inhibitor of macromolecular synthesis. The study entered 25 patients: n1= 14on the Homoharringtonin e arm and n2= 11 on the Caracemi de arm. The accrual goalwas 30 patients on each arm, bu t the study was terminated ea rly after being open for 43months due to poo r accrual. Unfortunately, there were no responses on either arm, that is,Y1= Y2= 0: Furthermore, there were Y1= 2 subjects with life-threatening toxicitieson the Homoharringtonine arm