An improved estimator of the logarithmic odds ratio for small sample sizes using a Bayesian approach

The logarithmic odds ratio is a well-known method for comparing binary data between two independent groups. Although various existing methods proposed for estimating a logarithmic odds ratio, most methods estimate two proportions in each group independently and then estimate the logarithmic odds ratio using the two estimated proportions. When using a logarithmic odds ratio, researchers are more interested in the logarithmic odds ratio than proportions for each group. Parameter estimations, generally, incur random and systematic errors. These errors in initially estimated parameter may affect later estimated parameter. We propose a Bayesian estimator to directly estimate a logarithmic odds ratio without using proportions for each group. Many existing methods need to estimate two parameters (two proportions in each group) to estimate a logarithmic odds ratio; however, the proposed method only estimates one parameter (logarithmic odds ratio). Therefore, the proposed estimator can be closer to the population's logarithmic odds ratio than existing estimators. Additionally, the validity of the proposed estimator is verified using numerical calculations and applications.