inference for $3\times2$ bilateral data is its ability to handle the nuisance parameter in Dallal’s model which complicates frequentist analyses.

There are no existing Bayesian methods for bilateral data in the literature yet. We present objective Bayesian inferences for three parameters of interest: the risk difference, the risk ratio, and the odds ratio. In Section 2, we present Dallal’s reduced model along with Dallal’s full model. Section 3 is dedicated to the derivation of the objective Bayesian modeling of bilateral data. We focus primarily on deriving Jeffreys’ prior and Bernardo’s reference prior. We then discuss the Bayes factor in the context of hypothesis testing as well as a simulation scheme for the joint posterior distribution. Section 4 presents results of an empirical comparison between Bayesian methods and frequentist methods. Section 5 presents two illustrative case studies. In Section 6, we present two families of Bayesian prior distributions, including Jeffreys’ or Bernardo’s reference priors as special cases. Section 7 concludes this paper.

Dallal’s Dependence Model {#sec2}
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Dallal’s model was presented for the first time as model 2 in @Dallal1988. Let “$i=1$" and “$i=0$" denote the treatment group and the control group, respectively. Denote by $Z_{ijk}$ a binary variable such that 