& \prod_{i=0}^1 {m_{+i} \choose
{m_{0i}, m_{1i}, m_{2i}}} \big[1-(1+\gamma)\lambda_i\big]^{m_{0i}}
(2\gamma\lambda_i)^{m_{1i}} \big[(1-\gamma)\lambda_i\big
]^{m_{2i}}\nonumber\\
& \propto& \gamma^{m_{10}+ m_{11}}
(1-\gamma)^{m_{20}+m_{21}}\lambda_0^{m_{10} + m_{20}} \lambda_1^{m_{11}
+ m_{21}}\;
\big[1-(1+\gamma)\lambda_0\big]^{m_{00}}\nonumber\\
& & \times\big[1-(1+\gamma)\lambda_1\big]^{m_{01}},\qquad\qquad\qquad
(\gamma,\lambda_0,\lambda_1)\in\Omega.\end{aligned}$$ In Appendix 1 of the Supplementary Web Appendix, we develop for the first time large-sample frequentist inferences for the risk difference, the risk ratio and the odds ratio under Dallal’s model.

Bayesian Analysis {#sec4}
=================

A key component in Bayesian analysis is the choice of the prior distribution. Traditionally, Bayesians have turned to conjugate priors. However, the concept of conjugate priors is only universal in standard univariate problems. In addition, when little or no prior information is available, conjugate priors become subjective. Thus, non-informative or objective priors are more widely accepted. The uniform distribution over the parameter space is an obvious non-informative prior. Jeffreys’ prior and Bernardo’s reference prior are alternative choices that are invariant under any parameterization or a larger class of parameterizations. Another alternative choice is to use a joint prior that is the compromise between an informative prior and a non-informative prior [@SunBerger1998]. We discuss four types of priors: the uniform prior, Jeffreys’ prior, Bernardo’s prior, and Sun and Berger’s reference prior in light of partial information. We derive the posterior 