groups, $\delta_0-\delta_1$, which is equal to $\Delta$ under Dallal’s reduced model. To date, the risk ratio, the odds ratio and the difference of excess risks have never been discussed in the bilateral data literature. The first two parameters add another dimension to the utility of bilateral data so that they can be collected under either a prospective study paradigm or a retrospective observational study paradigm, allowing for more applications than those under the current clinical setting. Although the addition of these new parameters poses more challenges for carrying out frequentist inference, no additional work is required in the Bayesian framework.

What makes the inferential process challenging in Dallal’s model is that one deals with a constrained parameter space. Indeed, the parameter space is $$\begin{aligned}
\Omega= \bigg\{ \big(\gamma, \lambda_0, \lambda_1\big): \; & & 0
< \gamma< 1 \mbox{ if } 0 < \max(\lambda_0, \lambda_1) \leq\frac
{1}{2};\nonumber\\
\; & & 0 <\gamma< \dfrac{1}{\max(\lambda_0, \lambda_1)}-1 \mbox{
if } \max(\lambda_0, \lambda_1)>\frac{1}{2} \bigg\}\;\;\end{aligned}$$ or equivalently $$\begin{aligned}
\label{eq1sec3}
\Omega= \bigg\{ \big(\gamma, \lambda_0, \lambda_1 \big): \;
& & 0<\gamma<1\;\mbox{and}\; 0 < \lambda_0, \lambda_1 <
\dfrac{1}{1+\gamma} \bigg\}.\end{aligned}$$ We adopt the second representation in the sequel. The likelihood function can be expressed as $$\begin{aligned}
L(\gamma, \lambda_0, \lambda_1) & = 