distributions of $\Delta$ and $R$ as well as $P\big(\Delta>\Delta_0\,|\,D\big)$ and $P\big(R>R_0\,|\,D\big)$ in Appendix 4. Although we do not provide the posterior distribution of $\psi$ (more complex), we discuss in Section \[sub4sec4\] how to sample from the posterior distributions of $\Delta
, R$ and $\psi$. We use the parameter values generated to compute posterior probabilities such as $P\big(\Delta>\Delta_0\,|\,D\big), P\big(R>R_0\,|\,D\big)$, $P\big(\psi
>\psi_0\,|\,D\big)$, and Bayesian credible intervals. The Bayes factors are introduced in Section \[sub5sec4\] to compare the models under the hypotheses (i) $H_0$: $\lambda_0=\lambda
_1$ and $H_1$: $\lambda_1\neq\lambda_0$ and (ii) $H^\ast_0$: $\gamma_1=\gamma_0$ versus $H^\ast_1$: $\gamma_1\neq
\gamma_0$.

The Uniform Prior Distribution {#sub1sec4}
------------------------------

The prior $\pi_U(\gamma,\lambda_0,\lambda_1) = 2$ for $(\gamma,\lambda
_0,\lambda_1)\in\Omega$ refers to the uniform distribution under Dallal’s model. This prior is proper and expresses a complete indifference of one vector of parameter values over another. The resulting posterior distribution is $$\begin{aligned}
\label{eq1sub1sec4}
\pi_U\big(\gamma, \lambda_0, \lambda_1|D\big)
& \propto& 2^{m_{1+}} \gamma^{m_{1+}}
(1-\gamma)^{m_{2+}}\lambda_0^{m_{10} + m_{20}}
\big[1-(1+\gamma)\lambda_0\big]^{m_{00}}\lambda_1^{m_{11} + m_{21}}
\nonumber\\
& & \times\big[1-(1+\gamma)\lambda_1\big]^{m_{01}},
\qquad\qquad(\gamma,\lambda_0,\lambda_1)\in\Omega,\qquad\qquad\end{aligned}$$ where $m_{1+}=m_{10}+ m_{11}$ and $m_{2+}=m_{20}+ m_{21}$. As a result, the marginal posterior distribution of the nuisance parameter $\gamma$ is $$\begin{aligned}
\label{eq2sub1sec4}
\pi_U\big(\gamma|D\big) = \dfrac{2^{m_{1+}+1}}{{\mbox{{\bf B}}}(m_{1+}+1,m_{2+}+1)}
\dfrac{\gamma^{m_{1+}}
(1-\gamma)^{m_{2+}}}{(1+\gamma)^{m_{1+}+m_{2+}+2}}, \qquad0<\gamma<1,\end{aligned}$$ and the conditional posterior distribution of $\lambda_i, i=0,1$ given $\gamma$ is $$\begin{aligned}
\label{eq3sub1sec4}
\pi_U\big(\lambda_i|\,\gamma, 