(\frac{1}{2},\frac{1}{2}\big)} \\
& \times\dfrac{{\mbox{{\bf B}}}\big(m_{2+}+m_{1+}+a,\,m_{0+}+\frac{1}{2})}{{\mbox{{\bf B}}}(a,\frac
{1}{2}\big)}.\end{aligned}$$ The ratio of these two marginal predictive distributions, $BF^\lambda
_{01}$, under the condition $P(H_1)=P(H_0)=\tfrac{1}{2}$ (the Bayes factor for testing $H_0$ vs $H_1$) satisfies $$\begin{aligned}
& \dfrac{1}{BF_\lambda}
= \dfrac{{\mbox{{\bf B}}}\big(m_{10}+m_{20}+\frac{1}{2}, m_{00}+\frac{1}{2}\big) {\mbox{{\bf B}}}\big(m_{11}+m_{21}+\frac{1}{2}, m_{01}+\frac{1}{2}\big) {\mbox{{\bf B}}}\big(a,\frac
{1}{2}\big)}
{{\mbox{{\bf B}}}\big(m_{2+}+m_{1+}+a, m_{0+}+\frac{1}{2}\big) {\mbox{{\bf B}}}\big(\frac
{1}{2},\frac{1}{2}\big) {\mbox{{\bf B}}}\big(\frac{1}{2},\frac{1}{2}\big)}\\
& \times\int_0^1 \int_0^1 \dfrac{(u+rv)^{d}}{K} \dfrac
{u^{m_{10}+m_{20}+\frac{1}{2}-1}
(1-u)^{m_{00}+\frac{1}{2}-1}}{{\mbox{{\bf B}}}(m_{10}+m_{20}+\frac{1}{2},m_{00}+\frac
{1}{2})} \dfrac{v^{m_{11}+m_{21}+\frac{1}{2}-1}
(1-v)^{m_{01}+\frac{1}{2}-1}}{{\mbox{{\bf B}}}(m_{11}+m_{21}+\frac{1}{2},m_{01}+\frac
{1}{2})} dv du.\end{aligned}$$ Under Jeffreys’ prior, the constant $K$ and the integral term in the Bayes factor are computed using computer simulation. Under the reference prior, the integral term disappears and the Bayes factor is computed exactly using only the Beta functions.

### $H^\ast_0$: $\gamma_1=\gamma_0$ versus $H^\ast_1$: $\gamma_1\neq\gamma_0$ {#sub2sub4sec4}

One of the statements made in Dallal’s model is that the parameter $\gamma$ is constant. As discussed earlier, this assumption can be relaxed to $P\big
(Z_{ijk}=1 \,|\, Z_{ij(3-k)}=1\big) = 1-\gamma_i$, giving rise to the full or saturated model. Therefore, it is important to test the hypothesis $H^\ast_0$: $\gamma_1=\gamma_0=\gamma$ (Dallal’s reduced model) versus the alternative hypothesis $H^\ast_1$: $\gamma_1\neq\gamma_0$ (Dallal’s full model). Under $H^\ast_1$, $U$ and $V$ are redefined as follows: $U=(1+\gamma_0)\lambda_0$ and $V=(1+\gamma
_1)\lambda_1$. Under $H^\ast_0$, the prior is $\pi_{H_1}(\gamma,u,v)$ defined in Section \[sub1sub4sec4\]. Under $H^\ast_1$, the family of 