Prove that (exp left(lambda_{111}^{x y z} ight)=frac{pi_{2,2,2} / pi_{1,2,2}}{pi_{2,1,2} / pi_{1,1,2}} / frac{pi_{2,2,1} / pi_{1,2,1}}{pi_{2,1,1} / pi_{1,1,1}})

Prove that \(\exp \left(\lambda_{111}^{x y z}\right)=\frac{\pi_{2,2,2} / \pi_{1,2,2}}{\pi_{2,1,2} / \pi_{1,1,2}} / \frac{\pi_{2,2,1} / \pi_{1,2,1}}{\pi_{2,1,1} / \pi_{1,1,1}}\) for a \(2 \times 2 \times 2\) three-way table.