Verify that the SLM version of the KW and HO tests are given by (9.47) with \(D\) defined in (9.50) and (9.51).

\[\begin{equation*}
S L M=\frac{L M_{1}-E\left(L M_{1}\right)}{\sqrt{\operatorname{var}\left(L M_{1}\right)}}=\frac{d-E(d)}{\sqrt{\operatorname{var}(d)}} \tag{9.47}
\end{equation*}\]

\[\begin{equation*}
D=\frac{1}{2} \frac{n}{\sqrt{M_{11}-n}}\left(\Delta_{1} \Delta_{1}^{\prime}\right)+\frac{1}{2} \frac{n}{\sqrt{M_{22}-n}}\left(\Delta_{2} \Delta_{2}^{\prime}\right) \tag{9.50}
\end{equation*}\]

\[\begin{equation*}
D=\frac{n}{\sqrt{2} \sqrt{M_{11}+M_{22}-2 n}}\left[\left(\Delta_{1} \Delta_{1}^{\prime}\right)+\left(\Delta_{2} \Delta_{2}^{\prime}\right)\right] \tag{9.51}
\end{equation*}\]