5.5 Wilks Lambda
The Hotellings \(T^2\) statistics was argued using its analogue \(t^2\) of students \(t\). One common procedure for
constructing test procedures is called the likelihood ratio method.
From our previous work we noted that the maximum of the multivariate normal likelihood has \(\underline {\mu }\) and \(\Sigma \)
are varied over other possible values is give by \begin {equation} \tag {1} \max \limits _{\mu , \Sigma } L\big (\underline {\mu },\Sigma \big ) = \big (2\pi \big )^{-np/2}\big |\widehat {\Sigma }\big |^{\frac {n}{p}}e^{-np/2} \end {equation} where \(\displaystyle {\widehat {\Sigma }=\frac {1}{n}\sum ^n_{j=1}\big (\underline {X}_j-\overline {\underline {X}}\big )\big (\underline {X}_j-\overline {\underline {X}}\big )'}\) and \(\displaystyle {\widehat {\mu }=\overline {X}=\frac {1}{n}\sum ^n_{j=1}\underline {X}_j}\) are the maximu likelihood
estimates.
Recall that the maximum likelihood estimates of \(\widehat {\mu }\) and \(\widehat {\Sigma }\) are those choices for \(\underline {\mu }\) and \(\Sigma \) that best explains the observed values for the random sample under the hypothesis \(H_0:\underline {\mu } = \underline {\mu }_0\) the normal likelihood becomes \begin {equation} \tag {2} L\big (\mu _0,\Sigma \big )= \big (2\pi \big )^{-np/2}\big |\Sigma \big |^{-n/2}\exp \Big \{-\frac {1}{2}\sum ^n_{j=1}\big (\underline {X}_j-\underline {\mu }_0\big )'\Sigma ^{-1}\big (\underline {X}_j-\underline {\mu }_0\big )\Big \} \end {equation}
The mean \(\underline {\mu }_0\) is fixed since it is known, but \(\Sigma \) can be varied to find the value that most likely with the
given data. The value is obtained by maximising \(L\big (\mu _0,\Sigma \big )\) to with respect to \(\Sigma \).
Using the steps of maximisation carried we have \begin {align*} -\frac {1}{2}\sum ^n_{j=1}\big (\underline {X}_j-\underline {\mu }_0\big )'\Sigma ^{-1}\big (\underline {X}_j-\underline {\mu }_0\big ) & = -\frac {1}{2}\sum ^n_{j=1}tra\Big \{\Sigma ^{-1}\big (\underline {X}_j-\underline {\mu }_0\big )\big (\underline {X}_j-\underline {\mu }\big )'\Big \}\\ & = -\frac {1}{2}tra\Big \{\Sigma ^{-1}\Big (\sum ^n_{j=1}\big (\underline {X}_j-\underline {\mu }_0\big )\big (\underline {X}_j-\underline {\mu }_0\big )'\Big )\Big \}\\ \end {align*}
Applying one results and \(\displaystyle {B= \sum ^n_{j=1}\big (\underline {X}_j-\underline {\mu }_0\big )\big (\underline {X}_j-\underline {\mu }_0\big )'}\,\) and \(b= \frac {n}{2}\) we have
\[\max \limits _{\Sigma }L\big (\mu _0,\Sigma \big ) = \frac {1}{(2\pi )^{np/2}\big |\widehat {\Sigma }_0\big |^{n/2}}e^{-np/2}\quad \cdots \cdot \cdots \quad (3)\]
with \(\displaystyle {\widehat {\Sigma }_0=\frac {1}{n}\sum ^n_{j=1}\big (\underline {X}_j-\underline {\mu }_0\big )\big (\underline {X}_j-\underline {\mu }_0\big )'}\).
To determine whether \(\Sigma _0\) is plausible value for \(\underline {\mu }\) the \(\max L\big (\mu _0,\Sigma \big )\) is compared with unrestricted maximum \(L\big (\underline {\mu },\Sigma \big )\). The resulting ratio is called the likelihood ratio statistic. Using (1) and (3) we have likelihood ratio \(\Delta \)
\begin {equation} \tag {4} \Delta = \frac {\max \limits _{\Sigma }L\big (\underline {\mu }_0,\Sigma \big )}{\max \limits _{\underline {\mu },\Sigma }L\big (\underline {\mu },\Sigma \big )}=\Bigg (\frac {\big |\hat {\Sigma }\big |}{\big |\hat {\Sigma _0}\big |}\Bigg )^{\frac {n}{2}} \end {equation}
The equivalent statistic is \(\Delta ^{2/n} = \frac {\big |\widehat {\Sigma }\big |}{\big |\widehat {\Sigma _0}\big |}\qquad (5)\) is called Wilks Lambda.
- 1.
- If the observed value of the likelihood ratio is too small, the hypothesis \(H_0: \underline {\mu } = \underline {\mu }_0\) is unlikely to be true and is, therefore, rejected.
- 2.
- \(\displaystyle {\Delta = \Bigg (\frac {\big |\widehat {\Sigma }\big |}{\big |\widehat {\Sigma }_0\big |}\Bigg )^{\frac {n}{2}}=\Bigg (\frac {\Big |\sum \limits _{j=1}^n\big (\underline {X}_j-\overline {\underline {X}}\big )\big (\underline {X}_j-\overline {\underline {X}}\big )'\Big |}{\Big |\sum \limits _{j=1}^n\big (\underline {X}_j-\underline {\mu }_0\big )\big (\underline {X}_j-\underline {\mu }_0\big )'\Big |}\Bigg )}\)
where \(C_{\alpha }\) is the lower \((100\alpha )\%\) of percentile of the distribution of \(\Delta \).
- 3.
- It can be shown that \begin {align*} \Delta _w & = \Delta ^{2/n}= \Bigg (1 + \frac {T^2}{n-1}\Bigg )^{-1}\\ & = \frac {\big |\widehat {\Sigma }\big |}{\big |\widehat {\Sigma }_0\big |}\\ \end {align*}
\begin {equation} \tag {7} \text {or}\quad T^2 = (n-1)\frac {\big |\widehat {\Sigma }\big |}{\big |\widehat {\Sigma }_0\big |}-(n-1) \end {equation}
- 4.
- Suppose that which to find \(\displaystyle { \Delta _{w,\alpha } = C_{\alpha } \ni Pr\big (\Delta _w < \Delta _{\alpha }\big ) = \alpha }\)
How do we proceed? \begin {align*} Pr\big (\Delta _w<C_{\alpha }\big ) & = Pr\Bigg (\Bigg (1 + \frac {T^2}{n-1}\Bigg )^{-1}<C_{\alpha }\Bigg )\\\\ & = Pr\Bigg (\frac {1}{C_{\alpha }}<1 + \frac {T^2}{n-1}\Bigg )\\ \end {align*}
\[Pr\Bigg (\Bigg (\frac {1}{C_{\alpha }}-1\Bigg )(n-1)<T^2\Bigg )\]
\[Pr\Bigg (\Bigg (\frac {1-C_{\alpha }}{C_{\alpha }}\Bigg )(n-1)<\frac {(n-1)p}{n-p}F_{p,n-p}\Bigg )\]
\[Pr\Bigg (\Bigg (\frac {1-C_{\alpha }}{C_{\alpha }}\Bigg )\frac {n-p}{p}<F_{p,n-p}\Bigg )<\alpha \]
\[\implies \quad F_{p,n-p,\alpha }=\frac {\big (1-C_{\alpha }\big )}{C_{\alpha }}\frac {(n-p)}{p}\]
\[\text {make}\quad C_{\alpha }\qquad \text {the subject}\]
\[C_{\alpha } = \frac {n-p}{n-p+pF_{p,n-p,\alpha }}\]
Example 5.22. Consider the ventilation value in the handout the \(F\) value calculated using SAS
is given as 2.9276
To calculate the \(T^2\) we observe that
\[p=t-1, n =8, t=6 \]
\begin {align*} T^2 & = \frac {(n-1)p}{n-p}F_{p,n-p,\alpha }\\ & = \frac {(n-1)(t-1)}{n-(t-1)}\times 2.9276\\ & = \frac {7\times 5}{8-6+1}\times 2.9276\\ & = \frac {35}{3}\times 2.9276 = 34.155 \end {align*}
is the observed value of Hotellings.
\(-\) The critical value is obtained in the similarly manner \[ F_{5,3,\alpha }=F_{5,3,0.05}=9.01\]
\[T^2_{0.05}= \frac {35}{3}\times 9.01=105.12\]
\(-\) Since \(T^2_{obse} = 34.155< T^2_{critical} =105.12\) We fail to reject \(H_0\) and conclude that there is not enough evidence evidence to suggest that
ventilation volume is affected by temperature.
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