7.2 Fishers Classification Rule

Fishers solution to the separation of two populations can also be used to assign a new observation \(\underline {X}_0\) to one of the populations.

Allocation Rule:
Suppose that \(\underline {x}_0\) is a new observation, Fishers allocation rule using the discrimination function is

1.

Allocate \(\underline {x}_0\) to population 1 \((\pi _1)\), if \begin {align*} y_0 & = \big (\overline {\underline {X}_1}-\overline {\underline {X}_2}\big ) S^{-1}_p\underline {x}_0\geq \\\\ \widehat {M} & = \frac {1}{2}\big (\overline {\underline {X}_1}-\overline {\underline {X}_2}\big )'S^{-1}_p\big (\overline {\underline {X}_1}+ \overline {\underline {X}_2}\big )\\\\ \text {or}\quad & y_0-\widehat {M}\geq 0\\ \end {align*}

2.

Allocate \(\underline {x}_0\) to population 2 \((\pi _2)\) if \(y_0 - \widehat {M} <0\qquad \text {or}\quad y_0<\widehat {M}\)

How should we classify this individual as far as gender is concerned.

\begin {align*} y_0 & =\big (\overline {\underline {X}_F} - \overline {\underline {X}_M}\big )'S^{-1}_p\underline {x}_0\\\\ & = \begin {pmatrix} -5.76265, & -404.761\\ \end {pmatrix} \begin {pmatrix} 16\\ \\ 0.46\\ \end {pmatrix}\\\\ & = -93.988 \end {align*}

\begin {align*} \widehat {M} & = \frac {1}{2}\big (\overline {\underline {X}_F}-\overline {\underline {X}_M}\big )'S^{-1}_p\big (\overline {\underline {X}_F}+\overline {\underline {X}_M}\big )\\\\ & = \begin {pmatrix} -5.76265, & -404.761\\ \end {pmatrix} \begin {pmatrix} 14.741\\ \\ 0.0432\\ \end {pmatrix}\\\\ & = -89.910\\ \end {align*}

Since \(y_0 < \widehat {M}\)
The observation \((16,0.46)\) belongs to a male participant.

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