7.4 More Than Two Population

Again Fishers suggestion was to look for a linear function \(a'X\) which maximised the ratio of between groups of sum of squares that is \begin {equation} \tag {5} \underline {Y}=X\underline {a}= \begin {pmatrix} X_1\underline {a}\\ \vdots \\ \vdots \\ X_g\underline {a}\\ \end {pmatrix} =\begin {pmatrix} Y_1\\ \vdots \\ \vdots \\ Y_g\\ \end {pmatrix} \end {equation}

where we assume there are \(g\) populations, be a linear combination of the columns of \(X\) , then \(Y\) has total sum of squares \begin {equation} \tag {6} \underline {Y}'H\underline {Y}=\underline {a}'X'HX\underline {a}= \underline {a}T\underline {a} \end {equation}

which can be partitioned as a sum of the within groups (SSS) \begin {equation} \tag {7} \sum ^g_{i=1}Y_i'H_iY_i = \sum ^g_{i=1}\underline {a}'X_i'H_iX_i\underline {a}_i \underline {a}'W\underline {a} \end {equation}

plus between groups sum of squares , each \(i\) for the specific group. Between groups of sum of squares can also be written as \begin {equation} \tag {8} \sum _in_i\big (\overline {Y}_i-\overline {Y}\big )^2= \sum _i n_i\big \{\underline {a}'\big (\underline {X}_i-\overline {X}\big \} = \underline {a}'B\underline {a} \end {equation}

where \(\overline {Y}_i\) is the mean of \(i^{\text {th}}\) sub vector of \(Y\) and \(H_i\) is the \(n_i\times n_i\) centering matrix for population \(i\).

The Ratio of interest is given by \(\displaystyle {\frac {\underline {a}'B\underline {a}}{\underline {a}W\underline {a}}\qquad (9)}\)

If \(\underline {a}\) is a vector which maximises (9) the linear combination \(\underline {a}'\underline {X}\) is called Fishers linear discrimination function.

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