Page 162 - Statistics and Data Analysis in Geology

Analysis of Multivariate Data

necessary to estimate the discriminant function coefficients:

S D h
[ 4311.640 747.058 ] ’ [ = [ -75.602 ]
- 783.442
59,098.305 4311.640
1::::!]
The set of h coefficients we have found are entries in the discriminant function
equation which has the form

Equation (6.17) is a linear function; that is, all the terms are added together to
yield a single number, the discriminant score, Ri. In a two-dimensional example,
we can plot the discriminant function as a line on the scatter diagram of the two
original variables. It is a line through the plot whose slope, a, is

a = h2Ihl (6.18)
Substitution of the midpoint between the two group means into the discriminant
function equation yields the discriminant index, Ro. That is, for each value of Xji
in Equation (6.17), we insert the terms
-
Aj +Bj
Xj. = - (6.19)
2
In our example,

Ro = (-783.442 * 0.335) + (-75.602 . 1.189)
= -352.146

The discriminant index, Ro, is the point along the discriminant function line
that is exactly halfway between the center of group A and the center of group B.
Next, we may substitute the multivariate mean of group A into the equation (that
is, we set Xj :Xj) to obtain RA and substitute the multivariate mean of group B
(setting Xj = Bj) to obtain RB. The centers of the two original groups projected
onto the axis defined by the discriminant function are RA and RB.
For group A,

RA = (-783.442 . 0.330) + (-75.602 * 1.167)
= -346.560

and for group B,

RB = (-783.442 * 0.340) + (-75.602 . 1.210)
= -357.732
The three points may be plotted as in Figure 6-3. In fact, every observation
in the analysis can be entered into the equation and its position along the discrim-
inant function located. These values are the raw discriminant scores. This has
been done on Figure 6-3; note that a few members of group A are located on the

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