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Clustering and Classification methods for Biologists

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Discriminant Analysis

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Discriminant Analysis : A brief mathematical background

This example uses data from 10 males and 10 females. Three variables were recorded: height (inches!), weight (pounds!) and age (years).

Sex	Height	Weight	Age
2	67	143	43
2	66	137	46
2	64	125	19
2	65	130	52
2	69	152	49
2	68	140	35
2	62	102	33
2	66	132	21
2	67	117	20
2	68	129	21
1	71	170	54
1	72	164	56
1	72	165	51
1	68	163	26
1	72	172	58
1	66	135	35
1	71	190	42
1	74	178	38
1	68	152	36
1	69	170	27

Data Summary

Variable	Male	Female	Difference
height	70.3	66.2	4.1
weight	165.9	130.7	35.2
age	42.3	33.9	8.4

A method is needed which is guaranteed to maximise the group differences, displayed by these three potentially discriminating variables, when they are combined into a single discriminating variable. This is achieved by calculating a Discriminant Function of the type:

score = w₁height + w₂weight + w₃age

The problem is finding suitable values for w_i.

First calculate the variance-covariance matrix A

		height	weight	age
Let A =	height	5.21	24.49	10.68
	weight	24.49	207.72	65.56
	age	10.68	64.56	155.17

 

Let w be a vector containing the unknown weights (we need these for the Discriminant Function).:

w = [w₁ w₂ w₃]

and d be a vector of the group differences (as shown above):

d = [ 4.1 35.2 8.4]

It can be shown that A.w = d

This is a relatively simple matrix algebra calculation since the equation be rewritten as w = A^-1.d and A^-1, the inverse of matrix A, is needed to solve the equation.

Using standardised variables.

w=	weight	variable
	0.0029	height
w =	1.028	weight
	-0.096	age

Hence;

discriminant score = 0.0029 height + 1.028 weight - 0.096 age

The group centroids ('mean scores') are females -1.23 and males +1.23

Consequently a positive score (>0) indicates a male, a negative score (<0) indicates a female. The means would not be symmetrical if the group sizes differed.

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