Statistical Methods in Medical Research 4ed

according to their values of this function. A different approach, which has been
developed over the last 20 years, is the construction of a binary decision tree. This
is a computer-intensive method. The method is more similar to diagnostic clinical
decision-making than is the construction of a discriminant function.
The basis of the method is that at the first stage, termed a node, the individuals
are divided into two sets, the branches, according to the value of one of the x
variables. This variable is chosen from the full set so that the subdivision
maximizes the separation of individuals in the groups between the two branches.
Also, if the variable is not already dichotomous, then it is converted to binary
form to achieve the same maximization. Each branch leads to a new node (Fig.
13.3) and the same procedure is carried out, using the remaining x variables, and
noting that the variable used for subdivision at each of the two nodes will most
probably be different. Some branches end at a terminal node, when the individuals
at that node are homogenous with respect to group, and the whole process
continues until all branches have terminated.
The choice of the final decision tree is a balance between its quality, defined
in terms of homogeneity of the terminal nodes, and a penalty for complexity.
This penalty is used to control making the tree unnecessarily complex whilst
achieving only minor improvements.
For a fuller description of the method of classification and regression trees,
CART TM , see Zhang et al. (1998).
Discrimination with more than two groups: MANOVA
13.3 Discriminant analysis 477
The linear discriminant function can be generalized to the situation where there
are k…>2† groups in two different ways. The first approach leads to what are
known as canonical variates. We saw from (13.5) that when k ˆ 2 the linear
discriminant function maximizes the ratio of the difference in means between the
groups to the standard deviation within groups. A natural generalization of this
criterion is to maximize the ratio of the sum of squares (SSq) between groups to
the SSq within groups. This requirement is found to lead to a standard technique
of matrix algebraÐthe calculation of eigenvalues or latent roots of a matrix. The
appropriate equation, in fact, has several solutions. One solution, corresponding
to the highest latent root, gives the coefficient in the linear function which
maximizes the ratio of SSqs. This is called the first canonical variate, W1. If
one wanted as good discrimination as possible from one linear function, this
would be the one to choose. The second canonical variate, W2, is the function
with the highest ratio of SSqs, subject to the condition that it is uncorrelated with
W1 both between and within groups. Similarly, W3 gives the highest ratio subject
to being uncorrelated with W1 and W2. The number of canonical variates is the
smaller of p or k 1. Thus, the linear discriminant function (13.4) is for k ˆ 2 the
first and only canonical variate.