*10.1.2 Classification Trees *

A classification tree is a rule for predicting the class of a nominal
(dichotomous or polytomous) dependent variable from
the values of its predictors.

#### Chi-square
Analysis

This method is used when analyzing one dependent
variable (nominal or ordinal) or a set of dichotomous variables with several
predictors. The method uses an impurity index, which can be written as the
classes of the dependent variable.

*i**
*(*t *)* = F {p *( *j *ç
*t)}* for *j* =
1,2,…, *J*

where (*j* ç
*t*) is the relative frequency of class *j* at node *t*.

*I *(*t *) is minimum (=0) when a node
contains individuals of only one class. The node is then called a pure node. *I
*(*t *) is maximum, when a node contains all
classes with equal relative frequency. Search uses entropy as a measure of
impurity of a node.

Entropy for group *g:*

where

*x*_{jgk} =
frequency (coded 0 to 1) of code *j* (or value of variable *j*) of
case *k* in group *g*.

For a given node and for all predictors and all admissible splits, the
predictor and splits are chosen that maximize the impurity reduction between
the parent node and its descendents.

This criterion is applied recursively to the descendents, which become the
parents of successive splits, and so on. The splitting process is continued
until the criteria of minimum reduction in impurity and minimum size of a node
are satisfied.