Reference Work Entry

Springer Handbook of Engineering Statistics

pp 551-570

Tree-Based Methods and Their Applications

  • Nan LinAffiliated withDepartment of Mathematics, Washington University in Saint Louis Email author 
  • , Douglas NoeAffiliated withDepartment of Statistics, University of Illinois at Urbana-Champaign Email author 
  • , Xuming HeAffiliated withDepartment of Statistics, University of Illinois at Urbana-Champaign Email author 


The first part of this chapter introduces the basic structure of tree-based methods using two examples. First, a classification tree is presented that uses e-mail text characteristics to identify spam. The second example uses a regression tree to estimate structural costs for seismic rehabilitation of various types of buildings. Our main focus in this section is the interpretive value of the resulting models.

This brief introduction is followed by a more detailed look at how these tree models are constructed. In the second section, we describe the algorithm employed by classification and regression tree (CART), a popular commercial software program for constructing trees for both classification and regression problems. In each case, we outline the processes of growing and pruning trees and discuss available options. The section concludes with a discussion of practical issues, including estimating a treeʼs predictive ability, handling missing data, assessing variable importance, and considering the effects of changes to the learning sample.

The third section presents several alternatives to the algorithms used by CART. We begin with a look at one class of algorithms – including QUEST, CRUISE, and GUIDE– which is designed to reduce potential bias toward variables with large numbers of available splitting values. Next, we explore C4.5, another program popular in the artificial-intelligence and machine-learning communities. C4.5 offers the added functionality of converting any tree to a series of decision rules, providing an alternative means of viewing and interpreting its results. Finally, we discuss chi-square automatic interaction detection (CHAID), an early classification-tree construction algorithm used with categorical predictors. The section concludes with a brief comparison of the characteristics of CART and each of these alternative algorithms.

In the fourth section, we discuss the use of ensemble methods for improving predictive ability. Ensemble methods generate collections of trees using different subsets of the training data. Final predictions are obtained by aggregating over the predictions of individual members of these collections. The first ensemble method we consider is boosting, a recursive method of generating small trees that each specialize in predicting cases for which its predecessors perform poorly. Next, we explore the use of random forests, which generate collections of trees based on bootstrap sampling procedures. We also comment on the tradeoff between the predictive power of ensemble methods and the interpretive value of their single-tree counterparts.

The chapter concludes with a discussion of tree-based methods in the broader context of supervised learning techniques. In particular, we compare classification and regression trees to multivariate adaptive regression splines, neural networks, and support vector machines.