from combining classifiers is valuable in assessing prediction confidence, which is usually difficult to obtain from a single classifier.

DECISION FOREST—A ROBUST CONSENSUS METHOD FOR DIAGNOSTIC CLASSIFICATION

Most consensus modeling relies on resampling approaches that use only a portion of the subjects for constructing the individual classifiers. Since we normally have a relatively small sample size, this approach will weaken individual classifiers’ predictive accuracy, which follows the reduction of the improvement in a combining system gained by the resampling approach.

A preferable consensus approach is to develop multiple classifiers using different sets of omics patterns. This approach takes full advantage of the available sample size as well as the redundant information presented in the data. Accordingly, we have developed the robust DF method (see Figure 4-1).

DF emphasizes the combining of heterogeneous yet comparable trees in order to better capture the association of omics profiles and disease outcomes. The heterogeneity requirement assures that each tree uniquely contributes to the combined prediction; whereas the quality comparability requirement assures that each tree equally contributes to the combined prediction. Since a certain degree of noise is always present in biologic data, optimizing a tree inherently risks over fitting the noise. DF attempts to minimize over fitting by maximizing the difference among individual trees.

There are three benefits associated with DF compared with other similar consensus modeling methods: (1) since the difference in individual trees is maximized, the best ensemble is usually realized by combining only a few trees (i.e., four or five), which consequentially reduces computational expense; (2) since DF is entirely reproducible, the disease-patterns associations are constant in their interpretability for biologic relevance; and (3) since all subjects are included in individual tree development, the information in the original data set is fully appreciated in the combining process.

For example, we develop a DF classifier on a proteomic data set to distinguish the prostate patients from healthy individuals. The data set consists of 326 samples, of which 167 samples are from the prostate cancer patients and the noncancer group contains 159 samples including both benign prostatic hyperplasia patients and healthy individuals. The



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