Unbiased Recursive Partitioning: A Conditional Inference Framework

Abstract

Recursive binary partitioning is a popular tool for regression analysis. Two fundamental problems of exhaustive search procedures usually applied to fit such models have been known for a long time: Overfitting and a selection bias towards covariates with many possible splits or missing values. While pruning procedures are able to solve the overfitting problem, the variable selection bias still seriously effects the interpretability of tree-structured regression models. For some special cases unbiased procedures have been suggested, however lacking a common theoretical foundation. We propose a unified framework for recursive partitioning which embeds tree-structured regression models into a well defined theory of conditional inference procedures. Stopping criteria based on multiple test procedures are implemented and it is shown that the predictive performance of the resulting trees is as good as the performance of established exhaustive search procedures. It turns out that the partitions and therefore the models induced by both approaches are structurally different, indicating the need for an unbiased variable selection. The methodology presented here is applicable to all kinds of regression problems, including nominal, ordinal, numeric, censored as well as multivariate response variables and arbitrary measurement scales of the covariates. Data from studies on animal abundance, glaucoma classification, node positive breast cancer and mammography experience are re-analyzed.

Bayesian Weibull tree models for survival analysis of clinico-genomic data

“…This has been stressed in the recent tree literature, e.g., [41,40,27]. Our models are not unbiased in the sense that variables with more splitting values are more likely to be selected in model building.…”

“…In anticipation of future studies we intend to perform further comparisons with existing methods [27,33] and further simulations to examine the impact of tuning parameters and prior assumptions on model performance. Our current approach to missing values is to perform imputation prior to modeling; however, we are considering adjusting our method to deal with missing values as these are common in realistic data analysis contexts.…”

“…This holds true for trees whether the model search is stochastic [67,7,35] or deterministic [10,44]. Our approach is a reflection of this finding and of the recent emphasis in the literature on ensemble methods for prediction [14,26,27]. A key aspect in our approach is the averaging of predictions over multiple candidate models, which we discuss in Section 3.…”

“…In the case of genomic data these combinations can then serve as a basis for further biological study. Recent additions to the survival tree modeling literature, including [26,27] and [33], reflect the importance of survival trees as an analytic technique for data sets with complex structure.…”

Seasonal Deposition Fluxes and Removal Efficiency of Atmospheric Polybrominated Diphenyl Ethers in a Large Urban Center: Importance of Natural and Anthropogenic Factors

“…Recoveries of the surrogate standards, i.e. 13 C-PCB-141, PCB-204, PCB-209, and 13 C-BDE-209, were 66 ± 19%, 64 ± 17%, 70 ± 18%, and 69 ± 18% for filtrated samples, whereas they were 68 ± 20%, 69 ± 17%, 71 ± 15%, and 68 ± 17% for particulate samples. In addition, the surrogate standard recoveries were 65 ± 14%, 67 ± 16%, 71 ± 19%, and 68 ± 15% for gas aerosol samples and were 66 ± 12%, 70 ± 15%, 72 ± 16%, and 69 ± 18% for particulate aerosol samples.…”