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1.  Letter to the Editor: On the term ‘interaction’ and related phrases in the literature on Random Forests 
Briefings in Bioinformatics  2014;16(2):338-345.
In an interesting and quite exhaustive review on Random Forests (RF) methodology in bioinformatics Touw et al. address—among other topics—the problem of the detection of interactions between variables based on RF methodology. We feel that some important statistical concepts, such as ‘interaction’, ‘conditional dependence’ or ‘correlation’, are sometimes employed inconsistently in the bioinformatics literature in general and in the literature on RF in particular. In this letter to the Editor, we aim to clarify some of the central statistical concepts and point out some confusing interpretations concerning RF given by Touw et al. and other authors.
PMCID: PMC4364067  PMID: 24723569
random forest; statistics; interaction; correlation; conditional inference trees; conditional variable importance
2.  An AUC-based permutation variable importance measure for random forests 
BMC Bioinformatics  2013;14:119.
The random forest (RF) method is a commonly used tool for classification with high dimensional data as well as for ranking candidate predictors based on the so-called random forest variable importance measures (VIMs). However the classification performance of RF is known to be suboptimal in case of strongly unbalanced data, i.e. data where response class sizes differ considerably. Suggestions were made to obtain better classification performance based either on sampling procedures or on cost sensitivity analyses. However to our knowledge the performance of the VIMs has not yet been examined in the case of unbalanced response classes. In this paper we explore the performance of the permutation VIM for unbalanced data settings and introduce an alternative permutation VIM based on the area under the curve (AUC) that is expected to be more robust towards class imbalance.
We investigated the performance of the standard permutation VIM and of our novel AUC-based permutation VIM for different class imbalance levels using simulated data and real data. The results suggest that the new AUC-based permutation VIM outperforms the standard permutation VIM for unbalanced data settings while both permutation VIMs have equal performance for balanced data settings.
The standard permutation VIM loses its ability to discriminate between associated predictors and predictors not associated with the response for increasing class imbalance. It is outperformed by our new AUC-based permutation VIM for unbalanced data settings, while the performance of both VIMs is very similar in the case of balanced classes. The new AUC-based VIM is implemented in the R package party for the unbiased RF variant based on conditional inference trees. The codes implementing our study are available from the companion website:
PMCID: PMC3626572  PMID: 23560875
Random forest; Conditional inference trees; Variable importance measure; Feature selection; Unbalanced data; Class imbalance; Area under the curve.
3.  An Introduction to Recursive Partitioning: Rationale, Application and Characteristics of Classification and Regression Trees, Bagging and Random Forests 
Psychological methods  2009;14(4):323-348.
Recursive partitioning methods have become popular and widely used tools for non-parametric regression and classification in many scientific fields. Especially random forests, that can deal with large numbers of predictor variables even in the presence of complex interactions, have been applied successfully in genetics, clinical medicine and bioinformatics within the past few years.
High dimensional problems are common not only in genetics, but also in some areas of psychological research, where only few subjects can be measured due to time or cost constraints, yet a large amount of data is generated for each subject. Random forests have been shown to achieve a high prediction accuracy in such applications, and provide descriptive variable importance measures reflecting the impact of each variable in both main effects and interactions.
The aim of this work is to introduce the principles of the standard recursive partitioning methods as well as recent methodological improvements, to illustrate their usage for low and high dimensional data exploration, but also to point out limitations of the methods and potential pitfalls in their practical application.
Application of the methods is illustrated using freely available implementations in the R system for statistical computing.
PMCID: PMC2927982  PMID: 19968396
4.  The behaviour of random forest permutation-based variable importance measures under predictor correlation 
BMC Bioinformatics  2010;11:110.
Random forests (RF) have been increasingly used in applications such as genome-wide association and microarray studies where predictor correlation is frequently observed. Recent works on permutation-based variable importance measures (VIMs) used in RF have come to apparently contradictory conclusions. We present an extended simulation study to synthesize results.
In the case when both predictor correlation was present and predictors were associated with the outcome (HA), the unconditional RF VIM attributed a higher share of importance to correlated predictors, while under the null hypothesis that no predictors are associated with the outcome (H0) the unconditional RF VIM was unbiased. Conditional VIMs showed a decrease in VIM values for correlated predictors versus the unconditional VIMs under HA and was unbiased under H0. Scaled VIMs were clearly biased under HA and H0.
Unconditional unscaled VIMs are a computationally tractable choice for large datasets and are unbiased under the null hypothesis. Whether the observed increased VIMs for correlated predictors may be considered a "bias" - because they do not directly reflect the coefficients in the generating model - or if it is a beneficial attribute of these VIMs is dependent on the application. For example, in genetic association studies, where correlation between markers may help to localize the functionally relevant variant, the increased importance of correlated predictors may be an advantage. On the other hand, we show examples where this increased importance may result in spurious signals.
PMCID: PMC2848005  PMID: 20187966
5.  Optimal classifier selection and negative bias in error rate estimation: an empirical study on high-dimensional prediction 
In biometric practice, researchers often apply a large number of different methods in a "trial-and-error" strategy to get as much as possible out of their data and, due to publication pressure or pressure from the consulting customer, present only the most favorable results. This strategy may induce a substantial optimistic bias in prediction error estimation, which is quantitatively assessed in the present manuscript. The focus of our work is on class prediction based on high-dimensional data (e.g. microarray data), since such analyses are particularly exposed to this kind of bias.
In our study we consider a total of 124 variants of classifiers (possibly including variable selection or tuning steps) within a cross-validation evaluation scheme. The classifiers are applied to original and modified real microarray data sets, some of which are obtained by randomly permuting the class labels to mimic non-informative predictors while preserving their correlation structure.
We assess the minimal misclassification rate over the different variants of classifiers in order to quantify the bias arising when the optimal classifier is selected a posteriori in a data-driven manner. The bias resulting from the parameter tuning (including gene selection parameters as a special case) and the bias resulting from the choice of the classification method are examined both separately and jointly.
The median minimal error rate over the investigated classifiers was as low as 31% and 41% based on permuted uninformative predictors from studies on colon cancer and prostate cancer, respectively. We conclude that the strategy to present only the optimal result is not acceptable because it yields a substantial bias in error rate estimation, and suggest alternative approaches for properly reporting classification accuracy.
PMCID: PMC2813849  PMID: 20025773
6.  Conditional variable importance for random forests 
BMC Bioinformatics  2008;9:307.
Random forests are becoming increasingly popular in many scientific fields because they can cope with "small n large p" problems, complex interactions and even highly correlated predictor variables. Their variable importance measures have recently been suggested as screening tools for, e.g., gene expression studies. However, these variable importance measures show a bias towards correlated predictor variables.
We identify two mechanisms responsible for this finding: (i) A preference for the selection of correlated predictors in the tree building process and (ii) an additional advantage for correlated predictor variables induced by the unconditional permutation scheme that is employed in the computation of the variable importance measure. Based on these considerations we develop a new, conditional permutation scheme for the computation of the variable importance measure.
The resulting conditional variable importance reflects the true impact of each predictor variable more reliably than the original marginal approach.
PMCID: PMC2491635  PMID: 18620558
7.  Bias in random forest variable importance measures: Illustrations, sources and a solution 
BMC Bioinformatics  2007;8:25.
Variable importance measures for random forests have been receiving increased attention as a means of variable selection in many classification tasks in bioinformatics and related scientific fields, for instance to select a subset of genetic markers relevant for the prediction of a certain disease. We show that random forest variable importance measures are a sensible means for variable selection in many applications, but are not reliable in situations where potential predictor variables vary in their scale of measurement or their number of categories. This is particularly important in genomics and computational biology, where predictors often include variables of different types, for example when predictors include both sequence data and continuous variables such as folding energy, or when amino acid sequence data show different numbers of categories.
Simulation studies are presented illustrating that, when random forest variable importance measures are used with data of varying types, the results are misleading because suboptimal predictor variables may be artificially preferred in variable selection. The two mechanisms underlying this deficiency are biased variable selection in the individual classification trees used to build the random forest on one hand, and effects induced by bootstrap sampling with replacement on the other hand.
We propose to employ an alternative implementation of random forests, that provides unbiased variable selection in the individual classification trees. When this method is applied using subsampling without replacement, the resulting variable importance measures can be used reliably for variable selection even in situations where the potential predictor variables vary in their scale of measurement or their number of categories. The usage of both random forest algorithms and their variable importance measures in the R system for statistical computing is illustrated and documented thoroughly in an application re-analyzing data from a study on RNA editing. Therefore the suggested method can be applied straightforwardly by scientists in bioinformatics research.
PMCID: PMC1796903  PMID: 17254353

Results 1-7 (7)