Improved highdimensional prediction with Random Forests by the use of codata
 Dennis E. te Beest^{1},
 Steven W. Mes^{3},
 Saskia M. Wilting^{4},
 Ruud H. Brakenhoff^{3} and
 Mark A. van de Wiel^{1, 2}Email author
https://doi.org/10.1186/s1285901719931
© The Author(s) 2017
Received: 3 July 2017
Accepted: 6 December 2017
Published: 28 December 2017
Abstract
Background
Prediction in high dimensional settings is difficult due to the large number of variables relative to the sample size. We demonstrate how auxiliary ‘codata’ can be used to improve the performance of a Random Forest in such a setting.
Results
Codata are incorporated in the Random Forest by replacing the uniform sampling probabilities that are used to draw candidate variables by codata moderated sampling probabilities. Codata here are defined as any type information that is available on the variables of the primary data, but does not use its response labels. These moderated sampling probabilities are, inspired by empirical Bayes, learned from the data at hand. We demonstrate the codata moderated Random Forest (CoRF) with two examples. In the first example we aim to predict the presence of a lymph node metastasis with gene expression data. We demonstrate how a set of external pvalues, a gene signature, and the correlation between gene expression and DNA copy number can improve the predictive performance. In the second example we demonstrate how the prediction of cervical (pre)cancer with methylation data can be improved by including the location of the probe relative to the known CpG islands, the number of CpG sites targeted by a probe, and a set of pvalues from a related study.
Conclusion
The proposed method is able to utilize auxiliary codata to improve the performance of a Random Forest.
Keywords
Background
Highdimensional prediction is inherently a difficult problem. In this paper we demonstrate how to improve the performance of the Random Forest (RF) on highdimensional (in particular genomics) data by guiding it with ‘codata’. Here, codata is defined as any type of qualitative or quantitative information on the variables that does not use the response labels of the primary data. The primary data may, for example, be a set of gene expression profiles with corresponding binary response labels. Examples of codata are: pvalues on the same genes in a external, related study, correlations with methylation or DNA copy number data, or simply the location on the genome. Guiding a prediction model by codata may lead to improved predictive performance and variable selection.
Several methods are able to incorporate codata during model training. A general multipenalty approach was suggested by [1], a weighted lasso by [2], and a groupregularized ridge by [3]. These methods are all based on penalized regression, with a penalty parameter that is allowed to vary depending on the codata, effectively rendering codata based weights. The grouplasso [4] and sparse grouplasso [5] are also regressionbased, but these methods apply a specific grouppenalty that can exclude entire groups of variables. Except for the groupregularized ridge, all these methods allow for only one type of codata. In addition, except for the weighted lasso, these methods require the codata to be specified as groups. The weighted lasso can handle one source of continuous codata, but requires an assumption about the functional form of the penalty weighting and the codata. For some types of codata this functional form is largely unknown. Hence, it may be desirable to learn it from the codata, and to enforce monotonous weights to ensure stability and interpretability.
The Random Forest (RF) is a learner that is popular due to its robustness to various types of data inputs, its ability to seamlessly handle nonlinearities, its invariance to data transformations, and its ease of use without any or much tuning [6]. The RF is suitable and computationally efficient for genomics data, with typically the number of variables, P, largely exceeding the sample size, n [7, 8]. Its scale invariance makes it a good candidate to analyse RNASeq data. Due to the skewed nature of such data, their analysis is less straightforward with penalized regression techniques and results depend strongly on the data transformation applied [9]. Our aim is to develop a codata moderated RF (CoRF) which allows the joint use of multiple types of codata, the use of continuous codata, and flexible modeling of the codata weights. Conveniently, these codata are only used when training the classifier; they are not required for applying the classifier to new samples.
The described methodology can in principle be used with any bagging classifier that uses the random subspace method [10], but in this paper we focus on the RF. The method is exemplified with two examples. First, we aim to predict the presence of a lymph node metastasis (LNM) for patients with head and neck squamous cell carcinoma (HNSCC) using TCGA RNAseq data. We show how the use of several types of codata, including DNA copy number, an external gene signature and mRNA microarray data from an independent patient cohort, improves the predictive performance, and validate these results on a second independent data set. The computational efficiency of the method is illustrated with a second example, where our aim is to predict the last precursor stage for cervical cancer based on methylation data with very large P≈350.000. The codata in this example consists of the location of the methylation site, the number of CpGs, and the external set of pvalues.
Methods and Results
Random forest
The aim of a supervised RF is to predict per sample i,i=1,…,n, an outcome Y _{ i } using a set of variables X _{ ij } where j=1,…,P indicates the variables. Here, we focus on binary outcome Y _{ i }, although the entire methodology and software also applies to continuous and censored (e.g. survival) outcomes. A RF consists of a large number of unpruned decision trees, where each tree is grown on a bootstrap sample of the data. At each node split in each tree only a random subset of the variables are candidates, its size denoted by mtry, typically set at \(\sqrt P\). In a standard RF, all variables have an equal probability of being candidates. Predictions are issued by majority voting across all trees, or on a fractional scale (fraction of trees predicting Y _{ i }=1). We will use the latter for assessing predictive performance. A RF is fitted to a bootstrap sample of the data implying that per tree the remaining fraction (on average 0.368) is outofbag (oob) and can be used to obtain an estimate of the prediction error. This leads to a computational advantage compared to methods that require crossvalidation for this purpose.
Groupspecific probabilities
We first briefly describe our method using one source of grouped codata only. Here, the basic idea is that, when an a priori grouping of variables is available (codata), we may sample the variables according to groupspecific probabilities, and these probabilities can be estimated empirically from the data. When the number of groups is limited, only a few parameters need to be estimated (the group specific probabilities). Especially when the difference in predictive power between groups of variables is large, the predictive performance may be enhanced.
where \(\hat {p}_{g}^{\normalsize {sel}}\) is the proportion of selected variables from group g across all trees divided by the size of group g and p ^{0}=1/P is the expected value of \(\hat {p}_{g}^{\normalsize {sel}}\) when the group structure is uninformative. Parameter γ can be used to tune the RF to adapt to groupsparsity by thresholding, but may also be set to one to avoid tuning. After normalizing w _{ g } such that these sum to one across variables, we obtain sampling probabilities \(\tilde {w_{g}}\). Then, a new RF is trained using these probabilities instead of the uniform ones, rendering the CoRF.
Modelbased probabilities
As earlier, γ can be set to 1 which provides a natural cutoff for p ^{0}, or γ may be tuned to more or less sparsity. Finally, we normalize w _{ j } to obtain the sampling probabilities \(\tilde {w_{j}}=w_{j}/\sum _{j} w_{j}\), which are then used to retrain the RF.
where X ^{ n } (X ^{ c }) denotes the submatrix of X containing the nominal (continuous) codata, and f _{ d }() represents a flexible function provided by the SCOP. To model f _{ d }, SCOP uses \(m(x)=\sum _{\ell =1}^{q} \theta _{\ell }B_{\ell }(x)\), where B _{ ℓ } is a Bspline basis function, which is monotonously increasing when θ _{ ℓ }≥θ _{ ℓ−1},ℓ=1,…,q [12]. The monotony in θ is enforced by defining \(\boldsymbol {\theta } =\Sigma \tilde {\boldsymbol {\theta }}\), where \(\tilde {\boldsymbol {\theta }} = [\tilde {\theta }_{1},\exp (\tilde {\theta }_{2}),\ldots,,\exp (\tilde {\theta }_{q})]^{T}\) and Σ _{ rs }=0 if r<s and Σ _{ rs }=1 if \(r \geqslant s\). Smoothness is enforced by penalisation of the squared differences analogous to [13]. Setting Σ _{ rs }=−1 for \(r \geqslant s\) renders a monotonically decreasing spline. Unrestricted splines can in principle also be used in the codata model, but are more liable to overfitting.
Instead of using the default of γ=1, this parameter can be tuned by a grid search. This requires calculating w _{ j } and refitting a RF for each gridvalue of γ. The optimal value of γ is then the one with the best oob performance. Note that tuning γ with the oob predictions may result in a degree of optimism. This may be solved by embedding the procedure in a crossvalidation loop. When γ is not tuned, the oob performance of CoRF may also be slightly optimistic, because the primary data was used to estimate the weights (4). However, when the regression model (2) is parsimonious, the overoptimism is likely small, as verified empirically in the Application section. To ensure that the codata model is parsimonious, it may be useful to perform codata selection to remove redundant codata sources, which also assists in assessing the relevance of the codata. The Additional file 1 supplies an heuristic procedure to do so.
CoRF algorithm
 1.
Fit a base RF with uniform sampling probabilities and obtain v _{ jk }.
 2.To disentangle the contributions of the various codata sources.
 3.
Obtain the predicted probabilities \(\hat {p}_{j}\) from the fitted codata model.
 4.
Calculate the sampling probabilities w _{ j } with threshold parameter γ. Default is to set γ=1, optionally γ can be tuned.
 5.
Refit the RF for each vector of \(\tilde {w_{j}}\).
 6.

If γ is not tuned (i.e. γ = 1), we directly obtain the CoRF, the base RF and their oob performances.

If γ is tuned, obtain \(\hat {\gamma }\) by maximizing the oob performance. Tuning γ may introduce a bias in the oob performance. Hence, the entire procedure is crossvalidated when γtuning is employed.

Implementation
The method as described here is implemented in a corresponding R package, called CoRF, and is available on GitHub. It depends on the R package randomForestSRC for fitting the RF [14–16]. A feature of this package that is of key importance for CoRF is the option to assign a sampling probability per variable. In addition, randomForestSRC applies to regression, classification and survival analysis, and by extension, so does CoRF.
For classification by the RF the recommended minimal node size is one. The node size can be tuned [17], but a RF is not very sensitive to the minimal node size. In CoRF the quality of the selected variables may influence the fit of the codata model. Variables that are used higher up in a tree are, on average, more relevant, and variables that split a node of size 2 are the least relevant. For CoRF, we believe it is better to slightly increase the minimal node size, improving the quality of the selected variables and as a result improve the quality of the codata model. As default in CoRF, we set the minimal node size for classification at 2.
Generally, CoRF will need a larger number of trees to fit than a base RF. A base RF needs enough trees to capture the underlying signal in the data. CoRF additionally needs an indication of the relevance of each variable, which feeds back to the codata model. Also, a codata model that contains splines generally needs more trees than a codata model with only linear effects. In the LNM example, described below, we set the number of trees at 15.000 to ensure convergence of both the RF/CoRF and the codata model. A lower number of trees, e.g. 2.000, gives a similar result in terms of predictive performance, but the variability between fits increases. When using CoRF, we recommend to use at least 5.000 trees to ensure a reliable, good fit of the codata model. An additional advantage of a large number of trees with tuning is that the variability between RF fits decreases, allowing for a more reliably selection of γ.
A RF is a computationally efficient algorithm to use for high dimensional data, primarily because at each node it selects only from \(\sqrt P\) variables. CoRF inherits this efficiency and when the default γ = 1 is used, only one RF refit is needed. Next to (re)fitting the RF, the only additional computation required for CoRF consists of fitting the codata model. Further tuning of γ may improve the performance, but also requires i) refitting a RF for each value for γ, and ii) an additional crossvalidation loop to assess performance, thereby increasing computational cost considerably.
Evaluating predictive performance
The predictive performance of CoRF and other classifiers was assessed on oob samples by two metrics: i) the areaundertheroc curve (AUC; [18]); and ii) the Brier score [19]. AUC is based on ranks and evaluates discrimination. It combines sensitivity and specificity, which are both important in a clinical setting. Moreover, it is a good indicator for the performance of a RF with unbalanced data [18]. Brier score is based on residuals and evaluates calibration. It equals the average Brier residual, i.e. B _{ i }=(Y _{ i }−q _{ i })^{2}, where q _{ i } is the fraction of trees predicting Y _{ i }=1. Brier score is reported in a relative sense, with the base RF as benchmark. For comparing CoRF with RF we implemented significance testing, both for ΔAUC: the difference between AUCs, using R’s pROC package [20] and for ΔBrier: the difference between Brier scores, using the onesided Wilcoxon signedrank test on paired Brier residuals \(\left (B^{\text {RF}}_{i}, B^{\text {CoRF}}_{i}\right)\). When performance was evaluated on the same data as used for training, we used multiple 2/3  1/3 splits, meaning that the power for testing, for which only 1/3 of the samples can be used, can be limited. These splits result in multiple pvalues, which we aggregate by applying the median, which was proven to control the type I error rate under mild conditions [21].
Similarly to the base RF, CoRF automatically renders oob predictions. CoRF is an empirical Bayestype classifier, which uses the relation between the codata and the primary data to estimate sampling weights. Such double use of data could lead so some degree over overoptimism, although this will likely be limited given that the codata model is parsimonious. In addition, when splines were used, the effective degreesoffreedom were reduced by imposing monotony. Nevertheless, in the examples below, we verified the oob performance of CoRF by crossvalidation when training and evaluation was applied on the same data.
Comparable methods
To our knowledge, there is only one highdimensional prediction method that can explicitly take multiple sources of codata into account: the groupregularized (logistic) ridge (GRridge [3]). CoRF provides several conceptual advantages over GRridge. First, unlike CoRF, GRridge requires discretisation of continuous codata. Second, CoRF fits the codata coefficients in one model, (2), instead of using the codata sources iteratively. Third, CoRF is computationally more efficient, because it a) inherits the better computational scalability of RF with respect to P; and b) requires very little tuning and no iterations. Finally, as with a base RF, CoRF is naturally able to incorporate categorical outcomes with > 2 groups (as demonstrated in the cervical cancer example). GRridge inherits the advantages of ridge regression, e.g. better interpretability of the model and the ability to include mandatory covariates. In the Application section we compare the performances of these two methods for the LNM example.
Applications
Predicting Lymph node metastasis with TCGA data
To exemplify the CoRF method, we use it to predict the presence of a lymph node metastasis (LNM) for patients with HPV negative oral cancer using RNASeqv2 data from TCGA [22]. We focus on the HPVnegatives, because these constitute the majority (approx. 90%) of the oral cancers, and HPVpositive tumors are known to have a different genomic etiology [23]. Early detection of LNM is important for assigning the appropriate treatment. Diagnosis of LNM with genomic markers could potentially improve diagnosis and treatment [24].
The primary data consists of normalized TCGA RNASeqv2 profiles of headandneck squamous cell carcinomas (HNSCC), which were downloaded together with the matching normalized DNA copy number codata from Broad GDAC Firehose using the R package TCGA2STAT. Of the 279 patients described in [22], we used the subset of 133 patients that had HPVnegative tumors in the oral cavity. Of these patients, 76 presented a LNM and 57 did not.
To enhance the prediction of the base RF, we consider three types of codata in this example: (1) DNA copy number; (2) pvalues from the external microarray data GSE30788/GSE85446; (3) a previously identified gene signature [24–26]. These three types of codata demonstrate the variety of codata sources that can be included in CoRF. The DNA copy number data are measurements on the same patients. We use the ciscorrelations between DNA copy number and the RNASeqv2 data. Given the nature of RNASeqv2 and DNA copy number data (discrete and ordinal, respectively), we applied Kendall’s τ to calculate the correlations, giving τ _{ j },j=1,…,P. Note that the DNA data are only used during training of the predictor; these are not required for test samples, which distinguishes this type of predictor from integrative predictors [27]. The pvalues of GSE30788/GSE85446 are derived from measurements of the same type of genomics features (mRNA gene expression), but measured on a different platform (microarray) than that of the primary RNAseq data and on a different set of patients. The gene signature is a published set of genes that were found to be important in a different study. Figure 1 illustrates how the various types of data are used within CoRF.
Each type of codata has its own characteristics that needs to be taken into account in the codata model. For the DNA copy number data, we a priori expect that a gene with a positive ciscorrelation is more likely to be of importance to the tumor [28]. We use a monotonically increasing spline f _{1} to model the relation between p _{ j } and \(X^{c}_{j1} = \tau _{j}\) (5). For the pvalues of GSE30788/GSE85446, we a priori expect that genes with a low pvalue are more likely to be important on the TCGA data, and we thus use a monotonically decreasing spline f _{2} to model the relation between p _{ j } and \(X^{c}_{j2} = \text {pval}_{j}\). The third type of codata, consisting of the published gene signature is included in the codata model (2) as a binary variable: \(X^{n}_{j1}=1\) when gene j is part of the signature, and 0 otherwise.
Data set GSE30788/GSE85446 consists of 150 Dutch patients (of which 60 presented a LNM and 90 did not) with a HPVnegative oral cancer tumor who are in that respect similar to the TCGA patients. Gene expression was measured by microarray, the pvalues on GSE30788/ GSE85446 were calculated using a Welch twosample ttest; further details on the study can be found in [29]. The differences between the TCGA and the Dutch data (notably the platform and the geographical location of the patients) preclude a straightforward metaanalytic data integration. Also, our focus here is on the TCGA data, which were measured on a more modern platform, but shared genomic features with the Dutch codata may enhance the weighted predictions.
After training the base RF and the CoRF, we validate these classifiers on an independent data set (GSE84846). GSE84846 contains microarray expression data of 97 HPVnegative oral cancer patients from Italy, of whom 49 had a LNM [29]. To directly apply the classifiers to the validation data, we need to account for the differences in scale between RNASeqv2 and microarray data. First, the TCGA RNASeqv2 data are transformed by the Anscombe transformation (i.e. \(\sqrt (x+3/8)\)). Next, both the TCGA RNASeqv2 and GSE84846 data are scaled to have zero mean and unit variance. We only included genes that could uniquely be matched between the two data sets (leaving 12838 genes). Since this validation does not require any retraining, the performance is directly assessed by comparing the predictions with the actual labels. As an alternative to this validation, we also use the relative frequency of variables used by the base RF and CoRF on the TCGA data as sampling probabilities in training a new RF on GSE84846 data, in which case the oobperformance was used.
We also assess the performance of the base RF and CoRF in terms of variable selection on both the training and validation data sets. For the TCGA training data set, we first select a set of genes (based on V _{ j }), retrain on this subset, and assess the performance with a 10fold crossvalidation. For the validation data set, we first select variables on the TCGA training data with the variablehuntvimp (vhvimp) as described by [30]. Roughly, vimp measures the importance of a variable by assessing the decrease in predictive performance when the values of the variable are ‘noised up’, e.g. randomly permuted across samples. Then, we refit with the selected set of variables on the TCGA training data, and evaluate the performance of the refitted model on the validation data using oobperformance. To assess the stability of variable selection with a base RF and CoRF, we repeatedly (20 times) sample 84 out of 133 TCGA cases without replacement and fit a base RF and CoRF to each sampled set. Note that the sampling fraction mimics the expected fraction of independent samples in a resampling scheme, 0.632. We preferred subsampling over resampling here, because the latter would lead to duplicate samples in the sampled set. Stability of gene selection was then assessed by calculating the average overlap between any combination of two sets of variables selected from the subsampled training sets, varying the sizes of the selection sets from 10,20,…,100 genes.
Performance on LNM example
For comparison with GRridge, we find a cvAUC of 0.682 on the training data and AUC of 0.689 on the validation data. With GRridge the global penalty parameter of the ridge regression was estimated using a 10fold crossvalidation and performance on the training data was assessed using a second 10fold crossvalidation. For the validation data we directly applied the resulting classifier. In performance this is comparable to CoRF, but note that CoRF is quicker, especially when we want an estimate of the prediction error (see the “Computational time” section).
Cervical cancer example
In this second example, our aim is to predict cervical (pre)cancer on a very highdimensional methylation data set. The methylation data consists of 365620 methylation sites, and contains 68 samples of which 28 correspond to normal cervical tissue, 36 have a highgrade precursor lesion (CIN3; CIN = cervical intraepithelial neoplasia) and 4 have cervical cancer. A diagnosis of either of the latter two stages usually implies surgery. The samples were taken using a selfsample test, implying a challenging diagnostic setting. The data used in this example originate from [31] where more information can be found on the clinical details and on the preprocessing of the methylation data. When training the RF, we consider the three separate categories, while for the final prediction we add up the votes for CIN3 and cancer, because of the small sample size for the latter. As before, our aim is to improve the prediction of the base RF by including codata (e.g. CoRF).
Finally, for the external pvalues we expect a different codata effect for methylation sites that were either up or downregulated in the external codata. We a priori expect a stronger effect of probes that are upregulated, because downregulated effects in the tissue data are hard to discover in selfsamples, due to likely contamination of affected samples by adjacent normal tissue. We accommodate this distinction by modeling the interaction between the pvalue and the direction of regulation (both obtained from the codata), essentially fitting two monotonically decreasing splines, one for up and one for downregulated methylation sites. We indeed find that there is an effect of the pvalues for the upregulated methylation sites (Fig. 5). For the the pvalues of the downregulated methylation sites, the codata model did indeed not identify an effect.
For this diagnostic setting, we observe there is an improvement by using CoRF (oobAUC base RF: 0.666, CoRF: 0.710) and a decrease in oobBrier score of 4.6%. Fig. 3 c shows the ROCcurves (specificity versus sensitivity), parametrized by a threshold for the proportion of trees predicting CIN3/Cancer. When using ten 2/3  1/3 splits, rendering an effective test sample size of only n _{test}=n/3≈23, the median pvalues equal \(\tilde {p}_{\Delta \text {AUC}} = 0.351\) and \(\tilde {p}_{\Delta \text {Brier}} = 0.065\), hence close to significant at α=0.05 for the latter. In terms of Brier residuals, predictions improved for 14.5 out of 23 test samples, on average across splits. With 10fold crossvalidation we find a similar increase in AUC (cvAUC base RF: 0.661, CoRF: 0.702). In this case tuning rendered a value of γ=1.7, which excluded all but 105 variables. Tuning increased the crossvalidated performance to AUC = 0.737 (See Additional file 1).
Computational time
When using 5000 trees and without tuning of γ, the LNM example (n=133, P=12838) runs in 1:18 min (single threaded on a E52660 cpu with 128 gb memory). The cervical cancer example (P≈350.000,n=68) runs 22:24 min, of which the codata model takes 8 min. By comparison, fitting a GRridge (R package GRridge) with a 10fold crossvalidation to estimate the global λ takes 2:07 min for the LNM example and 35:12 min for the cervical cancer example. To estimate the predictive error by crossvalidation with GRridge the respective times need to be multiplied by the number of folds, rendering it much slower than the default CoRF (γ=1), which does not require crossvalidation.
Discussion and conclusion
The LNM and cervical cancer examples demonstrate that CoRF is able to improve the base RF by using codata. Of course, the amount of improvement relates directly to the relevance of the codata for the data at hand. The codata are relevant if, for example, some of the codata groups contain a relatively large number of variables related to the outcome, or if a continuous source of codata (e.g. external pvalues or correlations with other genomic measurement) correlates strongly with the importance of a variable. Figures 2 and 5 show the relevance of the codata for our examples. Including additional informative codata could further increase the performance of CoRF. Hence, expert knowledge on the domain and available external data is crucial. In addition, stability of the set of selected variables increased by the use of CoRF. We argue that the use of codata provides a stronger foundation for the classifier, which may enhance generalization to other measurement platforms, which is sometimes problematic in omics settings.
If the codata are nonrelevant, CoRF provides a safeguard against overfitting. Firstly, the codata weights are estimated from a parsimonious (2) or smooth (5) model to ensure that they are stable. To further stimulate parsimony, one may conduct the codata selection procedure described in the Additional file 1, which removes redundant sources of codata. Practically, the use of CoRF is a bit more demanding than the use of RF: one needs to think about what codata could be of use, and invest time in processing such codata. On the other hand, this may also be perceived as an advantage: the classification process requires more involvement of the problem owner, e.g. a clinician or molecular biologist, instead of being a ‘black box’.
CoRF essentially aims at reducing the haystack of genomics variables by using codata. Of course, one could also use adhoc filtering methods to preselect variables on the basis of existing information, but this introduces a level of subjectivity and suboptimality when the threshold(s) are not chosen correctly. CoRF formalizes the weighting and thresholding process and lets the data decide on the importance of a given source of codata. We expect CoRF to be most useful in (very) highdimensional settings. In such settings, variables likely differ strongly in predictive ability while the size of the haystack complicates the search. In such situations our codata approach can assist in identifying the relevant variables. For P<n settings, the prediction model is trained with a (relatively small) selected set of features. This means that i) learners not supported by codata (e.g. base RF) are fairly well able to discriminate the important variables from the nonimportant ones; and ii) the small P complicates good estimation of our empirical Bayestype (sampling) weights. Hence, in such a situation, CoRF (and codata supported methods in general) are less likely to boost predictive performance. CoRF is weakly adaptive in that it learns the sampling weights from both the primary and the codata, in contrast to other adaptive methods like the enriched RF [33] or the adaptive lasso [34], where weights are inferred only from the primary data. In highdimensional applications such strong adaptation is more likely to lead to overfitting, unlike the codata moderated adaptation.
CoRF inherits its computational efficiency from the RF. When the tuningfree version is used (γ=1), we empirically found that the oob performance suffices and crossvalidation is not required. This makes the methodology very suitable for applications with extremely large P. Tuning of γ may slightly improve the predictive performance, but at a substantial computational cost, given the required grid search for γ and the additional CV loop. The CoRF methodology may be combined with any bagging classifier that uses the random subspace method, such as a random glm [35] or a random lasso [36]. If variable selection is more stringent for a particular method (i.e. less noisy), then identification of the relationships of the codata model may be easier. On the other hand, if most of the variables are not used, then we are unable to obtain an reliable assessment of the quality of those variables which may complicate fitting the codata model. One possible extension of CoRF could be to use the depth at which variables are used by the RF, for example through the average or minimal depth [30]. Variables that are used higher up in a tree are, on average, more relevant, and it could be beneficial to assign larger weights to these variables in the codata model. Another way of accomplishing this is to replace v _{ ij } by a measure that counts how often each variable is used in classifying the oob samples, analogous to the intervention in prediction measure [37, 38]. This measure naturally upweights variables that are often high up in a tree. We intend to investigate these matters in the future.
Declarations
Acknowledgements
We thank Wina Verlaat and Renske Steenbergen for their contributions with regard to the cervical cancer methylation data set.
Funding
This study received financial support from the European Union 7th Framework program (OraMod project, Grant Agreement no. 611425) and H2020 program (project BD2Decide, Grant Agreement no. 689715). This work was also supported by the European Research Council (ERC advanced 2012AdG, proposal 322986; Molecular Self Screening for Cervical Cancer Prevention, MASSCARE).
Availability of data and materials
All data corresponding to the lymph node metastasis example is publicly available. The TCGA RNASeqv2 data is available from: http://gdac.broadinstitute.org/runs/stddata__latest/data/HNSC/. The pvalue codata is available from GEO, accession numbers: GSE30788/GSE85446, and so is the validation data, accession number: GSE84846. All LNM data and codata is also contained in processed form in the CoRF package. Methylation data from the cervical cancer example will be shared publicly after publication of [31].
Software
The R package CoRF is available freely from GitHub: https://github.com/DennisBeest/CoRF.
Authors’ contributions
DtB and MvdW developed the method, DtB developed the R code, applied it to the examples and drafted the manuscript. MvdW revised the manuscript. SM and RB provided input for the LNM example. SW provided input for the cervical cancer example. All authors read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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