A comprehensive comparison of random forests and support vector machines for microarraybased cancer classification
 Alexander Statnikov^{1},
 Lily Wang^{2} and
 Constantin F Aliferis^{1, 2, 3, 4}Email author
DOI: 10.1186/147121059319
© Statnikov et al; licensee BioMed Central Ltd. 2008
Received: 24 January 2008
Accepted: 22 July 2008
Published: 22 July 2008
Abstract
Background
Cancer diagnosis and clinical outcome prediction are among the most important emerging applications of gene expression microarray technology with several molecular signatures on their way toward clinical deployment. Use of the most accurate classification algorithms available for microarray gene expression data is a critical ingredient in order to develop the best possible molecular signatures for patient care. As suggested by a large body of literature to date, support vector machines can be considered "best of class" algorithms for classification of such data. Recent work, however, suggests that random forest classifiers may outperform support vector machines in this domain.
Results
In the present paper we identify methodological biases of prior work comparing random forests and support vector machines and conduct a new rigorous evaluation of the two algorithms that corrects these limitations. Our experiments use 22 diagnostic and prognostic datasets and show that support vector machines outperform random forests, often by a large margin. Our data also underlines the importance of sound research design in benchmarking and comparison of bioinformatics algorithms.
Conclusion
We found that both on average and in the majority of microarray datasets, random forests are outperformed by support vector machines both in the settings when no gene selection is performed and when several popular gene selection methods are used.
Background
Gene expression microarrays are becoming increasingly promising for clinical decision support in the form of diagnosis and prediction of clinical outcomes of cancer and other complex diseases. In order to maximize benefits of this technology, researchers are continuously seeking to develop and apply the most accurate classification algorithms for the creation of gene expression patient profiles. Prior research suggests that among wellestablished and popular techniques for multicategory classification of microarray gene expression data, support vector machines (SVMs) have a predominant role, significantly outperforming knearest neighbours, backpropagation neural networks, probabilistic neural networks, weighted voting methods, and decision trees [1].
In the last few years substantial interest has developed within the bioinformatics community in the random forest algorithm [2] for classification of microarray and other highdimensional molecular data [3–5]. The random forest algorithm possesses a number of appealing properties making it wellsuited for classification of microarray data: (i) it is applicable when there are more predictors than observations, (ii) it performs embedded gene selection and it is relatively insensitive to the large number of irrelevant genes, (iii) it incorporates interactions between predictors, (iv) it is based on the theory of ensemble learning that allows the algorithm to learn accurately both simple and complex classification functions, (v) it is applicable for both binary and multicategory classification tasks, and (vi) according to its inventors it does not require much finetuning of parameters and the default parameterization often leads to excellent performance [2]. Recent work [5] reported an empirical evaluation of random forests in the cancer microarray gene expression domain and concluded that random forest classifiers have predictive performance comparable to that of the best performing alternatives (including SVMs) for classification of microarray gene expression data. In fact, the data in Table 2 of [5] suggests that random forests on average across 10 datasets slightly outperform SVMs as well as other methods. If true, this finding could be of great significance to the field, because combined with prior results about SVM performance (e.g., [1]), this suggests that random forests offer classification accuracy advantages over "best of class" classifier algorithms for this type of data.
However, closer inspection of this prior comparison [5] reveals several important data analytic biases that may have affected its conclusions: First, while the random forests were applied to datasets prior to gene selection, SVMs were applied with a subset of only 200 genes (the number 200 was chosen arbitrarily). Given that the number of optimal genes varies from dataset to dataset, and that SVMs are known to be fairly insensitive to a very large number of irrelevant genes, such application of SVMs likely biases down their performance. Second, a oneversusone SVM algorithm was applied for the multicategory classification tasks, while it is has been shown that in microarray gene expression domain this method is inferior to other multicategory SVM methods, such as oneversusrest [1, 6]. Third, the evaluation of [5] was limited only to linear SVMs without optimizing any algorithm parameters such as the penalty parameter C that balances data fit with insensitivity to outliers. Fourth, the performance metric used in [5], proportion of correct classifications, is sensitive to unbalanced distribution of classes and has lower power to discriminate among classification algorithms compared to existing alternatives such as area under the ROC curve and relative classifier information [7–10]. Fifth, no statistical comparison among classifiers has been performed. Finally, the prior comparison uses a .632+ bootstrap error estimator [11] which is not the most appropriate error estimator for microarray data where powerful classifiers such as SVMs and RFs typically achieve 0 training error and the .632+ bootstrap becomes equivalent to repeated holdout estimation that may suffer from the trainingsetsize bias as discussed in [12]. Furthermore, .632+ bootstrap is currently not developed for performance metrics other than proportion of correct classifications.
We hypothesize that these apparent methodological biases of prior work have compromised its conclusions and the question of whether random forests indeed outperform SVMs for classification of microarray gene expression data is not convincingly answered. In the present work we undertake a more methodologically rigorous comparison of the two algorithms to determine the relative errors when applied to a wide variety of datasets. We examine the algorithms both in the settings when no gene selection is performed and when several popular gene selection methods are used. To make our evaluation more relevant to practitioners, we focus not only on diagnostic datasets that are in general known to have strong predictive signals, but also include several outcome prediction datasets where the signals are weaker and larger gene sets are often required for optimal prediction.
Results
Using full set of genes
Comparison of classification performance of SVMs and RFs without gene selection.
Task & dataset  Classification performance metric  Classification performance  Nominally superior method  Pvalue  

SVM  RF  
DxAlon  AUC  0.867  0.867    1 
DxRamaswamy2  AUC  0.821  0.767  SVM  0.409 
DxShipp  AUC  0.992  0.973  SVM  0.500 
DxSingh  AUC  0.964  0.944  SVM  0.377 
PxBeer  AUC  0.798  0.646  SVM  0.032 
PxBhattacharjee  AUC  0.519  0.561  RF  0.546 
PxIizuka  AUC  0.663  0.763  RF  0.061 
PxPomeroy  AUC  0.692  0.600  SVM  0.235 
PxRosenwald  AUC  0.689  0.629  SVM  0.140 
PxVeer  AUC  0.747  0.754  RF  0.867 
PxYeoh  AUC  0.777  0.660  SVM  0.006 
DxAlizadeh  RCI  1.000  1.000    1 
DxArmstrong  RCI  0.944  0.894  SVM  0.658 
DxBhattacharjee  RCI  0.895  0.763  SVM  0.015 
DxGolub  RCI  0.939  0.934  SVM  1 
DxKhan  RCI  1.000  1.000    1 
DxNutt  RCI  0.775  0.733  SVM  0.498 
DxPomeroy  RCI  0.823  0.611  SVM  0.031 
DxRamaswamy  RCI  0.905  0.861  SVM  0.010 
DxStaunton  RCI  0.770  0.819  RF  0.249 
DxSu  RCI  0.958  0.910  SVM  0.004 
PxVeer2  RCI  0.451  0.304  SVM  0.004 
Using gene selection
Comparison of classification performance of SVMs and RFs with gene selection.
Task & dataset  Classification performance metric  Classification performance  Nominally superior method  Pvalue  

SVM  RF  
DxAlon  AUC  0.938  0.917  SVM  0.626 
DxRamaswamy2  AUC  0.821  0.781  SVM  0.624 
DxShipp  AUC  0.992  0.975  SVM  0.502 
DxSingh  AUC  0.964  0.972  RF  0.812 
PxBeer  AUC  0.798  0.648  SVM  0.016 
PxBhattacharjee  AUC  0.519  0.561  RF  0.550 
PxIizuka  AUC  0.713  0.763  RF  0.750 
PxPomeroy  AUC  0.692  0.629  SVM  0.506 
PxRosenwald  AUC  0.689  0.631  SVM  0.128 
PxVeer  AUC  0.758  0.754  SVM  0.954 
PxYeoh  AUC  0.777  0.716  SVM  0.082 
DxAlizadeh  RCI  1.000  1.000    1 
DxArmstrong  RCI  0.944  0.911  SVM  0.624 
DxBhattacharjee  RCI  0.895  0.817  SVM  0.125 
DxGolub  RCI  0.953  0.934  SVM  1 
DxKhan  RCI  1.000  1.000    1 
DxNutt  RCI  0.812  0.733  SVM  0.220 
DxPomeroy  RCI  0.823  0.688  SVM  0.079 
DxRamaswamy  RCI  0.911  0.880  SVM  0.066 
DxStaunton  RCI  0.876  0.856  SVM  0.626 
DxSu  RCI  0.958  0.922  SVM  0.078 
PxVeer2  RCI  0.451  0.371  SVM  0.262 
According to the results in Figure 2 and Table 2, in 17 datasets SVMs nominally outperform RFs, in 3 datasets RFs nominally outperform SVMs, and in 2 datasets algorithms perform the same. Furthermore, SVMs outperform RFs statistically significantly (at the 0.05 α level) in 1 dataset. There is no dataset where RFs outperform SVMs with statistically significant difference. The permutation test applied to all 22 datasets shows that SVMs statistically significantly outperform RFs on average over all datasets at the 0.05 α level (pvalue of the test = 0.001). A comparison of the average performance across datasets also suggests superiority of SVMs: the average performance of SVMs is 0.787 AUC and 0.875 RCI in binary and multicategory classification tasks, respectively; while the average performance of RFs in the same tasks is 0.759 AUC and 0.828 RCI.
Number of genes selected for each microarray dataset and gene selection method.
Task & dataset  No gene selection  RFE  RFVS1  RFVS2  KW  S2N 

DxAlizadeh  4026  12  62  73  19  15 
DxAlon  2000  105  16  3  15  13 
DxArmstrong  11225  74  709  57  106  48 
DxBhattacharjee  12600  289  27  15  1864  653 
DxGolub  5327  12  456  336  42  4 
DxKhan  2308  28  17  18  15  11 
DxNutt  10367  1598  126  101  476  926 
DxPomeroy  5920  186  34  16  70  435 
DxRamaswamy  15009  3346  966  411  8248  10277 
DxRamaswamy2  13247  1576  12  4  4129  1364 
DxShipp  5469  8  15  6  13  89 
DxSingh  10509  157  58  21  22  38 
DxStaunton  5726  169  152  73  93  97 
DxSu  12533  2429  845  320  1318  1927 
PxBeer  7129  201  15  7  953  1380 
PxBhattacharjee  12600  21  46  7  138  61 
PxIizuka  7070  103  38  7  168  185 
PxPomeroy  7129  70  29  13  445  439 
PxRosenwald  7399  2338  124  27  3201  3897 
PxVeer  24188  1056  124  20  5388  4405 
PxVeer2  24188  491  149  39  1194  1764 
PxYeoh  12240  1187  21  6  3077  1869 
Discussion
The results presented in this paper illustrate that SVMs offer classification performance advantages compared to RFs in diagnostic and prognostic classification tasks based on microarray gene expression data. We emphasize that when it comes to clinical applications of such models, because the size of the patient populations is typically very large, even very modest differences in performance (e.g., at the order of 0.01 AUC/RCI or even less) can result in very substantial differences in total clinical outcomes (e.g., number of lifeyears saved) [13].
The reasons for superior classification performance of one universal approximator classifier over the other in a domain where the generative functions are unknown are not trivial to decipher [2, 14]. We provide here as a starting point two plausible explanations supported by theory and a simulation experiment (in Additional File 2). We note that prior research has established that linear decision functions capture very well the underlying distributions in microarray classification tasks [15, 16]. In the following two paragraphs we first demonstrate that for such functions SVMs may be less sensitive to the choice of input parameters than RFs and then explain why SVMs model linear decision functions more naturally than RFs.
The simulation experiment described in Additional File 2 demonstrates high degree of sensitivity of RFs to the values of input parameters mtry (i.e., number of genes randomly selected at each node) and ntree (i.e., number of trees) even in the case of linear decision function when complicated decision surface modelling is not required. The experiment shows that the choice of RF parameters creates large variation in the classifier performance whereas the choice of the main SVM parameter has only minor effects on the error. In practical analysis of microarrays this means that finding the RFs with optimal error for the dataset may involve extensive model selection which in turn opens up the possibility for overfitting given the small sample sizes in validation datasets.
A second plausible explanation is that decision trees used as base learners in the RF algorithm cannot learn exactly many linear decision functions in the finite case. Specifically, if the generative linear decision function is not orthogonal to the coordinate axes, then a decision tree of infinite size is required to represent this function without error [17]. The voted decision function in RFs approximates linear functions based on rectangular partitioning of the input space, and this "staircase" approximation can capture a linear function exactly when the number of decision trees can grow without bound (assuming that each tree is of finite size). SVMs on the other hand use linear classifiers and thus can model such functions naturally, using a small number of free parameters (i.e., bounded by the available sample size).
We note that regardless of the specific reasons why RFs may have larger error on average in this domain, it is still important to be aware of the empirical performance differences when considering which classifier to use for building molecular signatures. It may take several years before the precise reasons of differences in empirical error are thoroughly understood, and in the meantime the empirical advantages and disadvantages of methods should be noted first by practitioners.
Data analysts should also be aware of a limitation of RFs imposed by its embedded random gene selection. In order for a RF classification model to overcome the trap of large variance, one has to use a large number of trees and build trees based on a large number of genes. The exact values of these parameters depend on both the complexity of the classification function and the number of genes in a microarray dataset. Therefore, in general, it is advisable to optimize these parameters by nested crossvalidation that accounts for the variability of the random forest model (e.g., the selected parameter configuration is the one that performs best on average over multiple validation sample sets).
Finally, it is worthwhile to mention the work by Segal [18] who questioned Breiman's empirical demonstration of the claim that random forests do not overfit as the number of trees grows [2]. In short, Segal showed that there exist some data distributions where maximal unpruned trees used in the random forests do not achieve as good performance as the trees with smaller number of splits and/or smaller node size. Thus, application of random forests in general requires careful tuning of the relevant classifier parameters. These observations may suggest future improvements of RFrelated analysis protocols.
Conclusion
The primary contribution of the present work is that we conducted the most comprehensive comparative benchmarking of random forests and support vector machines to date, using 22 diagnostic and outcome prediction datasets. Our hypothesis that in previously reported work, research design limitations may have biased the comparison of classifiers in favour of random forests, was verified. After removing these benchmarking limitations, we found that, both on average and in the majority of microarray datasets, random forests exhibit larger classification error than support vector machines both in the settings when no gene selection is performed and when several gene selection methods are used.
The quest for high performance classifiers with microarray gene expression and other "omics" data is ongoing. Random forests have appealing theoretical and practical characteristics, however our experiments show that currently they do not exhibit "best of class" performance. Our data also points to methodological limitations of prior evaluations and thus emphasizes the importance of careful design of bioinformatics algorithm evaluation studies.
Methods
Microarray datasets and classification tasks
Gene expression microarray datasets used in this study.
Task & dataset  Number of classes  Number of genes  Number of samples  Prediction task 

DxAlizadeh  3  4026  62  Diffuse large Bcell lymphoma, follicular lymphoma, chronic lymphocytic leukemia 
DxAlon  2  2000  62  Colon tumors and normal tissues 
DxArmstrong  3  11225  72  AML, ALL and mixedlineage leukemia (MLL) 
DxBhattacharjee  5  12600  203  4 lung cancer types and normal tissues 
DxGolub  3  5327  72  Acute myelogenous leukemia (AML), acute lymphoblastic leukemia (ALL) Bcell and ALL Tcell 
DxKhan  4  2308  83  Small, round blue cell tumors of childhood 
DxNutt  4  10367  50  4 malignant glioma types 
DxPomeroy  5  5920  90  5 human brain tumor types 
DxRamaswamy  26  15009  308  14 various human tumor types and 12 normal tissue types 
DxRamaswamy2  2  13247  76  Metastatic and primary tumors 
DxShipp  2  5469  77  Diffuse large Bcell lymphomas and follicular lymphomas 
DxSingh  2  10509  102  Prostate tumor and normal tissues 
DxStaunton  9  5726  60  9 various human tumor types 
DxSu  11  12533  174  11 various human tumor types 
PxBeer  2  7129  86  Lung adenocarcinoma survival 
PxBhattacharjee  2  12600  62  Lung adenocarcinoma 4year survival 
PxIizuka  2  7070  60  Hepatocellular carcinoma 1year recurrencefree survival 
PxPomeroy  2  7129  60  Medulloblastoma survival 
PxRosenwald  2  7399  240  NonHodgkin lymphoma survival 
PxVeer  2  24188  97  Breast cancer 5year metastasisfree survival 
PxVeer2  3  24188  115  Breast cancer 5year metastasisfree survival, metastasis within 5 years, germline BRCA1 mutation 
PxYeoh  2  12240  233  Acute lymphocytic leukemia relapsefree survival 
Crossvalidation design
We used 10fold crossvalidation to estimate the performance of the classification algorithms. In order to optimize algorithm parameters, we used another "nested" loop of crossvalidation by further splitting each of the 10 original training sets into smaller training sets and validation sets. For each combination of the classifier parameters, we obtained crossvalidation performance and selected the best performing parameters inside this inner loop of crossvalidation. Next, we built a classification model with the best parameters on the original training set and applied this model to the original testing set. Details about the "nested crossvalidation" procedure can be found in [19, 20]. Notice that the final performance estimate obtained by this procedure will be unbiased because each original testing set is used only once to estimate performance of a single classification model that was built by using training data exclusively.
Support vector machine classifiers
Several theoretical reasons explain the superior empirical performance of SVMs in microarray data: e.g., they are robust to the high variabletosample ratio and large number of variables, they can learn efficiently complex classification functions, and they employ powerful regularization principles to avoid overfitting [1, 21, 22]. Extensive applications literature in text categorization, image recognition and other fields also shows the excellent empirical performance of this classifier in many more domains. The underlying idea of SVM classifiers is to calculate a maximal margin hyperplane separating two classes of the data. To learn nonlinearly separable functions, the data are implicitly mapped to a higher dimensional space by means of a kernel function, where a separating hyperplane is found. New samples are classified according to the side of the hyperplane they belong to [22]. Many extensions of the basic SVM algorithm can handle multicategory data. The "oneversusrest" SVM works better for multiclass microarray data [1, 6], so we adopted this method for the analysis of multicategory datasets in the present study. In summary, this approach involves building a separate SVM model to classify each class against the rest, and then predicting the class of a new sample using the SVM model with the strongest vote.
We used SVM implementation in the libSVM software library [23]http://www.csie.ntu.edu.tw/~cjlin/libsvm with polynomial kernel. Recall that the SVM polynomial kernel can be defined as: K(x, y) = (γ·x^{T}y + r)^{ d }, where x and y are samples with gene expression values and γ, r, d are kernel parameters. The parameters γ and r were set to default value 1. The kernel degree d together with the SVM penalty parameter C were optimized by nested crossvalidation over d values {1, 2, 3} and C values {0.01, 1, 100}.
Random forest classifiers
Random forests (RF) is a classification algorithm that uses an ensemble of unpruned decision trees, each of which is built on a bootstrap sample of the training data using a randomly selected subset of variables [2]. As mentioned in the Background section, this algorithm possesses a number of properties making it an attractive technique for classification of microarray gene expression data.
We employed the stateoftheart implementation of RF available in the R package randomForest [24]. This implementation is based on the original Fortran code authored by Leo Breiman, the inventor of RFs. Following the suggestions of [24, 25] and http://www.stat.berkeley.edu/~breiman/RandomForests/, we considered different parameter configurations for the values of ntree = {500, 1000, 2000} (number of trees to build), mtryFactor ={0.5, 1, 2} (a multiplicative factor of the default value of mtry parameter denoting the number of genes randomly selected at each node; by default mtry = $\sqrt{number\cdot of\cdot genes}$), and nodesize = 1 (minimal size of the terminal nodes of the trees in a random forest) and selected the bestperforming configuration by nested crossvalidation. Note that the above parameter values are also consistent with the recommendations of the study [5].
Gene selection methods
Even though both SVM and RF classifiers are fairly insensitive to very large number of irrelevant genes, we applied the following widely used gene selection methods in order to further improve classification performance:

Random forestbased backward elimination procedure RFVS [5]: The RFVS procedure involves iteratively fitting RFs (on the training data), and at each iteration building a random forest after discarding genes with the smallest importance values. The returned subset of genes is the one with the smallest outofbag error. We used the varSelRF implementation of the RFVS method developed by its inventors and applied it with the recommended parameters: ntree = 2000, mtryFactor = 1, nodesize = 1, fraction.dropped = 0.2 (a parameter denoting fraction of genes with small importance values to be discarded during backward elimination procedure), and c.sd = 0 (a factor that multiplies the standard deviation of error for stopping iterations and choosing the best performing subset of genes). We refer to this method as "RFVS1."

RFVS procedure as described above, except for c.sd = 1 (denoted as "RFVS2"): This method differs from RFVS1 in that it performs statistical comparison to return the smallest subset of genes with performance statistically indistinguishable from the nominally best one.

SVMbased recursive feature elimination method RFE [26]: This is a stateoftheart procedure for gene selection from microarray data that involves iteratively fitting SVM classification models (on the training data) by discarding the genes with the small impact on classification and selecting the smallest subset of genes that participate in the best performing classification model (as assessed in the validation data). Even though RFE was originally introduced as a method for binary classification problems, it can be trivially extended to multiclass case by using binary SVM models in "oneversusrest" fashion (e.g., see [27]). Finally, to be comparable with the RFVS method, we used the fraction of genes that are discarded in the iterative SVM models equal to 0.2.

Backward elimination procedure based on univariate ranking of genes with "signaltonoise" ratio [1, 21, 28] (denoted as "S2N"): This procedure first ranks all genes according their signaltonoise value with the response variable, and then performs backward elimination using SVM classifier (fit on the training set and evaluated on the validation set) to determine the best performing smallest subset of genes. Similarly to RFE and RFVS, we perform backward elimination by discarding 0.2 proportion of genes at each iteration.

Backward elimination procedure based on univariate ranking of genes with KruskalWallis oneway nonparametric ANOVA [1] (denoted as "KW"): This procedure is applied similarly to the S2N method except for it uses different univariate ranking of genes.
We emphasize that all gene selection methods were applied during crossvalidation utilizing only the training data and splitting it into a smaller training and validation set if necessary.
Classification performance evaluation metrics
We used two classification performance metrics. For binary tasks, we used the area under the ROC curve (AUC) which was computed from continuous outputs of the classifiers (distances from separating hyperplane for SVMs and outcome probabilities for RFs) [8]. For multicategory tasks, where classical AUC is inapplicable, we employed the relative classifier information (RCI) [7]. RCI is an entropybased measure that quantifies how much the uncertainty of a decision problem is reduced by a classifier relative to classifying using only the prior probabilities of each class. We note that both AUC and RCI are more discriminative than the accuracy metric (also known as proportion of correct classifications) and are not sensitive to unbalanced distributions [7–10]. Both AUC and RCI take values on [0, 1], where 0 denotes worst possible classification and 1 denotes perfect classification.
Statistical comparison among classifiers
When comparing two classifiers, it is important to assess whether the observed difference in classification performance is statistically significant or simply due to chance. We assessed significance of differences in classification performance in individual datasets or in all datasets on average using a nonparametric permutation test [29] based on the theory of [30]. The null hypothesis of this test is no difference between performance of SVM and RF classifiers. The test was applied with 100,000 permutations and twosided pvalues were computed as described in [29]. We used a significance level α = 0.05 for this test.
Declarations
Acknowledgements
The work was in part supported by grant 2R56LM00794804A1.
Authors’ Affiliations
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