Volume 12 Supplement 10
Selecting a single model or combining multiple models for microarray-based classifier development? – A comparative analysis based on large and diverse datasets generated from the MAQC-II project
© Chen et al; licensee BioMed Central Ltd. 2011
Published: 18 October 2011
Genomic biomarkers play an increasing role in both preclinical and clinical application. Development of genomic biomarkers with microarrays is an area of intensive investigation. However, despite sustained and continuing effort, developing microarray-based predictive models (i.e., genomics biomarkers) capable of reliable prediction for an observed or measured outcome (i.e., endpoint) of unknown samples in preclinical and clinical practice remains a considerable challenge. No straightforward guidelines exist for selecting a single model that will perform best when presented with unknown samples. In the second phase of the MicroArray Quality Control (MAQC-II) project, 36 analysis teams produced a large number of models for 13 preclinical and clinical endpoints. Before external validation was performed, each team nominated one model per endpoint (referred to here as 'nominated models') from which MAQC-II experts selected 13 'candidate models' to represent the best model for each endpoint. Both the nominated and candidate models from MAQC-II provide benchmarks to assess other methodologies for developing microarray-based predictive models.
We developed a simple ensemble method by taking a number of the top performing models from cross-validation and developing an ensemble model for each of the MAQC-II endpoints. We compared the ensemble models with both nominated and candidate models from MAQC-II using blinded external validation.
For 10 of the 13 MAQC-II endpoints originally analyzed by the MAQC-II data analysis team from the National Center for Toxicological Research (NCTR), the ensemble models achieved equal or better predictive performance than the NCTR nominated models. Additionally, the ensemble models had performance comparable to the MAQC-II candidate models. Most ensemble models also had better performance than the nominated models generated by five other MAQC-II data analysis teams that analyzed all 13 endpoints.
Our findings suggest that an ensemble method can often attain a higher average predictive performance in an external validation set than a corresponding “optimized” model method. Using an ensemble method to determine a final model is a potentially important supplement to the good modeling practices recommended by the MAQC-II project for developing microarray-based genomic biomarkers.
Gene expression microarrays have been applied in various fields [1–6]. Despite widespread usage, the translation of basic findings to clinical utility such as diagnosis and prognosis has been slow. This is largely due to the fact that some clinical endpoints are difficult to predict with microarrays, such as prediction of drug-induced liver injury , and survival endpoints for many cancers . In addition, issues such as small sample size, low signal-to-noise ratio and lack of a fully annotated transcriptome contribute to the lack of success in developing biomarkers (i.e., predictive models or classifiers) with microarrays [9, 10].
The conventional procedure of developing a microarray-based biomarker involves a selection process to identify one classifier out of many others generated in this process for application to an external dataset. The selection is largely dependent on the accuracy estimation . Specifically, the “optimized” model is selected using the training set with, for example, cross-validation to estimate its predictive performance. Some authors argue that cross-validation can provide an unbiased estimate of performance when properly applied [12, 13] while others point out that the variability in the error estimation can be very high when cross-validation is applied to datasets with small sample sizes . Thus, there exists a great uncertainty that an accuracy-based model selection procedure will choose the best microarray-based classifier [12, 15].
Selecting a single optimized model is the most common approach to developing microarray-based predictive models [6, 16–18]. However, it is being challenged given the fact that many models with similar statistical performance are often identified for a studied endpoint. By reanalyzing the breast cancer prognosis dataset reported by van’t Veer et al., Ein-Dor et al. noticed that many gene sets gave nearly equal prediction accuracy . The question is whether the combination of these well performing models could be preferable to an accuracy-based selection of a single optimized model from among many.
Ensemble methods have been demonstrated its usage in some fields such as machine learning  and Quantitative Structure Activity Relationships (QSAR) . These investigations are carried out under the hypothesis that the methods likely capture a greater diversity of potentially informative features  that might improve the model robustness when included. Ensemble methods have similarly been explored in gene expression studies [22, 23]. It was found that enhanced prediction accuracy for ensemble methods compared to the single model selection method, especially for complex and/or heterogeneous endpoints. However, the comparative analysis was carried out on limited datasets sometimes having small sample sizes. A rigorous comparison where findings can be generalized is best achieved with a systematic comparative analysis using multiple datasets containing endpoints with different characteristics. The second phase of MicroArray Quality Control (MAQC-II) project, led by the U.S. Food and Drug Administration with broad participation from the large research community, offers the benchmark data to allow such a rigorous comparison.
One goal of the MAQC-II project was to develop baseline practices for the application of microarray technology to biomarker development . This process took nearly four years to enable a full investigation of the impact of modeling procedure choices on the quality of predictive models. The project provides the requisite datasets as well as a large number of validated models developed using diverse methods for comparison. Specifically, the 36 analysis teams generated more than 30,000 models across 13 endpoints from six datasets. Importantly, similar prediction performance was attained despite the use of different modeling algorithms and gene sets. However, the MAQC-II required each team to first nominate and then validate in blinded manner a single model (or nominated model) for each endpoint. A group of experts then selected 13 final models (one per endpoint) that were designated candidate models. The performance of these selected ‘optimized’ models (both nominate and candidate models) was assessed on blinded, independent validation sets. The comprehensive and disciplined process employed by this approach in selecting optimized models resulted in a set of nominate and candidate models constituting sound benchmarks for comparison of ensemble methods.
In this study we applied a simple ensemble approach of combining the top 50% of all the models from the selected MAQC-II team and compared them with the nominated and candidate models for each endpoint (More details can be found in Results.). In other words, we took the simplest way to generate ensemble models and then compare them with the optimized models generated from the most sophisticated and comprehensive approaches implemented in MAQC-II. Our study indicates that even such simple ensemble methods can achieve comparable if not better predictive performance in external validation sets than the corresponding single “optimized” model from MAQC-II.
The datasets used in MAQC-II project.
Lung tumorigen vs. non-tumorigen
Non-genotoxic hepatocarcinogen vs. non-carcinogen
Liver toxicants vs. non-toxicants
Pathologic complete response, pCR
Estrogen receptor status (ER +/-)
Male vs. female (positive control)
Random 2-class label (negative control)
Male vs. female (positive control)
Random 2-class label (negative control)
Data analysis protocol of NCTR models
Selection of NCTR nominated models
A complete 5-fold CV procedure was employed to determine the number of features and modeling parameters used to develop the final classifier. The complete CV embeds the entire modeling process including, batch correction, resizing training set and feature selection in each of the cross-validation steps. The average performance of the classifiers from the 50 CV runs was calculated and the parameters that resulted in the best classifier were used for developing the final classifier using the entire training set. As recommended by the MAQC-II consortium , MCC (Mathhews Correlation Coefficient) was the selected metric for assessing model performance. An MCC-guided method was used to identify the models to be submitted for each endpoint, which consisted of a hierarchical decision tree with a knowledge justification at each level of the decision. Specifically, the following step was used:
Step 1 – Decision based on the MCC value: The MCC value was adjusted to one decimal precision and models with the same MCC value were grouped. For example, models with MCC values of 0.89 and 0.91 were considered as performing equally and placed into the MCC > 0.9 group. The models in the group with the highest MCC value were passed to the next step.
Step 2 – Decision based on the number of features: Within a group of models with the same MCC value, more parsimonious models were given higher priority. However, if two models contained nearly the same number of features, then accuracy, sensitivity and specificity were used to choose the best performing model.
Step 3 – Decision based on the feature selection method: For equally well-performing models, those that used SAM for feature selection were chosen over those that used fold change plus p-value.
Step 4 – Decision based on the classification method: For equally well-performing models those created using KNN were selected over those created using a Naive Bayes classifier.
An ensemble model was developed for each endpoint for comparison to those submitted for MAQC-II evaluation. An ensemble model was derived by taking the 50% of the models from cross-validation with the highest MCC and using a voting process to make a final prediction about a sample. To begin, the average percentage of positive predictions for the 50% of models in the training set is recorded. For each sample in the validation set, the percentage of models producing a positive prediction is calculated. This percentage is then divided by the average percentage of positive predictions in the training set recorded earlier. If the ratio of these numbers is one or greater, the ensemble model will produce a positive prediction. Otherwise the ensemble model will give a negative prediction. External validation was done while blinded to the class of the external test sets as implemented in MAQC-II.
As one of the 36 analysis teams involved in the MAQC-II project, we generated 8320 models (7280 KNN models and 1040 Naïve Bayes models). As shown in Additional file 1, the correlation coefficient (r=0.927) of our submitted models in the external validation was higher than that from all the MAQC-II models (r=0.840) , indicating that the performance of our models was above the average among the 36 analysis teams. We also selected one model per endpoint (called the NCTR nominated models) from all the NCTR models using the accuracy-based selection method (refer to the Methods sections for more details). Meanwhile, each analysis team that participated in the project also nominated one model for each endpoint they analyzed according to the MAQC-II guidance. An MAQC-II expert committee then selected 13 candidate models, representing the best model for each endpoint from all the submitted models from the 36 teams, before external validation was performed. In this study, the ensemble models comprising of 50% of all the NCTR models with highest performance in cross-validation are against both the NCTR nominated models as well as the MAQC-II candidate models across all the 13 endpoints. To validate the findings from the NCTR-centric practice, the same analysis was carried out on the models generated by other analysis teams. Comparative assessment was based on the blinded external validation performance.
The NCTR ensemble models vs. the NCTR nominated models
The NCTR ensemble models vs. the MAQC-II candidate models
The comparative analysis of models from different MAQC-II analysis teams
Microarray-based models to predict preclinical and clinical endpoints have become routine in research. Most studies focus on the selection of a single “optimized” model, but it is never clear whether that model will provide acceptable performance on an external validation set. Given the fact that many models from the same training set could achieve similar predictive performance, we investigated a simple ensemble approach of combining the top 50% best performing models and compared it with the single model selection approach. We conducted the investigation using the MAQC-II results because the MAQC-II project (1) covers a diverse set of endpoints including both “disguised” positive and negative controls, offering an opportunity to examine the issue in a systematic fashion; (2) generated the results from the blinded validation sets with large sample sizes, an important criterion to ensure the validity of the investigation; (3) provides the nominated models from each of 36 analysis teams, which represents a broad range of model selection methods; and (4) yielded the MAQC-II candidate models, representing the “best practice” of developing classifiers using the model selection method.
Using the MAQC-II results from the NCTR team and validated by the results from other five MAQC-II data analysis teams, two important observations were made. First, within each team, the ensemble method consistently generated models performing better in the external datasets than the model selection methods implemented by different teams. Second, the ensemble method performed comparably to the MAQC-II candidate models that were chosen with considerable efforts. The results demonstrate that identification of a single best model solely based on the statistical performance is difficult as exemplified in the MAQC-II nominated models where knowledge and experience behind the model selection is crucial as practiced in the determination of the MAQC-II candidate models. The proposed ensemble approach is easy, objective and reproducible, and thus can be an alternative method to generate a robust model based on the training set.
Accuracy estimation of a classifier using only a training set is still a difficult issue due to over-fitting, which is one of the major limitations associated with predictive models. Models often have excellent performance in the training dataset but nonetheless poorly predict in external validation datasets, even when best modeling practices are employed. The inconsistent predictive performance between the training set and testing set stems from the influence of idiosyncratic associations between features and endpoints in the training set. Cross-validation is a common method to account for these idiosyncrasies and to estimate accurately the prediction error of the models. Simon et al. proposed that the cross-validation, if used properly , provides a nearly unbiased estimate of the true error of classification procedure, while incomplete cross-validation will result in a seriously biased underestimate of the error rate. From our experience in the MAQC-II consortium, we found that the accuracy based selection process, even using the complete cross-validation procedure, still lead to models that are apparently over-fit and perform poorly on the external datasets (Additional file 2). In other words, a degree of over-fitting still exists even after properly applying "complete" cross-validation. This demonstrates that reducing the risk of over-fitting is still an issue in the selection method that must be addressed in order to improve the performance of microarray-based predictive models.
In this study, during cross-validation it was observed that many models could attain similar performance, while the models that produced the best MCCs in the training sets did not necessarily provide the best MCCs in the external validation sets. Based on these observations, it is reasonable to assume that an ensemble modeling method could substantially mitigate the risk of over-fitting presented in the “optimized” model selection process, although the ensemble models could not always generate the best predictive model.
Ensemble models have been well studied in the machine learning area where they have been shown useful for improving prediction performance . Random forest is a representative algorithm that consists of many decision trees that vote to select class membership. Some authors also reported that ensemble methods have worked well in QSAR models [21, 36] and microarray-based studies [22, 23] with a small number of datasets, but a literature search did not produce any comprehensive evaluations of the utility of ensemble methods in microarray-based classifier development. The MAQC-II study participants did not determine a preferred approach to select a best model for each endpoint, leaving that selection as part of an individual team’s preference. The data reported here did support this conclusion; in 8 of the 11 non-random endpoints (i.e., excluding endpoints I and M) the ensemble models were ranked in the top 25% of MCC values from all of the developed models.
It should also be noted that the choice to use the top 50% models based on cross-validation for the ensemble models was arbitrary. The data from further experiments shown in Additional file 3 have suggested that the choice of the number of models to be combined does not greatly affect performance of an ensemble model as long as a sufficient number of models (e.g., > 10% models) are retained in the process. The combination of too many models will actually decease slightly the performance, likely because of the noise introduced by the models with relatively poor performance. In contrary, using too few models does not have too much value due to the lack of representative models in ensemble. Therefore, we suggest that a modest number of models should be retained for ensemble calculation.
Many factors affect the performance of the microarray-based classifiers. The MAQC-II consortium comprehensively evaluated most of these factors through a community-wide practice, and established good modeling practice guidelines . This study provides a follow-up and extension of the MAQC-II team efforts. We found that an ensemble modeling procedure can reduce the risk of over-fitting and provides stable and robust predictive power than those single “optimized” models. These findings provide a necessary supplement to the good modeling practices for developing microarray-based predictive classifiers developed in the MAQC-II process.
The views presented in this article do not necessarily reflect those of the US Food and Drug Administration.
List of abbreviations used
MicroArray Quality Control
National Center for Toxicological Research
Quantitative Structure Activity Relationship
Significance Analysis of Microarrays
Matthews Correlation Coefficient.
The MAQC-II project was funded in part by the FDA’s Office of Critical Path Program.
This article has been published as part of BMC Bioinformatics Volume 12 Supplement 10, 2011: Proceedings of the Eighth Annual MCBIOS Conference. Computational Biology and Bioinformatics for a New Decade. The full contents of the supplement are available online at http://www.biomedcentral.com/1471-2105/12?issue=S10.
- Waring JF, Ciurlionis R, Jolly RA, Heindel M, Ulrich RG: Microarray analysis of hepatotoxins in vitro reveals a correlation between gene expression profiles and mechanisms of toxicity. Toxicol Lett 2001, 120(1–3):359–368. 10.1016/S0378-4274(01)00267-3View ArticlePubMedGoogle Scholar
- Fielden MR, Zacharewski TR: Challenges and limitations of gene expression profiling in mechanistic and predictive toxicology. Toxicol Sci 2001, 60(1):6–10. 10.1093/toxsci/60.1.6View ArticlePubMedGoogle Scholar
- Golub TR, Slonim DK, Tamayo P, Huard C, Gaasenbeek M, Mesirov JP, Coller H, Loh ML, Downing JR, Caligiuri MA, et al.: Molecular classification of cancer: class discovery and class prediction by gene expression monitoring. Science 1999, 286(5439):531–537. 10.1126/science.286.5439.531View ArticlePubMedGoogle Scholar
- Moniaux N, Chakraborty S, Yalniz M, Gonzalez J, Shostrom VK, Standop J, Lele SM, Ouellette M, Pour PM, Sasson AR, et al.: Early diagnosis of pancreatic cancer: neutrophil gelatinase-associated lipocalin as a marker of pancreatic intraepithelial neoplasia. Br J Cancer 2008, 98(9):1540–1547. 10.1038/sj.bjc.6604329PubMed CentralView ArticlePubMedGoogle Scholar
- Huang F, Reeves K, Han X, Fairchild C, Platero S, Wong TW, Lee F, Shaw P, Clark E: Identification of candidate molecular markers predicting sensitivity in solid tumors to dasatinib: rationale for patient selection. Cancer Res 2007, 67(5):2226–2238. 10.1158/0008-5472.CAN-06-3633View ArticlePubMedGoogle Scholar
- van de Vijver MJ, He YD, van't Veer LJ, Dai H, Hart AA, Voskuil DW, Schreiber GJ, Peterse JL, Roberts C, Marton MJ, et al.: A gene-expression signature as a predictor of survival in breast cancer. N Engl J Med 2002, 347(25):1999–2009. 10.1056/NEJMoa021967View ArticlePubMedGoogle Scholar
- Kaplowitz N: Idiosyncratic drug hepatotoxicity. Nat Rev Drug Discov 2005, 4(6):489–499. 10.1038/nrd1750View ArticlePubMedGoogle Scholar
- van't Veer LJ, Dai H, van de Vijver MJ, He YD, Hart AA, Mao M, Peterse HL, van der Kooy K, Marton MJ, Witteveen AT, et al.: Gene expression profiling predicts clinical outcome of breast cancer. Nature 2002, 415(6871):530–536. 10.1038/415530aView ArticleGoogle Scholar
- Dupuy A, Simon RM: Critical review of published microarray studies for cancer outcome and guidelines on statistical analysis and reporting. J Natl Cancer Inst 2007, 99(2):147–157. 10.1093/jnci/djk018View ArticlePubMedGoogle Scholar
- Michiels S, Koscielny S, Hill C: Prediction of cancer outcome with microarrays: a multiple random validation strategy. Lancet 2005, 365(9458):488–492. 10.1016/S0140-6736(05)17866-0View ArticlePubMedGoogle Scholar
- Kohavi R: A study of cross-validation and bootstrap for accuracy estimation and model selection. International Joint Conference on Artificial Intelligence; Montreal IJCAI 1995. Unpaged UnpagedGoogle Scholar
- Simon R: Using DNA microarrays for diagnostic and prognostic prediction. Expert Rev Mol Diagn 2003, 3(5):587–595. 10.1586/1473718.104.22.1687View ArticlePubMedGoogle Scholar
- Simon R, Radmacher MD, Dobbin K, McShane LM: Pitfalls in the use of DNA microarray data for diagnostic and prognostic classification. J Natl Cancer Inst 2003, 95(1):14–18. 10.1093/jnci/95.1.14View ArticlePubMedGoogle Scholar
- Braga-Neto UM, Dougherty ER: Is cross-validation valid for small-sample microarray classification? Bioinformatics 2004, 20(3):374–380. 10.1093/bioinformatics/btg419View ArticlePubMedGoogle Scholar
- Varma S, Simon R: Bias in error estimation when using cross-validation for model selection. BMC Bioinformatics 2006, 7: 91. 10.1186/1471-2105-7-91PubMed CentralView ArticlePubMedGoogle Scholar
- Naderi A, Teschendorff AE, Barbosa-Morais NL, Pinder SE, Green AR, Powe DG, Robertson JF, Aparicio S, Ellis IO, Brenton JD, et al.: A gene-expression signature to predict survival in breast cancer across independent data sets. Oncogene 2007, 26(10):1507–1516. 10.1038/sj.onc.1209920View ArticlePubMedGoogle Scholar
- Sotiriou C, Neo SY, McShane LM, Korn EL, Long PM, Jazaeri A, Martiat P, Fox SB, Harris AL, Liu ET: Breast cancer classification and prognosis based on gene expression profiles from a population-based study. Proc Natl Acad Sci U S A 2003, 100(18):10393–10398. 10.1073/pnas.1732912100PubMed CentralView ArticlePubMedGoogle Scholar
- Wang Y, Klijn JG, Zhang Y, Sieuwerts AM, Look MP, Yang F, Talantov D, Timmermans M, Meijer-van Gelder ME, Yu J, et al.: Gene-expression profiles to predict distant metastasis of lymph-node-negative primary breast cancer. Lancet 2005, 365(9460):671–679.View ArticlePubMedGoogle Scholar
- Ein-Dor L, Kela I, Getz G, Givol D, Domany E: Outcome signature genes in breast cancer: is there a unique set? Bioinformatics 2005, 21(2):171–178. 10.1093/bioinformatics/bth469View ArticlePubMedGoogle Scholar
- Clemen R: Combining forecasts: A review and annotated bibliography. Journal of Forecasting 1989, 5: 559–583. 10.1016/0169-2070(89)90012-5View ArticleGoogle Scholar
- Gramatica P, Pilutti P, Papa E: Validated QSAR prediction of OH tropospheric degradation of VOCs: splitting into training-test sets and consensus modeling. J Chem Inf Comput Sci 2004, 44(5):1794–1802. 10.1021/ci049923uView ArticlePubMedGoogle Scholar
- Tan AC, Gilbert D: Ensemble machine learning on gene expression data for cancer classification. Appl Bioinformatics 2003, 2(3 Suppl):S75–83.PubMedGoogle Scholar
- Su Z, Hong H, Perkins R, Shao X, Cai W, Tong W: Consensus analysis of multiple classifiers using non-repetitive variables: diagnostic application to microarray gene expression data. Comput Biol Chem 2007, 31(1):48–56. 10.1016/j.compbiolchem.2007.01.001View ArticlePubMedGoogle Scholar
- Shi L, Campbell G, Jones WD, Campagne F, Wen Z, Walker SJ, Su Z, Chu TM, Goodsaid FM, Pusztai L, et al.: The MicroArray Quality Control (MAQC)-II study of common practices for the development and validation of microarray-based predictive models. Nat Biotechnol 2010, 28(8):827–838. 10.1038/nbt.1665View ArticlePubMedGoogle Scholar
- Thomas RS, Pluta L, Yang L, Halsey TA: Application of genomic biomarkers to predict increased lung tumor incidence in 2-year rodent cancer bioassays. Toxicol Sci 2007, 97(1):55–64. 10.1093/toxsci/kfm023View ArticlePubMedGoogle Scholar
- Fielden MR, Brennan R, Gollub J: A gene expression biomarker provides early prediction and mechanistic assessment of hepatic tumor induction by nongenotoxic chemicals. Toxicol Sci 2007, 99(1):90–100. 10.1093/toxsci/kfm156View ArticlePubMedGoogle Scholar
- Lobenhofer EK, Auman JT, Blackshear PE, Boorman GA, Bushel PR, Cunningham ML, Fostel JM, Gerrish K, Heinloth AN, Irwin RD, et al.: Gene expression response in target organ and whole blood varies as a function of target organ injury phenotype. Genome Biol 2008, 9(6):R100. 10.1186/gb-2008-9-6-r100PubMed CentralView ArticlePubMedGoogle Scholar
- Hess KR, Anderson K, Symmans WF, Valero V, Ibrahim N, Mejia JA, Booser D, Theriault RL, Buzdar AU, Dempsey PJ, et al.: Pharmacogenomic predictor of sensitivity to preoperative chemotherapy with paclitaxel and fluorouracil, doxorubicin, and cyclophosphamide in breast cancer. J Clin Oncol 2006, 24(26):4236–4244. 10.1200/JCO.2006.05.6861View ArticlePubMedGoogle Scholar
- Zhan F, Huang Y, Colla S, Stewart JP, Hanamura I, Gupta S, Epstein J, Yaccoby S, Sawyer J, Burington B, et al.: The molecular classification of multiple myeloma. Blood 2006, 108(6):2020–2028. 10.1182/blood-2005-11-013458PubMed CentralView ArticlePubMedGoogle Scholar
- Shaughnessy JD Jr., Zhan F, Burington BE, Huang Y, Colla S, Hanamura I, Stewart JP, Kordsmeier B, Randolph C, Williams DR, et al.: A validated gene expression model of high-risk multiple myeloma is defined by deregulated expression of genes mapping to chromosome 1. Blood 2007, 109(6):2276–2284. 10.1182/blood-2006-07-038430View ArticlePubMedGoogle Scholar
- Oberthuer A, Berthold F, Warnat P, Hero B, Kahlert Y, Spitz R, Ernestus K, Konig R, Haas S, Eils R, et al.: Customized oligonucleotide microarray gene expression-based classification of neuroblastoma patients outperforms current clinical risk stratification. J Clin Oncol 2006, 24(31):5070–5078. 10.1200/JCO.2006.06.1879View ArticlePubMedGoogle Scholar
- Tusher VG, Tibshirani R, Chu G: Significance analysis of microarrays applied to the ionizing radiation response. Proc Natl Acad Sci U S A 2001, 98(9):5116–5121. 10.1073/pnas.091062498PubMed CentralView ArticlePubMedGoogle Scholar
- Team RDC: R: A language and environment for statistical computing. R Foundation for Statistical computing Vienna, Austria ISBN 3–900051–07–0; 2010. [http://www.R-project.org]Google Scholar
- Weihs C, Ligges U, Luebke K, Rabbe N: klaR analyzing German business cycle. In Data Analysis and Decision Support. Edited by: Baier, D, Decker, R and Schmitd-Thieme, L. Springer-Verlag, Berlin; 2005:335–343.View ArticleGoogle Scholar
- Rokach L: Ensemble-based classifiers. The Artificial Intelligence Review 2010, 33(1–2):1–33. 10.1007/s10462-009-9124-7View ArticleGoogle Scholar
- Gramatica P, Giani E, Papa E: Statistical external validation and consensus modeling: a QSPR case study for Koc prediction. J Mol Graph Model 2007, 25(6):755–766. 10.1016/j.jmgm.2006.06.005View ArticlePubMedGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.