The topscoring ‘N’ algorithm: a generalized relative expression classification method from small numbers of biomolecules
 Andrew T Magis^{1, 2} and
 Nathan D Price^{1, 2}Email author
DOI: 10.1186/1471210513227
© Magis and Price; licensee BioMed Central Ltd. 2012
Received: 19 February 2012
Accepted: 3 September 2012
Published: 11 September 2012
Abstract
Background
Relative expression algorithms such as the topscoring pair (TSP) and the topscoring triplet (TST) have several strengths that distinguish them from other classification methods, including resistance to overfitting, invariance to most data normalization methods, and biological interpretability. The topscoring ‘N’ (TSN) algorithm is a generalized form of other relative expression algorithms which uses generic permutations and a dynamic classifier size to control both the permutation and combination space available for classification.
Results
TSN was tested on nine cancer datasets, showing statistically significant differences in classification accuracy between different classifier sizes (choices of N). TSN also performed competitively against a wide variety of different classification methods, including artificial neural networks, classification trees, discriminant analysis, kNearest neighbor, naïve Bayes, and support vector machines, when tested on the Microarray Quality Control II datasets. Furthermore, TSN exhibits low levels of overfitting on training data compared to other methods, giving confidence that results obtained during cross validation will be more generally applicable to external validation sets.
Conclusions
TSN preserves the strengths of other relative expression algorithms while allowing a much larger permutation and combination space to be explored, potentially improving classification accuracies when fewer numbers of measured features are available.
Keywords
Classification Topscoring pair Relative expression Cross validation Support vector machine Graphics processing unit MicroarrayBackground
Relative expression algorithms such as the topscoring pair (TSP)[1] and the topscoring triplet (TST)[2] represent powerful methods for disease classification, primarily focused on the creation of simple, yet effective classifiers. These algorithms have several strengths that distinguish them from other classification methods. First, only the ranks of the expression data are used, rather than the expression values directly, therefore these algorithms are invariant to data normalization methods that preserve rankorder. For example, quantile normalization is a rankpreserving common practice in microarray analysis to remove technical sources of variance between arrays[3]. It is therefore preferable that the classification algorithm be insensitive to such normalization procedures, particularly in metaanalyses combining data from multiple studies or in a clinical setting where additional measurements beyond the features used to build the classifier would be needed to apply the normalization step. Second, relative expression classifiers make use of only a few features to build each classifier, and require relatively little to no parameter tuning. As a result, the algorithms are generally resistant to overfitting, in which an algorithm learns to classify the noise of the training set rather than the true phenotypic signal of interest. Moreover, the small number of features in relative expression algorithms lends itself well to the development of inexpensive clinical tests[4]. Third, an underappreciated aspect of relative expression algorithms involves their potential for biological interpretation. The simplicity of these algorithms, in which the ranks of a few features shift positions in a predictable way between two phenotypic classes, suggests that the features participating in a highly accurate classifier may represent or reflect an underlying biological role for those features in the phenotypes being classified. Relative expression algorithms may therefore serve as hypothesis generators for additional study. This characteristic may become particularly relevant as classification methods move increasingly more into technologies such as secretomics and miRNA expression measurements that, at present, result in fewer measurements per sample than do transcriptomes.
In this paper we present a new formulation of the relative expression classification algorithm that generalizes the idea of pairwise rank comparisons (TSP) and triplet rank comparisons (TST) into generic permutation rank comparisons, where the size of the classifier is not defined a priori. This algorithm is called the topscoring ‘N’ (TSN), where N is a variable indicating the size of the classifier. As such, TSP and TST can be thought of as special cases of the general TSN algorithm (just with a fixed N = 2 or N = 3, respectively). Because the classifier size is unconstrained, TSN can explore a much larger permutation and combination space than that available to either TSP or TST. All of the results presented in this paper used no more than sixteen features from any of the training sets.
The classification accuracy of the existing relative expression algorithms has been demonstrated in several studies. Classifiers identified using relative expression algorithms have been used to distinguish multiple cancer types from normal tissue based on expression data[1, 2, 4, 5] as well as to predict cancer outcomes and model disease progression[6]. Furthermore, relative expression algorithms perform competitively when compared to other, often more complex, classification methods, including support vector machines[7], decision trees[8] and neural networks[9]. Relative expression algorithms have also been applied in a network context, illustrating the dysregulation of cellular pathways in disease phenotypes[10].
We first demonstrate that both TSP and TST are special cases of the TSN algorithm. We illustrate the performance of a range of TSN classifier sizes on a set of nine cancer datasets. Finally, we demonstrate that TSN performs competitively when compared to a broad range of classification models, including artificial neural networks, classification trees, and support vector machines, using data and results from the FDAsponsored Microarray Quality Control II project (MAQCII)[11].
Methods
Overview of relative expression algorithms TSP and TST
The TSP algorithm identifies the best pair of features for which the rank of x_{ i } falls lower than the rank of x_{ j } in most or all samples in class C_{ 1 }, and the rank of x_{ i } falls lower than the rank of x_{ j } in few or no samples of class C_{ 2 }. The max (Δ_{ i,j } = 1) indicates a perfect classifier on the training set in which no samples deviate from this pattern. Classification is performed by comparing the ordering of features {x_{ i }, x_{ j }} in each sample of the test set to the orderings associated with the two classes. A variant on this algorithm known as kTSP makes use of multiple disjoint pairs to improve classification accuracy[5].
The topscoring N algorithm
where σ_{ m } is the m th permutation of the classifier X. Recall that there are N! possible permutations of X. The permutation probability distribution for each class is determined by mapping the permutation of X for each training set sample to its corresponding factoradic, converting the factoradic to decimal representation, and using this as an index into a histogram of size N!. Once normalized by the number of samples in each class, the histogram represents the permutation probability distribution for that feature set on that training set class. When the two histograms are completely disjoint (i.e., there are no overlapping permutations between the two classes), the TSN score Δ_{ X } = 1.
where S is equal to the number of samples in the training set and N the size of the classifier X. R refers to the rank, and X(1) and X(N) are the first and last elements of the classifier, respectively. In the case of ties in the primary TSN score, the classifier chosen will have the largest distance in rank between the upper and lower elements of the classifier.
In the case where N = 2, the TSN algorithm simply reduces to the TSP algorithm, since X_{ 2 } = {x_{ i }, x_{ j }}, and Pr(σ_{1}) = Pr(x_{ i } < x_{ j }). In the case where N = 3, the TSN algorithm reduces to the TST algorithm, since X_{ 3 } = {x_{ i },x_{ j },x_{ k }} and Pr(σ_{ m }) = Pr(π_{ m }). Because the TSN algorithm uses factoradics to uniquely represent any permutation of any size classifier, it allows TSP and TSP classifiers to be used in concert as well as allowing for even larger classifiers to be explored.
The choice of N is clearly important in the determination of a new classifier for a training set. The simplest method is to choose the value of N with the greatest classification accuracy after iteration over a range of N. This method would reveal the apparently most effective classifier size. In this case the experimenter is artificially choosing the ‘best’ value of N for a given dataset. However, in fair comparisons with other classification methods it is important that the choice of N not be made a posteriori (once the best classifier and value of N have been determined) to avoid overly optimistic error estimates. We do not choose the value of N outside the cross validation loop, but rather dynamically select the value of N at each iteration of the cross validation loop; the choice is made based on the apparent accuracy of that value of N on the training set. We call this version of the algorithm dynamic N. Apparent accuracy is calculated by first finding the highest scoring classifier on the training set for each value of N in a range specified by the user. The value of N with the highest apparent accuracy on the training set is then applied to the test set. In the case of ties in apparent accuracy for multiple values of N, the algorithm chooses the smallest tied value of N for the classifier at that iteration of the cross validation loop. This process is repeated at each iteration of the cross validation loop. Note that this method does not preclude the user from artificially choosing the best value of N (outside of cross validation) for other purposes, but is rather a mechanism to avoid bias during cross validation. This allows us to make fair comparisons of the TSN algorithm with other classification methods without potentially biasing the results in our favor.
Classification with TSN
Implementation of TSN
While the TSN algorithm can theoretically explore a very large permutation space, the computational requirements of the algorithm rise very quickly and to avoid overfitting the number of permutations explored must be scaled to what is reasonable given available sample numbers. The complexity of TSN is$O\left(\left(N,M\right)N!\right)$, where M is the number of features and N is the size of the classifier. We have previously shown[12] that the graphics processing unit (GPU) is highly efficient when applied to easily parallelizable algorithms such as TSP and TST. Given that TSN preserves the parallel nature of the other relative expression algorithms, it is also easily applied to the GPU. However, given that GPU hardware is not yet widely available to many researchers, we are releasing the source code for both GPU and CPU implementations of the TSN algorithm. TSN has been implemented for both the GPU and the CPU in the MATLAB computing environment.
Results and discussion
Multiple values of N
TSN has been tested on nine cancer datasets that were used in the previous kTSP and TST papers[2, 5] for comparison between different values of N. These datasets represent a wide range of cancers, including colon[18], leukemia[19], central nervous system lymphoma (CNS)[20], diffuse large Bcell lymphoma (DLBCL)[21], prostate[22–24], and a global cancer map (GCM) dataset[25]. As discussed in the methods section, the TSN algorithm can be used in two different ways: the choice of N can be made a posteriori after all fixed values have been tested, or the choice of N can be made at each iteration of the cross validation loop (dynamic N) using apparent accuracy. Apparent accuracy is calculated by first finding the highest scoring classifier on the training set for each value of N in a range specified by the user. The value of N with the highest apparent accuracy on the training set is then applied to the test set. In order to directly compare the accuracies based on the number of permutations of features, we chose 16 features for N = 2, 10 features for N = 3, and 9 features for N = 4. This results in approximately 120 combinations for each value of N. The reason for choosing different numbers of features for each value of N is to equalize the combination space for each classifier size. For example, a classifier of size N = 2 given 16 features can explore 2! = 2 permutations over$\left(\underset{2}{16}\right)=120$ combinations. A classifier of size N = 3 given 10 features can explore 3! = 6 permutations over$\left(\underset{3}{10}\right)=120$ combinations. A classifier of size N = 4 given 9 features can explore 4! = 24 permutations over$\left(\underset{4}{9}\right)=126$ combinations. As a result, any difference in accuracy between these two classifiers depends primarily on the permutation space being explored and not the combination space (which is held relatively constant). The features were chosen to be the most differentially expressed genes based on the Wilcoxon rank sum test, again selected within each iteration of the cross validation loop to avoid overly optimistic estimates.
It is clear from Figure4 that the value of N can have a significant effect on the resulting accuracy of the classifier, which indicates that the larger permutation space afforded by larger values of N can be useful in identifying an effective classifier. In the Leukemia dataset, for example, N = 2 and N = 3 produced the apparently most effective classifiers; in the Lung dataset, N = 3 and N = 4 were the apparent best. In four of the nine datasets (DLBCL, Prostate2, Prostate3, and GCM), dynamic N yielded no significant difference in accuracy with the highestscoring fixed value of N. In two additional datasets (Leukemia and Lung), the dynamic N accuracy is statistically in between the highest and lowestscoring values of N. In the remaining three datasets (Colon, CNS, and Prostate1), the dynamic N accuracy is not significantly different from the lowestscoring fixed value of N. The dynamic N TSN result is the fair estimate of how well the algorithm would be expected to perform with optimization for N, without the bias that is introduced by choosing the apparently best N after the error estimate has been made.
Microarray quality control II datasets
The five MAQCII datasets, representing endpoints A through I that are available from the Gene Expression Omnibus
Dataset  Endpoint  Description  Platform 

Hamner  A  Lung tumorigen vs. nontumorigen  Affymetrix Mouse 430 2.0 
Iconix  B  Nongenotoxic liver carcinogens vs. noncarcinogens  Amersham Uniset Rat 1 Bioarray 
NIEHS  C  Liver toxicants vs. nontoxicants  Affymetrix Rat 230 2.0 
Breast Cancer  D  Preoperative treatment response  Affymetrix Human U133A 
E  Estrogen receptor status  
Multiple Myeloma  F  Overall survival milestone outcome  Affymetrix Human U133 Plus 2.0 
G  Eventfree survival milestone outcome  
H  Gender of patient (positive control)  
I  Random class labels (negative control) 
The participants that submitted models for every endpoint (original and swap) in the MAQCII study, and the classification methods used
Code  Name  Classification algorithm(s) used 

CAS  Chinese Academy of Sciences  Naïve Bayes, Support Vector Machine 
CBC  CapitalBio Corporation, China  kNearest Neighbor, Support Vector Machine 
Cornell  Weill Medical College of Cornell University  Support Vector Machine 
FBK  Fondazione Bruno Kessler, Italy  Discriminant Analysis, Support Vector Machine 
GeneGo  GeneGo, Inc.  Discriminant Analysis, Random Forest 
GHI  Golden Helix, Inc.  Classification Tree 
GSK  GlaxoSmithKline  Naïve Bayes 
NCTR  National Center for Toxicological Research, FDA  kNearest Neighbor, Naïve Bayes, Support Vector Machine 
NWU  Northwestern University  kNearest Neighbor, Classification Tree, Support Vector Machine 
SAI  Systems Analytics, Inc.  Discriminant Analysis, kNearest Neighbor, Machine Learning, Support Vector Machine, Logistic Regression 
SAS  SAS Institute, Inc.  Classification Tree, Discriminant Analysis, Logistic Regression, Partial Least Squares, Support Vector Machine 
Tsinghua  Tsinghua University, China  Classification Tree, kNearest Neighbor, Recursive Feature Elimination, Support Vector Machine 
UIUC  University of Illinois, UrbanaChampaign  Classification Tree, kNearest Neighbor, Naïve Bayes, Support Vector Machine 
USM  University of Southern Mississippi  Artificial Neural Network, Naïve Bayes, Sequential Minimal Optimization, Support Vector Machine 
ZJU  Zejiang University, China  kNearest Neighbor, Nearest Centroid 
If any of the sums in the denominator of the MCC are zero, the denominator is set to be one, resulting in an MCC equal to zero.
In addition to standard cross validation and validation set MCC, we also measured the statistical significance of different classifier sizes. As described with the cancer datasets above, we ran 100 iterations of TSN using fixed values of N = 2, N = 3, and N = 4, as well as dynamic N = {2,3,4} on all nine of the MAQCII training sets. For example, in endpoints A and B, N = 4 yields a statistically significant improvement over smaller classifier sizes. For endpoints C and E, N = 2 is the most effective classifier size. For endpoint G, there was no significant difference between any of the classifier sizes. In six out of the nine datasets (endpoints A, C, F, G, H, and I) there was no significant difference in MCC between dynamic N and the highestscoring fixed value of N. The complete results are available in Additional file1: Figure S5. All raw data is included in Additional file4.
For all analyses in this paper, up to sixteen differentially expressed genes were selected by the Wilcoxon rank sum test to input into the TSN algorithm. The fact that so few features were input to TSN in these analyses could explain the low levels of overfitting it exhibits. To test this, we ran all MAQCII training sets (except for the negative control endpoint I, which would bias the results of ΔMCC towards zero) over a range of input feature sizes. For N = 2, we input a range of 16 to 10,000 input features. For N = 3 we input a range of 10 to 670 input features. For N = 4 we input a range of 9 to 188 input features. These numbers were chosen to span approximately the same range of possible feature combinations for each value of N (approximately 120 combinations up to 50 million combinations). Finally we ran dynamic N for N = {2,3,4} over the same ranges of input feature sizes. ΔMCC values were calculated for each input feature size, and box plots of their distributions are shown in Additional file1: Figure S6. All raw data is included in Additional file5. While the mean ΔMCC values do increase as a function of input feature size, overall the levels of overfitting remain low for TSN despite the increase. The mean ΔMCC exhibited by dynamic N TSN at the largest input size of [10000, 760, 188], is 0.148. This is still among the smallest mean ΔMCC value observed in any of the participating groups; only three groups are smaller (GHI, GSK, and SAS).
Conclusions
The goal of relative expression classification algorithms is to identify simple yet effective classifiers that are resistant to data normalization procedures and overfitting, practical to implement in a clinical environment, and potentially biologically interpretable. The topscoring ‘N’ algorithm presented here retains these desirable properties while allowing a larger combination and permutation space to be searched than that afforded by earlier relative expression algorithms such as TSP and TST. TSN can also recommend the classifier size (N) most likely to result in effective classification based on the training set. Of course, more care must be taken to avoid overfitting with TSN, particularly on smaller datasets, given that the permutation space grows with the factorial numbers. However, the problem of overfitting can be well mitigated by choosing a suitably small number of features from which to build the classifier, or ensuring that the number of samples available is large enough to justify searching a larger combination space. All the results presented in this paper were performed using between nine and sixteen features of the microarray datasets. TSN is therefore well suited for datasets of emerging technologies that contain smaller numbers of features to begin with, such as secretomics and miRNAs. However, as Figure3 demonstrates, it is still possible to search tens of thousands of permutations in a relatively short amount of time, when justified by large sample sizes. The statistical significance of the resulting classifiers can then be determined though e.g. permutation tests of the class labels.
We have demonstrated the effectiveness of TSN in classification of the MAQCII datasets in comparison with many other classification strategies, including artificial neural networks, classification trees, discriminant analysis, kNearest neighbor, naïve Bayes, and support vector machines, as implemented by several universities and companies from around the world. We do not claim that TSN is necessarily the best or most effective classifier for every circumstance. For example, TSN performs relatively poorly on endpoint H, which as the positive control in which classes were simply assigned as the gender of the study participants, should be among the easiest to classify. A major strength of the algorithm is the level to which the MCC values for cross validation agree with the MCC values on the independent validation set (ΔMCC). Importantly, these results indicate a very low level of overfitting, and increase our confidence that results generated through cross validation on future datasets will be effective classifiers on independent validation sets. That is, when TSN works on a dataset it is relatively more likely to be true, and conversely, when it is going to fall short in independent validation it typically does not work well in cross validation and so can be discarded as a candidate diagnostic early in the process. Analyses over a range of input sizes indicate that overfitting remains low even as input feature numbers increase, given sufficient sample sizes.
Of all the MAQCII participants, including TSN, group SAS yielded the lowest mean ΔMCC score (0.074), indicating low levels of overfitting. Group SAI yielded the highest mean MCC (0.4893) for original and swap datasets, indicating high levels of validation set accuracy based on the training set. Both of these groups utilized multiple classification strategies across all endpoints. For example, group SAS used logistic regression for endpoints A, E, and I, support vector machines for endpoints B, G, and H, partial least squares regression for endpoints D and F, and a decision tree for endpoint C. Group SAI used support vector machines for endpoints A, B, E, F, G, and I, knearest neighbor for endpoints C and H, and a machine learning classifier for endpoint D. Group SAI also used a range of different feature selection methods for each endpoint. Both groups also used different classification strategies for the swap datasets. For example, group SAS used logistic regression for the original endpoint E data but partial least squares regression on swap endpoint E. Group SAI used a machine learning classifier for the original endpoint D, and discriminant analysis for swap endpoint D[11]. As a result, TSN is not only being compared to different classification strategies, but an ensemble of classification strategies that were chosen in an attempt to maximize success for each endpoint across both original and swap datasets. Given its advantages of relative simplicity, biological interpretability, and low levels of overfitting, the TSN algorithm can serve as a useful tool for hypothesis generation, particularly as next generation sequencing and proteomics technologies yield increasing sensitivity in biomolecule measurements.
Abbreviations
 CPU:

Central processing unit
 DEG:

Differentially expressed genes
 GPU:

Graphics processing unit
 MAQCII:

Microarray quality control II
 MCC:

Matthews correlation coefficient
 TSN:

Topscoring ‘N’
 TSP:

Topscoring pair
 TST:

Topscoring triplet.
Declarations
Acknowledgements
The authors thank Dr. Don Geman and Bahman Afsari for valuable discussions during the development of this paper. This work was supported by a National Institutes of Health Howard Temin Pathway to Independence Award in Cancer Research [R00 CA126184]; the Camille Dreyfus TeacherScholar Program, and the Grand Duchy of LuxembourgISB Systems Medicine Consortium.
Authors’ Affiliations
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