Open House, ISM in Tachikawa, 2010.7.9統計数理研究所オープンハウス（立川）、H22.7.9ポスター発

Osamu Komori

小森 理

English

RISM (Repository of the Institute of Statistical Mathematics) / 統計数理研究所 学術研究リポジトリ

Maximization of the Partial Area under theROC curve using a Boosting TechniqueOsamu Komori, Prediction and Knowledge Discovery Research Centerkomori@ism.ac.jpIntroduction This research attempts to improve the accuracy of discrimination of thediseased group from non-diseased group. The accuracy is often mea-sured by the partial area under the ROC curve (pAUC)[3]. The association between markers and outcome variable (disease or non-disease) is also important in medical or biological sciences.We have developed a new statistical method that aims to maximize thepAUC, using a boosting technique. The resultant score plots show us the association in a visually-apparentway.Status quo does not take into account the non-linear structure of the associationbetween markers and outcome variable[5];[6].Moreover, Eguchi and Copas [1] and McIntosh and Pepe [4] show theoptimal score function is derived from the likelihood ratio. That is, thelinearity of the association is not sucient in general.ROC curve and pAUC (x; y), x 2 Rp; y 2 f0; 1g F (x) : score function,c : threshold8><>: F (x)  c) positiveF (x) < c) negative+ROC(F ) = fFPR(c); TPR(c)g? The pAUC is more suitable for clinical setting in which a high true pos-itive rate is required with a low false positive rate.Theorem about pAUCTheorem 1. For a pair of xed 1 and 2, let	() = pAUCF + m ; 1; 2;where  is a scalar, (x) = g1(x)=g0(x) and m is a strictly increasingfunction. Then, 	() is strictly increasing function of , andsupFpAUC(F; 1; 2) = lim!1	() = pAUC ; 1; 2: As seen in Theorem 1, the approximate pAUC has no maximum, but asupremum. Hence, we will consider the penalty term in the objectivefunction in order to ensure the existence of the maximum and make thepAUCBoost algorithm numerically stable.Objective function Prepare a set of weak classiers, from which we construct a score func-tion F (x).F = ff (x) = Nk;l(xk)=Zk;lj k = 1; 2; : : : ; p; l = 1; 2; : : : ;mkg:The basis functions of the natural cubic spline for xk are dened asNk;l(xk) =8>>>><>>>>:1; l = 1;xk; l = 2;dl 2(xk)  dmk 1(xk); otherwise;wheredl(xk) =(xk   k;l 2)3+   (xk   k;mk)3+k;mk   k;l 2;and z+ denotes the positive part of z. The standardization factor Zk;lfor Nk;l(xk) is given asZk;l =8>>>><>>>>:1; l = 1;k;mk   k;1; l = 2;Nk;l(k;mk) Nk;l(k;l 2); otherwise;and k;l is one of mk knots (k;1 < k;2 < : : : < k;mk) for xk. the objective function we propose ispAUC;(F; 1; 2) = pAUC(F; 1; 2)  pXk=1Z F 00k (xk)	2dxk=1n0n1Xi2In Xj2JfanH(F (x1j)  F (x0i))+Xj2JrecH(F (x1j)  F (x0i))o pXk=1Z F 00k (xk)	2dxk;where F 00k (xk) is the second derivative of the k-th component of F (x),and  is a smoothing parameter that controls the smoothness of F (x).Without loss of generality, we can rewrite it aspAUC;(F; 1; 2) = pAUC1;(F; 1; 2) pAUC(F; 1; 2) Note that the maximizer of the objective function is shown to be thenatural cubic splines [2].pAUCBoost algorithm1. Start with a score function F0(x) = 0 and set each coecient 0(f )of weak classiers to be 1 or  1.2. For t = 1; :::; Ta. Calculate the values of thresholds c1 and c2 for each Ft 1 +t 1(f )f .b. Update t 1(f ) to t(f ) with a one-step Newton-Raphson itera-tion.c. Find the best weak classier ftft = argmaxfpAUC(Ft 1 + t(f )f; 1; 2)d. Update the score function asFt(x) = Ft 1(x) + t(ft)ft(x):3. Finally, output a nal score function F (x) =PTt=1 t(ft)ft(x).Breast cancer data Two types of data [7]clinical data: Age, Size, Grade, Angi, ERp, PRp and Lympgenomic data: gene expression proles (25000 genes) training data: 78 patients; test data 19 patientsWe apply AUCBoost to clinical data using natural cubic splines to Ageand Size (continuous markers), and decision stumps to the others (dis-crete markers)Agescore30 35 40 45 50 5501234Sizescore10 20 30 4001234Gradescore1 2 301234Angiscore0 101234ERpscore0 101234PRpscore0 101234Lympscore0 101234(a) (b)Figure 1. (a) Score plots of clinical markers; (b) Clustering cited from[7]. The resultant AUC is 0.882 and 0.869 for training and test data, resp. After the pAUC-based ltering process, we apply pAUCBoost with nat-ural cubic splines to the 11 genes. -1.0 -0.5 0.0Contig41613_RC02468score-0.2 0.0 0.2 0.4NM_00693102468score-2.0 -1.5 -1.0 -0.5 0.0 0.5Contig40831_RC02468score-0.4 -0.2 0.0 0.2 0.4Contig55574_RC02468score-2.0 -1.5 -1.0 -0.5 0.0AB02317302468score-2.0 -1.5 -1.0 -0.5 0.0 0.5Contig63649_RC02468score-0.4 -0.2 0.0 0.2 0.4NM_01896402468score-0.2 0.0 0.2AL13761502468score-0.2 0.0 0.2 0.4 0.6NM_00620102468score-1.0 -0.5 0.0 0.5NM_00171002468score-0.4 -0.2 0.0 0.2 0.4 0.6AA555029_RC02468score Figure 2. Score plots of the selected 11 genes. Results0 20 40 60 80 100T0.000.020.040.060.080.10pAUC(a)Figure 3. (a) The pAUCfor training data (black) andtest data (gray); (b) theresults of van't Veer [7] . (b) The resultant pAUCs for both training and test data are more than 3times bigger than their results: 0.025 and 0.0008, resp. [7].References[1]Eguchi, S. and Copas, J. (2002). A class of logistic-type dis-criminant functions. Biometrika 89, 1{22.[2]Hastie, T. and Tibshirani, R. (1990). Generalized AdditiveModels, Chapman & Hall.[3]Komori, O. and Eguchi, S. (2010). A boosting method for max-imizing the partial area under the ROC Curve. BMC Bioinformatics11, 314.[4]McIntosh, M. W. and Pepe, M. S. (2002). Combining severalscreening tests: Optimality of the risk score. Biometrics 58, 657{664.[5]Pepe, M. S. and Thompson, M. L. (2000). Combining diag-nostic test results to increase accuracy. Biostatistics 1, 123{140.[6]Pepe, M. S., Cai, T. and Longton, G. (2006). Combiningpredictors for classication using the area under the Receiver OperatingCharacteristic curve. Biometrics 62, 221{229.[7] van't Veer, L. J., Dai, H., van de Vijver, M. J., He,Y. D., Hart, A. A. M., Mao, M., Peterse, H. L., van derKooy, K., Marton, M. J., Witteveen, A. T., Schreiber,G. J., Kerkhoven, R. M., Roberts, C., Linsley, P. S.,Bernards, R. and Friend, S. H. (2002). Gene expression pro-ling predicts clinical outcome of breast cancer. Nature 415, 530{536.

Maximization of the partial area under the ROC curve

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Maximization of the partial area under the ROC curve

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Maximization of the partial area under the ROC curve

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