Original Article

Multi-Class Classification using Mixtures of Univariate and Multivariate ROC Curves


Introduction: Receiver Operating Characteristic (ROC) curve is one of the widely used supervised classification techniques to allocate/classify the individuals and also instrumental in comparing diagnostic tests. Generally to deal with classification problems we need to have knowledge of class labels. In most of the scenarios, data exhibit multi-model patterns in class labels which leads to multi-class classification problems.

Objective: The main aim of this study is to address the issue of constructing ROC models when there exist multi-models patterns in the class labels further, to classify the individuals for better diagnosis, and also to reduce the complexity of graphical representation of ROC curves in such classification problems.

Methods: A new version of univariate and multivariate ROC models are proposed in the framework of Finite Mixtures, due to the flexibility of identifying and modeling the subcomponents in the heterogeneous populations.

Results: Oral Glucose Tolerance Test and Disk Hernia datasets are used and simulation studies are also performed. Results show that the proposed models possess better accuracy when compared with Bi-Normal and MROC models with reasonable low 1-Specificity and higher Sensitivity. The ROC curves are depicted in a 2D space rather than a higher dimension for multi-class classification problems.

Conclusions: It is suggested that before one proceeds to model ROC curves, it is better to take a look at the density patterns of the study variable(s), which in turn help in explaining the true information between the classes and also provides a good amount of “true” accuracy.

1. Green DM, Swets JA. Signal detection theory and psychophysics. New York: Wiley. 1966.
2. Egan JP. Signal Detection Theory and ROC Analysis “Academic Press. New York. 1975.
3. England WL. An exponential model used for optimal threshold selection on ROC Curves. Med Deci Mak, 1988;8:120-131.
4. Campbell G, Ratnaparkhi MV. An application of Lomax distributions in receiver operating characteristic (ROC) curve analysis. Comm in Stat -Theory and Methods. 1993;22:1681-1687.
5. Hussain E. The bi-gamma ROC curve in a straightforward manner. J Bas & App Sci. 2012;8:309-314.
6. Mossman D, Peng H. Using dual beta distributions to create “proper” ROC curves based on rating category data. Med Deci Mak. 2016;36:349-365.
7. Balaswamy S, Vardhan RV, Sarma, KVS. The Hybrid ROC Curve and its Divergence Measures for Binary Classification. Int J Stat in Med Res. 2015a;4(1):94–102.
8. Balaswamy S, Vardhan RV. Estimation of Confidence Intervals of a GHROC Curve in the presence of Scale and Shape parameters. Res J Math and Stat Sci. 2015b;3(10):4-11.
9. Balaswamy S, Vardhan RV. ROC curve Estimation using the combination of Generalized Half Normal and Weibull distributions. J Indian Society for Prob and Stat. 2016a;17(1):11-23.
10. Balaswamy S, Vishnu Vardhan R. An Anthology of Parametric ROC Models. Research & Reviews: J Sta. 2016b;5(2):32-46.
11. Su JQ, Liu JS. Linear combinations of multiple diagnostic markers. J Ame Stat Assoc. 1993;88:1350-1355.
12. Schisterman EF, Faraggi D, Reiser B. Adjusting the generalized ROC curve for covariates. Stat in Med. 2004;23:3319-3331.
13. Yuan Z, Ghosh D. Combining multiple biomarker models in logistic regression. Biometrics. 2008;64:431-439.
14. Yin J, Tian L. Optimal linear combinations of multiple diagnostic biomarkers based on Youden index. Stat in Med. 2014;33:1426-1440.
15. Sameera G, Vardhan, RV, Sarma KVS. Binary classification using multivariate receiver operating characteristic curve for continuous data. J Biopha Stat. 2016;26:421-431.
16. Lasko TA, Bhagwat JG, Zou KH, Ohno-Machado L. The use of receiver operating characteristic curves in biomedical informatics. J Biomed Inf. 2005;38(5):404-415.
17. Dempster AP, Laird NM, Rubin DB. Maximum likelihood from incomplete data via the EM algorithm. J Royal Stat Soc: Ser B (Meth). 1977;39:1-22.
18. Titterington DM, Smith AFM, Makov UE. Statistical analysis of finite mixture distributions (Vol. 198). John Wiley & Sons Inc. 1985.
19. McLachlan G, Peel D. Finite Mixture Models. John Wiley and Sons, Inc., New York. 2000.
20. Krzanowski WJ, Hand DJ. ROC curves for continuous data. Crc Press. 2009.
21. Anderson TW, Bahadur RR. Classification into two multivariate normal distributions with different covariance matrices. The Ann Math Sta. 1962;33:420-431.
22. Mossman D. Three-way rocs. Med Deci Mak. 1999;19:78-89.
23. Srinivasan A. Note on the location of optimal classifiers in n-dimensional ROC space. 1999.
24. Hand DJ, Till RJ. A simple generalisation of the area under the ROC curve for multiple class classification problems. Mach lear, 2001;45:171-186.
25. Ferri C, Hernandez-Orallo J, Salido MA. Volume under the ROC surface for multi-class problems. In Eur conf mach lear, Spri, Berlin, Heidelberg, 2003:108-120.
26. He X, Frey EC. The meaning and use of the volume under a three-class ROC surface (VUS). IEEE Trans Med Imag. 2008;27(5):577-588.
27. Kang L, Tian L. Estimation of the volume under the ROC surface with three ordinal diagnostic categories. Comp Stat & Dat Ana. 2013;62:39-51.
28. Liu S, Zhu H, Yi K, Sun X, Xu W, Wang C. Fast and Unbiased Estimation of Volume Under Ordered Three-Class ROC Surface (VUS) With Continuous or Discrete Measurements. IEEE Access. 2020;8:136206-136222.
29. Youden WJ. Index for rating diagnostic tests. Cancer. 1950;3:32-35.
30. Perkins NJ, Schisterman EF. The Youden Index and the optimal cut‐point corrected for measurement error. J Math Meth in Biosci. 2005;47:428-441.
31. Perkins NJ, Schisterman EF. The inconsistency of “optimal” cutpoints obtained using two criteria based on the receiver operating characteristic curve. Ame J Epid. 2006;163:670-675.
32. Leal12 J, Oliveira M, Sanches12 JM. Analysis of cut-off criteria in ROC curve for endarterectomy decision making. 2011.
33. da Rocha Neto AR, de Alencar Barreto G. On the application of ensembles of classifiers to the diagnosis of pathologies of the vertebral column: A comparative analysis. IEEE Lat Am Trans. 2009;7:487-496.
IssueVol 8 No 2 (2022) QRcode
SectionOriginal Article(s)
DOI https://doi.org/10.18502/jbe.v8i2.10418
Mixture models Bi-Normal ROC Multivariate ROC Multi-class classification Area under the Curve.

Rights and permissions
Creative Commons License This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
How to Cite
Gajjalavari S, R VV. Multi-Class Classification using Mixtures of Univariate and Multivariate ROC Curves. JBE. 2022;8(2):208-233.