Cluster validity indices for mixture hazards regression models

Yi-Wen Chang; Kang-Ping Lu; Shao-Tung Chang; Yi-Wen Chang; Kang-Ping Lu; Shao-Tung Chang

doi:10.3934/mbe.2020085

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 2: 1616-1636. doi: 10.3934/mbe.2020085

Previous Article Next Article

Research article Special Issues

Cluster validity indices for mixture hazards regression models

1.
Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan
2.
Department of Applied Statistics, National Taichung University of Science and Technology, Taichung, Taiwan

Received: 30 August 2019 Accepted: 22 November 2019 Published: 10 December 2019

In the analysis of survival data, the problems of competing risks arise frequently in medical applications where individuals fail from multiple causes. Semiparametric mixture regression models have become a prominent approach in competing risks analysis due to their flexibility and easy interpretation of resultant estimates. The literature presents several semiparametric methods on the estimations for mixture Cox proportional hazards models, but fewer works appear on the determination of the number of model components and the estimation of baseline hazard functions using kernel approaches. These two issues are important because both incorrect number of components and inappropriate baseline functions can lead to insufficient estimates of mixture Cox hazard models. This research thus proposes four validity indices to select the optimal number of model components based on the posterior probabilities and residuals resulting from the application of an EM-based algorithm on a mixture Cox regression model. We also introduce a kernel approach to produce a smooth estimate of the baseline hazard function in a mixture model. The effectiveness and the preference of the proposed cluster indices are demonstrated through a simulation study. An analysis on a prostate cancer dataset illustrates the practical use of the proposed method.

Keywords:

Citation: Yi-Wen Chang, Kang-Ping Lu, Shao-Tung Chang. Cluster validity indices for mixture hazards regression models[J]. Mathematical Biosciences and Engineering, 2020, 17(2): 1616-1636. doi: 10.3934/mbe.2020085

Related Papers:

[1]	Zindoga Mukandavire, Abba B. Gumel, Winston Garira, Jean Michel Tchuenche . Mathematical analysis of a model for HIV-malaria co-infection. Mathematical Biosciences and Engineering, 2009, 6(2): 333-362. doi: 10.3934/mbe.2009.6.333
[2]	Peter Witbooi, Gbenga Abiodun, Mozart Nsuami . A model of malaria population dynamics with migrants. Mathematical Biosciences and Engineering, 2021, 18(6): 7301-7317. doi: 10.3934/mbe.2021361
[3]	Pride Duve, Samuel Charles, Justin Munyakazi, Renke Lühken, Peter Witbooi . A mathematical model for malaria disease dynamics with vaccination and infected immigrants. Mathematical Biosciences and Engineering, 2024, 21(1): 1082-1109. doi: 10.3934/mbe.2024045
[4]	Yilong Li, Shigui Ruan, Dongmei Xiao . The Within-Host dynamics of malaria infection with immune response. Mathematical Biosciences and Engineering, 2011, 8(4): 999-1018. doi: 10.3934/mbe.2011.8.999
[5]	Zhong-Kai Guo, Hai-Feng Huo, Hong Xiang . Global dynamics of an age-structured malaria model with prevention. Mathematical Biosciences and Engineering, 2019, 16(3): 1625-1653. doi: 10.3934/mbe.2019078
[6]	Bassidy Dembele, Abdul-Aziz Yakubu . Controlling imported malaria cases in the United States of America. Mathematical Biosciences and Engineering, 2017, 14(1): 95-109. doi: 10.3934/mbe.2017007
[7]	Jinliang Wang, Jingmei Pang, Toshikazu Kuniya . A note on global stability for malaria infections model with latencies. Mathematical Biosciences and Engineering, 2014, 11(4): 995-1001. doi: 10.3934/mbe.2014.11.995
[8]	Khadiza Akter Eme, Md Kamrujjaman, Muntasir Alam, Md Afsar Ali . Vaccination and combined optimal control measures for malaria prevention and spread mitigation. Mathematical Biosciences and Engineering, 2025, 22(8): 2039-2071. doi: 10.3934/mbe.2025075
[9]	Zhilan Feng, Carlos Castillo-Chavez . The influence of infectious diseases on population genetics. Mathematical Biosciences and Engineering, 2006, 3(3): 467-483. doi: 10.3934/mbe.2006.3.467
[10]	Kento Okuwa, Hisashi Inaba, Toshikazu Kuniya . An age-structured epidemic model with boosting and waning of immune status. Mathematical Biosciences and Engineering, 2021, 18(5): 5707-5736. doi: 10.3934/mbe.2021289

Abstract

References

[1]	D. R. Cox, Regression models and life-tables with discussion, J. R. Stat. Soc. Ser. B-Stat. Methodol., 34 (1972), 187-220.
[2]	G. Escarela, and R. Bowater, Fitting a semi-parametric mixture model for competing risks in survival data, Commun. Stat.-Theory Methods, 37 (2008), 277-293.
[3]	S. C. Cheng, J. P. Fine, and L. J. Wei, Prediction of cumulative incidence function under the proportional hazards model, Biometrics 54 (1998), 219-228.
[4]	G. J. McLachlan and D. Peel, Finite mixture models, Wiley Series in Probability and Statistics: Applied Probability and Statistics, Wiley-Interscience, New York, 2000.
[5]	S. K. Ng and G. J. McLachlan, An EM-based semi-parametric mixture model approach to the regression analysis of competing risks data, Stat. Med., 22 (2003), 1097-1111.
[6]	I. S. Chang, C. A. Hsiung, C. C. Wen and W. C. Yang, Non-parametric maximum likelihood estimation in a semiparametric mixture model for competing risks data, Scand. J. Stat., 34 (2007), 870-895.
[7]	W. Lu and L. Peng, Semiparametric analysis of mixture regression models with competing risks data, Lifetime Data Anal., 14 (2008), 231-252.
[8]	S. Choi and X. Huang, Maximum likelihood estimation of semiparametric mixture component models for competing risks data, Biometrics, 70 (2014), 588-598.
[9]	A. K. Jain and R. C. Dubes, Algorithms for clustering data, Prentice-Hall, Englewood Cliffs, NJ, 1988.
[10]	Y. G. Tang, F. C. Sun and Z. Q. Sun, Improved validation index for fuzzy clustering, in American Control Conference, June 8-10, 2005. Portland, OR, USA, (2005), 1120-1125.
[11]	W. Wang and Y. Zhang, On fuzzy cluster validity indices, Fuzzy Sets Syst., 158 (2007), 2095-2117.
[12]	K. L. Wu, M. S. Yang and J. N. Hsieh, Robust cluster validity indexes, Pattern Recognit., 42 (2009), 2541-2550.
[13]	K. L. Zhou, S. Ding, C. Fu and S. L. Yang, Comparison and weighted summation type of fuzzy cluster validity indices, Int. J. Comput. Commun. Control, 9 (2014), 370-378.
[14]	J. M. Henson, S. P. Reise and K. H. Kim, Detecting mixtures from structural model differences using latent variable mixture modeling: A comparison of relative model fit statistics, Struct. Equ. Modeling, 14 (2007), 202-226.
[15]	J. R. Busemeyer, J. Wang, J. T. Townsend and A. Eidels, The Oxford Handbook of Computational and Mathematical Psychology. Oxford University Press 2015.
[16]	R. Bender, T. Augustin and M. Blettner, Generating survival times to simulate Cox proportional hazards models, Stat. Med., 24 (2005), 1713-1723.
[17]	P. Royston, Estimating a smooth baseline hazard function for the Cox model. Technical Report No. 314. University College London, London 2011. Available from: https://www.semanticscholar.org/paper/Estimating-a-smooth-baseline-hazard-function-for-Royston/2f329b48f674a74253eb428b71ff237365fd4051.
[18]	A. Guilloux, S. Lemler and M. L. Taupin, Adaptive kernel estimation of the baseline function in the Cox model with high-dimensional covariates, J. Multivar. Anal., 148 (2016) 141-159.
[19]	M. Zhou, Empirical likelihood method in survival analysis, CRC Press 2016
[20]	I. Horova, J. Kolacek, and J. Zelinka, Kernel smoothing in Matlab theory and practice of kernel smoothing, World Scientific Publishing, 2012.
[21]	P. N. Patil, Bandwidth choice for nonparametric hazard rate estimation, J. Stat. Plan. Infer., 35 (1993), 15-30.
[22]	J. C. Bezdek, Numerical taxonomy with fuzzy sets, J. Math. Biol., 7 (1974), 57-71.
[23]	R. N. Dave, Validating fuzzy partition obtained through c-shells clustering, Pattern Recognit. Lett., 17 (1996), 613-623.
[24]	J. C. Bezdek, Cluster validity with fuzzy sets, Journal of Cybernetics, 3 (1974), 58-73. DOI: 10.1080/01969727308546047.
[25]	J. C. Dunn, Indices of partition fuzziness and the detection of clusters in large data sets, in Fuzzy Automata and Decision Processes, (ed. M.M. Gupta, Elsevier), NY, 1977.
[26]	D. P. Byar and S. B. Green, The choice of treatment for cancer patients based on covariate information: application to prostate cancer, Bull. Cancer, 67 (1980), 477-490.
[27]	D. F. Andrews and A. M. Herzberg, Data: a collection of problems from many fields for the student and research worker, Springer-Verlag, New York, (1985), 261-274.
[28]	R. Kay, Treatment effects in competing-risks analysis of prostate cancer data, Biometrics, 42 (1986), 203-211.

This article has been cited by:

1.	Yoram Vodovotz, Gregory Constantine, James Faeder, Qi Mi, Jonathan Rubin, John Bartels, Joydeep Sarkar, Robert H. Squires, David O. Okonkwo, Jörg Gerlach, Ruben Zamora, Shirley Luckhart, Bard Ermentrout, Gary An, Translational Systems Approaches to the Biology of Inflammation and Healing, 2010, 32, 0892-3973, 181, 10.3109/08923970903369867
2.	Jinhu Xu, Yicang Zhou, Hopf bifurcation and its stability for a vector-borne disease model with delay and reinfection, 2016, 40, 0307904X, 1685, 10.1016/j.apm.2015.09.007
3.	Jia Li, Modelling of transgenic mosquitoes and impact on malaria transmission, 2011, 5, 1751-3758, 474, 10.1080/17513758.2010.523122
4.	Kazeem O. Okosun, Ouifki Rachid, Nizar Marcus, Optimal control strategies and cost-effectiveness analysis of a malaria model, 2013, 111, 03032647, 83, 10.1016/j.biosystems.2012.09.008
5.	Liming Cai, Xuezhi Li, Necibe Tuncer, Maia Martcheva, Abid Ali Lashari, Optimal control of a malaria model with asymptomatic class and superinfection, 2017, 288, 00255564, 94, 10.1016/j.mbs.2017.03.003
6.	Liming Cai, Necibe Tuncer, Maia Martcheva, How does within-host dynamics affect population-level dynamics? Insights from an immuno-epidemiological model of malaria, 2017, 40, 01704214, 6424, 10.1002/mma.4466
7.	Li-Ming Cai, Abid Ali Lashari, Il Hyo Jung, Kazeem Oare Okosun, Young Il Seo, Mathematical Analysis of a Malaria Model with Partial Immunity to Reinfection, 2013, 2013, 1085-3375, 1, 10.1155/2013/405258
8.	Yang Li, Jia Li, Discrete-time model for malaria transmission with constant releases of sterile mosquitoes, 2019, 13, 1751-3758, 225, 10.1080/17513758.2018.1551580
9.	K.O. Okosun, Rachid Ouifki, Nizar Marcus, Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity, 2011, 106, 03032647, 136, 10.1016/j.biosystems.2011.07.006
10.	O.D. Makinde, K.O. Okosun, Impact of Chemo-therapy on Optimal Control of Malaria Disease with Infected Immigrants, 2011, 104, 03032647, 32, 10.1016/j.biosystems.2010.12.010
11.	Yijun Lou, Xiao-Qiang Zhao, The periodic Ross–Macdonald model with diffusion and advection, 2010, 89, 0003-6811, 1067, 10.1080/00036810903437804
12.	Yijun Lou, Xiao-Qiang Zhao, A Climate-Based Malaria Transmission Model with Structured Vector Population, 2010, 70, 0036-1399, 2023, 10.1137/080744438
13.	DYNAMICS OF STAGE-STRUCTURED DISCRETE MOSQUITO POPULATION MODELS, 2011, 1, 2156-907X, 53, 10.11948/2011005
14.	Hongyan Yin, Cuihong Yang, Xin'an Zhang, Jia Li, Dynamics of malaria transmission model with sterile mosquitoes, 2018, 12, 1751-3758, 577, 10.1080/17513758.2018.1498983
15.	Jia Li, Simple discrete-time malarial models, 2013, 19, 1023-6198, 649, 10.1080/10236198.2012.672566
16.	A. K. Misra, Anupama Sharma, Jia Li, A mathematical model for control of vector borne diseases through media campaigns, 2013, 18, 1553-524X, 1909, 10.3934/dcdsb.2013.18.1909
17.	Shangbing Ai, Jia Li, Junliang Lu, Mosquito-Stage-Structured Malaria Models and Their Global Dynamics, 2012, 72, 0036-1399, 1213, 10.1137/110860318
18.	Arman Rajaei, Amin Vahidi‐Moghaddam, Amir Chizfahm, Mojtaba Sharifi, Control of malaria outbreak using a non‐linear robust strategy with adaptive gains, 2019, 13, 1751-8652, 2308, 10.1049/iet-cta.2018.5292
19.	Yanyuan Xing, Zhiming Guo, Jian Liu, Backward bifurcation in a malaria transmission model, 2020, 14, 1751-3758, 368, 10.1080/17513758.2020.1771443
20.	Rashid Jan, Sultan Alyobi, Mustafa Inc, Ali Saleh Alshomrani, Muhammad Farooq, A robust study of the transmission dynamics of malaria through non-local and non-singular kernel, 2023, 8, 2473-6988, 7618, 10.3934/math.2023382
21.	S.Y. Tchoumi, H. Rwezaura, J.M. Tchuenche, A mathematical model with numerical simulations for malaria transmission dynamics with differential susceptibility and partial immunity, 2023, 3, 27724425, 100165, 10.1016/j.health.2023.100165
22.	Ge Zhang, Ying Peng, Ruoheng Wang, Cuihong Yang, Xin'an Zhang, The impact of releasing sterile mosquitoes on the dynamics of competition between different species of mosquitoes, 2024, 0, 1531-3492, 0, 10.3934/dcdsb.2024016

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)