Survival Time Analysis of Multiple Myeloma Patients using Type 1 Censored Exponential Distribution Parameter Estimation

Cahya Arsyika Wahyu Subekti; Inas Nilasari; Devi Mufidah Syarif; Idrus Syahzaqi; Ardi Kurniawan

doi:10.31764/jtam.v9i3.32179

Survival Time Analysis of Multiple Myeloma Patients using Type 1 Censored Exponential Distribution Parameter Estimation

Cahya Arsyika Wahyu Subekti, Inas Nilasari, Devi Mufidah Syarif, Idrus Syahzaqi, Ardi Kurniawan

Abstract

Multiple myeloma is a type of blood cancer that attacks plasma cells in the bone marrow and affects the immune system. This study analyzes the survival time of patients with multiple myeloma using Type 1 censored exponential distributed parameter estimation. The data, consisting of 47 patients (35 uncensored and 12 censored), were tested for exponential distribution fit using the Anderson-Darling test, yielding a p-value of 0.495, confirming the suitability of the exponential model. The maximum likelihood estimation method was applied, resulting in a parameter estimate (θ ̂) of approximately 54.028 days, representing the mean survival time. Hypothesis testing and confidence intervals were conducted, with the 95% confidence interval for θ_0 ranging between 32 and 53 days. The findings suggest that the exponential distribution effectively models the survival data, providing insights into patient survival trends and supporting clinical decision-making.

Keywords

Multiple Myeloma; Survival Analysis; Exponential Distribution; Type 1 Censoring; Parameter Estimation.

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References

Accioly, M. A., Bezerra, H. de M., Barreto, L. S., Castro, M. F., & da Silva, H. B. (2024). Multiple myeloma: A literature review. In Navigating through the knowledge of education V.2. Seven Editora. https://doi.org/10.56238/sevened2024.015-016

Akbar, M. H., Ali, S., Shah, I., & Alqifari, H. N. (2024). Sample size determination for time-to-event endpoints in randomized selection trials with generalized exponential distribution. Heliyon, 10(5). https://doi.org/10.1016/j.heliyon.2024.e27013

Alomari, H. M. (2023). A Comparison of Four Methods of Estimating the Scale Parameter for the Exponential Distribution. Journal of Applied Mathematics and Physics, 11(10), 2838–2847. https://doi.org/10.4236/jamp.2023.1110186

Alvares, D., Barrett, J. K., Mercier, F., Roumpanis, S., Yiu, S., Castro, F., Schulze, J., & Zhu, Y. (2024). A Bayesian joint model of multiple nonlinear longitudinal and competing risks outcomes for dynamic prediction in multiple myeloma: Joint estimation and corrected two-stage approaches. Statistics in Medicine, 44(3-4), Article e10322. https://doi.org/10.1002/sim.10322

Ashour, S. K., & Nassar, M. M. A. (2014). Analysis of exponential distribution under adaptive type-I progressive hybrid censored competing risks data. Pakistan Journal of Statistics and Operation Research, 10(2), 229–245. https://doi.org/10.18187/pjsor.v10i2.705

Bakker, L., Thielen, F., Redekop, W., Groot, C. U. de, & Blommestein, H. (2023). Extrapolating empirical long-term survival data: The impact of updated follow-up data and parametric extrapolation methods on survival estimates in multiple myeloma. BMC Medical Research Methodology, 23(1), 132. https://doi.org/10.1186/s12874-023-01952-2

Barrajón, E., & Barrajón, L. (2020). Effect of right censoring bias on survival analysis. ArXiv Preprint ArXiv:2012.08649. http://arxiv.org/abs/2012.08649

Caroni, F., Sammartano, V., Pacelli, P., Sicuranza, A., Malchiodi, M., Dragomir, A., Ciofini, S., Raspadori, D., Bocchia, M., & Gozzetti, A. (2025). Minimal residual disease significance in multiple myeloma patients treated with anti-CD38 monoclonal antibodies. Pharmaceuticals, 18(2), 159. https://doi.org/10.3390/ph18020159

Dimopoulos, M. A., Merlini, G., Bridoux, F., Leung, N., Mikhael, J., Harrison, S. J., Kastritis, E., Garderet, L., Gozzetti, A., Van De Donk, N. W. C. J., Weisel, K. C., Badros, A. Z., Beksac, M., Hillengass, J., Mohty, M., Ho, J., Ntanasis-Stathopoulos, I., Mateos, M.-V., Richardson, P., … Terpos, E. (2023). Management of multiple myeloma-related renal impairment: recommendations from the International Myeloma Working Group. The Lancet Oncology, 24(7), e293–e311. https://doi.org/10.1016/S1470-2045(23)00223-1

Dutta, S., & Kayal, S. (2022a). Estimation of parameters of the logistic exponential distribution under progressive type-I hybrid censored sample. Quality Technology and Quantitative Management, 19(2), 234–258. https://doi.org/10.1080/16843703.2022.2027601

Dutta, S., & Kayal, S. (2022b). Estimation of parameters of the logistic exponential distribution under progressive type-I hybrid censored sample. Quality Technology and Quantitative Management, 19(2), 234–258. https://doi.org/10.1080/16843703.2022.2027601

Etikan, İ. (2018). Choosing statistical tests for survival analysis. Biometrics & Biostatistics International Journal, 7(5), 477–481. https://doi.org/10.15406/bbij.2018.07.00249

Faiman, B. M., Mangan, P., Spong, J., & Tariman, J. D. (2011). Renal complications in multiple myeloma and related disorders: Survivorship care plan of the international myeloma foundation nurse leadership board. Clinical Journal of Oncology Nursing, 15(SUPPL.), 66–76. https://doi.org/10.1188/11.CJON.S1.66-76

Indrayan, A., & Tripathi, C. B. (2022). Survival Analysis: Where, Why, What and How?. Indian Pediatrics, 59(1), 74–79. https://doi.org/10.1007/s13312-022-2425-5

Insan Firsawan, D., Nessyana Debataraja, N., & Wira Rizki, S. (2022). Analisis survival pada data tersensor tipe I dengan metode Kaplan Meier. Buletin Ilmiah Matematika, Statistika dan Terapannya (Bimaster), 11(1), 19–26. https://doi.org/10.26418/bbimst.v11i1.51592

Jäntschi, L., & Bolboacă, S. D. (2018). Computation of probability associated with Anderson-Darling statistic. Mathematics, 6(6). https://doi.org/10.3390/math6060088

Jayakodi, G., Sundaram, N., & Venkatesan, P. (2022). An application of Exponential-Lindley distribution in modelling cancer survival data. Indian Journal of Science and Technology, 15(46), 2579–2588. https://doi.org/10.17485/IJST/v15i46.1870

Jia, X., Nadarajah, S., & Guo, B. (2018). Exact Inference on Weibull Parameters with Multiply Type-I Censored Data. IEEE Transactions on Reliability, 67(2), 432–445. https://doi.org/10.1109/TR.2018.2799967

Job, O., & Solomon Ogunsanya, A. (2022). Weibull Log Logistic {Exponential} distribution: Some properties and application to survival data. International Journal of Statistical Distributions and Applications, 8(1), 1–13. https://doi.org/10.11648/j.ijsd.20220801.11

Lin, Y. H., Chen, S. Y., Lin, P. H., Tai, A. S., Pan, Y. C., Hsieh, C. E., & Lin, S. H. (2020). Assessing user retention of a mobile app: Survival analysis. JMIR MHealth and UHealth, 8(11). https://doi.org/10.2196/16309

Magister, M. W., Pendidikan, A., Kristen, U., & Wacana, S. (2023). Pendekatan penelitian pendidikan: Metode penelitian kualitatif, metode penelitian kuantitatif dan metode penelitian kombinasi (mixed method). Jurnal Pendidikan Tambusai, 7(1), 2896–2910. https://jptam.org/index.php/jptam/article/view/6187

Makalic, E., & Schmidt, D. F. (2021). Minimum message length inference of the exponential distribution with type i censoring. Entropy, 23(11). https://doi.org/10.3390/e23111439

Mamudu, L., & Tsokos, C. P. (2020). Parametric and Non-Parametric Analysis of the Survival Times of Patients with Multiple Myeloma Cancer. Open Journal of Applied Sciences, 10(04), 118–134. https://doi.org/10.4236/ojapps.2020.104010

Mamudu, L., & Tsokos, C. P. (2021). A New Statistical Modeling Approach for Survival Analysis of Cancer Patients-Multiple Myeloma Cancer. Open Journal of Applied Sciences, 11, 365–378. https://doi.org/10.4236/ojapps.2021.114027

McDonald, Z., Taylor, P., Liyasova, M., Liu, Q., & Ma, B. (2021). Mass Spectrometry Provides a Highly Sensitive Noninvasive Means of Sequencing and Tracking M-Protein in the Blood of Multiple Myeloma Patients. Journal of Proteome Research, 20(8), 4176–4185. https://doi.org/10.1021/acs.jproteome.0c01022

Nayabuddin. (2025). Estimation of Parameters of Lomax Exponential Distribution. International Journal of Analysis and Applications, 23. https://doi.org/10.28924/2291-8639-23-2025-16

Pradana, A. N. I., & Sofro, A. (2019). Penerapan analisis survival pada pasien multiple myeloma. Jurnal Ilmiah Matematika (MATHunesa), 7(2), 46–50. https://ejournal.unesa.ac.id/index.php/mathunesa/article/view/28314

Proust, M. (2024). Reliability and Survival Methods. https://www.jmp.com/thirdpartysoftware.

Pushpanjali, K., & Vijayalakshmi, G. (2022). On some aspects of an exponential distribution as life model approach. International Journal of Statistics and Applied Mathematics, 7(6), 129–133. https://doi.org/10.22271/maths.2022.v7.i6b.907

Raje, N., Singhal, S., Lurie, R. H., Somlo, G., Stockerl-Goldstein, K., Treon, S. P., Weber, D., Yahalom, J., Gasparetto, C., Hernandez-Ilizaliturri, F., Ann Huff, C., Krishnan, A. Y., Liedtke, M., Lunning, M., Buffett, P., Meredith, R., Anderson, K. C., Alsina, M., Sybil Biermann, J., … Shead, D. A. (2014). NCCN Guidelines Index Multiple Myeloma Table of Contents Discussion NCCN Guidelines Panel Disclosures NCCN Guidelines Version 2.2014 Multiple Myeloma-Panel Members.

Su, S. (2015). Flexible modelling of survival curves for censored data. Journal of Statistical Distributions and Applications, 3(1). https://doi.org/10.1186/s40488-016-0045-0

Triana, Y., & Purwadi, J. (2019). Exponential Distribution Parameter Estimation with Bayesian SELF Method in Survival Analysis. Journal of Physics: Conference Series, 1373(1). https://doi.org/10.1088/1742-6596/1373/1/012050

Turkson, A. J., Ayiah-Mensah, F., & Nimoh, V. (2021). Handling censoring and censored data in survival analysis: A standalone systematic literature review. International Journal of Mathematics and Mathematical Sciences, 2021(1), 1–17. https://doi.org/10.1155/2021/9307475

Utsu, Y., Isono, Y., Masuda, S. I., Arai, H., Shimoji, S., Matsumoto, R., Tsushima, T., Tanaka, K., Matsuo, K., Kimeda, C., Konno, S., Yano, Y., Kuramoto, N., & Aotsuka, N. (2025). Time-dependent recovery of renal impairment in patients with newly diagnosed multiple myeloma. Annals of Hematology, 104(1), 573–579. https://doi.org/10.1007/s00277-025-06201-8

Wijnands, C., Karel, P. G. A., Gloerich, J., Armony, G., Tzasta, A., de Kat Angelino, C. M., Di Stefano, L., Bonifay, V., Luider, T. M., VanDuijn, M. M., Croockewit, S. J., de Kort, E. A., Castelijn, D. A. R., Stege, C. A. M., Wessels, H. J. C. T., van Gool, A. J., van de Donk, N. W. C. J., & Jacobs, J. F. M. (2025). Monitoring M-Protein, therapeutic antibodies, and polyclonal antibodies in a multiparametric mass spectrometry assay provides insight into therapy response kinetics in patients with multiple myeloma. Pharmaceutics, 17(1), 135. https://doi.org/10.3390/pharmaceutics17010135

DOI: https://doi.org/10.31764/jtam.v9i3.32179