Variable Selection for Length-Biased and Interval-Censored Failure Time Data
Abstract
:1. Introduction
2. Notation, Model and Penalized Likelihood
3. Estimation Procedure
4. Asymptotic Properties
- (C1)
- The true regression parameter lies in a compact set in , and the true cumulative baseline hazard function is continuously differentiable and positive with for all , where is the union of the support of and . In addition, we assume that .
- (C2)
- The covariate vector is bounded with probability one and the covariance matrix of is positive definite.
- (C3)
- The number of examination times, M, is positive and . Additionally, there exists a positive constant such that . Furthermore, there exists a probability measure in such that the bivariate distribution function of conditional on is dominated by , and its Radon–Nikodym derivative, denoted by , can be expanded to a positive and has twice-continuous derivatives with respect to u and v when and are continuously differentiable with respect to .
- 1.
- (Sparsity) ;
- 2.
- (Asymptotic normality) , where is the upper-left sub-matrix of the efficient Fisher information matrix for β, denoted by .
5. A Simulation Study
6. An Application
6.1. Background and Analysis Methods
6.2. Results
7. Discussion and Conclusions
8. Suggestions for Future Work
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
EM | expectation-maximization |
LASSO | the least absolute shrinkage and selection operator penalty |
ALASSO | the adaptive LASSO penalty |
SCAD | the smoothly clipped absolute deviation penalty |
SELO | the seamless- penalty |
SICA | the smooth integration of counting and absolute deviation penalty |
MCP | the minimax concave penalty |
BAR | the broken adaptive ridge penalty |
Appendix A. Proofs of The Asymptotic Results
Appendix A.1. Proof of Theorem 1
Appendix A.2. Proof of Theorem 2
Appendix B. The Penalized Conditional Likelihood Method
References
- Sun, J. The Statistical Analysis of Interval-Censored Failure Time Data; Springer: New York, NY, USA, 2006. [Google Scholar]
- Huang, J. Efficient estimation for the proportional hazards model with interval censoring. Ann. Stat. 1996, 24, 540–568. [Google Scholar] [CrossRef]
- Shen, X. Proportional odds regression and sieve maximum likelihood estimation. Biometrika 1998, 85, 165–177. [Google Scholar] [CrossRef]
- Zeng, D.; Cai, J.; Shen, Y. Semiparametric additive risks model for interval-censored data. Stat. Sin. 2006, 16, 287–302. [Google Scholar]
- Zhang, Y.; Hua, L.; Huang, J. A spline-based semiparametric maximum likelihood estimation method for the Cox model with interval-censored data. Scand. J. Stat. 2010, 37, 338–354. [Google Scholar] [CrossRef]
- Wang, L.; McMahan, C.S.; Hudgens, M.G.; Qureshi, Z.P. A flexible, computationally efficient method for fitting the proportional hazards model to interval-censored data. Biometrics 2016, 72, 222–231. [Google Scholar] [CrossRef] [PubMed]
- Zeng, D.; Mao, L.; Lin, D.Y. Maximum likelihood estimation for semiparametric transformation models with interval-censored data. Biometrika 2016, 103, 253–271. [Google Scholar] [CrossRef]
- Zhou, Q.; Hu, T.; Sun, J. A sieve semiparametric maximum likelihood approach for regression analysis of bivariate interval-censored failure time data. J. Am. Stat. Assoc. 2017, 112, 664–672. [Google Scholar] [CrossRef]
- Prorok, P.C.; Andriole, G.L.; Bresalier, R.S.; Buys, S.S.; Chia, D.; Crawford, E.D.; Fogel, R.; Gelmann, E.P.; Gilbert, F.; Gohagan, J.K. Design of the prostate, lung, colorectal and ovarian (PLCO) cancer screening trial. Control. Clin. Trials 2000, 21, 273S–309S. [Google Scholar] [CrossRef]
- Wang, M.C. Nonparametric estimation from cross-sectional survival data. J. Am. Stat. Assoc. 1991, 86, 130–143. [Google Scholar] [CrossRef]
- Shen, Y.; Ning, J.; Qin, J. Analyzing length-biased data with semiparametric transformation and accelerated failure time models. J. Am. Stat. Assoc. 2009, 104, 1192–1202. [Google Scholar] [CrossRef]
- Ning, J.; Qin, J.; Shen, Y. Semiparametric accelerated failure time model for length-biased data with application to dementia study. Stat. Sin. 2014, 24, 313–333. [Google Scholar] [CrossRef] [PubMed]
- Qin, J.; Shen, Y. Statistical methods for analyzing right-censored length-biased data under Cox model. Biometrics 2010, 66, 382–392. [Google Scholar] [CrossRef] [PubMed]
- Qin, J.; Ning, J.; Liu, H.; Shen, Y. Maximum likelihood estimations and EM algorithms with length-biased data. J. Am. Stat. Assoc. 2011, 106, 1434–1449. [Google Scholar] [CrossRef] [PubMed]
- Gao, F.; Chan, K.C.G. Semiparametric regression analysis of length-biased interval-censored data. Biometrics 2019, 75, 121–132. [Google Scholar] [CrossRef]
- Shen, P.S.; Peng, Y.; Chen, H.J.; Chen, C.M. Maximum likelihood estimation for length-biased and interval-censored data with a nonsusceptible fraction. Lifetime Data Anal. 2022, 28, 68–88. [Google Scholar] [CrossRef]
- Tibshirani, R. Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B (Methodol.) 1996, 58, 267–288. [Google Scholar] [CrossRef]
- Fan, J.; Li, R. Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 2001, 96, 1348–1360. [Google Scholar] [CrossRef]
- Zou, H. The adaptive lasso and its oracle properties. J. Am. Stat. Assoc. 2006, 101, 1418–1429. [Google Scholar] [CrossRef]
- Lv, J.; Fan, Y. A unified approach to model selection and sparse recovery using regularized least squares. Ann. Stat. 2009, 37, 3498–3528. [Google Scholar] [CrossRef]
- Dicker, L.; Huang, B.; Lin, X. Variable selection and estimation with the seamless-L-0 penalty. Stat. Sin. 2013, 23, 929–962. [Google Scholar]
- Zhang, C.H. Nearly unbiased variable selection under minimax concave penalty. Ann. Stat. 2010, 38, 894–942. [Google Scholar] [CrossRef]
- Liu, Z.; Li, G. Efficient regularized regression with penalty for variable selection and network construction. Comput. Math. Methods Med. 2016, 2016, 3456153. [Google Scholar] [CrossRef]
- Dai, L.; Chen, K.; Sun, Z.; Liu, Z.; Li, G. Broken adaptive ridge regression and its asymptotic properties. J. Multivar. Anal. 2018, 168, 334–351. [Google Scholar] [CrossRef] [PubMed]
- Fan, J.; Li, R.; Zhang, C.H.; Zou, H. Statistical Foundations of Data Science; Chapman and Hall/CRC: New York, NY, USA, 2020. [Google Scholar]
- Garavand, A.; Salehnasab, C.; Behmanesh, A.; Aslani, N.; Zadeh, A.; Ghaderzadeh, M. Efficient Model for Coronary Artery Disease Diagnosis: A Comparative Study of Several Machine Learning Algorithms. J. Healthc. Eng. 2022, 2022, 5359540. [Google Scholar] [CrossRef] [PubMed]
- Hosseini, A.; Eshraghi, M.A.; Taami, T.; Sadeghsalehi, H.; Hoseinzadeh, Z.; Ghaderzadeh, M.; Rafiee, M. A mobile application based on efficient lightweight CNN model for classification of B-ALL cancer from non-cancerous cells: A design and implementation study. Inform. Med. Unlocked 2023, 39, 101244. [Google Scholar] [CrossRef]
- Garavand, A.; Behmanesh, A.; Aslani, N.; Sadeghsalehi, H.; Ghaderzadeh, M. Towards Diagnostic Aided Systems in Coronary Artery Disease Detection: A Comprehensive Multiview Survey of the State of the Art. Int. J. Intell. Syst. 2023, 2023, 6442756. [Google Scholar] [CrossRef]
- Ghaderzadeh, M.; Aria, M. Management of Covid-19 Detection Using Artificial Intelligence in 2020 Pandemic. In Proceedings of the ICMHI ’21: 5th International Conference on Medical and Health Informatics, Kyoto, Japan, 14–16 May 2021; pp. 32–38. [Google Scholar]
- Chen, L.P. Variable selection and estimation for the additive hazards model subject to left-truncation, right-censoring and measurement error in covariates. J. Stat. Comput. Simul. 2020, 90, 3261–3300. [Google Scholar] [CrossRef]
- He, D.; Zhou, Y.; Zou, H. High-dimensional variable selection with right-censored length-biased data. Stat. Sin. 2020, 30, 193–215. [Google Scholar] [CrossRef]
- Li, C.; Pak, D.; Todem, D. Adaptive lasso for the Cox regression with interval censored and possibly left truncated data. Stat. Methods Med. Res. 2020, 29, 1243–1255. [Google Scholar] [CrossRef]
- Withana Gamage, P.; McMahan, C.; Wang, L. Variable selection in semiparametric nonmixture cure model with interval-censored failure time data. Stat. Med. 2019, 38, 3026–3039. [Google Scholar]
- Li, S.; Peng, L. Instrumental Variable Estimation of Complier Causal Treatment Effect with Interval-Censored Data. Biometrics 2023, 79, 253–263. [Google Scholar] [CrossRef]
- Withana Gamage, P.; McMahan, C.; Wang, L. A flexible parametric approach for analyzing arbitrarily censored data that are potentially subject to left truncation under the proportional hazards model. Lifetime Data Anal. 2023, 29, 188–212. [Google Scholar] [CrossRef]
- Huang, C.Y.; Qin, J. Semiparametric estimation for the additive hazards model with left-truncated and right-censored data. Biometrika 2013, 100, 877–888. [Google Scholar] [CrossRef]
- Turnbull, B.W. The empirical distribution function with arbitrarily grouped, censored and truncated data. J. R. Stat. Soc. Ser. (Methodol.) 1976, 38, 290–295. [Google Scholar] [CrossRef]
- Zhang, H.H.; Lu, W. Adaptive Lasso for Cox’s proportional hazards model. Biometrika 2007, 94, 691–703. [Google Scholar] [CrossRef]
- Li, S.; Wu, Q.; Sun, J. Penalized estimation of semiparametric transformation models with interval-censored data and application to Alzheimer’s disease. Stat. Methods Med. Res. 2020, 29, 2151–2166. [Google Scholar] [CrossRef] [PubMed]
- Zou, H.; Li, R. One-step sparse estimates in nonconcave penalized likelihood models. Ann. Stat. 2008, 36, 1509–1533. [Google Scholar]
- Andriole, G.L.; Crawford, E.D.; Grubb, R.L.; Buys, S.S.; Chia, D.; Church, T.R.; Fouad, M.N.; Isaacs, C.; Prorok, P. Prostate cancer screening in the randomized prostate, lung, colorectal, and ovarian cancer screening trial: Mortality results after 13 years of follow-up. J. Natl. Cancer Inst. 2012, 104, 125–132. [Google Scholar] [CrossRef] [PubMed]
- Meister, K. Risk Factors for Prostate Cancer; American Council on Science and Health: New York, NY, USA, 2002. [Google Scholar]
- Pierce, B.L. Why are diabetics at reduced risk for prostate cancer? A review of the epidemiologic evidence. Urol. Oncol. Semin. Orig. Investig. 2012, 30, 735–743. [Google Scholar] [CrossRef]
- Lu, T.; Li, S.; Sun, L. Combined estimating equation approaches for the additive hazards model with left-truncated and interval-censored data. Lifetime Data Anal. 2023, 29, 672–697. [Google Scholar] [CrossRef]
- Sun, L.; Li, S.; Wang, L.; Song, X.; Sui, X. Simultaneous variable selection in regression analysis of multivariate interval-censored data. Biometrics 2022, 78, 1402–1413. [Google Scholar] [CrossRef] [PubMed]
- Murphy, S.A.; Van Der Vaart, A.W. On profile likelihood. J. Am. Stat. Assoc. 2000, 95, 449–465. [Google Scholar] [CrossRef]
- Huang, J.; Wellner, J.A. Interval censored survival data: A review of recent progress. In Proceedings of the First Seattle Symposium in Biostatistics; Lin, D.Y., Fleming, T.R., Eds.; Springer: New York, NY, USA, 1997; pp. 123–169. [Google Scholar]
Method | Penalty | ||||||
---|---|---|---|---|---|---|---|
TP | FP | MMSE (SD) | TP | FP | MMSE (SD) | ||
Proposed method | LASSO | 3 | 1.27 | 0.163 (0.107) | 3 | 1.15 | 0.092(0.060) |
ALASSO | 3 | 0.12 | 0.051 (0.058) | 3 | 0.11 | 0.025 (0.022) | |
SCAD | 3 | 0.09 | 0.025 (0.046) | 3 | 0.07 | 0.012 (0.014) | |
SELO | 3 | 0.14 | 0.030 (0.042) | 3 | 0.07 | 0.013 (0.016) | |
SICA | 3 | 0.16 | 0.030 (0.042) | 3 | 0.07 | 0.013 (0.015) | |
MCP | 3 | 0.16 | 0.025 (0.042) | 3 | 0.07 | 0.012 (0.015) | |
BAR | 3 | 0.13 | 0.032 (0.040) | 3 | 0.11 | 0.014 (0.016) | |
Oracle | - | - | 0.024 (0.041) | - | - | 0.011 (0.014) | |
Without VS | - | - | 0.057 (0.055) | - | - | 0.026 (0.020) | |
PCL method | LASSO | 3 | 1.42 | 0.163 (0.162) | 3 | 1.29 | 0.089 (0.074) |
ALASSO | 3 | 0.20 | 0.081 (0.108) | 3 | 0.13 | 0.033 (0.041) | |
SCAD | 3 | 0.12 | 0.056 (0.116) | 3 | 0.08 | 0.025 (0.041) | |
SELO | 3 | 0.18 | 0.065 (0.087) | 3 | 0.10 | 0.021 (0.038) | |
SICA | 3 | 0.18 | 0.066 (0.087) | 3 | 0.10 | 0.021 (0.037) | |
MCP | 3 | 0.13 | 0.054 (0.116) | 3 | 0.10 | 0.025 (0.043) | |
BAR | 3 | 0.19 | 0.065 (0.089) | 3 | 0.13 | 0.023 (0.035) |
Method | Penalty | ||||||
---|---|---|---|---|---|---|---|
TP | FP | MMSE (SD) | TP | FP | MMSE (SD) | ||
Proposed method | LASSO | 3 | 0.80 | 0.047 (0.040) | 3 | 0.73 | 0.024(0.024) |
ALASSO | 3 | 0.25 | 0.027 (0.024) | 3 | 0.12 | 0.009 (0.010) | |
SCAD | 3 | 0.36 | 0.030 (0.028) | 3 | 0.17 | 0.010 (0.011) | |
SELO | 3 | 0.23 | 0.017 (0.019) | 3 | 0.10 | 0.008 (0.009) | |
SICA | 2.99 | 0.22 | 0.017 (0.021) | 3 | 0.07 | 0.008 (0.008) | |
MCP | 2.98 | 0.25 | 0.018 (0.025) | 3 | 0.10 | 0.007 (0.009) | |
BAR | 3 | 0.22 | 0.015 (0.018) | 3 | 0.12 | 0.008 (0.009) | |
Oracle | - | - | 0.013 (0.014) | - | - | 0.007 (0.007) | |
Without VS | - | - | 0.043 (0.031) | - | - | 0.020 (0.013) | |
PCL method | LASSO | 2.93 | 1.18 | 0.065 (0.075) | 3 | 0.72 | 0.039 (0.040) |
ALASSO | 2.85 | 0.50 | 0.060 (0.062) | 2.98 | 0.05 | 0.015 (0.028) | |
SCAD | 2.86 | 0.55 | 0.076 (0.063) | 2.99 | 0.15 | 0.017 (0.026) | |
SELO | 2.87 | 0.47 | 0.053 (0.060) | 2.98 | 0.06 | 0.018 (0.028) | |
SICA | 2.88 | 0.48 | 0.053 (0.059) | 2.98 | 0.07 | 0.015 (0.027) | |
MCP | 2.84 | 0.43 | 0.072 (0.068) | 2.98 | 0.05 | 0.015 (0.028) | |
BAR | 2.87 | 0.37 | 0.052 (0.059) | 2.97 | 0.08 | 0.014 (0.028) |
Method | Penalty | ||||||
---|---|---|---|---|---|---|---|
TP | FP | MMSE (SD) | TP | FP | MMSE (SD) | ||
Proposed method | LASSO | 3 | 1.59 | 0.307 (0.143) | 3 | 1.55 | 0.184 (0.082) |
ALASSO | 3 | 0.45 | 0.085 (0.075) | 3 | 0.13 | 0.028 (0.030) | |
SCAD | 3 | 0.26 | 0.041 (0.073) | 3 | 0.15 | 0.011 (0.017) | |
SELO | 3 | 0.35 | 0.046 (0.051) | 3 | 0.12 | 0.014 (0.019) | |
SICA | 3 | 0.31 | 0.045 (0.051) | 3 | 0.12 | 0.015 (0.019) | |
MCP | 3 | 0.14 | 0.030 (0.043) | 3 | 0.13 | 0.012 (0.017) | |
BAR | 3 | 0.46 | 0.042 (0.044) | 3 | 0.23 | 0.015 (0.018) | |
Oracle | - | - | 0.026 (0.035) | - | - | 0.010 (0.017) | |
Without VS | - | - | 0.216 (0.148) | - | - | 0.087 (0.043) | |
PCL method | LASSO | 3 | 1.87 | 0.362 (0.223) | 3 | 1.33 | 0.220 (0.129) |
ALASSO | 3 | 0.85 | 0.114 (0.124) | 3 | 0.24 | 0.027 (0.046) | |
SCAD | 2.97 | 0.41 | 0.080 (0.134) | 3 | 0.14 | 0.028 (0.038) | |
SELO | 2.99 | 0.80 | 0.094 (0.106) | 3 | 0.20 | 0.025 (0.035) | |
SICA | 3 | 0.50 | 0.091 (0.112) | 3 | 0.18 | 0.024 (0.035) | |
MCP | 2.98 | 0.44 | 0.073 (0.146) | 3 | 0.15 | 0.021 (0.044) | |
BAR | 2.99 | 0.77 | 0.100 (0.136) | 3 | 0.37 | 0.026 (0.037) |
Method | Penalty | ||||||
---|---|---|---|---|---|---|---|
TP | FP | MMSE (SD) | TP | FP | MMSE (SD) | ||
Proposed method | LASSO | 3 | 1.54 | 0.394 (0.117) | 3 | 1.33 | 0.269 (0.091) |
ALASSO | 3 | 0.45 | 0.130 (0.076) | 3 | 0.21 | 0.037 (0.033) | |
SCAD | 3 | 0.29 | 0.038 (0.061) | 3 | 0.03 | 0.014 (0.023) | |
SELO | 3 | 0.27 | 0.049 (0.041) | 3 | 0.14 | 0.018 (0.022) | |
SICA | 3 | 0.27 | 0.048 (0.047) | 3 | 0.20 | 0.016 (0.020) | |
MCP | 3 | 0.14 | 0.028 (0.046) | 3 | 0.06 | 0.014 (0.017) | |
BAR | 3 | 0.58 | 0.045 (0.044) | 3 | 0.37 | 0.017 (0.020) | |
Oracle | - | - | 0.021 (0.030) | - | - | 0.011 (0.016) | |
Without VS | - | - | 0.637 (0.453) | - | - | 0.169 (0.070) | |
PCL method | LASSO | 3 | 1.74 | 0.469 (0.263) | 3 | 1.58 | 0.297 (0.139) |
ALASSO | 2.99 | 0.82 | 0.155 (0.160) | 3 | 0.38 | 0.037 (0.045) | |
SCAD | 2.99 | 0.61 | 0.101 (0.152) | 3 | 0.20 | 0.024 (0.035) | |
SELO | 2.99 | 0.70 | 0.088 (0.105) | 3 | 0.54 | 0.023 (0.051) | |
SICA | 2.99 | 0.65 | 0.089 (0.103) | 3 | 0.57 | 0.022 (0.051) | |
MCP | 2.99 | 0.56 | 0.093 (0.157) | 3 | 0.29 | 0.025 (0.041) | |
BAR | 3 | 1.02 | 0.097 (0.164) | 3 | 0.62 | 0.036 (0.042) |
Method | Covariate | LASSO | ALASSO | SCAD | SELO | SICA | MCP | BAR | Without VS |
---|---|---|---|---|---|---|---|---|---|
Proposed | Race | 0.332 (0.084 ) | 0.364 * (0.088) | 0.444 * (0.072) | 0.408 * (0.082) | 0.408 * (0.083) | 0.444 * (0.072) | 0.412 * (0.081) | 0.458 * (0.064) |
method | Education | 0.047 (0.027 ) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0.070 * (0.031) |
Cancer | 0.032 (0.029 ) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0.047 (0.032 ) | |
ProsCancer | 0.346 * (0.053 ) | 0.377 * (0.058) | 0.421 * (0.049) | 0.397 * (0.053) | 0.398 * (0.056) | 0.421 * (0.049) | 0.405 * (0.052) | 0.394 * (0.049) | |
Diabetes | −0.310 * (0.060 ) | −0.336 * (0.068) | −0.416 * (0.059) | −0.373 * (0.065) | −0.374 * (0.068) | −0.416 * (0.059) | −0.384 * (0.062) | −0.402 * (0.064) | |
Stroke | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | −0.244 * (0.109 ) | |
Gallblad | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | −0.064 (0.058 ) | |
PCL | Race | 0.307 * (0.087 ) | 0.399 * (0.114) | 0.420 * (0.102) | 0.385 * (0.102) | 0.394 * (0.105) | 0.420 * (0.102) | 0.377 * (0.114) | 0.430 * (0.067) |
method | Education | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | −0.005 (0.032 ) |
Cancer | 0.075 * (0.033 ) | 0.065 (0.049) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0.089 * (0.033 ) | |
ProsCancer | 0.360 * (0.062 ) | 0.407 * (0.062) | 0.453 * (0.062) | 0.436 * (0.063) | 0.441* (0.064) | 0.453 * (0.062) | 0.437 * (0.063) | 0.407 * (0.051) | |
Diabetes | −0.216 * (0.073 ) | −0.286 * (0.078) | −0.316 * (0.077) | −0.266 * (0.089) | −0.279 * (0.093) | −0.316 * (0.077) | −0.266 * (0.087) | −0.320 * (0.065 ) | |
Stroke | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | −0.011 (0.109) | |
Gallblad | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0 (-) | 0.090 (0.060) |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Feng, F.; Cheng, G.; Sun, J. Variable Selection for Length-Biased and Interval-Censored Failure Time Data. Mathematics 2023, 11, 4576. https://doi.org/10.3390/math11224576
Feng F, Cheng G, Sun J. Variable Selection for Length-Biased and Interval-Censored Failure Time Data. Mathematics. 2023; 11(22):4576. https://doi.org/10.3390/math11224576
Chicago/Turabian StyleFeng, Fan, Guanghui Cheng, and Jianguo Sun. 2023. "Variable Selection for Length-Biased and Interval-Censored Failure Time Data" Mathematics 11, no. 22: 4576. https://doi.org/10.3390/math11224576
APA StyleFeng, F., Cheng, G., & Sun, J. (2023). Variable Selection for Length-Biased and Interval-Censored Failure Time Data. Mathematics, 11(22), 4576. https://doi.org/10.3390/math11224576