带治愈组右删失数据的模型平均研究

doi:10.6040/j.issn.1671-9352.0.2022.465

摘要/Abstract

摘要：

在已有生存分析研究中, 大多直接假设响应变量与指定协变量的模型形式, 进而估计协变量效应, 但当模型假设错误时, 对应的结论可能是错误的。因此, 为了避免指定协变量构建模型引起的不准确性, 考虑使用一种基于模型平均方法的加速失效时间模型来对带治愈组的右删失数据进行刻画。在极大似然估计的框架下, 采用基于信息准则的模型选择和模型平均方法进行统计推断研究。数值模拟结果显示, 在带治愈组的右删失数据下基于模型平均方法的加速失效时间(accelerated failure time, AFT)模型估计及预测精度高于模型选择方法。最后通过黑色素瘤临床试验数据的分析, 对所提方法的可行性和实用性进行验证。

关键词: 右删失数据, 模型平均, 混合治愈模型, 加速失效时间模型, 极大似然估计

Abstract:

In the existing survival analysis studies, most of them directly assume the model form of response variables and specified covariates, and then estimate the covariate effect. However, when the assumption of the model is wrong, the corresponding conclusion may be wrong. Under the right-censored data with a cured group, in order to avoid the inaccuracy caused by the construction of the model with specified covariates, an accelerated failure time model (AFT model) based on the model averaging method is proposed. In the framework of maximum likelihood estimation, model selection and model averaging based on information criteria are used for statistical inference. The numerical simulation results show that under the right-censored data with a cured group, the estimation and prediction accuracy of the AFT model based on the model averaging method is higher than that of model selection method. Finally, the feasibility and practicability of the proposed method are verified by analysis of trial data on melanoma clinical.

Key words: right-censored data, model averaging, mixture cure model, accelerated failure time model, maximum likelihood estimation

中图分类号:

O212

王淑影,张亚男,程云飞,周丽芳. 带治愈组右删失数据的模型平均研究[J]. 《山东大学学报(理学版)》, 2024, 59(4): 108-116.

Shuying WANG,Yanan ZHANG,Yunfei CHENG,Lifang ZHOU. Model averaging study with a cured group right-censored data[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2024, 59(4): 108-116.

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参考文献 29

1	BOAG J W . Maximum likelihood estimates of the proportion of patients cured by cancer therapy[J]. Journal of the Royal Statistical Society: Series B (Methodological), 1949, 11 (1): 15- 44. doi: 10.1111/j.2517-6161.1949.tb00020.x
2	BERKSON J , GAGE R P . Survival curve for cancer patients following treatment[J]. Journal of the American Statistical Association, 1952, 47 (259): 501- 515. doi: 10.1080/01621459.1952.10501187
3	PENG Y , DEAR K B G , DENHAM J W . A generalized F mixture model for cure rate estimation[J]. Statistics in Medicine, 1998, 17 (8): 813- 830. doi: 10.1002/(SICI)1097-0258(19980430)17:8<813::AID-SIM775>3.0.CO;2-#
4	SY J P , TAYLOR J M G . Estimation in a Cox proportional hazards cure model[J]. Biometrics, 2000, 56 (1): 227- 236. doi: 10.1111/j.0006-341X.2000.00227.x
5	PENG Y W , DEAR K B G . A nonparametric mixture model for cure rate estimation[J]. Biometrics, 2000, 56 (1): 237- 243. doi: 10.1111/j.0006-341X.2000.00237.x
6	LI C S , TAYLOR J M G . A semi-parametric accelerated failure time cure model[J]. Statistics in Medicine, 2002, 21 (21): 3235- 3247. doi: 10.1002/sim.1260
7	ZHANG J J , PENG Y W . A new estimation method for the semiparametric accelerated failure time mixture cure model[J]. Statistics in Medicine, 2007, 26 (16): 3157- 3171. doi: 10.1002/sim.2748
8	BATES J M , GRANGER C W J . The combination of forecasts[J]. Journal of the Operational Research Society, 1969, 20 (4): 451- 468. doi: 10.1057/jors.1969.103
9	BUCKLAND S , BURNHAM K , AUGUSTIN N . Model selection: an integral part of inference[J]. Biometrics, 1997, 53 (2): 603- 618. doi: 10.2307/2533961
10	CLAESKENS G , HJORT N L . The focused information criterion[J]. Journal of the American Statistical Association, 2003, 98 (464): 900- 916. doi: 10.1198/016214503000000819
11	HANSEN B E . Least squares model averaging[J]. Econometrica, 2007, 75 (4): 1175- 1189. doi: 10.1111/j.1468-0262.2007.00785.x
12	ZHANG X Y , LIANG H . Focused information criterion and model averaging for generalized additive partial linear models[J]. The Annals of Statistics, 2011, 39 (1): 174- 200.
13	ZHANG X Y , ZOU G H , LIANG H . Model averaging and weight choice in linear mixed-effects models[J]. Biometrika, 2014, 101 (1): 205- 218. doi: 10.1093/biomet/ast052
14	张翊, 王秀丽. 响应变量缺失下线性模型的模型平均[J]. 山东师范大学学报(自然科学版), 2020, 35 (2): 178- 182.
	ZHANG Yi , WANG Xiuli . Model averaging procedure for linear model with missing responses[J]. Journal of Shandong Normal University (Natural Science), 2020, 35 (2): 178- 182.
15	胡国治, 程维虎, 曾婕. 协变量缺失下部分线性模型的模型选择和模型平均[J]. 应用数学学报, 2020, 43 (3): 535- 554.
	HU Guozhi , CHENG Weihu , ZENG Jie . Model selection and model averaging for partially linear models with missing covariates[J]. Acta Mathematicae Applicatae Sinica, 2020, 43 (3): 535- 554.
16	DING X W , XIE J H , YAN X D . Model averaging for multiple quantile regression with covariates missing at random[J]. Journal of Statistical Computation and Simulation, 2021, 91 (11): 2249- 2275. doi: 10.1080/00949655.2021.1890733
17	祝恒坤, 张海丽. 基于逆概率加权和插补的Mallows模型平均方法[J]. 系统科学与数学, 2022, 42 (4): 1032- 1059.
	ZHU Hengkun , ZHANG Haili . Mallows model averaging based on inverse probability weighting and imputation[J]. Journal of Systems Science and Mathematical Sciences, 2022, 42 (4): 1032- 1059.
18	王苗苗. 基于线性模型平均估计的置信区间[J]. 系统科学与数学, 2020, 40 (10): 1866- 1881.
	WANG Miaomiao . Confidence interval based on model average estimator for linear regression[J]. Journal of Systems Science and Mathematical Sciences, 2020, 40 (10): 1866- 1881.
19	尹潇潇, 鲁筠, 石磊. Meta分析中的频率模型平均估计[J]. 数理统计与管理, 2021, 40 (2): 233- 241.
	YIN Xiaoxiao , LU Jun , SHI Lei . Frequentist model averaging in Meta-analysis[J]. Journal of Applied Statistics and Management, 2021, 40 (2): 233- 241.
20	FENG Y , LIU Q , YAO Q , et al. Model averaging for nonlinear regression models[J]. Journal of Business & Economic Statistics, 2022, 40 (2): 785- 798.
21	LI J , YU T H , LYU J , et al. Semiparametric model averaging prediction for lifetime data via hazards regression[J]. Journal of the Royal Statistical Society Series C: Applied Statistics, 2021, 70 (5): 1187- 1209.
22	SCOLAS S , El GHOUCH A , LEGRAND C , et al. Variable selection in a flexible parametric mixture cure model with interval-censored data[J]. Statistics in Medicine, 2016, 35 (7): 1210- 1225.
23	HJORT N L , CLAESKENS G . Focused information criteria and model averaging for the Cox hazard regression model[J]. Journal of the American Statistical Association, 2006, 101 (476): 1449- 1464.
24	HJORT N L , CLAESKENS G . Frequentist model average estimators[J]. Journal of the American Statistical Association, 2003, 98 (464): 879- 899.
25	朱容, 邹国华. 半参数模型平均估计的渐近理论[J]. 中国科学(数学), 2018, 48 (8): 1019- 1052.
	ZHU Rong , ZOU Guohua . The asymptotic theory for model averaging in general semiparametric models[J]. Scientia Sinica (Mathematica), 2018, 48 (8): 1019- 1052.
26	朱容, 邹国华, 张新雨. 部分函数线性模型的模型平均方法[J]. 系统科学与数学, 2018, 38 (7): 777- 800.
	ZHU Rong , ZOU Guohua , ZHANG Xinyu . Optimal model averaging estimation for partial functional linear models[J]. Journal of Systems Science and Mathematical Sciences, 2018, 38 (7): 777- 800.
27	KIRKWOOD J M , STRAWDERMAN M H , ERNSTOFF M S , et al. Interferon Alfa-2b adjuvant therapy of high-risk resected cutaneous melanoma: the Eastern Cooperative Oncology Group Trial EST 1684[J]. Journal of Clinical Oncology, 1996, 14 (1): 7- 17.
28	BRADFORD P T , GOLDSTEIN A M , MCMASTER M L , et al. Acral lentiginous melanoma: incidence and survival patterns in the United States, 1986-2005[J]. Archives of Dermatology, 2009, 145 (4): 427- 434.
29	OMER M E A M E , ABU BAKAR M R , ADAM M B , et al. Cure models with exponentiated Weibull exponential distribution for the analysis of melanoma patients[J]. Mathematics, 2020, 8 (11): 1926.

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参数	n=200								n=400
参数	η₀	η₀	β₁	γ₁	γ₂	γ₂	μ	σ	η₀	η₀	β₁	γ₁	γ₂	γ₂	μ	σ
设置A	0.6	0.3	0.3	-0.5	0.3	-0.3	0.5	0.4	0.6	0.3	0.3	-0.5	0.3	-0.3	0.5	0.4
设置B	0.6	0.3	0.3	-0.5	-0.3	0.3	0.5	0.4	0.6	0.3	0.3	-0.5	-0.3	0.3	0.5	0.4
设置C	0.6	0.3	-0.3	-0.5	0.3	0.3	0.5	0.4	0.6	0.3	-0.3	-0.5	0.3	0.3	0.5	0.4

评价指标	样本量	感兴趣指标	S-AIC	AIC	S-BIC	BIC
E_MS	n=200	(a)	0.222 5	0.418 2	0.223 2	0.453 6
		(b)	0.460 4	0.842 7	0.462 0	0.913 5
		(c)	0.168 1	0.314 9	0.168 3	0.341 7
	n=400	(a)	0.222 5	0.418 2	0.223 2	0.453 6
		(b)	0.460 4	0.842 7	0.462 0	0.913 5
		(c)	0.168 1	0.314 9	0.168 3	0.3417
E_MSP	n=200	(d)	0.000 4	0.000 4	0.000 4	0.000 4
	n=200	(e)	0.000 2	0.000 2	0.000 2	0.000 2
	n=400	(d)	0.000 2	0.000 2	0.000 2	0.000 2
	n=400	(e)	0.000 1	0.000 1	0.000 1	0.000 1

评价指标	样本量	感兴趣指标	S-AIC	AIC	S-BIC	BIC
E_MS	n=200	(a)	0.229 4	0.434 8	0.229 8	0.466 1
		(b)	0.474 9	0.876 3	0.475 8	0.938 9
		(c)	0.173 3	0.327 2	0.173 3	0.350 8
	n=400	(a)	0.110 8	0.326 9	0.111 0	0.361 2
		(b)	0.229 3	0.657 1	0.229 6	0.725 7
		(c)	0.083 7	0.245 8	0.083 7	0.271 7
E_MSP	n=200	(d)	0.000 4	0.000 5	0.000 4	0.000 5
	n=200	(e)	0.000 2	0.000 2	0.000 2	0.000 2
	n=400	(d)	0.000 2	0.000 2	0.000 2	0.000 2
	n=400	(e)	0.000 1	0.000 1	0.000 1	0.000 1

评价指标	样本量	感兴趣指标	S-AIC	AIC	S-BIC	BIC
E_MS	n=200	(a)	0.225 1	0.413 4	0.226 0	0.456 0
		(b)	0.466 3	0.833 2	0.468 2	0.918 4
		(c)	0.170 2	0.311 3	0.170 5	0.343 5
	n=400	(a)	0.114 7	0.354 1	0.115 0	0.388 2
		(b)	0.237 1	0.711 2	0.237 7	0.779 7
		(c)	0.086 7	0.266 1	0.086 7	0.2918
E_MSP	n=200	(d)	0.000 3	0.000 5	0.000 3	0.000 5
	n=200	(e)	0.000 2	0.000 2	0.000 2	0.000 2
	n=400	(d)	0.000 2	0.000 2	0.000 2	0.000 2
	n=400	(e)	0.000 1	0.000 1	0.000 1	0.000 1

参数	估计	S-AIC	S-BIC	AIC/BIC
μ₁	估计值(标准差)	0.740 3(0.392 7)	0.740 2(0.392 7)	-0.448 5(0.254 2)
μ₁	95%置信区间	(-0.029 5, 1.510 0)	(-0.029 5, 1.510 0)	(-0.946 8, 0.049 8)
μ₂	估计值(标准差)	0.562 8(0.263 0)	0.562 9(0.263 0)	0.309 1(0.185 6)
μ₂	95%置信区间	(0.047 3, 1.078 3)	(0.047 4, 1.078 3)	(-0.054 7, 0.672 9)
μ₃	估计值(标准差)	0.002 3(0.008 1)	0.002 3(0.008 1)	—
μ₃	95%置信区间	(-0.013 6, 0.018 2)	(-0.013 6, 0.018 2)	—
μ₄	估计值(标准差)	-2.6250e-05(2.5502e-05)	-4.5440e-06(5.6240e-06)	-0.038 6(0.188 0)
μ₄	95%置信区间	(-7.6333e-05, 2.3734e-05)	(-1.5567e-05, 6.4800e-06)	(-0.407 1, 0.330 0)
μ₅	估计值(标准差)	1.305 4(0.663 9)	1.305 4(0.663 9)	-0.178 0(0.627 9)
μ₅	95%置信区间	(0.004 2, 2.606 6)	(0.004 2, 2.606 6)	(-1.408 7, 1.052 7)