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Webb A, Ma J. Cox models with time-varying covariates and partly-interval censoring-A maximum penalised likelihood approach. Stat Med 2023; 42:815-833. [PMID: 36585040 PMCID: PMC10107645 DOI: 10.1002/sim.9645] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/20/2022] [Revised: 09/30/2022] [Accepted: 12/20/2022] [Indexed: 01/01/2023]
Abstract
Time-varying covariates can be important predictors when model based predictions are considered. A Cox model that includes time-varying covariates is usually referred to as an extended Cox model. When only right censoring is presented in the observed survival times, the conventional partial likelihood method is still applicable to estimate the regression coefficients of an extended Cox model. However, if there are interval-censored survival times, then the partial likelihood method is not directly available unless an imputation, such as the middle point imputation, is used to replaced the left- and interval-censored data. However, such imputation methods are well known for causing biases. This paper considers fitting of the extended Cox models using the maximum penalised likelihood method allowing observed survival times to be partly interval censored, where a penalty function is used to regularise the baseline hazard estimate. We present simulation studies to demonstrate the performance of our proposed method, and illustrate our method with applications to two real datasets from medical research.
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Affiliation(s)
- Annabel Webb
- Department of Mathematics and StatisticsMacquarie UniversityMacquarie ParkNew South WalesAustralia
| | - Jun Ma
- Department of Mathematics and StatisticsMacquarie UniversityMacquarie ParkNew South WalesAustralia
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2
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Beyene KM, El Ghouch A. Time-dependent ROC curve estimation for interval-censored data. Biom J 2022; 64:1056-1074. [PMID: 35523738 DOI: 10.1002/bimj.202000382] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/16/2020] [Revised: 10/20/2021] [Accepted: 11/07/2021] [Indexed: 11/10/2022]
Abstract
The receiver-operating characteristic (ROC) curve is the most popular graphical method for evaluating the classification accuracy of a diagnostic marker. In time-to-event studies, the subject's event status is time-dependent, and hence, time-dependent extensions of ROC curve have been proposed. However, in practice, the calculation of this curve is not straightforward due to the presence of censoring that may be of different types. Existing methods focus on the more standard and simple case of right-censoring and neglect the general case of mixed interval-censored data that may involve left-, right-, and interval-censored observations. In this context, we propose and study a new time-dependent ROC curve estimator. We also consider some summary measures (area under the ROC curve and Youden index) traditionally associated with ROC as well as the Youden-based cutoff estimation method. The proposed method uses available data very efficiently. To this end, the unknown status (positive or negative) of censored subjects are estimated from the data via the estimation of the conditional survival function given the marker. For that, we investigate both model-based and nonparametric approaches. We also provide variance estimates and confidence intervals using Bootstrap. A simulation study is conducted to investigate the finite sample behavior of the proposed methods and to compare their performance with a competitor. Globally, we observed better finite sample performances for the proposed estimators. Finally, we illustrate the methods using two data sets one from a hypobaric decompression sickness study and the other from an oral health study. The proposed methods are implemented in the R package cenROC.
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Affiliation(s)
- Kassu Mehari Beyene
- ISBA, UCLouvain, Louvain la Neuve, Belgium.,Department of Statistics, College of Natural Sciences, Wollo University, Dessie, Ethiopia
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3
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Mehrotra DV, Zhang Y. Hazard ratio estimation and inference in clinical trials with many tied event times. Stat Med 2018; 37:3547-3556. [PMID: 29900572 DOI: 10.1002/sim.7843] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/03/2018] [Revised: 04/13/2018] [Accepted: 05/12/2018] [Indexed: 11/08/2022]
Abstract
The medical literature contains numerous examples of randomized clinical trials with time-to-event endpoints in which large numbers of events accrued over relatively short follow-up periods, resulting in many tied event times. A generally common feature across such examples was that the logrank test was used for hypothesis testing and the Cox proportional hazards model was used for hazard ratio estimation. We caution that this common practice is particularly risky in the setting of many tied event times for two reasons. First, the estimator of the hazard ratio can be severely biased if the Breslow tie-handling approximation for the Cox model (the default in SAS and Stata software) is used. Second, the 95% confidence interval for the hazard ratio can include one even when the corresponding logrank test p-value is less than 0.05. To help establish a better practice, with applicability for both superiority and noninferiority trials, we use theory and simulations to contrast Wald and score tests based on well-known tie-handling approximations for the Cox model. Our recommendation is to report the Wald test p-value and corresponding confidence interval based on the Efron approximation. The recommended test is essentially as powerful as the logrank test, the accompanying point and interval estimates of the hazard ratio have excellent statistical properties even in settings with many tied event times, inferential alignment between the p-value and confidence interval is guaranteed, and implementation is straightforward using commonly used software.
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4
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Cure rate proportional odds models with spatial frailties for interval-censored data. COMMUNICATIONS FOR STATISTICAL APPLICATIONS AND METHODS 2017. [DOI: 10.29220/csam.2017.24.6.605] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/29/2022]
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5
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Dugué AE, Pulido M, Chabaud S, Belin L, Gal J. How to Deal with Interval-Censored Data Practically while Assessing the Progression-Free Survival: A Step-by-Step Guide Using SAS and R Software. Clin Cancer Res 2016; 22:5629-5635. [PMID: 27663596 DOI: 10.1158/1078-0432.ccr-16-1017] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/21/2016] [Revised: 08/04/2016] [Accepted: 08/12/2016] [Indexed: 11/16/2022]
Abstract
We describe how to estimate progression-free survival while dealing with interval-censored data in the setting of clinical trials in oncology. Three procedures with SAS and R statistical software are described: one allowing for a nonparametric maximum likelihood estimation of the survival curve using the EM-ICM (Expectation and Maximization-Iterative Convex Minorant) algorithm as described by Wellner and Zhan in 1997; a sensitivity analysis procedure in which the progression time is assigned (i) at the midpoint, (ii) at the upper limit (reflecting the standard analysis when the progression time is assigned at the first radiologic exam showing progressive disease), or (iii) at the lower limit of the censoring interval; and finally, two multiple imputations are described considering a uniform or the nonparametric maximum likelihood estimation (NPMLE) distribution. Clin Cancer Res; 22(23); 5629-35. ©2016 AACR.
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Affiliation(s)
| | - Marina Pulido
- Clinical and Epidemiological Research Unit, Bergonié Cancer Institute, Bordeaux, France
| | - Sylvie Chabaud
- Biostatistics and Therapeutics Evaluation Unit, Léon Bérard Cancer Center, Lyon, France
| | - Lisa Belin
- Department of Biostatistics, Curie Cancer Institute, Paris, France
| | - Jocelyn Gal
- Epidemiology and Biostatistics Unit, Antoine Lacassagne Cancer Center, Nice, France
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6
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Pan C, Cai B, Wang L, Lin X. Bayesian semiparametric model for spatially correlated interval-censored survival data. Comput Stat Data Anal 2014. [DOI: 10.1016/j.csda.2013.11.016] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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7
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Chen DG, Yu L, Peace KE, Lio YL, Wang Y. Approximating the baseline hazard function by taylor series for interval-censored time-to-event data. J Biopharm Stat 2013; 23:695-708. [PMID: 23611204 DOI: 10.1080/10543406.2012.756497] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
Abstract
In many oncology clinical trials, time-to-event data are generated from scanning for cancer within a specific interval, resulting in interval censoring along with complete-time and right-left-censored time-to-event data. A common practice in analyzing data from this type of trial is to impute the interval-censored event time using the midpoint or right endpoint (i.e., the first observed time) of the interval so that well-known statistical methods developed for right-censored time-to-event data, such as Cox regression, may be used for the requisite analyses. This may introduce bias and lead to erroneous conclusions. In this paper, a Taylor series is proposed to approximate the log baseline hazard function in Cox proportional hazards regression to mitigate the bias arising from analyzing the imputed time-to-event data. With this formulation, the likelihood ratio test can be used to select an appropriate order for this Taylor series approximation and maximum likelihood techniques used to estimate model parameters and provide statistical inference, for example, on treatment effect. The application of this novel method is demonstrated by a simulation study and application to data from a breast cancer clinical trial.
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Affiliation(s)
- Ding-Geng Chen
- Department of Biostatistcs and Computational Biology, University of Rochester Medical Center , Rochester , NY, USA.
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8
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Gong Q, Fang L. Comparison of different parametric proportional hazards models for interval-censored data: A simulation study. Contemp Clin Trials 2013; 36:276-83. [DOI: 10.1016/j.cct.2013.07.012] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/21/2013] [Revised: 07/26/2013] [Accepted: 07/28/2013] [Indexed: 11/25/2022]
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9
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Bushnell W, Stone A. Research outcomes and recommendations for the assessment of progression in cancer clinical trials. J Biopharm Stat 2013; 23:949-50. [PMID: 23957508 DOI: 10.1080/10543406.2013.813718] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
Affiliation(s)
- Will Bushnell
- AstraZeneca Pharmaceuticals LP, Wilmington, Delaware 19850 , USA.
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Denne JS, Stone AM, Bailey-Iacona R, Chen TT. Missing Data and Censoring in the Analysis of Progression-Free Survival in Oncology Clinical Trials. J Biopharm Stat 2013; 23:951-70. [DOI: 10.1080/10543406.2013.813515] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
Affiliation(s)
- J. S. Denne
- a Eli Lilly & Co. , Indianapolis , Indiana , USA
| | - A. M. Stone
- b AstraZeneca , Macclesfield , United Kingdom
| | | | - T.-T. Chen
- d Bristol Myers Squibb Wallingford Connecticut,, and Columbia University , New York , New York , USA
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11
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Sun X, Li X, Chen C, Song Y. A Review of Statistical Issues with Progression-Free Survival as an Interval-Censored Time-to-Event Endpoint. J Biopharm Stat 2013; 23:986-1003. [DOI: 10.1080/10543406.2013.813524] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
Affiliation(s)
- Xing Sun
- a Merck & Co., Merck Research Labs , North Wales , Pennsylvania , USA
| | - Xiaoyun Li
- a Merck & Co., Merck Research Labs , North Wales , Pennsylvania , USA
| | - Cong Chen
- a Merck & Co., Merck Research Labs , North Wales , Pennsylvania , USA
| | - Yang Song
- a Merck & Co., Merck Research Labs , North Wales , Pennsylvania , USA
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12
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Fay MP, Shih JH. Weighted logrank tests for interval censored data when assessment times depend on treatment. Stat Med 2012; 31:3760-72. [PMID: 22786795 DOI: 10.1002/sim.5447] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/07/2011] [Revised: 04/27/2012] [Accepted: 04/30/2012] [Indexed: 11/11/2022]
Abstract
We consider weighted logrank tests for interval censored data when assessment times may depend on treatment, and for each individual, we only use the two assessment times that bracket the event of interest. It is known that treating finite right endpoints as observed events can substantially inflate the type I error rate under assessment-treatment dependence (ATD), but the validity of several other implementations of weighted logrank tests (score tests, permutation tests, multiple imputation tests) has not been studied in this situation. With a bounded number of unique assessment times, the score test under the grouped continuous model retains the type I error rate asymptotically under ATD; however, although the approximate permutation test based on the permutation central limit theorem is not asymptotically valid under every ATD scenario, we show through simulation that in many ATD scenarios, it retains the type I error rate better than the score test. We show a case where the approximate permutation test retains the type I error rate when the exact permutation test does not. We study and modify the multiple imputation logrank tests of Huang, Lee, and Yu (2008, Statistics in Medicine, 27: 3217-3226), showing that the distribution of the rank-like scores asymptotically does not depend on the assessment times. We show through simulations that our modifications of the multiple imputation logrank tests retain the type I error rate in all cases studied, even with ATD and a small number of individuals in each treatment group. Simulations were performed using the interval R package.
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Affiliation(s)
- Michael P Fay
- National Institute of Allergy and Infectious Diseases, 6700-B Rockledge Dr, MSC 7630, Bethesda, MD 20892-7630, USA.
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13
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Research outcomes and recommendations for the assessment of progression in cancer clinical trials from a PhRMA working group. Eur J Cancer 2011; 47:1763-71. [DOI: 10.1016/j.ejca.2011.02.011] [Citation(s) in RCA: 37] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2010] [Accepted: 02/16/2011] [Indexed: 11/20/2022]
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