Change-Point Analysis of Survival Data with Application in Clinical Trials
Affiliation(s)
1 Department of Biostatistics and Programming, Sanofi, Beijing, China. 2 Department of Mathematics and Statistics, American University, Washington DC, USA. 3 Department of Mathematical Sciences, University of Texas at Dallas, Richardson, USA.
1 Department of Biostatistics and Programming, Sanofi, Beijing, China. 2 Department of Mathematics and Statistics, American University, Washington DC, USA. 3 Department of Mathematical Sciences, University of Texas at Dallas, Richardson, USA.
ABSTRACT
Effects of many medical procedures appear after a time lag, when a significant change occurs in subjects’ failure rate. This paper focuses on the detection and estimation of such changes which is important for the evaluation and comparison of treatments and prediction of their effects. Unlike the classical change-point model, measurements may still be identically distributed, and the change point is a parameter of their common survival function. Some of the classical change-point detection techniques can still be used but the results are different. Contrary to the classical model, the maximum likelihood estimator of a change point appears consistent, even in presence of nuisance parameters. However, a more efficient procedure can be derived from Kaplan-Meier estimation of the survival function followed by the least-squares estimation of the change point. Strong consistency of these estimation schemes is proved. The finite-sample properties are examined by a Monte Carlo study. Proposed methods are applied to a recent clinical trial of the treatment program for strong drug dependence.
Effects of many medical procedures appear after a time lag, when a significant change occurs in subjects’ failure rate. This paper focuses on the detection and estimation of such changes which is important for the evaluation and comparison of treatments and prediction of their effects. Unlike the classical change-point model, measurements may still be identically distributed, and the change point is a parameter of their common survival function. Some of the classical change-point detection techniques can still be used but the results are different. Contrary to the classical model, the maximum likelihood estimator of a change point appears consistent, even in presence of nuisance parameters. However, a more efficient procedure can be derived from Kaplan-Meier estimation of the survival function followed by the least-squares estimation of the change point. Strong consistency of these estimation schemes is proved. The finite-sample properties are examined by a Monte Carlo study. Proposed methods are applied to a recent clinical trial of the treatment program for strong drug dependence.
KEYWORDS
Change-Point Problem, Failure Rate, Kaplan-Meier Estimation, Least Squares Estimation, Maximum Likelihood Estimation, Strong Consistency, Survival Function
Change-Point Problem, Failure Rate, Kaplan-Meier Estimation, Least Squares Estimation, Maximum Likelihood Estimation, Strong Consistency, Survival Function
Cite this paper
Chen, X. , Baron, M. (2014) Change-Point Analysis of Survival Data with Application in Clinical Trials. Open Journal of Statistics, 4, 663-677. doi: 10.4236/ojs.2014.49062.
Chen, X. , Baron, M. (2014) Change-Point Analysis of Survival Data with Application in Clinical Trials. Open Journal of Statistics, 4, 663-677. doi: 10.4236/ojs.2014.49062.
References
[1] Desmond, R.A., Weiss, H.L., Arani, R.B., Soong, S.-J., Wood, M.J., Fiddian, P., Gnann, J. and Whitley, R.J. (2002) Clinical Applications for Change-Point Analysis of Herpes Zoster Pain. Journal of Pain and Symptom Management, 23, 510-516.
http://dx.doi.org/10.1016/S0885-3924(02)00393-7 [2] Zucker, D.M. and Lakatos, E. (1990) Weighted Log Rank Type Statistics for Comparing Survival Curves When There Is a Time Lag in the Effectiveness of Treatment. Biometrika, 77, 853-864.
http://dx.doi.org/10.1093/biomet/77.4.853 [3] Goodman, M.S., Li, Y. and Tiwari, R.C. (2011) Detecting Multiple Change Points in Piecewise Constant Hazard Functions. Journal of Applied Statistics, 38, 2523-2532.
http://dx.doi.org/10.1080/02664763.2011.559209 [4] He, P., Kong, G. and Su, Z. (2013) Estimating the Survival Functions for Right-Censored and Interval-Censored Data with Piecewise Constant Hazard Functions. Contemporary Clinical Trials, 36, 122-127.
http://dx.doi.org/10.1016/j.cct.2013.04.009 [5] Loader, C.R. (1991) Inference for a Hazard Rate Change Point. Biometrika, 78, 749-757.
http://dx.doi.org/10.1093/biomet/78.4.749 [6] Matthews, D.E. and Farewell, V.T. (1982) On Testing for a Constant Hazard against a Change-Point Alternative. Biometrics, 38, 463-468.
http://dx.doi.org/10.2307/2530460 [7] Muller, H.G. and Wang, J.L. (1990) Nonparametric Analysis of Changes in Hazard Rates for Censored Survival Data: An Alternative to Change-Point Models. Biometrika, 77, 305-314.
http://dx.doi.org/10.1093/biomet/77.2.305 [8] Nguyen, H.T., Rogers, G.S. and Walker, E.A. (1984) Estimation in Change-Point Hazard Rate Models. Biometrika, 71, 299-304.
http://dx.doi.org/10.1093/biomet/71.2.299 [9] Sertkaya, D. and Sözer, M.T. (2003) A Bayesian Approach to the Constant Hazard Model with a Change Point and an Application to Breast Cancer Data. Hacettepe Journal of Mathematics and Statistics, 32, 33-41.? [10] Ahsanullah, M., Rukhin, A.L. and Sinha, B. (1995) Applied Change Point Problems in Statistics. Nova Science Publishers, New York. [11] Basseville, M. and Nikiforov, I.V. (1993) Detection of Abrupt Changes: Theory and Application. PTR Prentice-Hall, Inc., Englewood Cliffs. [12] Bhattacharya, P.K. (1995) Some Aspects of Change-Point Analysis. In: Change-Point Problems, IMS Lecture Notes-Monograph Series (Vol. 23), 28-56. [13] Chen, J. and Gupta, A.K. (2012) Parametric Statistical Change Point Analysis: With Applications to Genetics, Medicine, and Finance. Birkh?user, Boston.
http://dx.doi.org/10.1007/978-0-8176-4801-5 [14] Poor, H.V. and Hadjiliadis, O. (2009) Quickest Detection. Cambridge University Press, Cambridge. [15] Worsley, K.J. (1988) Exact Percentage Points of the Likelihood-Ratio Test for a Change-Point Hazard-Rate Model. Biometrics, 44, 259-263.
http://dx.doi.org/10.2307/2531914 [16] Kaplan, E.L. and Meier, P. (1958) Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association, 53, 457-481.
http://dx.doi.org/10.1080/01621459.1958.10501452 [17] Urschel, H.C., Hanselka, L.L., Gromov, I., White, L. and Baron, M. (2007) Open-Label Study of a Proprietary Treatment Program Targeting Type a γ-Aminobutyric Acid Receptor Dysregulation in Methamphetamine Dependence. Mayo Clinic Proceedings, 82, 1170-1178.
http://dx.doi.org/10.4065/82.10.1170 [18] Yao, Y.-C. (1987) Approximating the Distribution of the Maximum Likelihood Estimate of the Change-Point in a Sequence of Independent Random Variables. Annals of Statistics, 15, 1321-1328.
http://dx.doi.org/10.1214/aos/1176350509 [19] Barndorff-Nielsen, O.E. and Cox, D.R. (1984) Bartlett Adjustments to the Likelihood Ratio Statistic and the Distribution of the Maximum Likelihood Estimator. Journal of the Royal Statistical Society: Series B, 46, 483-495. [20] Dinwoodie, I.H. (1993) Large Deviations for Censored Data. Annals of Statistics, 21, 1608-1620.
http://dx.doi.org/10.1214/aos/1176349274 [21] Billingsley, P. (1995) Probability and Measure. 3rd Edition, Wiley, New York. [22] Chen, X. (2009) Change-Point Analysis of Survival Data with Application in Clinical Trials. Ph.D. Dissertation, The University of Texas at Dallas, Richardson.
[1] Desmond, R.A., Weiss, H.L., Arani, R.B., Soong, S.-J., Wood, M.J., Fiddian, P., Gnann, J. and Whitley, R.J. (2002) Clinical Applications for Change-Point Analysis of Herpes Zoster Pain. Journal of Pain and Symptom Management, 23, 510-516.
http://dx.doi.org/10.1016/S0885-3924(02)00393-7 [2] Zucker, D.M. and Lakatos, E. (1990) Weighted Log Rank Type Statistics for Comparing Survival Curves When There Is a Time Lag in the Effectiveness of Treatment. Biometrika, 77, 853-864.
http://dx.doi.org/10.1093/biomet/77.4.853 [3] Goodman, M.S., Li, Y. and Tiwari, R.C. (2011) Detecting Multiple Change Points in Piecewise Constant Hazard Functions. Journal of Applied Statistics, 38, 2523-2532.
http://dx.doi.org/10.1080/02664763.2011.559209 [4] He, P., Kong, G. and Su, Z. (2013) Estimating the Survival Functions for Right-Censored and Interval-Censored Data with Piecewise Constant Hazard Functions. Contemporary Clinical Trials, 36, 122-127.
http://dx.doi.org/10.1016/j.cct.2013.04.009 [5] Loader, C.R. (1991) Inference for a Hazard Rate Change Point. Biometrika, 78, 749-757.
http://dx.doi.org/10.1093/biomet/78.4.749 [6] Matthews, D.E. and Farewell, V.T. (1982) On Testing for a Constant Hazard against a Change-Point Alternative. Biometrics, 38, 463-468.
http://dx.doi.org/10.2307/2530460 [7] Muller, H.G. and Wang, J.L. (1990) Nonparametric Analysis of Changes in Hazard Rates for Censored Survival Data: An Alternative to Change-Point Models. Biometrika, 77, 305-314.
http://dx.doi.org/10.1093/biomet/77.2.305 [8] Nguyen, H.T., Rogers, G.S. and Walker, E.A. (1984) Estimation in Change-Point Hazard Rate Models. Biometrika, 71, 299-304.
http://dx.doi.org/10.1093/biomet/71.2.299 [9] Sertkaya, D. and Sözer, M.T. (2003) A Bayesian Approach to the Constant Hazard Model with a Change Point and an Application to Breast Cancer Data. Hacettepe Journal of Mathematics and Statistics, 32, 33-41.? [10] Ahsanullah, M., Rukhin, A.L. and Sinha, B. (1995) Applied Change Point Problems in Statistics. Nova Science Publishers, New York. [11] Basseville, M. and Nikiforov, I.V. (1993) Detection of Abrupt Changes: Theory and Application. PTR Prentice-Hall, Inc., Englewood Cliffs. [12] Bhattacharya, P.K. (1995) Some Aspects of Change-Point Analysis. In: Change-Point Problems, IMS Lecture Notes-Monograph Series (Vol. 23), 28-56. [13] Chen, J. and Gupta, A.K. (2012) Parametric Statistical Change Point Analysis: With Applications to Genetics, Medicine, and Finance. Birkh?user, Boston.
http://dx.doi.org/10.1007/978-0-8176-4801-5 [14] Poor, H.V. and Hadjiliadis, O. (2009) Quickest Detection. Cambridge University Press, Cambridge. [15] Worsley, K.J. (1988) Exact Percentage Points of the Likelihood-Ratio Test for a Change-Point Hazard-Rate Model. Biometrics, 44, 259-263.
http://dx.doi.org/10.2307/2531914 [16] Kaplan, E.L. and Meier, P. (1958) Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association, 53, 457-481.
http://dx.doi.org/10.1080/01621459.1958.10501452 [17] Urschel, H.C., Hanselka, L.L., Gromov, I., White, L. and Baron, M. (2007) Open-Label Study of a Proprietary Treatment Program Targeting Type a γ-Aminobutyric Acid Receptor Dysregulation in Methamphetamine Dependence. Mayo Clinic Proceedings, 82, 1170-1178.
http://dx.doi.org/10.4065/82.10.1170 [18] Yao, Y.-C. (1987) Approximating the Distribution of the Maximum Likelihood Estimate of the Change-Point in a Sequence of Independent Random Variables. Annals of Statistics, 15, 1321-1328.
http://dx.doi.org/10.1214/aos/1176350509 [19] Barndorff-Nielsen, O.E. and Cox, D.R. (1984) Bartlett Adjustments to the Likelihood Ratio Statistic and the Distribution of the Maximum Likelihood Estimator. Journal of the Royal Statistical Society: Series B, 46, 483-495. [20] Dinwoodie, I.H. (1993) Large Deviations for Censored Data. Annals of Statistics, 21, 1608-1620.
http://dx.doi.org/10.1214/aos/1176349274 [21] Billingsley, P. (1995) Probability and Measure. 3rd Edition, Wiley, New York. [22] Chen, X. (2009) Change-Point Analysis of Survival Data with Application in Clinical Trials. Ph.D. Dissertation, The University of Texas at Dallas, Richardson.