Trees and splines in survival analysis

Abstract

During the past few years several nonparametric alternatives to the Cox proportional hazards model have appeared in the literature. These methods extend techniques that are well known from regression analysis to the analysis of censored survival data. In this paper we discuss methods based on (partition) trees and (polynomial) splines, analyse two datasets using both Survival Trees and HARE, and compare the strengths and weaknesses of the two methods. One of the strengths of HARE is that its model fitting procedure has an implicit check for proportionality of the underlying hazards model. It also provides an explicit model for the conditional hazards function, which makes it very convenient to obtain graphical summaries. On the other hand, the tree-based methods automatically partition a dataset into groups of cases that are similar in survival history. Results obtained by survival trees and HARE are often complementary. Trees and splines in survival analysis should provide the data analyst with two useful tools when analysing survival data.

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References

Breiman L. , Friedman JH , Olshen RA , Stone CJ Classification and regression trees. Pacific Grove, CA : Wadsworth, 1984.

Friedman JH Multivariate regression splines (with discussion). The Annals of Statistics 1988; 19: 1-141.

Kooperberg C. , Stone CJ Logspline density estimation for censored data. Journal of Computational and Graphical Statistics 1992; 1: 301-28.

Kooperberg C. , Stone CJ , Truong YK Logspline estimation of a possibly mixed spectral distribution. Journal of Time Series Analysis 1995; 16: 359-88.

Bose S. , Kooperberg C. , Stone CJ Polychotomous regression. University of Washington, Department of Statistics, Technical Report, No. 288, 1995 .

Cox DR Regression models and life tables (with discussion). Journal of the Royal Statistical Society, Series B 1972; 34: 187-220.

Kwak, LW , Halpern J. , Olshen RA , Homing SJ Prognostic significance of actual dose intensity in diffuse large cell lymphoma: Results of tree-structured survival analysis. Journal of Clinical Oncology 1990; 8: 963-77.

Carmelli D. , Halpern J. , Swan GE et al. 27-Year mortality in the Western Collaborative Group Study: construction of risk groups by recursive partitioning. Journal of Clinical Epidemiology 1991; 44: 1341-51.

Piette JD , Intrator O. , Zierler S. , Mor V. , Stein M. Differences in case fatality rates for Aids patients: application of a new methodology for survival research. Epidemiology 1992; 3: 310-18.

10.

Mor V. , Intrator O. , Laliberte LL Factors affecting conversion rates to Medicaid among new admissions to nursing homes. Health Services Research 1993; 28: 1-25.

11.

Schumacher M. , Schmoor C. , Sauerbrei W. et al. The prognostic effect of histological tumor grade in node negative breast cancer patients. Breast Cancer Research and Treatment 1993; 25: 235-45.

12.

Curran WJ , Scott CB , Horton J. et al. Recursive partitioning analysis of prognostic factors in three radiation therapy oncology group malignant glioma trials. Journal of the National Cancer Institute 1993; 85(9): 704-10.

13.

Gordon L. , Olshen RA Tree-structured survival analysis. Cancer Treatment Reports 1985; 69: 1065-69.

14.

Ciampi A. , Chang, CH , Hogg S. , McKinney S. Recursive partitioning: a versatile method for exploratory data analysis in Biostatistics. In: MacNeil IB , Umphrey G eds. Proceedings from Joshi Festschrift . Amsterdam: North Holland, 1987: 23-50.

15.

Davis RB , Anderson JR Exponential survival trees. Statistics in Medicine 1989; 8: 947-61.

16.

LeBlanc M. , Crowley J. Relative risk trees for censored data. Biometrics 1992; 48: 411-25.

17.

Zhang HP Splitting criteria in survival trees. In: 10th Workshop on Statistical Modeling. Lecture Notes in Statistics Series New York: Springer-Verlag, 1995: 305-13.

18.

Intrator 0. Exploratory trees for semi-markov processes. Proceedings of the 23rd symposium on the interface of computing science and statistics 1991; 352-55.

19.

Segal MR Regression trees for censored data. Biometrics 1988; 44: 35-47.

20.

Segal MR , Bloch DA A comparison of estimated proportional hazards models and regression trees . Statistics in Medicine 1989; 8: 539-50.

21.

Butler JH , Gilpin E. , Gordon L. , Olshen RA Tree structured survival analysis, II. Technical Report, Stanford University, 1991.

22.

LeBlanc M. , Crowley J. A review of tree-based prognostic models. In: New Advances in the Design and Analysis of Clinical Trials Data. Kluwer Academic Publishers, 1995: 113-24.

23.

Ciampi A. , Thiffault J. , Nakache J-P. , Asselain B. Stratification by stepwise regression, correspondence analysis and recursive partition. Computational Statistics and Data Analysis 1986; 4: 185-204.

24.

Harrington DP , Fleming TR A class of rank test procedures for censored survival data. Biometrika 1982; 69: 553-66.

25.

Fleming TR , Augustine GA , Elcombe SA , Offord KP The SURVDIF procedure. SAS/ SUGI Users manual, version 5, 1986 .

26.

Fleming TR , Harrington DP , O'Sullivan M. Supremum versions of the log-rank and generalized Wilcoxon statistics. Journal of the American Statistical Association 1987; 82: 312-20.

27.

Breslow N. Contribution to the discussion of a paper by DR Cox. Journal of the Royal Statistical Society, Series B 1972; 34: 216-17.

28.

Nelson, W. On estimating the distribution of a random vector when only the coordinate is observable. Technometrics 1969; 12: 923-24.

29.

Eubank RL Spline smoothing and nonparametric regression. New York : Marcel Dekker, 1988.

30.

Hastie TJ , Tibshirani R. Generalized additive models. London: Chapman and Hall, 1990.

31.

Wahba G. Spline models for observational data. Philadelphia: SIAM, 1990.

32.

Anderson JA , Senthilselvan A. Smooth estimates for the hazard function. Journal of the Royal Statistical Society, Series B 1980; 42: 322-27.

33.

Whittemore AS , Keller JB Survival estimation using splines. Biometrics 1986; 42: 495-506.

34.

Senthilselvan A. Penalized likelihood estimation of hazard and intensity functions. Journal of the Royal Statistical Society, Series B 1987; 49: 170-74.

35.

O'Sullivan F. Nonparametric estimation of relative risk using splines and cross-validation. SIAM Journal of Scientific and Statistical Computing 1988; 9: 531-42.

36.

O'Sullivan F. Fast computation of fully automated log-density and log-hazard estimators. SIAM Journal of Scientific and Statistical Computing 1988; 9: 363-79.

37.

Gray RJ Flexible methods for analyzing survival data using splines, with applications to breast cancer prognosis. Journal of the American Statistical Association 1992; 87: 942-51.

38.

Hastie TJ , Tibshirani R. Varying-coefficient models (with discussion). Journal of the Royal Statistical Society, Series B 1993; 55: 757-800.

39.

Gu C. Penalized likelihood hazard estimation. Technical Report No. 91-58, Department of Statistics, Purdue University, 1991.

40.

Gu C. Structural multivariate function estimation: some automatic density and hazard estimates. Purdue University, Department of Statistics, Technical Report, 1994.

41.

Smith PL Curve fitting and modeling with splines using statistical variable selection methods. NASA Report 166034, Langley Research Center, Hampla, VA: NASA, 1982.

42.

Etezadi-Amoli J. , Ciampi A. Extended hazard regression for censored survival data with covariates: a spline approximation for the baseline hazard function . Biometrics 1987; 43: 181-92.

43.

Efron B.

Logistic regression, survival analysis and the Kaplan-Meier curve. Journal of

American Statistical Association 1988; 83: 414-25.

44.

Abrahamowicz M. , Ciampi A. , Ramsay JO Nonparametric density estimation for censored survival data: Regression-spline approach. The Canadian Journal of Statistics 1991; 20: 171-85.

45.

Kooperberg C. , Stone CJ , Truong YK Hazard regression. Journal of the American Statistical Association 1995; 90: 78-94.

46.

Kooperberg C. Hazard regression for interval censored data . In: The Proceedings of the International Biometrics Conference, Hamilton, Ontario, 1994: 81-96.

47.

Kooperberg C. , Stone CJ , Truong YK The L2 rate of convergence of hazard regression. Scandinavian Journal of Statistics 1995; 22: 143-57.

48.

Miller RG Survival analysis. New York: Wiley , 1981.

49.

Rao CR Linear statistical inference and its applications, second edition. New York: Wiley, 1973.

50.

Chambers JM , Hastie TJ Statistical models in S. Pacific Grove, CA: Wadsworth, 1992.

51.

Becker RA , Chambers JM , Wilks AR The new S language. Pacific Grove, CA: Wadsworth, 1988.

52.

Lindley DV Contribution to the discussion of Regression models and life tables. Journal of the Royal Statistical Society, Series B 1972; 34: 208-209.

53.

Rosenman RH , Friedman M. , Strauss R. et al. A predictive study of coronary heart disease: The Western Collaborative Group Study. Journal of the American Medical Association 1964; 189: 15-22.

54.

Rosenman RH , Brand RJ , Sholtz RI , Friedman M. Multivariate prediction of coronary heart disease during 8.5 year follow-up in the Western Collaborative Group Study . American Journal of Cardiology 1976; 37: 903-10.

55.

Ragland DR , Brand RJ Coronary heart disease mortality in the Western Collaborative Group Study: follow-up experience of 22 years. American Journal of Epidemiology 1988; 127: 462-75.

56.

Cox DR , Oakes D. Survival analysis. London: Chapman and Hall, 1983.