Weighted Estimation in Cox Regression Model: An Application to Breast Cancer Data

Cox regression is a well-known approach for modeling censored survival data. However, the model has an assumption of proportional hazards which requires an attention. In this study, we examine weighted estimation in Cox regression model under nonproportional hazards. Our aim is to propose various weighting functions that are more appropriate than existing ones. The proposed and existing weighting functions are applied to a data set in breast cancer in order to analyze their effects on the analysis results. In order to analyze the performance and effect of proposed and existing weighting functions, a wide simulation study, covering different censoring rates and tied observations, is carried out. Keywords:Cox regression model, Log-rank tests, Nonproportional hazards, Weighted estimation.

___

  • Altshuler, B., “Theory for the measurement of in competing Mathematical Biosciences, 6: 1-11, (1970). animal experiments”,
  • Andersen, P.K., Borgan, Gill, R.D., Keiding, N., “Linear nonparametric tests for comparison of counting process with application to censored data”, International Statistical Review, 50:219-258, (1982).
  • Arjas, E., “A graphical method for assessing goodness of fit in Cox’s proportional hazards model”, Journal of the American Statistical Association, 83:204-212, (1988).
  • Ata, N., “Survival Models for Nonproportional Hazards”, Unpublished Doctoral Thesis, Hacettepe University, Institute of Science, Ankara, (2010).
  • Breslow, N.E., “Covariance analysis of censored survival data”, Biometrics, 30, 89-99, (1974).
  • Collett, D., “Modelling Survival Data in Medical Research”, Chapman and Hall, New York, (2003).
  • Cox, D.R., “Regression models and life-tables”, Journal of the Royal Statistical Society Series B, 34, 187-220, (1972).
  • Erdoğan, A., “Proportional Hazards Model”, Unpublished Master of Science Thesis, Hacettepe University, Institute of Science, Ankara, (1993).
  • Gehan, E.A., “A generalized Wilcoxon test for comparing arbitrarily singly-censored samples”, Biometrika, 52: 203-223, (1965).
  • Gill, R.D., Schumacher, M., “A simple test for the proportional hazards assumption”, Biometrika, 74: 289-300, (1987).
  • Harris E.K., Albert A., Survivorship analysis for clinical studies, Marcel Dekker, New York. (1991).
  • Kaplan, E.L., Meier, P., “Nonparametric estimation from incomplete observations”, Journal of the American Statistical Association, 53: 457-481, (1958).
  • Leton, E., Zuluaga, P., “Equivalence between score and Communications in Statistics-Theory and Methods, 30(4): 591-608, (2001). survival curves”,
  • Moreau, T., Maccario, J., Lellouch, J., Huber, C., “Weighted log rank statistics for comparing two distributions”, Biometrika, 79: 195-198, (1992).
  • Perperoglou, A., Keramopoullos, A., Houwelingen, H.C., “Approaches in modelling long-term survival: An application to breast cancer”, Statistics in Medicine, 26(13): 2666-2685, (2007).
  • Persson, I., “Essays on the assumption of proportional http://www.diva-portal.org, (2002). in Cox regression”,
  • Persson, I, Khamis H., “Bias of the Cox model hazard ratio”, Journal of Modern Applied Statistical Methods, 4(1): 90-99, (2005).
  • Prentice, R.L., “Linear rank tests with right censored data”, Biometrika, 65: 167-179, (1978).
  • Prentice, R.L., Marek, P., “A qualitative discrepancy between censored data rank tests”, Biometrics, 35: 861-867, (1979).
  • Schemper, M., “Cox analysis of survival data with hazards nonproportional Statistician, 41: 455-465, (1992). functions”, The
  • Schemper, M., Wakounig, S., Heinze, G., “The estimation of average hazard ratios by weighted Cox regression”, Statistics in Medicine, 28: 2473-2489, (2009).
  • Schoenfeld, D., “Partial proportional hazards model”, Biometrika, 69: 551- 555, (1982). residuals for the
  • Therneau, T.M., Grambsch, P.M., “Modelling Survival Data: Extending the Cox model”, Springer- Verlang, New York, (2000).