On the Convergence of Observed Partial Likelihood under Incomplete Data with Two Class Possibilities

Author(s)
Tomoyuki Sugimoto

ABSTRACT

In this paper, we discuss the theoretical validity of the observed partial likelihood (OPL) constructed in a Coxtype model under incomplete data with two class possibilities, such as missing binary covariates, a cure-mixture model or doubly censored data. A main result is establishing the asymptotic convergence of the OPL. To reach this result, as it is difficult to apply some standard tools in the survival analysis, we develop tools for weak convergence based on partial-sum processes. The result of the asymptotic convergence shown here indicates that a suitable order of the number of Monte Carlo trials is less than the square of the sample size. In addition, using numerical examples, we investigate how the asymptotic properties discussed here behave in a finite sample.

Cite this paper

T. Sugimoto, "On the Convergence of Observed Partial Likelihood under Incomplete Data with Two Class Possibilities,"*Open Journal of Statistics*, Vol. 4 No. 2, 2014, pp. 118-136. doi: 10.4236/ojs.2014.42012.

T. Sugimoto, "On the Convergence of Observed Partial Likelihood under Incomplete Data with Two Class Possibilities,"

References

[1] D. R. Cox, “Regression Models and Life Tables (with Discussion),” Journal of the Royal Statistical Society, Series B, Vol. 34, No. 2, 1972, pp. 187-220.

[2] J. S. Kim, “Maximum Likelihood Estimation for the Proportional Hazards Model with Partly Interval-Censored Data,” Journal of the Royal Statistical Society, Series B, Vol. 65, No. 2, 2003, pp. 489-502.

http://dx.doi.org/10.1111/1467-9868.00398

[3] M. C. Paik and W.-Y. Tsai, “On Using the Cox Proportional Hazards Model with Missing Covariates,” Biometrika, Vol. 84, No. 3, 1997, pp. 579-593. http://dx.doi.org/10.1093/biomet/84.3.579

[4] H. Y. Chen and R. J. A. Little, “Proportional Hazards Regression with Missing Covariates,” Journal of the American Statistical Association, Vol. 94, No. 447, 1999, pp. 896-908.

http://dx.doi.org/10.1080/01621459.1999.10474195

[5] A. H. Herring and J. G. Ibrahim, “Likelihood-Based Methods for Missing Covariates in the Cox Proportional Hazards Model,” Journal of the American Statistical Association, Vol. 96, No. 453, 2001, pp. 292-302.

http://dx.doi.org/10.1198/016214501750332866

[6] S. A. Murphy and A. W. van der Vaart, “On Profile Likelihood (with Discussion),” Journal of the American Statistical Association, Vol. 95, No. 450, 2000, pp. 449-465.

http://dx.doi.org/10.1080/01621459.2000.10474219

[7] D. R. Cox, “Partial Likelihood,” Biometrika, Vol. 62, No. 2, 1975, pp. 269-276.

http://dx.doi.org/10.1093/biomet/62.2.269

[8] R. Gill, “Marginal Partial Likelihood,” Scandinavian Journal of Statistics, Vol. 19, No. 2, 1992, pp. 133-137.

[9] M. R. Kosorok, “Introduction to Empirical Processes and Semiparametric Inference,” Springer, Berlin, 2008.

http://dx.doi.org/10.1007/978-0-387-74978-5

[10] J. P. Sy and J. M. G. Taylor, “Estimation in a Cox Proportional Hazards Cure Model,” Biometrics, Vol. 56, No. 1, 2000, pp. 227-236. http://dx.doi.org/10.1111/j.0006-341X.2000.00227.x

[11] T. Sugimoto, “A Large Sample Study of Marginal Partial Likelihood in a Cox Cure-Mixture Regression Model,” Unpublished.

[12] A. Y. C. Kuk and C.-H. Chen, “A Mixture Model Combining Logistic Regression with Proportional Hazards Regression,” Biometrika, Vol. 79, No. 3, 1992, pp. 531-541.

http://dx.doi.org/10.1093/biomet/79.3.531

[13] Y. Peng and K. B. G. Dear, “A Nonparametric Mixture Model for Cure Rate Estimation,” Biometrics, Vol. 56, No. 1, 2000, pp. 237-243. http://dx.doi.org/10.1111/j.0006-341X.2000.00237.x

[14] W. Lu and Z. Ying, “On Semiparametric Transformation Cure Models,” Biometrika, Vol. 91, No. 2, 2004, pp. 331-343.

http://dx.doi.org/10.1093/biomet/91.2.331

[15] T. Sugimoto, T. Hamasaki and M. Goto, “Estimation from Pseudo Partial Likelihood in a Semiparametric Cure Model,” Journal of the Japanese Society of Computational Statistics, Vol. 18, No. 1, 2005, pp. 33-46.

[16] B. W. Turnbull, “Nonparametric Estimation of a Survivorship Function with Doubly Censored Data,” Journal of the American Statistical Association, Vol. 69, No. 345, 1974, pp. 169-173. http://dx.doi.org/10.1080/01621459.1974.10480146

[17] B. W. Turnbull, “The Empirical Distribution Function with Arbitrarily Grouped, Censored and Truncated Data,” Journal of the Royal Statistical Society, Series B, Vol. 38, No. 3, 1976, pp. 290-295.

[18] J. D. Kalbleisch and R. L. Prentice, “Marginal Likelihoods Based on Cox’s Regression and Life Model,” Biometrika, Vol. 60, No. 2, 1973, pp. 267-278. http://dx.doi.org/10.1093/biomet/60.2.267

[19] P. K. Andersen and R. D. Gill, “Cox’s Regression Model for Counting Processes: A Large Sample Study,” Annals of Statistics, Vol. 10, No. 3, 1982, pp. 1100-1120.

http://dx.doi.org/10.1214/aos/1176345976

[20] D. Collett, “Modelling Survival Data in Medical Research,” 2nd Edition, Chapman & Hall/CRC, London, 2003.

[21] A. W. van der Vaart and J. A. Wellner, “Weak Convergence and Empirical Processes,” Springer-Verlag, New York, 1996.

http://dx.doi.org/10.1007/978-1-4757-2545-2

[1] D. R. Cox, “Regression Models and Life Tables (with Discussion),” Journal of the Royal Statistical Society, Series B, Vol. 34, No. 2, 1972, pp. 187-220.

[2] J. S. Kim, “Maximum Likelihood Estimation for the Proportional Hazards Model with Partly Interval-Censored Data,” Journal of the Royal Statistical Society, Series B, Vol. 65, No. 2, 2003, pp. 489-502.

http://dx.doi.org/10.1111/1467-9868.00398

[3] M. C. Paik and W.-Y. Tsai, “On Using the Cox Proportional Hazards Model with Missing Covariates,” Biometrika, Vol. 84, No. 3, 1997, pp. 579-593. http://dx.doi.org/10.1093/biomet/84.3.579

[4] H. Y. Chen and R. J. A. Little, “Proportional Hazards Regression with Missing Covariates,” Journal of the American Statistical Association, Vol. 94, No. 447, 1999, pp. 896-908.

http://dx.doi.org/10.1080/01621459.1999.10474195

[5] A. H. Herring and J. G. Ibrahim, “Likelihood-Based Methods for Missing Covariates in the Cox Proportional Hazards Model,” Journal of the American Statistical Association, Vol. 96, No. 453, 2001, pp. 292-302.

http://dx.doi.org/10.1198/016214501750332866

[6] S. A. Murphy and A. W. van der Vaart, “On Profile Likelihood (with Discussion),” Journal of the American Statistical Association, Vol. 95, No. 450, 2000, pp. 449-465.

http://dx.doi.org/10.1080/01621459.2000.10474219

[7] D. R. Cox, “Partial Likelihood,” Biometrika, Vol. 62, No. 2, 1975, pp. 269-276.

http://dx.doi.org/10.1093/biomet/62.2.269

[8] R. Gill, “Marginal Partial Likelihood,” Scandinavian Journal of Statistics, Vol. 19, No. 2, 1992, pp. 133-137.

[9] M. R. Kosorok, “Introduction to Empirical Processes and Semiparametric Inference,” Springer, Berlin, 2008.

http://dx.doi.org/10.1007/978-0-387-74978-5

[10] J. P. Sy and J. M. G. Taylor, “Estimation in a Cox Proportional Hazards Cure Model,” Biometrics, Vol. 56, No. 1, 2000, pp. 227-236. http://dx.doi.org/10.1111/j.0006-341X.2000.00227.x

[11] T. Sugimoto, “A Large Sample Study of Marginal Partial Likelihood in a Cox Cure-Mixture Regression Model,” Unpublished.

[12] A. Y. C. Kuk and C.-H. Chen, “A Mixture Model Combining Logistic Regression with Proportional Hazards Regression,” Biometrika, Vol. 79, No. 3, 1992, pp. 531-541.

http://dx.doi.org/10.1093/biomet/79.3.531

[13] Y. Peng and K. B. G. Dear, “A Nonparametric Mixture Model for Cure Rate Estimation,” Biometrics, Vol. 56, No. 1, 2000, pp. 237-243. http://dx.doi.org/10.1111/j.0006-341X.2000.00237.x

[14] W. Lu and Z. Ying, “On Semiparametric Transformation Cure Models,” Biometrika, Vol. 91, No. 2, 2004, pp. 331-343.

http://dx.doi.org/10.1093/biomet/91.2.331

[15] T. Sugimoto, T. Hamasaki and M. Goto, “Estimation from Pseudo Partial Likelihood in a Semiparametric Cure Model,” Journal of the Japanese Society of Computational Statistics, Vol. 18, No. 1, 2005, pp. 33-46.

[16] B. W. Turnbull, “Nonparametric Estimation of a Survivorship Function with Doubly Censored Data,” Journal of the American Statistical Association, Vol. 69, No. 345, 1974, pp. 169-173. http://dx.doi.org/10.1080/01621459.1974.10480146

[17] B. W. Turnbull, “The Empirical Distribution Function with Arbitrarily Grouped, Censored and Truncated Data,” Journal of the Royal Statistical Society, Series B, Vol. 38, No. 3, 1976, pp. 290-295.

[18] J. D. Kalbleisch and R. L. Prentice, “Marginal Likelihoods Based on Cox’s Regression and Life Model,” Biometrika, Vol. 60, No. 2, 1973, pp. 267-278. http://dx.doi.org/10.1093/biomet/60.2.267

[19] P. K. Andersen and R. D. Gill, “Cox’s Regression Model for Counting Processes: A Large Sample Study,” Annals of Statistics, Vol. 10, No. 3, 1982, pp. 1100-1120.

http://dx.doi.org/10.1214/aos/1176345976

[20] D. Collett, “Modelling Survival Data in Medical Research,” 2nd Edition, Chapman & Hall/CRC, London, 2003.

[21] A. W. van der Vaart and J. A. Wellner, “Weak Convergence and Empirical Processes,” Springer-Verlag, New York, 1996.

http://dx.doi.org/10.1007/978-1-4757-2545-2