Survival Model Inference Using Functions of Brownian Motion

Author(s)
John O’Quigley

ABSTRACT

A family of tests for the presence of regression effect under proportional and non-proportional hazards models is described. The non-proportional hazards model, although not completely general, is very broad and includes a large number of possibilities. In the absence of restrictions, the regression coefficient,*β*(*t*), can be any real function of time. When *β*(*t*) = *β*, we recover the proportional hazards model which can then be taken as a special case of a non-proportional hazards model. We study tests of the null hypothesis; *H*_{0}:*β*(*t*) = 0 for all *t* against alternatives such as; *H*_{1}:∫*β*(*t*)d*F*(*t*) ≠ 0 or *H*_{1}:*β*(*t*) ≠ 0 for some t. In contrast to now classical approaches based on partial likelihood and martingale theory, the development here is based on Brownian motion, Donsker’s theorem and theorems from O’Quigley [1] and Xu and O’Quigley [2]. The usual partial likelihood score test arises as a special case. Large sample theory follows without special arguments, such as the martingale central limit theorem, and is relatively straightforward.

A family of tests for the presence of regression effect under proportional and non-proportional hazards models is described. The non-proportional hazards model, although not completely general, is very broad and includes a large number of possibilities. In the absence of restrictions, the regression coefficient,

Cite this paper

J. O’Quigley, "Survival Model Inference Using Functions of Brownian Motion,"*Applied Mathematics*, Vol. 3 No. 6, 2012, pp. 641-651. doi: 10.4236/am.2012.36098.

J. O’Quigley, "Survival Model Inference Using Functions of Brownian Motion,"

References

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[2] R. Xu and J. O’Quigley, “Proportional Hazards Estimate of the Conditional Survival Function,” Journal of the Royal Statistical Society: Series B, Vol. 62, No. 4, 2000, pp. 667-680. doi:10.1111/1467-9868.00256

[3] D. R. Cox, “Regression Models and Life Tables (with discussion),” Journal of the Royal Statistical Society: Series B, Vol. 34, 1972, pp. 187-220.

[4] T. Lancaster and S. Nickell, “The Analysis of Re-Employment Probabilities for the Unemployed,” Journal of the Royal Statistical Society, Series A, Vol. 143, No. 2, 1980, pp. 141-165. doi:10.2307/2981986

[5] M. H. Gail, S. Wieand and S. Piantadosi, “Biased Estimates of Treatment Effect in Randomized Experiments with Nonlinear Regressions and Omitted Covariates,” Biometrika, Vol. 71, No. 3, 1984, pp. 431-444. doi:10.1093/biomet/71.3.431

[6] J. Bretagnolle and C. Huber-Carol, “Effects of Omitting Covariates in Cox’s Model for Survival Data,” Scandinavian Journal of Statistics, Vol. 15, 1988, pp. 125-138.

[7] J. O’Quigley and F. Pessione, “Score Tests for Homogeneity of Regression Effect in the Proportional Hazards Model,” Biometrics, Vol. 45, 1989, pp. 135-144. doi:10.2307/2532040

[8] J. O’Quigley and F. Pessione, “The Problem of a Covariate-Time Qualitative Interaction in a Survival Study,” Biometrics, Vol. 47, 1991, pp. 101-115. doi:10.2307/2532499

[9] G. L. Anderson and T. R. Fleming, “Model Misspecification in Proportional Hazards Regression,” Biometrika, Vol. 82, No. 3, 1995, pp. 527-541.

[10] I. Ford, J. Norrie and S. Ahmadi, “Model Inconsistency, Illustrated by the Cox Proportional Hazards Model,” Statistics in Medicine, Vol. 14, No. 8, 1995, pp. 735-746. doi:10.1002/sim.4780140804

[11] R. Xu and J. O’Quigley, “Estimating Average Regression Effect under Non Proportional Hazards,” Biostatistics, Vol. 1, 2000, pp. 23-39. doi:10.1093/biostatistics/1.4.423

[12] J. O’Quigley and J. Stare, “Proportional Hazard Models with Frailties and Random Effects,” Statistics in Medicine, Vol. 21, 2003, pp. 3219-3233. doi:10.1002/sim.1259

[13] J. Kalbfleisch and R. L. Prentice, “The Statistical Analysis of Failure Time Data,” Wiley, New York, 1980.

[14] L. J. Wei, “Testing Goodness of Fit for Proportional Hazards Model with Censored Observations,” Journal of the American Statistical Association, Vol. 79, 1984, pp. 649-652.

[15] D. Y. Lin, L. J. Wei and Z. Ying, “Checking the Cox Model with Cumulative Sums of Martingale-Based Residuals,” Biometrika, Vol. 80, No. 3, 1993, pp. 557-572. doi:10.1093/biomet/80.3.557

[16] D. R. Cox, “Partial Likelihood,” Biometrika, Vol. 63, 1975, pp. 269-276. doi:10.1093/biomet/62.2.269

[17] P. K. Andersen and R. D. Gill, “Cox’s Regression Model for Counting Processes: A Large Sample Study,” Annals of Statistics, Vol. 10, No. 4, 1982, pp. 1100-1121. doi:10.1214/aos/1176345976

[18] P. Billingsley, “Convergence of Probability Measures,” Wiley, New York, 1968.

[19] R. B. Davies, “Hypothesis Testing When a Nuisance Parameter Is Present Only under the Alternative,” Biometrika, Vol. 64, No. 2, 1977, pp. 247-254. doi:10.2307/2335690

[20] R. B. Davies, “Hypothesis Testing When a Nuisance Parameter Is Present Only under the Alternative,” Biometrika, Vol. 74, 1987, pp. 33-43.

[21] B. Efron, “Censored Data and the Bootstrap,” Journal of the American Statistical Association, Vol. 76, No. 374, 1981, pp. 312-319.

[22] B. Efron, “Nonparametric Estimates of Standard Error: The Jacknife, the Bootstrap and Other Resampling Methods,” Biometrika, Vol. 68, 1981, pp. 589-599. doi:10.1093/biomet/68.3.589

[23] J. O’Quigley and L. Natarajan, “Erosion of Regression Effect in a Survival Study,” Biometrics, Vol. 60, No. 2, 2004, pp. 344-351. doi:10.1111/j.0006-341X.2004.00178.x

[1] J. O’Quigley, “Khmaladze-Type Graphical Evaluation of the Proportional Hazards Assumption,” Biometrika, Vol. 90, No. 3, 2003, pp. 577-584. doi:10.1093/biomet/90.3.577

[2] R. Xu and J. O’Quigley, “Proportional Hazards Estimate of the Conditional Survival Function,” Journal of the Royal Statistical Society: Series B, Vol. 62, No. 4, 2000, pp. 667-680. doi:10.1111/1467-9868.00256

[3] D. R. Cox, “Regression Models and Life Tables (with discussion),” Journal of the Royal Statistical Society: Series B, Vol. 34, 1972, pp. 187-220.

[4] T. Lancaster and S. Nickell, “The Analysis of Re-Employment Probabilities for the Unemployed,” Journal of the Royal Statistical Society, Series A, Vol. 143, No. 2, 1980, pp. 141-165. doi:10.2307/2981986

[5] M. H. Gail, S. Wieand and S. Piantadosi, “Biased Estimates of Treatment Effect in Randomized Experiments with Nonlinear Regressions and Omitted Covariates,” Biometrika, Vol. 71, No. 3, 1984, pp. 431-444. doi:10.1093/biomet/71.3.431

[6] J. Bretagnolle and C. Huber-Carol, “Effects of Omitting Covariates in Cox’s Model for Survival Data,” Scandinavian Journal of Statistics, Vol. 15, 1988, pp. 125-138.

[7] J. O’Quigley and F. Pessione, “Score Tests for Homogeneity of Regression Effect in the Proportional Hazards Model,” Biometrics, Vol. 45, 1989, pp. 135-144. doi:10.2307/2532040

[8] J. O’Quigley and F. Pessione, “The Problem of a Covariate-Time Qualitative Interaction in a Survival Study,” Biometrics, Vol. 47, 1991, pp. 101-115. doi:10.2307/2532499

[9] G. L. Anderson and T. R. Fleming, “Model Misspecification in Proportional Hazards Regression,” Biometrika, Vol. 82, No. 3, 1995, pp. 527-541.

[10] I. Ford, J. Norrie and S. Ahmadi, “Model Inconsistency, Illustrated by the Cox Proportional Hazards Model,” Statistics in Medicine, Vol. 14, No. 8, 1995, pp. 735-746. doi:10.1002/sim.4780140804

[11] R. Xu and J. O’Quigley, “Estimating Average Regression Effect under Non Proportional Hazards,” Biostatistics, Vol. 1, 2000, pp. 23-39. doi:10.1093/biostatistics/1.4.423

[12] J. O’Quigley and J. Stare, “Proportional Hazard Models with Frailties and Random Effects,” Statistics in Medicine, Vol. 21, 2003, pp. 3219-3233. doi:10.1002/sim.1259

[13] J. Kalbfleisch and R. L. Prentice, “The Statistical Analysis of Failure Time Data,” Wiley, New York, 1980.

[14] L. J. Wei, “Testing Goodness of Fit for Proportional Hazards Model with Censored Observations,” Journal of the American Statistical Association, Vol. 79, 1984, pp. 649-652.

[15] D. Y. Lin, L. J. Wei and Z. Ying, “Checking the Cox Model with Cumulative Sums of Martingale-Based Residuals,” Biometrika, Vol. 80, No. 3, 1993, pp. 557-572. doi:10.1093/biomet/80.3.557

[16] D. R. Cox, “Partial Likelihood,” Biometrika, Vol. 63, 1975, pp. 269-276. doi:10.1093/biomet/62.2.269

[17] P. K. Andersen and R. D. Gill, “Cox’s Regression Model for Counting Processes: A Large Sample Study,” Annals of Statistics, Vol. 10, No. 4, 1982, pp. 1100-1121. doi:10.1214/aos/1176345976

[18] P. Billingsley, “Convergence of Probability Measures,” Wiley, New York, 1968.

[19] R. B. Davies, “Hypothesis Testing When a Nuisance Parameter Is Present Only under the Alternative,” Biometrika, Vol. 64, No. 2, 1977, pp. 247-254. doi:10.2307/2335690

[20] R. B. Davies, “Hypothesis Testing When a Nuisance Parameter Is Present Only under the Alternative,” Biometrika, Vol. 74, 1987, pp. 33-43.

[21] B. Efron, “Censored Data and the Bootstrap,” Journal of the American Statistical Association, Vol. 76, No. 374, 1981, pp. 312-319.

[22] B. Efron, “Nonparametric Estimates of Standard Error: The Jacknife, the Bootstrap and Other Resampling Methods,” Biometrika, Vol. 68, 1981, pp. 589-599. doi:10.1093/biomet/68.3.589

[23] J. O’Quigley and L. Natarajan, “Erosion of Regression Effect in a Survival Study,” Biometrics, Vol. 60, No. 2, 2004, pp. 344-351. doi:10.1111/j.0006-341X.2004.00178.x