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)dF(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.

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

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

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

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

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.

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

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

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

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

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

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.

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

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

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

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

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