Tests for proportional hazards assumption concerning specified covariates or groups of covariates are proposed. The class of alternatives is wide: log-hazard rates under different values of covariates may cross, approach, go away. The data may be right censored. The limit distribution of the test statistic is derived. Power of the test against approaching alternatives is given. Real data examples are considered.

The most known semi-parametric model for analysis of failure time regression data is the proportional hazards (PH) model. There are many tests for the PH model from right censored data given by Cox [

We consider tests for proportional hazards assumption concerning specified covariates or groups of covariates. The class of alternatives is wide: log-hazard rates under different values of covariates may cross, approach, go away. The data may be right censored. The limit distribution of the test statistics is derived. Power of the test under approaching alternatives is given. Real examples are considered.

Let

Let us consider the proportional hazards (PH) model:

Under the PH model the ratios of hazard functions under any two different explanatory variables are constant over time, i.e. they are proportional.

Suppose that proportionality of hazard functions with respect to a specified covariate

We propose a model of the form

The PH model is a particular case of this model when

Indeed, suppose that two different values

We do not discuss application of the model for analysis of survival regression data (which is a subject of another article). Such analysis could be done if the PH model would be rejected. Here the model is used only as a generator of a wide class of alternatives to the PH model.

Let us consider right censored failure time regression data:

Set

Suppose that the survival distribution of the

Suppose that the multiplicative intensities model is verified: the compensators of the counting processes

Information about time-to-failure distribution contains the points, where the counting processes

The multiplicative intensities model is verified in the case of type I, type II, independent random censoring.

In the parametric case with known

Let us consider asymptotic distribution of the statistic (

the covariates

the matrix

Assumption a) can be weakened considerably but we avoid writing complicated formulas for easier reading. Assumption b) simply means that at some finite time moment (perhaps very remote) observations are stopped. This is a usual assumption for asymptotic results to hold in survival analysis. Assumption c) also can be weakened. Assumption d) means that if censoring would be absent then units might survive after the moment

Under Assumptions

Assumption

Convergences (

Under the Cox model,

The predictable variation of the local martingale

Under Assumptions

The test: the null hypothesis

Let us consider the alternative model (

So the model

Let us consider the stochastic process

Note that the derivative with respect to

Fix

Assumption

Note that

Set

The first term converges weakly to

Let us consider the norm of the difference:

The mean value theorem and consistency of the estimator

Set

Set

Note that the non-martingale part is

The power function of the test against approaching alternatives is

The proportional hazards hypothesis for several covariates

Set

Replacing

So

The test statistic

Stablein and Koutrouvelis [

The number of patients

The data are given in [

The results of testing hypothesis for all covariates: the value of the test statistic

The results for each covariate are given in Table

Prisoners data. The values of test statistics and

Covar. | FIN | AGE | RACE | WEX | MAR | PAR | PRI | Glob. test |

Stat. | 0.162 | 2.464 | 1.423 | −2.033 | −1.017 | −0.222 | 0.672 | 17.58 |

0.872 | 0.014 | 0.155 | 0.042 | 0.309 | 0.824 | 0.502 | 0.014 |

Let us consider right censored UIS data set given in [

UIS was a 5-year research project comprised of two concurrent randomized trials of residential treatment for drug abuse. The purpose of the study was to compare treatment programs of different planned durations designed to reduce drug abuse and to prevent high-risk HIV behavior. The UIS sought to determine whether alternative residential treatment approaches are variable in effectiveness and whether efficacy depends on planned program duration. The time variable is time to return to drug use (measured from admission). The individuals who did not returned to drug use are right censored. We use the model with 10 covariates (which support PH assumption) given by Hosmer, Lemeshow and May (2008). The covariates are: AGE (years); Beck depression score (beckt; 0 – 54);

The results of testing hypothesis for all covariates: the value of the test statistic

The results for each covariate are given in Table

UIS dataset. The values of test statistics and

Covariate | AGE | beckt | NDR1 | NDR2 | IVHX_3 | RACE |

Stat. | −0.061 | 1.085 | 0.182 | 0.118 | 0.912 | −1.278 |

0.952 | 0.278 | 0.856 | 0.906 | 0.362 | 0.201 |

Covariate | TREAT | SITE | AGEXS | RACEXS | Global test |

Stat. | 0.792 | 1.016 | −0.378 | 6.781 | −0.107 |

0.429 | 0.309 | 0.705 | 0.746 | 0.915 |

The assumption of the PH assumption for individual covariates is not rejected.

We thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions.