Reisensburg 1997: Abstract Utikal
Statistical Computing '97 - Schloß Reisensburg

Goodness-of-Fit Tests to Distinguish Between Additive and Proportional Hazards Models

Klaus Utikal

Fakultät für Wirtschaftswissenschaften, Wirtschaftstheorie II
Universität Bonn

The additive risk model, introduced by Aalen (1980) and abbreviated AdR model, has gained popularity as a plausible, conceptually simple alternative to the Cox proportional hazards model (PH model). It allows (unlike the PH model) covariate influence to be time-dependent; as family of surfaces, these can be intersecting. In the analysis of censored survival data it can lead to substantially different conclusions.

In the present paper we propose a test of the AdR model against the alternative of the PH model. Note that the PH and AdR models represent separate (i.e. non-nested) hypotheses, except in the case of degenerate or bivariate covariates. The approach chosen here is to compare the maximum partial likelihood estimator of the vector of regression weights in the PH model with an estimate of the expectation of this estimate under the AdR model.

This idea for testing separate hypotheses, by comparing an estimator of a key feature of an alternative model with its expectation under the null hypothesis, is not new and goes back at least to the work of Cox (1961, 1962). It has been well studied in the econometric literature in recent years (see, e.g., Mizon and Richard, 1986) where it is called the "encompassing principle." According to this principle, a model should be able to explain the salient features of rival models.

We also consider the "dual" problem of assessing goodness-of-fit for the PH model with the AdR model as the alternative. In this case we compare the Huffer-McKeague (1991) estimator of the vector of cumulative hazard components for the AdR model with an estimate of its expectation under the PH model.


  1. Aalen, O. O. (1980). A model for nonparametric regression analysis of counting processes. Lecture Notes in Statistics, 2.
  2. Cox, D.R. (1961). Tests of separate families of hypotheses, Proc. 4th Berkely Symp., 1, 105-123.
  3. Cox, D.R. (1962). Further results on tests of separate families of hypotheses. JRSS, B, 24, 406-424.
  4. Mizon, G. E. and Richard, J. F. (1986). The encompassing principle and its application to testing non-nested hypotheses. Econometrica 54, pp. 657-678.
  5. Huffer, F.W. and McKeague, I.W. (1991). Weighted least squares estimation for Aalen's additive risk model. J. Amer. Statist. Assoc. 86, pp. 114-126.

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