Proportional hazards tests and diagnostics based on weighted residuals

Patricia M. Grambsch, Terry M. Therneau

Research output: Contribution to journalArticlepeer-review

3254 Scopus citations


SUMMARY: Nonproportional hazards can often be expressed by extending the Cox model to include time varying coefficients; e.g., for a single covariate, the hazard function for subject i is modelled as exp {ß(t)Zi(t)}. A common example is a treatment effect that decreases with time. We show that the function ßi(t) can be directly visualized by smoothing an appropriate residual plot. Also, many tests of proportional hazards, including those of Cox (1972), Gill & Schumacher (1987), Harrell (1986), Lin (1991), Moreau, O'Quigley & Mesbah (1985), Nagelkerke, Oosting & Hart (1984), O'Quigley & Pessione (1989), Schoenfeld (1980) and Wei (1984) are related to time-weighted score tests of the proportional hazards hypothesis, and can be visualized as a weighted least-squares line fitted to the residual plot.

Original languageEnglish (US)
Pages (from-to)515-526
Number of pages12
Issue number3
StatePublished - Sep 1 1994


  • Cox model
  • Loess
  • Schoenfeld residuals
  • Weighted regression

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Agricultural and Biological Sciences (miscellaneous)
  • Agricultural and Biological Sciences(all)
  • Statistics, Probability and Uncertainty
  • Applied Mathematics


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