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Biometrika 1994 81(3):515-526; doi:10.1093/biomet/81.3.515
© 1994 by Biometrika Trust
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Proportional hazards tests and diagnostics based on weighted residuals

PATRICIA M. GRAMBSCH1 and TERRY M. THERNEAU2

1 Division of Biostatistics, School of Public Health, University of Minnesota Minneapolis, Minnesota 55455, U.S.A.
2 Department of Health Science Research Mayo Clinic, Rochester, Minnesota 55905, U.S.A.

Received for publication 1 March 1993. Revision received 1 January 1994.
   Abstract

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 {β(t)Zi(t)}. A common example is a treatment effect that decreases with time. We show that the function βi(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.

Key Words: Cox model • Loess • Schoenfeld residuals • Weighted regression


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M. J. O'Connell, M. E. Campbell, R. M. Goldberg, A. Grothey, J.-F. Seitz, J. K. Benedetti, T. Andre, D. G. Haller, and D. J. Sargent
Survival Following Recurrence in Stage II and III Colon Cancer: Findings From the ACCENT Data Set
J. Clin. Oncol., May 10, 2008; 26(14): 2336 - 2341.
[Abstract] [Full Text] [PDF]


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CMAJHome page
L. Pilote MD PhD, M. Abrahamowicz PhD, M. Eisenberg MD MPH, K. Humphries DSc, H. Behlouli MSc PhD, and J. V. Tu MD PhD
Effect of different angiotensin-converting-enzyme inhibitors on mortality among elderly patients with congestive heart failure
Can. Med. Assoc. J., May 6, 2008; 178(10): 1303 - 1311.
[Abstract] [Full Text] [PDF]


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CirculationHome page
G. A. Wellenius, K. J. Mukamal, A. Kulshreshtha, S. Asonganyi, and M. A. Mittleman
Depressive Symptoms and the Risk of Atherosclerotic Progression Among Patients With Coronary Artery Bypass Grafts
Circulation, May 6, 2008; 117(18): 2313 - 2319.
[Abstract] [Full Text] [PDF]


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Am. J. Clin. Nutr.Home page
E. J Brunner, A. Mosdol, D. R Witte, P. Martikainen, M. Stafford, M. J Shipley, and M. G Marmot
Dietary patterns and 15-y risks of major coronary events, diabetes, and mortality
Am. J. Clinical Nutrition, May 1, 2008; 87(5): 1414 - 1421.
[Abstract] [Full Text] [PDF]


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Eur Heart JHome page
A. Torabi, J. G.F. Cleland, N. K. Khan, P. H. Loh, A. L. Clark, F. Alamgir, J. L. Caplin, A. S. Rigby, and K. Goode
The timing of development and subsequent clinical course of heart failure after a myocardial infarction
Eur. Heart J., April 1, 2008; 29(7): 859 - 870.
[Abstract] [Full Text] [PDF]


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Nephrol Dial TransplantHome page
M. Li, G. Tomlinson, G. Naglie, W. L Cook, and S. V. Jassal
Geriatric comorbidities, such as falls, confer an independent mortality risk to elderly dialysis patients
Nephrol. Dial. Transplant., April 1, 2008; 23(4): 1396 - 1400.
[Abstract] [Full Text] [PDF]


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NEJMHome page
S. Kathiresan, O. Melander, D. Anevski, C. Guiducci, N. P. Burtt, C. Roos, J. N. Hirschhorn, G. Berglund, B. Hedblad, L. Groop, et al.
Polymorphisms Associated with Cholesterol and Risk of Cardiovascular Events
N. Engl. J. Med., March 20, 2008; 358(12): 1240 - 1249.
[Abstract] [Full Text] [PDF]


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JNCI J Natl Cancer InstHome page
F. Di Costanzo, S. Gasperoni, L. Manzione, G. Bisagni, R. Labianca, S. Bravi, E. Cortesi, P. Carlini, R. Bracci, S. Tomao, et al.
Adjuvant Chemotherapy in Completely Resected Gastric Cancer: A Randomized Phase III Trial Conducted by GOIRC
J Natl Cancer Inst, March 19, 2008; 100(6): 388 - 398.
[Abstract] [Full Text] [PDF]


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Clin. Cancer Res.Home page
M. C.U. Cheang, D. Voduc, C. Bajdik, S. Leung, S. McKinney, S. K. Chia, C. M. Perou, and T. O. Nielsen
Basal-Like Breast Cancer Defined by Five Biomarkers Has Superior Prognostic Value than Triple-Negative Phenotype
Clin. Cancer Res., March 1, 2008; 14(5): 1368 - 1376.
[Abstract] [Full Text] [PDF]


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Antimicrob. Agents Chemother.Home page
K. H. Fife, T.-C. Meng, D. G. Ferris, and P. Liu
Effect of Resiquimod 0.01% Gel on Lesion Healing and Viral Shedding When Applied to Genital Herpes Lesions
Antimicrob. Agents Chemother., February 1, 2008; 52(2): 477 - 482.
[Abstract] [Full Text] [PDF]


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Eur Heart JHome page
C. Tribouilloy, D. Rusinaru, H. Mahjoub, V. Souliere, F. Levy, M. Peltier, M. Slama, and Z. Massy
Prognosis of heart failure with preserved ejection fraction: a 5 year prospective population-based study
Eur. Heart J., February 1, 2008; 29(3): 339 - 347.
[Abstract] [Full Text] [PDF]


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Hum ReprodHome page
J. W. van der Steeg, P. Steures, M. J.C. Eijkemans, J. D. F. Habbema, P. G.A. Hompes, J. M. Burggraaff, G. J. E. Oosterhuis, P. M.M. Bossuyt, F. van der Veen, and B. W.J. Mol
Obesity affects spontaneous pregnancy chances in subfertile, ovulatory women
Hum. Reprod., February 1, 2008; 23(2): 324 - 328.
[Abstract] [Full Text] [PDF]


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BloodHome page
E. Hoster, M. Dreyling, W. Klapper, C. Gisselbrecht, A. van Hoof, H. C. Kluin-Nelemans, M. Pfreundschuh, M. Reiser, B. Metzner, H. Einsele, et al.
A new prognostic index (MIPI) for patients with advanced-stage mantle cell lymphoma
Blood, January 15, 2008; 111(2): 558 - 565.
[Abstract] [Full Text] [PDF]


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NEJMHome page
K. Miller, M. Wang, J. Gralow, M. Dickler, M. Cobleigh, E. A. Perez, T. Shenkier, D. Cella, and N. E. Davidson
Paclitaxel plus Bevacizumab versus Paclitaxel Alone for Metastatic Breast Cancer
N. Engl. J. Med., December 27, 2007; 357(26): 2666 - 2676.
[Abstract] [Full Text] [PDF]


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NEJMHome page
T. G. Liou, F. R. Adler, D. R. Cox, and B. C. Cahill
Lung Transplantation and Survival in Children with Cystic Fibrosis
N. Engl. J. Med., November 22, 2007; 357(21): 2143 - 2152.
[Abstract] [Full Text] [PDF]


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JCOHome page
E. O. Hanrahan, A. M. Gonzalez-Angulo, S. H. Giordano, R. Rouzier, K. R. Broglio, G. N. Hortobagyi, and V. Valero
Overall Survival and Cause-Specific Mortality of Patients With Stage T1a,bN0M0 Breast Carcinoma
J. Clin. Oncol., November 1, 2007; 25(31): 4952 - 4960.
[Abstract] [Full Text] [PDF]


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J. Lipid Res.Home page
K.-L. Chien, H.-C. Hsu, T.-C. Su, M.-F. Chen, Y.-T. Lee, and F. B. Hu
Apolipoprotein B and non-high density lipoprotein cholesterol and the risk of coronary heart disease in Chinese
J. Lipid Res., November 1, 2007; 48(11): 2499 - 2505.
[Abstract] [Full Text] [PDF]


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Clin. Cancer Res.Home page
P. Mathew, P. F. Thall, C. D. Bucana, W. K. Oh, M. J. Morris, D. M. Jones, M. M. Johnson, S. Wen, L. C. Pagliaro, N. M. Tannir, et al.
Platelet-Derived Growth Factor Receptor Inhibition and Chemotherapy for Castration-Resistant Prostate Cancer with Bone Metastases
Clin. Cancer Res., October 1, 2007; 13(19): 5816 - 5824.
[Abstract] [Full Text] [PDF]


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JCOHome page
M. C. Pinder, Z. Duan, J. S. Goodwin, G. N. Hortobagyi, and S. H. Giordano
Congestive Heart Failure in Older Women Treated With Adjuvant Anthracycline Chemotherapy for Breast Cancer
J. Clin. Oncol., September 1, 2007; 25(25): 3808 - 3815.
[Abstract] [Full Text] [PDF]



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