Patricia M. Grambsch

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 { fl(t)Zi(t)}. A common example is a treatment effect that decreases with time. We show that the function /3(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.

Graphical methods based on the analysis of residuals are considered for the setting of the highly-used Cox (1972) regression model and for the Andersen-Gill (1982) generalization of that model. We start with a class of martingale-based residuals as proposed by Barlow & Prentice (1988). These residuals and/or their transforms are useful for investigating the functional form of a covariate, the proportional hazards assumption, the leverage of each subject upon the estimates of β, and the lack of model fit to a given subject.

The ideal mathematical model for predicting survival for individual patients with primary biliary cirrhosis should be based on a small number of inexpensive, noninvasive measurements that are universally available. Such a model would be useful in medical management by aiding in the selection of patients for and timing of orthotopic liver transplantation. This paper describes the development, testing and use of a mathematical model for predicting survival. The Cox regression method and comprehensive data from 312 Mayo Clinic patients with primary biliary cirrhosis were used to derive a model based on patient's age, total serum bilirubin and serum albumin concentrations, prothrombin time and severity of edema. When cross-validated on an independent set of 106 Mayo Clinic primary biliary cirrhosis patients, the model predicted survival accurately. Our model was found to be comparable in quality to two other primary biliary cirrhosis survival models reported in the literature and to have the advantage of not requiring liver biopsy.

The natural history of primary sclerosing cholangitis was assessed in 174 patients; 37 were asymptomatic and 137 had symptoms related to underlying liver disease. At the time of diagnosis, the mean age was 39.9 years, 66% of the primary sclerosing cholangitis patients were male and 71% had associated inflammatory bowel disease, most commonly chronic ulcerative colitis. Long-term follow-up (mean: 6.0 years; range: 2.7 to 15.5 years) was available in all patients. During follow-up, 59 (34%) of the patients died: 55 in the symptomatic group and four in the asymptomatic group. Median survival from the time of diagnosis of primary sclerosing cholangitis at the Mayo Clinic was 11.9 years. Survival in the asymptomatic group was significantly decreased compared with that in a control population matched for age, race and sex. Multivariate analysis (Cox proportional hazards regression modeling) revealed that age, serum bilirubin concentration, blood hemoglobin concentration, presence or absence of inflammatory bowel disease and histologic stage on liver biopsy were independent predictors of high risk of dying. The development of a multivariate statistical survival model is a major step in identifying individual primary sclerosing cholangitis patients at low, moderate and high risk of dying. Such models will be useful for stratifying patients in therapeutic trials, in patient counseling and in patient selection and timing of liver transplantation.