Proportional hazards regression

The most popular method for analysis of covariance is the proportional hazards model. This model, originally developed by Cox (1972), is now used extensively in the analysis of survival data to incorporate and adjust for both centre and covariate effects. The model assumes that the hazard ratio for the treatment effect is constant. 

The method provides a model for the hazard function. As in Section 6.6, let z be an indicator variable for treatment taking the value one for patients in the active group and zero for patients in the control group and let Xj, X2, etc. denote the covariates. If we let t) denote the hazard rate as a function of t (time), the main effects model takes the form  

As before, the coefficient c measures the effect of treatment on the hazard rate. If c 0 then the log hazard rate, and therefore the hazard rate itself, in the active group is lower than the hazard rate in the control group. If c 0 then the reverse 

Example 13.2 Genasence in the treatment of advance malignant melanoma 

In this study reported by Bedikian et al. (2006), several potential baseline prognostic factors were included in a proportional hazards model. These factors were  

---

TABLE 5. Multivariate proportional hazard regression analysis of relative risk... 

In this light, Robinson et al. (206) performed a study of 104 first-episode patients who were followed for a minimum of 2 months (mean, 207 weeks). The protocol was later limited to a maximal 5-year follow-up. Patients who wished to discontinue drug could do so. The rate of relapse was determined by the Cox Proportional Hazard Regression model. The cumulative life table relapse rate at 5 years was 82%. Most importantly, despite use of medication by many, there were only four unrelapsed patients after 5 years. Although medication was not controlled, the patients who discontinued had a fivefold increase in the relapse rate. This finding suggests that almost all first admission patients will relapse within the next 5 years, suggesting that most should be on maintenance medication. 

Ahn, H., and Loh, W.-Y. (1994). Tree-strucmred proportional hazards regression modeling. Biometrics, 50 471-485. 

Tumorigenidty Occupational and animal studies have shown an increased risk of cancer after exposure to coal tar, which contains several carcinogenic compounds, but the risk of cancer is unclear. In 13200 patients with psoriasis and eczema, proportional hazards regression was used to evaluate differences in cancer risks [5. Patients who used coal tar ointments were compared with patients who used topical glucocorticoids. The median duration of exposure was 6 (range 1-300) months. Coal tar did not increase the risk of non-skin malignancies (HR=0.92 95% Cl = 0.8, 1.1) or the risk of skin cancers (HR= 1.1 95% CI=0.69,1.7). 

The difference in overall survival between paracetamol-related and non-paracetamol-related acute liver failure was not statistically significant Patients with paracetamol-related acute liver failure required dialysis before transplantation more often than all the other groups (27% versus 3-10%). Cox proportional hazards regression analysis showed that the independent pre-transplantation predictors of death after transplantation were being on life support, liver failure due to antiepileptic drugs at age under 18 years, and a raised serum creatinine concentration. 

Risk Ratios Adjusted by Proportional Hazards Regression ... 

Using regression analysis based on Cox s proportional hazards model, Ott and Zober (1996) found evidence of association between 2,3,7,8-TCDD exposure and digestive cancer (conditional risk ratio of 1.46 95% 0=1.13-1.89) the primary tumor sites were the liver, stomach, and pancreas. 

Cox regression. Another name for the proportional hazards model originally proposed by Sir David Cox in 1972. 

Proportional hazards model. Generally, a regression model used in survival analysis whereby it is assumed that the hazard functions of individuals under study are proportional to each other over time. (This is equivalent to assuming that the log-hazard functions differ by a constant.) More specifically, and in the form originally proposed by Cox (1972), no further definite assumption is made about the hazard functions themselves. One of the most commonly used statistical techniques in survival analysis. 

The appeal is the ease of computation and applicability. The resulting statistics or p-values for the chosen filter method are then ranked and a cutoff chosen to select the most significant features. Examples of filter methods are t-tests, Wdcoxon rank-sum or signed-rank tests, Pearson correlation estimates, log-rank tests, and univariate regression techniques such as linear, logistic, or Cox proportional hazards. 

Liao, H., W. Zhao and H. Guo (2006). Predicting remaining useful life of an individual unit using proportional hazards model and logistic regression model. In Reliability and Maintainability Symposium, 2006. RAMS 06. Annual, pp. 127-132. 

The observations of the censoring variable Wi come from the Poisson distribution, a member of the exponential family. The logarithm of the parameter /r, is linked to the linear predictor rj. The observations are independent. Clearly the proportional hazards model is a generalized linear model and can be analyzed the same way as the Poisson regression model. 

The proportional hazards model is used to regress censored survival data on when the values of a set of predictor variables are known for each observation. For each observation we record f j, its time of death, Wi the censoring variable which equals 1 if we observe the true death ti, and 0 when the observation is censored and all we know the true lifetime is greater than ti, and the values of the predictors Xji,..., Xip for that observation. 

---