Cox proportional hazards model
Cox regression, or sometimes referred to as proportional hazards regression, is a multivariate regression technique used to model survival analysis data.1 This technique is most commonly utilized when investigators are looking at time-to-event outcomes. Unlike other types of regression models, the only outcome reported for Cox regression is a hazard ratio. As a clinical informaticist, it is always a goal to improve patient outcomes. Cox regression is an invaluable tool to accomplish this. It allows the investigation of survival time of patients and the relationship to a series of continuous and/or binary predictors (covariates).
Cox regression assumptions
• Proportional hazards assumption Cox regression is sometimes referred to as proportional hazards regression because it is required that the assumption of proportional hazards is met. Simply stated, the assumption states that the hazard ratio for any two individuals in the study needs to be constant over time. There are multiple ways to evaluate the proportional hazards assumption but below 4 are listed.
1. Examine log(-log(S(t)) plots
2. Include interaction with time in the model
3. Plot Schoenfield residuals
4. Regress Schoenfeld residuals against time to test for independence between residuals and time
• Schoenfeld residual – a separate residual for each individual for each covariate.2
• Hazard ratio – an estimated ratio of hazard rates for the treated arm versus the control arm. A measure of how often an event happens in one group compared to how often the event occurs in another group, over time.3
o HR > 1 indicates an increase in risk
o HR < 1 indicates a decrease in risk
o HR = 0 indicates no change in risk
In 1972 Sir David Cox came out with his paper titled, “Regression Models and Life-Tables" outlining the proportional hazards model which was subsequently named after him.1 The model led to countless medical studies on survival time and various patient exposures/attributes such as age, diet, and drug exposure. He was knighted by Queen Elizabeth II in 1985.4
Tyring S. et al. “Famciclovir for the treatment of acute herpes zoster: effects on acute disease and postherpetic neuralgia. A randomized, double-blind, placebo-controlled trial. Collaborative Famciclovir Herpes Zoster Study Group”. Ann Intern Med. 1995 Jul 15;123(2):89-96. o The study was using Cox Regression and hazard ratios to investigate the outcomes of a specific drug treatment regimen. This model is commonly used when determining the effects of a drug on disease prognosis while incorporating multiple other predictors into the model.
Being able to identify variables that will influence the outcome of a patient’s prognosis is a critical role of all healthcare providers and clinical informaticists. Cox regression allows investigators to evaluate the effects of multiple predictors simultaneously and gain a better understanding of the effect size for each predictor. With the advent of EMR’s and dramatically increased access to data, healthcare providers can utilize Cox regression models to help guide therapies, clinical decision making, and prognostic criteria.
1. Cox, D. (1972). "Regression Models and Life-Tables". Journal of the Royal Statistical Society, Series B. 34 (2): 187–220.
2. Schoenfeld D. (1982) Residuals for the proportional hazards regression model. Biometrika, 69(1):239-241.
3. Spruance, Spotswood L et al. “Hazard ratio in clinical trials.” Antimicrobial agents and chemotherapy vol. 48,8 (2004): 2787-92. doi:10.1128/AAC.48.8.2787-2792.2004
4. “No. 50221”. The London Gazette. 6 August 1985. P. 10815.
Submitted by (Matthew Hill)