Cox Proportional Hazards Model

Why does the time to readmission differ among patients who had the same heart surgery? Are patients with comorbidities readmitted sooner? Can we also use the information from patients who were not readmitted by the end of observation?

The method for analyzing the relationship between the time until an event and risk factors is the Cox proportional hazards model.

If you only compare whether patients were readmitted, you lose the information about how many days after surgery they were readmitted. The Cox model uses the time of the event and the length of follow-up, and it also handles censored data, where the event was not observed during follow-up. The effect of a variable is expressed as a hazard ratio (HR), which compares the instantaneous risk at a given time.

It is used not only in medical research but in any situation where the timing of an event matters, such as machine failure, customer churn, or the time it takes to find a job. In this lesson we use readmission after heart surgery as an example to cover censored data, interpreting hazard ratios, how the model differs from Kaplan–Meier and AFT models, the proportional hazards assumption, and reporting the results.

The Cox proportional hazards model is a survival analysis method that analyzes the relationship between the timing of patients’ readmission and risk factors. Rather than simply comparing whether patients were readmitted, it also uses the length of follow-up and the information from patients who have not yet been readmitted.

The model estimates how much the hazard, the instantaneous event rate among people who have not yet had the event at a given time, changes with the explanatory variables.

If a patient readmitted 30 days after surgery and a patient not readmitted over 120 days are treated the same, important time information is lost. Some patients may also drop out of the study during follow-up or reach the end of observation.

The Cox model expresses the hazard function as h(t|X) = h₀(t) × exp(β₁X₁ + ··· + βₖXₖ). h₀(t) is the baseline hazard function, and the exponential of each coefficient, exp(β), is the hazard ratio (HR).

The Cox model is a semiparametric method that estimates the regression coefficients using the partial likelihood, without specifying the shape of the baseline hazard function in advance.

Suppose 300 hypothetical heart surgery patients were followed for up to 120 days after surgery. For each patient, we record the last date they were seen and whether they were readmitted. The numbers below are hypothetical results to illustrate the flow of the analysis, not the results of fitting real patient data.

HR = 1 means the two conditions being compared have the same risk. HR > 1 means the risk is higher, and HR < 1 means it is lower. For example, HR = 1.85 is the model’s statement that, adjusting for other factors, the instantaneous risk of readmission is about 85% higher at any point in time.

Censored patients provide the information that the event had not occurred by the time of censoring. It is not assumed that they were never readmitted after censoring. Also, if patients can die before being readmitted, consider whether treating this as simple censoring is valid, and consider a competing risks analysis.

In these hypothetical results, age and comorbidity are associated with a higher risk of readmission. Surgery type B has an HR above 1, but its confidence interval includes 1, so it is hard to conclude there is a clear difference. These are teaching numbers set to be consistent with one another, not estimates fitted to data.

The Cox model describes the instantaneous hazard ratio (HR) of the event when other conditions are equal. The accelerated failure time (AFT) model focuses on how much the explanatory variables stretch or shorten the time until the event.

The two models rest on different assumptions, so rather than choosing one just because it is easier to interpret, choose based on the data and the research question.

Proportional hazards assumption: assumes that each explanatory variable’s HR is constant over time. Check Schoenfeld residuals, interactions with time and so on; if the assumption does not hold, consider time-varying effects or a stratified Cox model.

Linearity: for continuous variables such as age, check whether the relationship with the log hazard is linear. You can also use splines.

Censoring and event definitions: clearly define the reasons for dropout, what counts as readmission, the start of follow-up, and competing events such as death.

Causation: hazard ratios from observational data show adjusted associations and do not remove the influence of unmeasured confounders.

When writing up the results, present the study population, the follow-up period, the definitions of the event and censoring, the covariates, the hazard ratios and confidence intervals, and the results of checking the proportional hazards assumption.

A real paper should add information produced by the actual analysis, such as the number of events, the number censored, and the results of checking the proportional hazards assumption. This example does not claim that those diagnostics were carried out.

The Cox proportional hazards model estimates the relationship between explanatory variables and the instantaneous risk, taking into account not only whether an event happened but when. In the example of readmission after heart surgery, it described the relationship of patients’ age, comorbidity and type of surgery with the time until readmission in terms of hazard ratios (HR).

A hazard ratio does not directly mean the probability of readmission over a given period or the average time to readmission. When interpreting it, check the confidence interval and the definitions of the event and censoring, and examine the proportional hazards assumption that the hazard ratio is constant over time. If the assumption does not hold, also consider methods that allow effects to change over time.