Cox Proportional Hazards Calculator

The Cox proportional hazards model is a regression method used in survival analysis to examine the relationship between covariates (predictors) and the time until an event occurs (e.g., death, disease recurrence). It estimates a hazard ratio indicating how much a covariate changes the risk of the event. Unlike parametric models, it does not assume a specific distribution for survival times, making it widely used in medical and clinical research.

A hazard ratio is the ratio of the hazard rate in the treatment/exposed group to the hazard rate in the control/unexposed group. An HR of 1 means no difference in risk between groups. An HR greater than 1 indicates increased hazard (higher risk) in the treatment group, while an HR less than 1 indicates decreased hazard (lower risk or protective effect). The HR is interpreted as the instantaneous relative risk of the event occurring at any given time point.

If the HR is, say, 0.60, the treatment group has a 40% lower instantaneous risk of the event compared to the control group at any time point. If the 95% confidence interval does not include 1.0 and the p-value is below your significance threshold (commonly 0.05), the result is statistically significant. An HR above 1 with a significant p-value suggests the covariate is associated with increased hazard.

A risk ratio compares the cumulative probability of an event over a fixed time period between two groups, while a hazard ratio compares instantaneous rates of the event at any given moment throughout the follow-up period. In survival analysis, the hazard ratio is preferred because it accounts for censored observations and time-varying risk. They can be numerically similar for rare events but diverge when events are common or when the follow-up period is long.

A censored observation is one where the event of interest had not yet occurred by the end of the study period, or the subject was lost to follow-up. Censoring is marked as 0 (event = 0), while subjects who experienced the event are marked as 1. The Cox model handles censored data appropriately, which is one of its key advantages over simple comparison methods. Censored subjects still contribute information about survival up to their last observed time point.

Person-time (also called total follow-up time or exposure time) is the sum of the time each subject was observed until either the event occurred or censoring happened. For example, if 10 subjects are each followed for an average of 5 months, the total person-time is 50 person-months. It reflects the actual time at risk across all subjects and is used to calculate the hazard rate (events ÷ person-time) for each group.

The confidence interval is calculated on the log scale using the standard error of the log hazard ratio. The log HR ± (Z-critical × SE) gives the lower and upper bounds on the log scale, which are then exponentiated back to the HR scale. For a 95% two-sided CI, the Z-critical value is approximately 1.96. Wider intervals indicate more uncertainty, often due to smaller sample sizes or fewer observed events.

A common rule of thumb is that you need at least 10–15 events per predictor variable (EPV) included in the Cox model to ensure stable coefficient estimates and reliable p-values. For this simplified two-group calculator, having at least 10 events in each group is recommended. Small event counts lead to wide confidence intervals and unreliable estimates. Increasing the follow-up period or sample size improves statistical power.