Cox Proportional Hazards Regression

Cox proportional hazards regression models the relationship between one or more predictors and the hazard of an event over time. Unlike Kaplan-Meier analysis, which estimates survival curves without predictors, Cox regression allows you to evaluate how each covariate is associated with the rate at which events occur, while controlling for the others.

When to use Cox regression

Cox regression is appropriate when:

• The outcome is time to an event with censoring
• You want to estimate the effect of one or more predictors on the hazard rate
• You want to report hazard ratios with confidence intervals and p-values

If you want to estimate and visualize survival curves without modeling covariates, Kaplan-Meier analysis is simpler — see Kaplan-Meier Analysis.

Data requirements

Your dataset needs:

• A time column — duration from the start of follow-up to the event or censoring
• An event indicator column — binary variable indicating whether the event occurred
• One or more covariate columns — the predictors to include in the model

Licklider cannot determine automatically whether the chosen time origin, event definition, censoring process, or covariate set is scientifically appropriate for your study. In particular, it cannot infer from the table alone whether censoring is informative, whether competing risks should be modeled differently, whether a predictor has a time-varying effect, or whether collinearity and omitted covariates make the hazard ratio hard to interpret.

These limits matter because a Cox model can fit and return hazard ratios even when the survival question has been posed poorly. In that case, the HR, confidence interval, and proportional-hazards check may look technically complete while still supporting a misleading scientific conclusion.

How to request it

Describe the analysis in the Chat. For example:

• "Run a Cox regression with treatment group and age as covariates"
• "Fit a proportional hazards model"
• "Show the hazard ratio for each predictor"

Specify the covariates you want to include. If no covariates are named, Licklider will ask which columns to include.

What the results include

Coefficient table

One row per covariate. Each row shows:

• Hazard ratio (HR) — the exponentiated coefficient. An HR greater than 1 indicates that higher values of the predictor are associated with a higher rate of the event. An HR less than 1 indicates a lower rate.
• 95% confidence interval on the hazard ratio
• Coefficient (log hazard ratio)
• Standard error
• z-statistic
• p-value

Model summary

• Number of observations and number of events
• Concordance index — a measure of how well the model discriminates between subjects who experienced the event sooner versus later. A value of 0.5 indicates no better than chance; 1.0 indicates perfect discrimination.
• Ties handling method used

The concordance index is included to give you a quick sense of how well the fitted model separates earlier from later events. The ties handling method is reported because the estimated coefficients can depend slightly on how identical event times were handled, especially when ties are common.

Proportional hazards assumption

The Cox model assumes that the hazard ratio for each predictor is constant over time — the proportional hazards (PH) assumption. If this assumption is violated, the estimated hazard ratios are averaged across time in a way that may misrepresent the actual relationship.

Licklider evaluates the PH assumption after fitting the model and reports one of three states in the Inspector:

No PH violation detected

The statistical test did not find evidence of a PH assumption violation. The hazard ratio estimates can be interpreted as approximately constant over the follow-up period.

Potential PH violation detected

The test found evidence that the proportional hazards assumption may not hold for one or more covariates. This means the hazard ratio may change over time. Consider reporting time-varying effects or stratifying by the covariate in question.

Check not available

The PH check could not be run for this dataset — for example, because the dataset is too large (the check is available for datasets with 1,000 observations or fewer), because there are too few events, or because the model configuration does not support it.

Ties handling

When multiple observations have the same recorded event time, different methods exist for handling these ties. Licklider supports the Breslow, Efron, and exact methods. The method used is reported in the results.

Design Rationale & References

This page follows a simple rule: a survival regression result should report not only effect estimates, but also the key assumptions that make those estimates interpretable. That is why Licklider reports hazard ratios with confidence intervals, surfaces the proportional hazards check, and records the ties handling method used by the model.

Cox regression is used here because it is the standard semi-parametric approach for relating covariates to time-to-event outcomes without requiring the baseline hazard function to be specified in advance [1]. The proportional hazards check is highlighted because a hazard ratio is most readable when it can be treated as approximately constant over follow-up; when that assumption fails, a single HR can mask time-varying effects [2].

The concordance index is shown to give a practical summary of how well the model orders earlier versus later events, while the ties method is kept visible because identical event times are common in discretized clinical follow-up and can affect estimation details. The "check not available" state is also intentional: it is better to state that the PH diagnostic was not run under the current conditions than to imply that the assumption has been cleared automatically.

1. Cox, D. R. (1972). Regression Models and Life-Tables. Journal of the Royal Statistical Society: Series B (Methodological), 34(2), 187-220. https://rss.onlinelibrary.wiley.com/doi/10.1111/j.2517-6161.1972.tb00899.x
2. Grambsch, P. M., & Therneau, T. M. (1994). Proportional Hazards Tests and Diagnostics Based on Weighted Residuals. Biometrika, 81(3), 515-526.

What this page does not cover

• Survival curves without covariates — see Kaplan-Meier Analysis
• How survival data is detected automatically — see Survival Data Detection Guard