Cox Proportional Hazards Regression

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What is Cox Proportional Hazards Regression?

Cox Proportional Hazards Regression (also called Cox regression or the Cox model) is a semi-parametric regression model for time-to-event data. It models how covariates (predictors) affect the hazard function, which is the instantaneous rate of the event occurring at any given time.

The model is "semi-parametric" because it does not specify the shape of the baseline hazard function. It only models the relative effect of covariates on the hazard, expressed as hazard ratios. This makes it very flexible: you do not need to assume the survival distribution is exponential, Weibull, or any other specific shape.

Cox Proportional Hazards Regression is also called the Cox Model, Cox Regression, Cox PH Model, or the Proportional Hazards Model.

Assumptions for Cox Proportional Hazards Regression

Every statistical method has assumptions. Assumptions mean that your data must satisfy certain properties in order for statistical method results to be accurate.

The assumptions for Cox Proportional Hazards Regression include:

1. Proportional Hazards
2. Time-to-Event Outcome
3. Non-Informative Censoring
4. Linear Covariate Effects (on the log-hazard scale)
5. Independent Observations

Proportional Hazards

The ratio of hazards between any two groups must be constant over time. If Group A has twice the hazard of Group B at day 1, it should have approximately twice the hazard at day 30 and day 90 as well.

Test this assumption using Schoenfeld residuals. If it is violated, consider stratified models, time-varying coefficients, or alternative approaches like the Accelerated Failure Time model.

Time-to-Event Outcome

Your dependent variable is the time from a defined starting point to an event. All the standard survival analysis requirements apply: a clear origin, a well-defined event, and proper handling of censoring.

Non-Informative Censoring

Censored subjects must not be systematically different from those still at risk. If subjects drop out because they are about to experience the event, the model estimates will be biased.

Linear Covariate Effects

Cox regression assumes a linear relationship between covariates and the log hazard. For continuous predictors, check for non-linearity using martingale residuals or by testing polynomial or spline terms.

Independent Observations

Observations must be independent. If subjects are clustered (e.g., patients within hospitals, users within companies), use a frailty model or robust standard errors.

If you want to compare survival curves without covariates, use the Log-Rank Test instead. If you want to estimate a survival curve, use the Kaplan-Meier Estimator.

When to use Cox Proportional Hazards Regression?

You should use Cox Proportional Hazards Regression in the following scenario:

1. Your outcome is time until an event (churn, conversion, failure)
2. You want to understand which factors affect the rate of the event
3. You have multiple covariates to control for
4. You have censored observations
5. The hazard ratios are approximately constant over time

Multivariate Survival Analysis

The main advantage of Cox regression over the log-rank test is the ability to include multiple covariates simultaneously. While the log-rank test compares groups, Cox regression quantifies the effect of each predictor while controlling for the others.

Censored Data with Covariates

Like all survival methods, Cox regression handles censoring correctly. Unlike logistic regression, it uses the full timing information rather than reducing the outcome to a binary indicator.

If the proportional hazards assumption is violated, consider the Accelerated Failure Time (AFT) model. If your outcome is binary (event within a fixed window: yes/no), Logistic Regression may be more appropriate.

Cox Proportional Hazards Example

Outcome: Time from free-trial signup to paid conversion. Covariates: Plan tier viewed, number of features used in first week, company size, referral source.

We follow a cohort of free-trial users and track when each one converts to a paid plan. Some users never convert during the observation period and are censored.

We fit a Cox model and obtain hazard ratios for each covariate. For example:

• Features used in first week: HR = 1.12 (each additional feature used increases conversion hazard by 12%)
• Enterprise plan viewed: HR = 0.75 (enterprise prospects convert more slowly, likely due to longer procurement cycles)
• Referred by existing customer: HR = 1.45 (referral users convert 45% faster)

Each hazard ratio tells us how the covariate changes the rate of conversion while controlling for all other variables. A $p$-value $\le 0.05$ for any covariate means its effect is statistically significant.