Cox Proportional Hazards Model

The defining constraint of the Cox model: the hazard ratio between any two individuals is constant over time. This means the effect of a predictor variable (e.g., subscription tier) on churn risk is multiplicative and does not change as time progresses.

• Validation: Tested using Schoenfeld residuals.
• Violation Fix: Introduce time-varying covariates or stratification.
• Interpretation: A coefficient of 0.5 implies a 65% increase in the hazard rate for a one-unit increase in the predictor.

The model is semi-parametric, meaning it factors into two distinct components:

1. Non-Parametric Baseline Hazard h₀(t): The shared, unspecified risk of churn over time when all covariates are zero. It is left completely unmodeled.
2. Parametric Linear Predictor exp(βX): The multiplicative effect of the covariates, estimated using Partial Likelihood maximization.

This split allows the model to focus purely on ranking risk without needing to specify if churn risk follows a Weibull or exponential distribution.

Instead of a full likelihood function, the Cox model maximizes a partial likelihood that only considers the ordering of event times, not the gaps between them.

• Mechanism: At each distinct churn time, the model calculates the probability that the specific customer who churned was the one to experience the event, out of all customers still 'at risk'.
• Advantage: This elegantly eliminates the unknown baseline hazard h₀(t) from the estimation process.
• Handling Ties: Requires approximation methods like the Efron or Breslow methods when multiple customers churn at the exact same timestamp.

A core strength is the native handling of right-censored data, which is ubiquitous in CLV forecasting. Not all customers have churned by the analysis cutoff date.

• Right-Censored: A customer is still active at the end of the observation window. Their exact survival time is unknown, only that it exceeds the study period.
• Contribution: Censored observations contribute to the risk set up to their last known active time, informing the model about who survives, without biasing the hazard ratios.
• Contrast: Simple regression models that discard censored users introduce significant survivorship bias.

The standard model assumes covariates are fixed at baseline, but an extension allows for time-dependent variables that change during the observation period.

• Example: A customer's 'monthly support tickets filed' or 'logins per week' can be updated dynamically.
• Counting Process Format: Data must be structured in (start_time, stop_time, event) intervals, where each row represents a period where covariates remain constant.
• Interpretation Shift: The hazard ratio now reflects the instantaneous risk associated with the current value of the covariate, not just the starting value.

When the proportional hazards assumption fails for a categorical variable (e.g., geographic region), a stratified model can be used.

• Mechanism: The data is split into strata, and a separate, unique baseline hazard h₀ₖ(t) is estimated for each stratum.
• Constraint: The coefficients β for all other covariates are assumed to be identical across all strata.
• Trade-off: This fixes the proportionality violation but prevents the model from estimating the main effect of the stratification variable itself, as it is absorbed into the distinct baseline hazards.