Survival Analysis

The defining feature of survival analysis is its ability to incorporate censored observations. A data point is censored if the terminal event (e.g., churn, purchase) is not observed during the study period.

• Right Censoring: The most common type. A user hasn't churned by the end of the observation window, so their true survival time is unknown but bounded below.
• Left Censoring: The event occurred before observation began.
• Interval Censoring: The event occurred within a known time interval.

Ignoring censored data by treating them as non-events or dropping them introduces significant bias.

The core output of a survival model is the survival function S(t), which defines the probability that a subject survives longer than time t.

• Formula: S(t) = P(T > t), where T is the random variable for event time.
• Properties: S(0) = 1 (everyone survives at the start). S(t) is a non-increasing function that approaches 0 as time goes to infinity.
• Interpretation: If S(30 days) = 0.85, there is an 85% probability a user remains active for more than 30 days.

This function provides a complete temporal risk profile, not just a single point prediction.

The hazard function represents the instantaneous rate of event occurrence at time t, conditional on survival up to that moment. It describes the immediate risk of failure.

• Formula: λ(t) = lim Δt→0 P(t ≤ T < t + Δt | T ≥ t) / Δt
• Interpretation: A high hazard rate indicates a peak period of churn risk. A constant hazard implies a memoryless process.
• Relationship: The hazard function uniquely defines the survival function via S(t) = exp(-∫₀ᵗ λ(u) du).

Modeling the hazard directly allows for dynamic risk assessment over a user's lifecycle.

The Kaplan-Meier estimator is a non-parametric statistic used to estimate the survival function from lifetime data without assuming an underlying distribution.

• Mechanism: It calculates the probability of survival at each distinct event time by multiplying the conditional survival probabilities sequentially.
• Step Function: The resulting curve is a step function that drops only at observed event times, remaining flat during intervals with no events.
• Comparison: The log-rank test is used to statistically compare the Kaplan-Meier curves of two different cohorts (e.g., treatment vs. control).

This is the foundational tool for exploratory survival data visualization.

The Cox Proportional Hazards (Cox PH) model is a semi-parametric regression technique for assessing the effect of multiple covariates on the hazard rate.

• Core Assumption: The hazard for an individual is a product of a baseline hazard λ₀(t) and an exponential function of covariates: λ(t|X) = λ₀(t) exp(βX).
• Proportional Hazards: The ratio of hazards for two individuals is constant over time, independent of the baseline hazard.
• Coefficients: A positive β coefficient indicates an increased hazard (higher risk), while a negative coefficient indicates a protective effect.

It is widely used for churn prediction to identify the behavioral features that accelerate abandonment.

Standard survival models assume covariates are fixed at baseline. Time-dependent covariates allow features to change value during the observation period, capturing dynamic behavioral shifts.

• Extended Cox Model: The hazard becomes λ(t|X(t)) = λ₀(t) exp(βX(t)), where X(t) is the covariate value at time t.
• Use Cases: Modeling the effect of a recent price change, a cumulative click count, or a live session intensity metric on immediate churn risk.
• Counting Process Format: Data must be structured into intervals with start and stop times to accommodate changing feature values.

This extension is critical for integrating real-time behavioral streams into survival frameworks.

A defining characteristic of survival data where the exact event time is unknown for some subjects during the study period.

• Right-censoring: The most common type; the event occurs after the observation window ends (e.g., a user hasn't churned by the end of the analysis period).
• Left-censoring: The event occurred before the study began.
• Interval-censoring: The event is known only to have occurred within a specific time interval.

Ignoring censored data leads to biased estimates of the true time-to-event distribution.

A non-parametric statistic used to estimate the survival function from lifetime data, even in the presence of censored observations.

• Calculates the probability of surviving past a specific time t.
• Produces a characteristic step function that drops only at observed event times.
• Allows visual comparison of survival curves between different groups (e.g., users acquired via organic vs. paid channels) using the log-rank test.

A semi-parametric regression model that assesses the effect of multiple covariates on the hazard rate without requiring specification of the baseline hazard function.

• The hazard ratio quantifies the multiplicative effect of a one-unit increase in a predictor on the instantaneous risk of the event.
• The proportional hazards assumption requires that the ratio of hazards between any two individuals is constant over time.
• Widely used to identify key drivers of churn, such as session_frequency or support_ticket_count.

The instantaneous potential per unit time for the event to occur, given that the individual has survived up to that moment.

• Unlike the survival function (which always decreases), the hazard rate can increase, decrease, or remain constant over time.
• A bathtub-shaped hazard is common in user retention: high initial risk of abandonment, a period of low stable risk, followed by increasing risk as the product ages.
• Directly models the concept of "imminent risk" in next-event prediction.

A parametric alternative to the Cox model that directly models the logarithm of survival time as a linear function of covariates.

• Assumes covariates accelerate or decelerate the time to event by a constant factor.
• Requires specifying an explicit distribution for the error term (e.g., Weibull, log-normal, log-logistic).
• Interpretation is direct: a coefficient of 0.5 means the covariate halves the expected survival time.

A framework for scenarios where an individual can experience one of several mutually exclusive event types, and the occurrence of one precludes the others.

• In e-commerce, a session can end via purchase, bounce, or abandonment—each a competing risk.
• The Cumulative Incidence Function (CIF) estimates the probability of a specific event occurring before time t in the presence of competing events.
• Treating competing events as independent censoring leads to overestimation of the true event probability.