Survival Analysis

The defining capability of survival analysis is its ability to incorporate right-censored observations—shipments still in transit at the time of analysis. Unlike ordinary least squares regression, which would discard these incomplete records or introduce bias by treating them as failures, survival models use the Kaplan-Meier estimator and partial likelihood methods to extract information from the fact that a delivery has not yet occurred by a given timestamp. This is critical in supply chains where 30-60% of open purchase orders may be in-flight during any analysis window.

The hazard function λ(t) represents the instantaneous risk of delivery at time t, given that the shipment has survived up to that point. This reveals non-intuitive dynamics:

• Bathtub hazard: High early delivery risk (expedited orders), a stable period, then rising risk as deadlines approach
• Increasing hazard: Perishable goods where delivery pressure escalates over time
• Decreasing hazard: Bulk shipments where initial consolidation delays dominate

The Cox Proportional Hazards model allows quantification of how covariates like carrier type, season, or port congestion multiplicatively shift this baseline hazard.

The survival function S(t) = P(T > t) gives the probability that a shipment's delivery time T exceeds a given duration t. This directly answers the planner's question: What is the probability this order will arrive after day 14?

Key estimation methods include:

• Kaplan-Meier: Non-parametric, step-function estimator that handles censoring without distributional assumptions
• Parametric models: Weibull, exponential, or log-normal distributions fitted when the underlying failure process is understood
• Nelson-Aalen: Cumulative hazard estimator used as an alternative when hazard interpretation is preferred

Supply chain conditions change while shipments are in transit. The extended Cox model accommodates time-dependent covariates that evolve during the observation period:

• Port congestion indices that fluctuate daily
• Weather severity scores along the route
• Carrier performance metrics updated in real-time

This prevents the immortal time bias that would occur if a covariate measured at delivery were incorrectly treated as fixed from origin. The model segments each shipment's timeline into intervals where covariates remain constant, recalculating risk at each change point.

A delivery can fail to arrive on time for multiple, mutually exclusive reasons—a competing risks scenario. The cause-specific hazard λⱼ(t) estimates the instantaneous risk of failure from cause j (e.g., customs hold, carrier bankruptcy, weather delay) while treating other failure modes as censoring events.

The cumulative incidence function (CIF) then estimates the probability of failing from a specific cause by time t, accounting for the fact that earlier failure from one cause precludes observing others. This enables planners to decompose delay risk by root cause and allocate mitigation resources accordingly.

While Cox models focus on the hazard rate, Accelerated Failure Time (AFT) models directly model the effect of covariates on the survival time itself. The model assumes covariates either accelerate or decelerate the time to delivery by a multiplicative factor:

• Weibull AFT: log(T) = βX + σW, where W follows an extreme value distribution
• Log-logistic AFT: Handles non-monotonic hazards common in logistics
• Log-normal AFT: Appropriate when delivery times are products of many small independent factors

AFT models produce directly interpretable coefficients: a coefficient of 0.3 means a 35% increase in expected delivery time (e^0.3 ≈ 1.35).

The defining statistical advantage of survival analysis. Right-censoring occurs when a shipment is still in transit at the time of analysis—its true delivery time is unknown, but we know it has survived at least this long. Unlike regression models that would discard these observations, survival models incorporate partial information from censored records, preventing survivorship bias and producing unbiased lead time estimates across the entire order book.

A semi-parametric workhorse for lead time analysis. The Cox model estimates the hazard function—the instantaneous probability of delivery at time t, given survival until t—without requiring assumptions about the baseline delivery distribution. Key capabilities:

• Quantifies how covariates like carrier type, season, or origin port multiply risk
• Produces hazard ratios: e.g., air freight may have 3.2x the delivery hazard of ocean freight
• Handles time-varying covariates for dynamic risk assessment

The non-parametric foundation for visualizing delivery probability over time. The Kaplan-Meier curve plots the estimated probability that a shipment has not yet been delivered at each time point, creating a descending staircase function. Essential applications:

• Comparing delivery performance across carriers or lanes using log-rank tests
• Estimating median lead time directly from the survival curve
• Identifying when 95% of orders have been fulfilled

Two complementary lenses on delivery timing. The survival function S(t) answers: 'What is the probability a shipment takes longer than t days?' The hazard function h(t) answers: 'Given it has not arrived by day t, what is the instantaneous risk of delivery right now?' In logistics, a bathtub-shaped hazard is common—high early delivery probability, a stable middle period, then increasing hazard as deadlines approach and expediting kicks in.

Static features like planned carrier are insufficient for dynamic supply chains. Survival models with time-varying covariates update risk assessments as conditions change mid-transit:

• A port congestion index that spikes while the vessel is en route
• Weather severity scores that change along the journey path
• Real-time AIS data showing vessel speed deviations This transforms a static prediction into a living risk model that updates with every new telemetry event.

Shipments don't just deliver or not—they face multiple mutually exclusive outcomes. The competing risks model extends survival analysis to handle distinct event types:

• Delivered on time vs. delivered late vs. lost in transit vs. returned to sender
• Each risk has its own cause-specific hazard function
• Produces cumulative incidence curves showing the probability of each outcome over time Critical for calculating true OTIF probabilities rather than naive delivery estimates.