Cox Proportional Hazards

The model estimates the hazard rate h(t)—the instantaneous probability that a delivery will be completed at time t, given it has survived up to that point. Unlike simple regression, it models the timing of events, not just their occurrence. The hazard is decomposed into two parts: a non-parametric baseline hazard h₀(t) representing the natural risk over time, and a parametric component exp(βX) capturing the multiplicative effect of covariates like carrier type or seasonality.

The model's defining constraint is that the hazard ratio between any two subjects is constant over time. If using Carrier A versus Carrier B reduces delivery risk by 30% on day one, it must also reduce risk by 30% on day ten. This is tested using Schoenfeld residuals—if the assumption is violated, the effect of a covariate changes with time, requiring extensions like time-varying coefficients or stratified models.

A critical advantage in supply chain applications is the model's native ability to handle right-censored observations—shipments still in transit at the time of analysis. Rather than discarding these incomplete records or imputing arbitrary delivery times, the Cox model incorporates them through the partial likelihood function, which only considers the risk set of orders still at risk at each observed event time. This produces unbiased coefficient estimates even when 40-60% of orders are undelivered.

Unlike full parametric models, the Cox model estimates regression coefficients β using maximum partial likelihood rather than full likelihood. This elegant mathematical approach eliminates the need to specify the baseline hazard h₀(t), making the model semi-parametric. The estimation only considers the ordering of event times, not their exact values, which provides robustness against outliers in delivery duration while still quantifying the relative impact of each covariate.

The standard Cox model can be extended to incorporate time-dependent predictors that change during the observation period. In logistics, this allows modeling dynamic features such as:

• Accumulated port congestion that worsens over a shipment's journey
• Real-time weather severity indices along the transit route
• Carrier capacity utilization that fluctuates weekly This extension transforms the model from static risk assessment to a dynamic, continuously updating risk score as new operational data arrives.

The model outputs hazard ratios (exp(β)) that quantify multiplicative risk effects. A hazard ratio of 2.0 for a 'peak season' indicator means shipments during peak periods have twice the instantaneous risk of delay at any given moment. A ratio of 0.5 for 'premium carrier' means the risk is halved. These ratios are directionally intuitive for supply chain planners: values above 1.0 indicate risk acceleration, while values below 1.0 indicate protective factors that extend expected delivery survival time.

A branch of statistics for analyzing the expected duration until an event occurs. Unlike standard regression, it explicitly handles censored data—observations where the event hasn't happened yet, such as shipments still in transit. Key outputs include the survival function S(t) and the hazard function h(t), which the Cox model directly estimates. In supply chains, it's used to model time-to-delivery, time-to-failure, and supplier response times.

Statistical techniques for managing incomplete observations where the exact event time is unknown. Right-censoring occurs when a shipment hasn't arrived by the analysis cutoff date—we know it took at least X days, but not the final value. The Cox model's partial likelihood function elegantly incorporates this partial information without bias. Ignoring censored data leads to systematically underestimating true lead times.

The exponentiated coefficient exp(β) from a Cox model, representing the multiplicative effect of a covariate on the instantaneous risk. A hazard ratio of 2.0 for 'Expedited Shipping' means that choosing this mode doubles the probability of delivery at any given moment compared to standard shipping. A ratio of 0.5 for 'Peak Season' indicates a 50% reduction in the delivery rate, meaning longer wait times. This is the primary interpretable output of the model.

The foundational assumption of the Cox model: the ratio of hazards between any two individuals is constant over time. For example, if Carrier A has twice the delivery risk of Carrier B on Day 1, it must also have twice the risk on Day 10. Violations are diagnosed using Schoenfeld residuals and visual checks of log-log survival plots. When violated, consider stratified Cox models or time-varying covariates.

A non-parametric statistic used to estimate the survival function from lifetime data. It computes the probability of surviving past time t by multiplying successive conditional survival probabilities. In logistics, a Kaplan-Meier curve visually displays the proportion of shipments still undelivered over time. It serves as the baseline comparison for evaluating how much a Cox model improves predictive accuracy over a naive average.

An extension to the standard Cox model that allows predictor values to change over the observation period. Standard covariates like 'Carrier Type' are fixed, but 'Port Congestion Index' or 'Accumulated Weather Delay' fluctuate daily. Implementing time-varying covariates requires restructuring data into counting process format with start-stop intervals. This captures dynamic risk factors that static models miss.