Propensity Score Matching

The propensity score is the conditional probability of receiving a treatment given a set of observed covariates: e(X) = P(T=1 | X). It is a balancing score, meaning that at each value of the propensity score, the distribution of covariates is identical between treated and untreated groups. This reduces a high-dimensional matching problem to a single scalar value, enabling the estimation of the Average Treatment Effect on the Treated (ATT) in non-randomized supply chain studies.

The common support or overlap condition requires that for each treated unit, there exists an untreated unit with a similar propensity score. Without sufficient overlap, comparisons become extrapolations rather than valid matches. Key diagnostics include:

• Histograms of propensity scores by treatment group
• Trimming extreme propensity scores (e.g., below 0.1 or above 0.9)
• Assessing the region of common support before estimating treatment effects

Several algorithms pair treated and control units based on propensity score proximity:

• Nearest Neighbor Matching: Pairs each treated unit with the untreated unit having the closest propensity score, with or without a caliper (maximum allowable distance)
• Kernel Matching: Uses a weighted average of all untreated units, with weights inversely proportional to distance
• Stratification Matching: Divides the propensity score range into blocks and compares outcomes within each block

After matching, covariate balance must be assessed to verify that the matching procedure successfully eliminated systematic differences between groups. Standard diagnostics include:

• Standardized Mean Differences (SMD): Values below 0.1 indicate adequate balance
• Variance Ratios: Comparing the variance of covariates between groups
• Love Plots: Visualizing covariate balance before and after matching
• t-tests and Kolmogorov-Smirnov tests for distributional equivalence

PSM only controls for observed confounders. Sensitivity analysis assesses how robust the estimated treatment effect is to unobserved confounding or hidden bias. The Rosenbaum bounds method quantifies how strongly an unmeasured confounder would need to influence treatment assignment to nullify the estimated effect. This is critical in supply chain disruption analysis where latent factors like supplier relationship quality may be unmeasured.

When analyzing the impact of a logistics disruption (e.g., port closure), PSM constructs a valid counterfactual by matching affected shipping lanes with unaffected lanes that had similar pre-disruption characteristics:

• Pre-treatment covariates: Historical volume, transit time variability, carrier concentration
• Treatment: Exposure to the disruption event
• Outcome: Post-disruption delivery delay or cost increase This isolates the causal impact of the disruption from confounding factors like seasonal demand shifts.

A variable that influences both the treatment assignment and the outcome, creating a spurious association. In supply chains, a confounder might be supplier size—larger suppliers are both more likely to receive preferential treatment and have better on-time delivery rates. PSM aims to balance these confounders across treated and control groups to isolate the true causal effect.

The estimand most commonly targeted by PSM. ATT measures the mean difference in outcomes for units that actually received the treatment, compared to what would have happened had they not. Key distinction from ATE:

• ATE: Effect across entire population
• ATT: Effect only on treated units
• PSM naturally estimates ATT when matching treated units to controls

The region where the propensity score distributions of treated and control groups overlap. PSM requires sufficient overlap to find valid matches. Without it, comparisons extrapolate beyond the data. Diagnostics include:

• Visual inspection of propensity score histograms
• Trimming observations outside the common support region
• Assessing the trade-off between bias and variance when restricting the sample

An alternative to matching that uses propensity scores as weights rather than for pairing. Each observation is weighted by the inverse of its probability of receiving the treatment it actually received. Comparison with PSM:

• IPTW: Uses all data, weights create a pseudo-population
• PSM: Discards unmatched observations, creates a balanced subset
• IPTW is more sensitive to extreme propensity scores near 0 or 1

The critical diagnostic step after performing PSM. Researchers must verify that matching successfully balanced the distribution of covariates between groups. Standard metrics include:

• Standardized Mean Difference: Should be < 0.1 after matching
• Variance Ratios: Should approach 1.0
• Love Plots: Visual comparison of balance before and after matching Poor balance indicates the propensity model was misspecified.

The systematic error introduced when treatment assignment is non-random. In supply chain disruption analysis, selection bias occurs when the factors that determine whether a supplier receives an intervention also influence the outcome. PSM addresses observable selection bias—bias from measured confounders—but cannot correct for hidden bias from unmeasured confounders without sensitivity analysis.