📊Causal Inference Unit 5 – Matching and propensity scores

Key Concepts and Definitions

• Matching is a non-parametric method for estimating causal effects by pairing treated and control units with similar observed characteristics
• Treatment group consists of units that receive the intervention or exposure of interest
• Control group consists of units that do not receive the intervention or exposure and serve as a comparison
• Confounding variables are factors that influence both treatment assignment and the outcome, potentially biasing causal estimates if not accounted for
• Propensity score is the probability of receiving treatment given observed covariates, used to balance treatment and control groups
• Common support refers to the overlap in propensity scores between treatment and control groups, ensuring comparability
• Covariate balance assesses the similarity of observed characteristics between matched treatment and control units
• Average treatment effect (ATE) is the expected difference in outcomes between treatment and control groups across the entire population

Theoretical Foundations

• Matching methods rely on the assumption of unconfoundedness or ignorability, which states that treatment assignment is independent of potential outcomes given observed covariates
  • This assumption implies that there are no unobserved confounders influencing both treatment and outcome
• Stable unit treatment value assumption (SUTVA) requires that the potential outcomes for each unit are unaffected by the treatment assignment of other units and that there is only one version of the treatment
• Matching aims to mimic a randomized experiment by creating balanced treatment and control groups based on observed characteristics
• Matching can be viewed as a form of preprocessing data before estimating causal effects, reducing model dependence and increasing robustness
• Matching methods are particularly useful when there is limited overlap in covariate distributions between treatment and control groups
• The choice of matching variables should be guided by substantive knowledge and the assumed causal structure of the problem

Types of Matching Methods

• Exact matching pairs treatment and control units with identical values for all covariates, ensuring perfect balance but potentially discarding many units
• Nearest neighbor matching selects the control unit with the smallest distance (e.g., Mahalanobis distance or propensity score) to each treated unit
  • Variants include 1:1 matching, k:1 matching, and matching with replacement or without replacement
• Caliper matching imposes a maximum distance or tolerance for matching, preventing poor matches and improving balance
• Stratification or subclassification divides the propensity score into strata and estimates causal effects within each stratum
• Kernel matching uses a weighted average of all control units to construct the counterfactual outcome for each treated unit, with weights based on the distance between propensity scores
• Coarsened exact matching (CEM) temporarily coarsens covariates into discrete categories, performs exact matching on the coarsened data, and then retains the original values for analysis

Propensity Score Basics

• Propensity score is a balancing score that summarizes the information from multiple covariates into a single scalar value
• Propensity scores are typically estimated using logistic regression, with treatment as the dependent variable and covariates as predictors
• The estimated propensity score is the predicted probability of receiving treatment given the observed covariates
• Propensity score matching pairs treated and control units with similar propensity scores, creating balanced groups
• Propensity score stratification divides the propensity score into subclasses and estimates causal effects within each subclass
• Inverse probability of treatment weighting (IPTW) uses the propensity score to weight observations and create a pseudo-population where treatment assignment is independent of covariates
• Propensity score methods assume that the propensity score model is correctly specified and includes all relevant confounders

Implementing Matching Techniques

• Select the relevant covariates to include in the matching procedure based on substantive knowledge and the assumed causal structure
• Estimate the propensity score using logistic regression or other suitable methods
• Choose the appropriate matching method (e.g., nearest neighbor, caliper, stratification) based on the data and research question
• Specify the matching parameters, such as the distance metric, caliper width, or number of strata
• Perform the matching procedure and assess the resulting covariate balance between matched treatment and control groups
• Estimate the causal effect on the matched sample using appropriate methods (e.g., difference in means, regression adjustment)
• Conduct sensitivity analyses to assess the robustness of the results to unobserved confounding or alternative matching specifications

Assessing Balance and Diagnostics

• Standardized mean differences (SMD) compare the means of each covariate between treatment and control groups, with values close to zero indicating good balance
  • SMD is calculated as the difference in means divided by the pooled standard deviation
• Visual diagnostics, such as propensity score distributions, histograms, or jitter plots, can help assess the overlap and balance of propensity scores
• Kolmogorov-Smirnov test or other statistical tests can be used to assess the equality of covariate distributions between matched groups
• Variance ratios compare the variances of each covariate between treatment and control groups, with values close to one indicating good balance
• Absolute standardized mean differences (ASMD) provide a standardized measure of covariate balance, with values below 0.1 or 0.2 often considered acceptable
• Diagnostic plots, such as love plots or cobweb plots, can summarize the balance of multiple covariates simultaneously
• Assessing balance helps determine the success of the matching procedure and the credibility of the causal estimates

Limitations and Challenges

• Matching methods rely on the assumption of unconfoundedness, which is untestable and may not hold in practice if important confounders are unmeasured
• The choice of matching variables and the specification of the propensity score model can impact the results and should be carefully considered
• Matching can lead to reduced sample size and loss of statistical power, especially when using exact matching or strict caliper widths
• The estimated causal effects may be sensitive to the choice of matching method and parameters, requiring sensitivity analyses
• Matching methods may not perform well when there is limited overlap in covariate distributions between treatment and control groups
• Matching does not account for unobserved confounding, and the results may be biased if important confounders are omitted
• The interpretation of causal effects from matching methods is limited to the matched sample and may not generalize to the entire population

Advanced Topics and Extensions

• Doubly robust estimation combines propensity score methods with outcome regression to provide consistent estimates if either the propensity score or outcome model is correctly specified
• Matching methods can be extended to handle multiple treatments, continuous treatments, or time-varying treatments
• Genetic matching uses a genetic algorithm to optimize the balance of covariates between treatment and control groups
• Prognostic score matching uses the predicted outcome under the control condition as a balancing score, similar to the propensity score
• Matching can be combined with other causal inference methods, such as instrumental variables or regression discontinuity designs, to strengthen causal claims
• Sensitivity analysis techniques, such as Rosenbaum bounds or simulation-based approaches, can assess the robustness of the results to unobserved confounding
• Machine learning methods, such as decision trees or random forests, can be used to estimate propensity scores or improve covariate balance
• Matching methods can be applied to longitudinal or clustered data, accounting for the dependence structure of the observations