Multi-Pseudo Propensity Score Framework

• Multi-pseudo propensity score framework is a statistical methodology that integrates dimension reduction, balancing scores, and doubly robust techniques to estimate causal effects.
• It simplifies high-dimensional covariate adjustment by condensing variables into a scalar propensity variable, often derived using the Fisher linear discriminant.
• The framework yields robust causal estimates via AIPW estimators that remain valid even if either the propensity model or outcome regression is misspecified.

A multi-pseudo propensity score framework refers to a class of statistical methodologies in causal inference that extend the conventional propensity score approach to accommodate dimension reduction, improve precision, address complex data structures such as clustering, multiple error-prone covariates, and enhance robustness through doubly robust estimation. This framework unifies various forms of "pseudo" balancing scores—such as minimal sufficient variables, estimated propensity variables, and their empirical or sample-based counterparts—for estimating causal effects from observational data, particularly under high-dimensional or heterogeneous covariate structures (Guo et al., 2015).

1. Foundational Concepts: Sufficient Covariate and Propensity Variable

Central to the framework is the notion of a sufficient covariate. A variable $X$ is regarded as a covariate if its distribution remains unchanged across different regimes, formalized as $X \perp F_T$. $X$ is a sufficient covariate if, in addition, the conditional distribution of the response $Y$ given $(X, T)$ is invariant across regimes: $Y \perp F_T \mid (X, T)$. This mirrors the strongly ignorable treatment assignment or "no unobserved confounders" postulate. If, for every value $x$ of $X$, both treatment groups are present (positivity), $X$ is also strongly sufficient.

The framework then focuses on the propensity variable and propensity score: the propensity score $\pi(X) = P(T = 1 \mid X)$ is a widely-used balancing score. The "minimal treatment-sufficient" reduction refers to a scalar function of $X$ that acts as a sufficient summary for $T$—a concept formalized via the likelihood ratio statistic $\Lambda$:

$$\pi = \frac{\theta \Lambda}{1 - \theta + \theta \Lambda}$$

where $\theta$ is the marginal probability of treatment and $\Lambda$ is the likelihood ratio of $X$ under treated versus control. In the normal linear model, the Fisher linear discriminant provides this scalar propensity variable.

2. Statistical Modeling and Dimension Reduction

The multi-pseudo propensity score framework centers on dimensionality reduction. Adjusting for high-dimensional $X$ is often impractical, but the framework shows that one may replace $X$ with a lower-dimensional variable $V$ as long as $V$ is also a sufficient covariate; i.e., $Y \perp F_T \mid (V, T)$. Two canonical reduction strategies are discussed:

• Response-sufficient reduction: $V$ captures all the predictive information about the response $Y$.
• Treatment-sufficient reduction: $V$ captures all the information in $X$ about $T$, specifically satisfying $T \perp X \mid (V, F_T = \emptyset)$.

In the case of the normal linear model, $V$ is a linear function of $X$ (the Fisher discriminant), simplifying adjustment from a multivariate vector to a univariate score.

3. Augmented Inverse Probability Weighted (AIPW) Estimation and Double Robustness

The framework encompasses estimators for the average causal effect (ACE) that combine response regression models and propensity score models. The augmented inverse probability weighted (AIPW) estimator takes the form:

$$\bar{E}_{1, m} = m(X) + \frac{T}{\pi(X)} (Y - m(X))$$

where $m(X)$ is a fitted response model and $\pi(X)$ the propensity score. An analogous estimator is formed for the control group ($T = 0$); the difference yields the ACE. The AIPW is doubly robust: it provides consistent estimation if either the propensity model or the outcome regression model is correctly specified, not necessarily both. This property affords practical robustness to model misspecification.

4. Sample-based Versus Population-based Precision: The “Paradox” of Estimated Propensity Variables

A key insight from the framework is the precision paradox observed in sample-based versus population-based propensity variable adjustment. In the linear normal model with homoscedastic errors, the estimated coefficient of $T$ in a regression of $Y$ on $(T, X)$ is algebraically identical to that obtained from $Y$ regressed on $(T, V)$, where $V$ is the (Fisher) discriminant. However, the use of a sample-estimated $V$ (rather than the "true" population $V$) may, in some circumstances, yield lower variance in causal estimates. This phenomenon arises due to empirical error-cancellation and shrinkage effects when $V$ is estimated. The effect is illustrated in simulation studies and is most pronounced when $V$ is weakly correlated with the optimal predictor of $Y$.

5. Mathematical Formulation and Theoretical Guarantees

Foundational formulas articulated in the framework include:

• Identification of the average causal effect: $$\mathrm{ACE} = \mathbb{E}(Y \mid F_T = 1) - \mathbb{E}(Y \mid F_T = 0) = \mathbb{E}_{\emptyset}\{\mathbb{E}(Y \mid X, T = 1) - \mathbb{E}(Y \mid X, T = 0)\}$$
• Sufficient covariate conditions: $X \perp F_T, \quad Y \perp F_T \mid (X, T)$
• Propensity score definition: $\pi(X) = P_{\emptyset}(T = 1 \mid X)$
• In the normal linear outcome model: $Y \sim d + \delta T + b'X + \text{error}$ where $\delta$ is the ACE, and $b$ is a vector of covariate coefficients.

For the likelihood ratio statistic: $$\log \Lambda = -\frac{1}{2}\left[\mu_1' \Sigma^{-1} \mu_1 - \mu_0' \Sigma^{-1} \mu_0\right] + \gamma X, \qquad \gamma = \Sigma^{-1} (\mu_1 - \mu_0), \quad V = \gamma X$$

6. Comparative Evaluation with Multivariate Adjustment

The framework rigorously compares multivariate regression on $X$ and regression adjustment on the scalar propensity variable $V$. Results establish:

• In the normal linear model with homoscedasticity, the estimated ACE from both approaches is identical.
• Adjustment for $V$ is computationally advantageous and may yield practical benefits in finite samples, especially as the dimensionality of $X$ rises.
• Under certain circumstances, empirical estimation of the balancing score $V$ locally enhances exchangeability across treatment groups and slightly reduces variance.

However, full multivariate adjustment is theoretically optimal when model assumptions precisely hold; differences in precision depend on model misspecification, sample size, and the particular correlation structure between covariates and outcome.

7. Practical Implications and Applications

Practitioners are encouraged to apply the multi-pseudo propensity score framework when:

• Dimension reduction is needed: replacing high-dimensional $X$ with a propensity variable simplifies modeling and improves computational tractability.
• Robust causal effect estimation is required: AIPW estimators provide double robustness.
• Complex data structures (e.g., clustered data, measurement error in covariates) are present, as the flexible framework and its extensions accommodate these situations.

Simulation studies in the framework's development highlight that in “realistic” empirical circumstances, estimated propensity variables may be preferable to theoretical ones, while the theoretical guarantee of equivalence holds under ideal model specifications.

This framework substantiates, mathematically and empirically, the central role of sufficient covariate reduction via the propensity variable and the practical value of doubly robust estimation, providing a comprehensive approach for causal effect estimation in modern observational studies (Guo et al., 2015).