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The covariance of causal effect estimators for binary v-structures

Jack Kuipers, Giusi Moffa

TL;DR

This work analyzes the covariance structure of two binary-v-structure causal-effect estimators, $R$ (from raw conditionals) and $M$ (from marginalisation), and shows that a convex combination $P(\alpha)=\alpha R+(1-\alpha)M$ can achieve strictly lower variance than either estimator individually. The authors derive the covariance $C[R,M]$ via generating-function methods, obtain an optimal $\alpha^*$ that minimizes $V[P(\alpha)]$, and prove that variance reduction persists whenever $C[R,M] < \min(V[R],V[M])$, including in block-randomised designs where $X$ is fixed with proportions $N^0$ and $N^1$. Extensive numerical checks, asymptotic analyses, and fixed-$X$ extensions confirm that the combined estimator consistently outperforms both $R$ and $M$ across a wide range of parameter settings, offering a practical route to more precise causal inference in binary DAGs. The results are accompanied by open-source code and visualizations that illustrate the robustness and scaling behavior, highlighting the method's potential for improved precision in applied causal analyses.

Abstract

Previously [Journal of Causal Inference, 10, 90-105 (2022)], we computed the variance of two estimators of causal effects for a v-structure of binary variables. Here we show that a linear combination of these estimators has lower variance than either. Furthermore, we show that this holds also when the treatment variable is block randomised with a predefined number receiving treatment, with analogous results to when it is sampled randomly.

The covariance of causal effect estimators for binary v-structures

TL;DR

This work analyzes the covariance structure of two binary-v-structure causal-effect estimators, (from raw conditionals) and (from marginalisation), and shows that a convex combination can achieve strictly lower variance than either estimator individually. The authors derive the covariance via generating-function methods, obtain an optimal that minimizes , and prove that variance reduction persists whenever , including in block-randomised designs where is fixed with proportions and . Extensive numerical checks, asymptotic analyses, and fixed- extensions confirm that the combined estimator consistently outperforms both and across a wide range of parameter settings, offering a practical route to more precise causal inference in binary DAGs. The results are accompanied by open-source code and visualizations that illustrate the robustness and scaling behavior, highlighting the method's potential for improved precision in applied causal analyses.

Abstract

Previously [Journal of Causal Inference, 10, 90-105 (2022)], we computed the variance of two estimators of causal effects for a v-structure of binary variables. Here we show that a linear combination of these estimators has lower variance than either. Furthermore, we show that this holds also when the treatment variable is block randomised with a predefined number receiving treatment, with analogous results to when it is sampled randomly.

Paper Structure

This paper contains 15 sections, 65 equations, 5 figures.

Table of Contents

  1. Recap
  2. A combined estimator
  3. Computing the covariance
  4. Numerical checks
  5. Relative difference in variances
  6. Asymptotics
  7. Conclusions
  8. Variances when $X$ is fixed
  9. Raw Conditionals
  10. Marginalisation
  11. The covariance of $R$ and $M$
  12. Numerical checks
  13. Relative difference plots
  14. Asymptotics
  15. Combined estimator plot

Figures (5)

  • Figure 1: A v-structure on 3 nodes.
  • Figure 2: Relative difference in variance of the combined estimator to the better of the other two.
  • Figure 3: Relative difference in variance of the two estimators. (a) $N=100$, (b) $N=400$.
  • Figure 4: Relative difference in variance of the two estimators for $N=100$ with interactions. (a) $D=\frac{1}{8}$, (b) $D=\frac{1}{4}$.
  • Figure 5: Relative difference in variance of the combined estimator to the better of the other two when $X$ is fixed. This is the analogue of Figure \ref{['fig:deltacoplot']} in the main text.