🔎 When paired outcomes go missing

Pairwise randomized experiments face a common problem: some units have missing outcomes. One common fix is to delete only the units with missing outcomes (the unitwise deletion estimator, UDE). However, if attrition is nonignorable, the UDE produces biased estimates.

🧾 What the alternative does and why it matters

• The pairwise deletion estimator (PDE) removes both a missing unit and its pairmate.
• This approach changes the estimand and the sampling properties of the estimator, with trade-offs between bias and efficiency.

📐 Analytic results and technical contribution

• Formal proofs show the PDE can still be biased under some forms of nonignorable attrition, so it is not universally unbiased.
• Despite potential bias, the PDE can be more efficient than the UDE in finite samples.
• Surprisingly, the conventional variance estimator commonly applied to the PDE is unbiased in a super-population setting.
• A new variance estimator for the UDE is proposed to improve interpretation and inference for analyses that delete only missing units.

📊 Key takeaways

• UDE: Biased under nonignorable attrition; requires careful variance estimation.
• PDE: Can be biased but often more efficient; its conventional variance estimator is unbiased in a super-population.
• The PDE is argued to yield a more straightforward causal interpretation than the UDE because it conditions on the same paired structure that generated treatment assignment.

⚖️ Recommendation for practice

Given the bias–efficiency trade-offs, the analytical results, and the interpretability advantages, the PDE is recommended over the UDE for handling missing outcomes in pairwise randomized experiments, with attention to the conditions under which bias may still arise.