Index: Rosenman 2020 - Combining Observational and Experimental Datasets Using Shrinkage Estimators

Rosenman 2020 - Combining Observational and Experimental Datasets Using Shrinkage Estimators

Routing Summary

This folder fully captures Rosenman, Basse, Owen & Baiocchi (2020), Combining Observational and Experimental Datasets Using Shrinkage Estimators (arXiv:2002.06708): setup and bias structure, the generalised unbiased risk estimate, the ${\kappa }_1$ and ${\kappa }_2$ shrinkage estimators with dominance and asymptotic optimality, SIPW and variance estimation, the implied-$\Gamma$ sensitivity link, simulations, and a vault synthesis (hierarchical Bayesian bias model for E103). It contains 7 notes plus this index.

• Need the paper on one page or the section-to-note map? → Rosenman 2020 - Combining Observational and Experimental Datasets - Overview
• Need Assumptions 1-2, the loss, or the bias ${\xi }_k$? → Rosenman 2020 - Setup Assumptions and Bias Structure
• Need Theorem 1 (URE), ${\kappa }_1$, ${\lambda }_{opt}$, Lemmas 1-2 or Theorem 2? → Rosenman 2020 - Unbiased Risk Estimate and Common Shrinkage
• Need ${\kappa }_2$, SIPW, or variance estimation? → Rosenman 2020 - Variance-Weighted Shrinkage and Practical Estimation
• Need the Zhao et al. $\Gamma$ link and the implied-$\Gamma$ algorithm? → Rosenman 2020 - Sensitivity Analysis and Implied Gamma
• Need simulation design and results (Figs. 1-4) or the conclusion? → Rosenman 2020 - Simulation Results
• Need a PyMC hierarchical bias model with a few holdouts (E103)? → Rosenman 2020 - Hierarchical Bias Model Sketch for Few Holdouts

Concept Map

Concept	Note	Type	Depends On	Key Result
Common treatment effect (Asm. 2)	Rosenman 2020 - Setup Assumptions and Bias Structure	definition	stratification	same ${\tau }_k$ in RCT and observational data
Stratum bias ${\xi }_k$	Rosenman 2020 - Setup Assumptions and Bias Structure	theorem	Asm. 1	$s_{tk}/{\bar{p}}_k+s_{ck}/(1-{\bar{p}}_k)$
Theorem 1 (URE)	Rosenman 2020 - Unbiased Risk Estimate and Common Shrinkage	theorem	Strawderman et al. 2003	risk of $Z+\Sigma g$
${\kappa }_1$	Rosenman 2020 - Unbiased Risk Estimate and Common Shrinkage	definition	Thm 1	$\lambda =\text{Tr}({\Sigma }_{r}D)/{\Delta }^{T}D\Delta$
Lemma 1	Rosenman 2020 - Unbiased Risk Estimate and Common Shrinkage	theorem	Thm 1	dominance if $4\max d_k{\sigma }_{rk}^2\le \sum d_k{\sigma }_{rk}^2$
Theorem 2	Rosenman 2020 - Unbiased Risk Estimate and Common Shrinkage	theorem	Lemmas 4-5	${\kappa }_{1+}$ asymptotically optimal
${\kappa }_2$ and Thm 3	Rosenman 2020 - Variance-Weighted Shrinkage and Practical Estimation	theorem	Thm 1	variance-weighted shrinkage
SIPW	Rosenman 2020 - Variance-Weighted Shrinkage and Practical Estimation	definition	propensity score	reduces bias at little variance cost
Implied $\Gamma$	Rosenman 2020 - Sensitivity Analysis and Implied Gamma	concept	Zhao et al. 2019	binary search for $\Gamma$ matching ${\lambda }_{1+}$
Simulation results	Rosenman 2020 - Simulation Results	example	all estimators	1%/5% (unadjusted) vs 25%/40% (SIPW) risk reduction
Hierarchical bias model	Rosenman 2020 - Hierarchical Bias Model Sketch for Few Holdouts	example	${\xi }_k$, ${\lambda }_{opt}$	Bayesian weight equals $1-{\lambda }_{opt}$ with ${\xi }^2\to {\tau }_{\xi }^2$

Notes

• Rosenman 2020 - Combining Observational and Experimental Datasets - Overview — CONTAINS: research question, contributions, Source Map, E103/E101/E102 links, toy simulation.
• Rosenman 2020 - Setup Assumptions and Bias Structure — CONTAINS: Sec. 1-3: related work, Green-Strawderman estimators, Asm. 1-2, loss, estimators, bias ${\xi }_k$.
• Rosenman 2020 - Unbiased Risk Estimate and Common Shrinkage — CONTAINS: Sec. 4.1-4.2, App. 8: Thm 1 + proof, ${\kappa }_1$, ${\lambda }_{opt}$, Lemma 1 + proof, Thm 2 + proof sketch, corrected ${\kappa }_1^{\ast }$, Lemma 2.
• Rosenman 2020 - Variance-Weighted Shrinkage and Practical Estimation — CONTAINS: Sec. 4.3-4.4, App. 9: ${\kappa }_2$, Lemma 3, Thm 3, variance estimator, SIPW.
• Rosenman 2020 - Sensitivity Analysis and Implied Gamma — CONTAINS: Sec. 5: marginal sensitivity model, worst-case bias/variance, implied-$\Gamma$ algorithm.
• Rosenman 2020 - Simulation Results — CONTAINS: Sec. 6-7: design, Figs. 1-4 findings, conclusion and extensions, miniature replication.
• Rosenman 2020 - Hierarchical Bias Model Sketch for Few Holdouts — CONTAINS: vault synthesis: marginal model, posterior-weight equivalence, PyMC sketch, numeric example.

Sources

• Rosenman 2020 - Combining Observational and Experimental Datasets Using Shrinkage Estimators — Rosenman E, Basse G, Owen A, Baiocchi M (2020), arXiv:2002.06708v2; 29 pp. Sole source.

See Also

• Cinelli Hazlett 2020 - Making Sense of Sensitivity - Overview, Franks 2019 - Flexible Sensitivity Analysis Without Observable Implications - Overview — cluster siblings.
• James-Stein Estimator, Empirical Bayes Interpretation of Shrinkage, Hierarchical Models, Experimental Benchmarks for Observational Ad Measurement.