Transfer Learning for Meta-analysis Under Covariate Shift

Zilong Wang; Ali Abdeen; Turgay Ayer

stat.MLApr 3, 2026

Transfer Learning for Meta-analysis Under Covariate Shift

Authors: Zilong Wang, Ali Abdeen, Turgay Ayer

Organizations: Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia, USA

Abstract

Randomized controlled trials often do not represent the populations where decisions are made, and covariate shift across studies can invalidate standard IPD meta-analysis and transport estimators. We propose a placebo-anchored transport framework that treats source-trial outcomes as abundant proxy signals and target-trial placebo outcomes as scarce, high-fidelity gold labels to calibrate baseline risk. A low-complexity (sparse) correction anchors proxy outcome models to the target population, and the anchored models are embedded in a cross-fitted doubly robust learner, yielding a Neyman-orthogonal, target-site doubly robust estimator for patient-level heterogeneous treatment effects when target treated outcomes are available. We distinguish two regimes: in connected targets (with a treated arm), the method yields target-identified effect estimates; in disconnected targets (placebo-only), it reduces to a principled screen--then--transport procedure under explicit working-model transport assumptions. Experiments on synthetic data and a semi-synthetic IHDP benchmark evaluate pointwise CATE accuracy, ATE error, ranking quality for targeting, decision-theoretic policy regret, and calibration. Across connected settings, the proposed method is best or near-best and improves substantially over proxy-only, target-only, and transport baselines at small target sample sizes; in disconnected settings, it retains strong ranking performance for targeting while pointwise accuracy depends on the strength of the working transport condition.

Figures & tables

	Performance				Calibration
Method	PEHE	ATE	$\tau$	Regret	Slope	R 2	ECE
TargetOnly	6.9	3.3	7.0	6.8	8.3	7.0	6.4
ProxyOnly	8.6	8.2	8.6	8.6	7.0	8.5	8.3
IPW-Transport	4.5	6.6	4.4	4.5	4.5	4.4	5.2
OM-Transport	3.7	5.7	3.5	3.7	3.2	3.4	4.2
EntropyBal	5.3	6.3	5.2	5.3	5.4	5.2	5.7

TABLE I: Average rank per metric across all sweeps (1 = best). Calibration: Slope (ideal=1), R 2 , ECE (Expected Calibration Error).

$p$	Method	50/0	150/100	550/500
10	TargetOnly	–	1.43 $\pm$ 0.39	1.38 $\pm$ 0.45
	ProxyOnly	2.34 $\pm$ 0.67	1.96 $\pm$ 0.66	2.10 $\pm$ 0.83
	IPW-Transport	1.80 $\pm$ 0.83	1.60 $\pm$ 0.70	1.82 $\pm$ 1.06
	OM-Transport	1.80 $\pm$ 0.83	1.60 $\pm$ 0.70	1.81 $\pm$ 1.06
	EntropyBal	1.81 $\pm$ 0.83	1.60 $\pm$ 0.70	1.82 $\pm$ 1.06
	AnchorOnly	–	1.46 $\pm$ 0.41	1.39 $\pm$ 0.46

TABLE II: PEHE across budget

(m_{0},m_{1})

and dimensionality

p

. Lower is better.

$C$	Method	50/0	150/100	550/500
2	TargetOnly	–	4.95 $\pm$ 1.18	4.46 $\pm$ 0.91
	ProxyOnly	5.67 $\pm$ 1.05	5.55 $\pm$ 1.37	5.19 $\pm$ 1.22
	IPW-Transport	3.64 $\pm$ 1.45	3.78 $\pm$ 1.85	3.74 $\pm$ 1.97
	OM-Transport	3.54 $\pm$ 1.45	3.59 $\pm$ 1.89	3.52 $\pm$ 1.98
	EntropyBal	3.76 $\pm$ 1.45	3.79 $\pm$ 1.85	3.73 $\pm$ 1.98
	AnchorOnly	–	5.01 $\pm$ 1.26	4.52 $\pm$ 0.94

TABLE III: PEHE across source sites

C

and budget. Lower is better.

$s/p$	Method	$\Gamma$ =0	0.5	1.0
0.05	TargetOnly	4.41 $\pm$ 0.70	4.25 $\pm$ 0.53	4.30 $\pm$ 0.57
	ProxyOnly	4.97 $\pm$ 0.89	4.72 $\pm$ 0.68	4.84 $\pm$ 0.72
	IPW-Transport	2.79 $\pm$ 1.25	2.13 $\pm$ 1.02	2.20 $\pm$ 1.11
	OM-Transport	2.67 $\pm$ 1.26	2.04 $\pm$ 1.03	2.10 $\pm$ 1.13
	EntropyBal	2.80 $\pm$ 1.25	2.14 $\pm$ 1.02	2.21 $\pm$ 1.11
	AnchorOnly	4.45 $\pm$ 0.70	4.28 $\pm$ 0.53	4.34 $\pm$ 0.56

TABLE IV: PEHE under A5 violations:

s/p

=sparsity ratio,

\Gamma

=nonlinearity. Lower is better.

$m_{0}=25$
Method	$m_{1}{=}25$	$m_{1}{=}50$	$m_{1}{=}100$
TargetOnly	4.22 $\pm$ 6.07	5.45 $\pm$ 7.12	3.09 $\pm$ 3.97
ProxyOnly	3.33 $\pm$ 4.71	4.21 $\pm$ 5.59	2.36 $\pm$ 2.75
AnchorOnly	3.12 $\pm$ 4.32	3.88 $\pm$ 4.87	2.37 $\pm$ 2.57
IPW-Transport	3.69 $\pm$ 6.23	3.76 $\pm$ 5.55	2.29 $\pm$ 2.21
EntropyBal	4.87 $\pm$ 8.67	6.41 $\pm$ 13.96	3.55 $\pm$ 4.33

TABLE VI: IHDP connected regime: PEHE across target budgets (mean

\pm

SD, 50 realizations).

Method	25	50	100	200
ProxyOnly	2.64 $\pm$ 2.65	3.46 $\pm$ 4.88	3.50 $\pm$ 4.27	3.79 $\pm$ 4.53
IPW-Transport	2.30 $\pm$ 3.08	3.36 $\pm$ 5.84	4.87 $\pm$ 6.95	4.79 $\pm$ 6.95
EntropyBal	2.82 $\pm$ 3.02	3.61 $\pm$ 5.65	5.47 $\pm$ 7.99	5.63 $\pm$ 8.62
OM-Transport	2.28 $\pm$ 3.41	3.37 $\pm$ 6.43	4.18 $\pm$ 5.96	4.50 $\pm$ 6.64
Proposed-B	2.11 $\pm$ 3.44	3.19 $\pm$ 6.23	3.90 $\pm$ 5.88	4.33 $\pm$ 6.59

TABLE VII: IHDP disconnected regime (

m_{1}=0

): PEHE across placebo budgets (mean

\pm

SD, 50 realizations).

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

$p$	Method	(50,0)	(100,50)	(150,100)	(250,200)	(550,500)
10	TargetOnly	–	1.69 $\pm$ 0.49	1.43 $\pm$ 0.39	1.42 $\pm$ 0.38	1.38 $\pm$ 0.45
	ProxyOnly	2.34 $\pm$ 0.67	2.09 $\pm$ 0.66	1.96 $\pm$ 0.66	2.08 $\pm$ 0.68	2.10 $\pm$ 0.83
	IPW-Transport	1.80 $\pm$ 0.83	1.80 $\pm$ 0.75	1.60 $\pm$ 0.70	1.78 $\pm$ 0.77	1.82 $\pm$ 1.06
	OM-Transport	1.80 $\pm$ 0.83	1.81 $\pm$ 0.75	1.60 $\pm$ 0.70	1.78 $\pm$ 0.77	1.81 $\pm$ 1.06
	EntropyBal	1.81 $\pm$ 0.83	1.81 $\pm$ 0.75	1.60 $\pm$ 0.70	1.78 $\pm$ 0.77	1.82 $\pm$ 1.06
	AnchorOnly	–	1.71 $\pm$ 0.53	1.46 $\pm$ 0.41	1.44 $\pm$ 0.41	1.39 $\pm$ 0.46

Appendix

TABLE VIII: PEHE (mean

\pm

SD) across budget and

p

. Lower is better.

$p$	Method	(50,0)	(100,50)	(150,100)	(250,200)	(550,500)
10	TargetOnly	–	0.19 $\pm$ 0.15	0.14 $\pm$ 0.10	0.10 $\pm$ 0.07	0.07 $\pm$ 0.06
	ProxyOnly	0.69 $\pm$ 0.52	0.56 $\pm$ 0.45	0.53 $\pm$ 0.48	0.69 $\pm$ 0.48	0.67 $\pm$ 0.48
	IPW-Transport	0.77 $\pm$ 0.56	0.74 $\pm$ 0.50	0.52 $\pm$ 0.42	0.64 $\pm$ 0.50	0.70 $\pm$ 0.57
	OM-Transport	0.77 $\pm$ 0.56	0.74 $\pm$ 0.50	0.52 $\pm$ 0.42	0.64 $\pm$ 0.49	0.70 $\pm$ 0.57
	EntropyBal	0.77 $\pm$ 0.56	0.74 $\pm$ 0.51	0.52 $\pm$ 0.42	0.64 $\pm$ 0.50	0.70 $\pm$ 0.57
	AnchorOnly	–	0.21 $\pm$ 0.16	0.13 $\pm$ 0.11	0.10 $\pm$ 0.09	0.08 $\pm$ 0.06

Appendix

TABLE IX: ATE (mean

\pm

SD) across budget and

p

. Lower is better.

$p$	Method	(50,0)	(100,50)	(150,100)	(250,200)	(550,500)
10	TargetOnly	–	0.80 $\pm$ 0.06	0.85 $\pm$ 0.04	0.86 $\pm$ 0.03	0.88 $\pm$ 0.03
	ProxyOnly	0.64 $\pm$ 0.15	0.68 $\pm$ 0.16	0.72 $\pm$ 0.15	0.71 $\pm$ 0.14	0.71 $\pm$ 0.20
	IPW-Transport	0.81 $\pm$ 0.17	0.77 $\pm$ 0.21	0.80 $\pm$ 0.19	0.78 $\pm$ 0.19	0.78 $\pm$ 0.25
	OM-Transport	0.81 $\pm$ 0.17	0.77 $\pm$ 0.21	0.80 $\pm$ 0.19	0.78 $\pm$ 0.19	0.78 $\pm$ 0.25
	EntropyBal	0.81 $\pm$ 0.17	0.77 $\pm$ 0.21	0.80 $\pm$ 0.19	0.78 $\pm$ 0.19	0.78 $\pm$ 0.25
	AnchorOnly	–	0.80 $\pm$ 0.07	0.85 $\pm$ 0.05	0.86 $\pm$ 0.04	0.87 $\pm$ 0.03

Appendix

TABLE X: Spearman (mean

\pm

SD) across budget and

p

. Higher is better.

$p$	Method	(50,0)	(100,50)	(150,100)	(250,200)	(550,500)
10	TargetOnly	–	0.20 $\pm$ 0.09	0.14 $\pm$ 0.05	0.13 $\pm$ 0.05	0.12 $\pm$ 0.05
	ProxyOnly	0.43 $\pm$ 0.25	0.36 $\pm$ 0.22	0.32 $\pm$ 0.22	0.35 $\pm$ 0.21	0.35 $\pm$ 0.27
	IPW-Transport	0.27 $\pm$ 0.25	0.28 $\pm$ 0.24	0.22 $\pm$ 0.20	0.27 $\pm$ 0.24	0.29 $\pm$ 0.37
	OM-Transport	0.27 $\pm$ 0.25	0.28 $\pm$ 0.24	0.22 $\pm$ 0.20	0.27 $\pm$ 0.25	0.29 $\pm$ 0.37
	EntropyBal	0.27 $\pm$ 0.24	0.28 $\pm$ 0.25	0.22 $\pm$ 0.20	0.27 $\pm$ 0.24	0.29 $\pm$ 0.37
	AnchorOnly	–	0.19 $\pm$ 0.08	0.14 $\pm$ 0.06	0.13 $\pm$ 0.06	0.12 $\pm$ 0.06

Appendix

TABLE XI: Regret (mean

\pm

SD) across budget and

p

. Lower is better.

$C$	Method	(50,0)	(100,50)	(150,100)	(250,200)	(550,500)
2	TargetOnly	–	5.33 $\pm$ 0.92	4.95 $\pm$ 1.18	4.70 $\pm$ 0.75	4.46 $\pm$ 0.91
	ProxyOnly	5.67 $\pm$ 1.05	5.62 $\pm$ 1.06	5.55 $\pm$ 1.37	5.42 $\pm$ 1.16	5.19 $\pm$ 1.22
	IPW-Transport	3.64 $\pm$ 1.45	4.17 $\pm$ 1.88	3.78 $\pm$ 1.85	4.03 $\pm$ 1.94	3.74 $\pm$ 1.97
	OM-Transport	3.54 $\pm$ 1.45	4.01 $\pm$ 1.93	3.59 $\pm$ 1.89	3.86 $\pm$ 1.93	3.52 $\pm$ 1.98
	EntropyBal	3.76 $\pm$ 1.45	4.23 $\pm$ 1.87	3.79 $\pm$ 1.85	4.04 $\pm$ 1.94	3.73 $\pm$ 1.98
	AnchorOnly	–	5.39 $\pm$ 1.00	5.01 $\pm$ 1.26	4.75 $\pm$ 0.78	4.52 $\pm$ 0.94

Appendix

TABLE XII: PEHE (mean

\pm

SD) across sources

C

and budget. Lower is better.

$C$	Method	(50,0)	(100,50)	(150,100)	(250,200)	(550,500)
2	TargetOnly	–	0.49 $\pm$ 0.41	0.40 $\pm$ 0.35	0.31 $\pm$ 0.24	0.21 $\pm$ 0.15
	ProxyOnly	1.29 $\pm$ 1.05	1.16 $\pm$ 0.97	1.08 $\pm$ 0.96	1.20 $\pm$ 1.20	1.13 $\pm$ 0.94
	IPW-Transport	0.79 $\pm$ 0.63	1.13 $\pm$ 0.95	0.88 $\pm$ 0.64	1.12 $\pm$ 0.97	0.79 $\pm$ 0.66
	OM-Transport	0.79 $\pm$ 0.64	1.11 $\pm$ 0.93	0.88 $\pm$ 0.66	1.12 $\pm$ 0.97	0.80 $\pm$ 0.67
	EntropyBal	0.82 $\pm$ 0.63	1.14 $\pm$ 0.96	0.88 $\pm$ 0.64	1.13 $\pm$ 0.97	0.79 $\pm$ 0.66
	AnchorOnly	–	0.54 $\pm$ 0.46	0.38 $\pm$ 0.33	0.30 $\pm$ 0.23	0.21 $\pm$ 0.14

Appendix

TABLE XIII: ATE (mean

\pm

SD) across sources

C

and budget. Lower is better.

$C$	Method	(50,0)	(100,50)	(150,100)	(250,200)	(550,500)
2	TargetOnly	–	0.47 $\pm$ 0.08	0.60 $\pm$ 0.06	0.66 $\pm$ 0.05	0.68 $\pm$ 0.04
	ProxyOnly	0.34 $\pm$ 0.09	0.41 $\pm$ 0.10	0.45 $\pm$ 0.10	0.49 $\pm$ 0.10	0.49 $\pm$ 0.12
	IPW-Transport	0.78 $\pm$ 0.15	0.74 $\pm$ 0.19	0.78 $\pm$ 0.16	0.75 $\pm$ 0.19	0.77 $\pm$ 0.20
	OM-Transport	0.79 $\pm$ 0.15	0.75 $\pm$ 0.19	0.80 $\pm$ 0.17	0.77 $\pm$ 0.19	0.79 $\pm$ 0.20
	EntropyBal	0.77 $\pm$ 0.15	0.73 $\pm$ 0.19	0.78 $\pm$ 0.16	0.75 $\pm$ 0.19	0.77 $\pm$ 0.20
	AnchorOnly	–	0.46 $\pm$ 0.08	0.59 $\pm$ 0.06	0.65 $\pm$ 0.04	0.67 $\pm$ 0.05

Appendix

TABLE XIV: Spearman (mean

\pm

SD) across sources

C

and budget. Higher is better.

$C$	Method	(50,0)	(100,50)	(150,100)	(250,200)	(550,500)
2	TargetOnly	–	1.18 $\pm$ 0.24	0.92 $\pm$ 0.28	0.79 $\pm$ 0.18	0.71 $\pm$ 0.19
	ProxyOnly	1.58 $\pm$ 0.51	1.50 $\pm$ 0.45	1.39 $\pm$ 0.56	1.34 $\pm$ 0.60	1.23 $\pm$ 0.50
	IPW-Transport	0.50 $\pm$ 0.42	0.67 $\pm$ 0.55	0.55 $\pm$ 0.56	0.63 $\pm$ 0.56	0.56 $\pm$ 0.57
	OM-Transport	0.48 $\pm$ 0.41	0.63 $\pm$ 0.55	0.51 $\pm$ 0.55	0.60 $\pm$ 0.57	0.52 $\pm$ 0.59
	EntropyBal	0.52 $\pm$ 0.42	0.68 $\pm$ 0.54	0.55 $\pm$ 0.56	0.63 $\pm$ 0.56	0.55 $\pm$ 0.57
	AnchorOnly	–	1.20 $\pm$ 0.30	0.93 $\pm$ 0.28	0.82 $\pm$ 0.17	0.75 $\pm$ 0.21

Appendix

TABLE XV: Regret (mean

\pm

SD) across sources

C

and budget. Lower is better.

$s/p$	Method	$\Gamma=0.0$	$\Gamma=0.5$	$\Gamma=1.0$
0.05	TargetOnly	4.41 $\pm$ 0.70	4.25 $\pm$ 0.53	4.30 $\pm$ 0.57
	ProxyOnly	4.97 $\pm$ 0.89	4.72 $\pm$ 0.68	4.84 $\pm$ 0.72
	IPW-Transport	2.79 $\pm$ 1.25	2.13 $\pm$ 1.02	2.20 $\pm$ 1.11
	OM-Transport	2.67 $\pm$ 1.26	2.04 $\pm$ 1.03	2.10 $\pm$ 1.13
	EntropyBal	2.80 $\pm$ 1.25	2.14 $\pm$ 1.02	2.21 $\pm$ 1.11
	AnchorOnly	4.45 $\pm$ 0.70	4.28 $\pm$ 0.53	4.34 $\pm$ 0.56

Appendix

TABLE XVI: PEHE (mean

\pm

SD) under A5 violations. Lower is better.

$s/p$	Method	$\Gamma=0.0$	$\Gamma=0.5$	$\Gamma=1.0$
0.05	TargetOnly	0.22 $\pm$ 0.14	0.21 $\pm$ 0.17	0.16 $\pm$ 0.12
	ProxyOnly	0.92 $\pm$ 0.75	0.91 $\pm$ 0.74	0.98 $\pm$ 0.79
	IPW-Transport	0.80 $\pm$ 0.60	0.65 $\pm$ 0.51	0.61 $\pm$ 0.47
	OM-Transport	0.79 $\pm$ 0.58	0.65 $\pm$ 0.50	0.60 $\pm$ 0.47
	EntropyBal	0.80 $\pm$ 0.60	0.65 $\pm$ 0.51	0.61 $\pm$ 0.46
	AnchorOnly	0.20 $\pm$ 0.14	0.19 $\pm$ 0.14	0.15 $\pm$ 0.12

Appendix

TABLE XVII: ATE (mean

\pm

SD) under A5 violations. Lower is better.

$s/p$	Method	$\Gamma=0.0$	$\Gamma=0.5$	$\Gamma=1.0$
0.05	TargetOnly	0.68 $\pm$ 0.04	0.67 $\pm$ 0.04	0.67 $\pm$ 0.04
	ProxyOnly	0.54 $\pm$ 0.09	0.55 $\pm$ 0.08	0.55 $\pm$ 0.08
	IPW-Transport	0.87 $\pm$ 0.11	0.91 $\pm$ 0.08	0.91 $\pm$ 0.08
	OM-Transport	0.88 $\pm$ 0.11	0.92 $\pm$ 0.08	0.92 $\pm$ 0.08
	EntropyBal	0.87 $\pm$ 0.11	0.91 $\pm$ 0.08	0.91 $\pm$ 0.08
	AnchorOnly	0.67 $\pm$ 0.04	0.66 $\pm$ 0.04	0.66 $\pm$ 0.04

Appendix

TABLE XVIII: Spearman (mean

\pm

SD) under A5 violations. Higher is better.

$s/p$	Method	$\Gamma=0.0$	$\Gamma=0.5$	$\Gamma=1.0$
0.05	TargetOnly	0.71 $\pm$ 0.15	0.68 $\pm$ 0.14	0.70 $\pm$ 0.15
	ProxyOnly	1.11 $\pm$ 0.40	1.02 $\pm$ 0.33	1.08 $\pm$ 0.31
	IPW-Transport	0.32 $\pm$ 0.29	0.20 $\pm$ 0.19	0.22 $\pm$ 0.22
	OM-Transport	0.30 $\pm$ 0.28	0.19 $\pm$ 0.19	0.20 $\pm$ 0.22
	EntropyBal	0.32 $\pm$ 0.29	0.20 $\pm$ 0.20	0.22 $\pm$ 0.22
	AnchorOnly	0.73 $\pm$ 0.16	0.71 $\pm$ 0.14	0.72 $\pm$ 0.15

Appendix

TABLE XIX: Regret (mean

\pm

SD) under A5 violations. Lower is better.

	p=10			p=20			p=50			p=100
Method	Slope	R 2	ECE	Slope	R 2	ECE	Slope	R 2	ECE	Slope	R 2	ECE
TargetOnly	1.18	0.72	0.40	1.38	0.57	0.76	1.56	0.37	1.21	1.45	0.20	1.47
ProxyOnly	1.13	0.54	0.73	1.36	0.40	1.03	1.57	0.24	1.52	1.53	0.13	1.79
IPW-Transport	0.94	0.69	0.65	0.95	0.70	0.87	0.96	0.75	0.99	0.96	0.75	1.18
OM-Transport	0.94	0.69	0.66	0.95	0.70	0.87	0.97	0.75	0.98	0.97	0.76	1.14
EntropyBal	0.94	0.69	0.66	0.94	0.70	0.87	0.95	0.74	0.99	0.91	0.71	1.30

Appendix

TABLE XX: CATE Calibration: Dim Sweep at budget (150,100). Slope (ideal=1), R 2 (higher=better), ECE (lower=better).

	C=2			C=5			C=10			C=20			C=50
Method	Slope	R 2	ECE	Slope	R 2	ECE	Slope	R 2	ECE	Slope	R 2	ECE	Slope	R 2	ECE
TargetOnly	1.48	0.36	1.24	1.50	0.35	1.20	1.48	0.36	1.20	1.54	0.35	1.23	1.50	0.35	1.20
ProxyOnly	1.43	0.22	1.48	1.47	0.23	1.34	1.57	0.24	1.54	1.50	0.22	1.44	1.52	0.22	1.54
IPW-Transport	0.84	0.66	1.32	0.88	0.69	1.18	0.97	0.71	1.14	0.97	0.76	0.86	0.99	0.74	0.96
OM-Transport	0.87	0.69	1.25	0.92	0.71	1.09	0.99	0.73	1.11	0.98	0.77	0.84	0.99	0.74	0.95
EntropyBal	0.84	0.66	1.32	0.87	0.68	1.20	0.95	0.70	1.15	0.95	0.75	0.88	0.99	0.73	0.97

Appendix

TABLE XXI: CATE Calibration: Sources Sweep at budget (150,100). Slope (ideal=1), R 2 (higher=better), ECE (lower=better).

		$\Gamma$ =0			$\Gamma$ =0.5			$\Gamma$ =1
$s/p$	Method	Slope	R 2	ECE	Slope	R 2	ECE	Slope	R 2	ECE
0.05	TargetOnly	1.69	0.48	1.33	1.75	0.47	1.32	1.68	0.47	1.26
	ProxyOnly	1.51	0.32	1.37	1.54	0.33	1.36	1.53	0.32	1.45
	IPW-Transport	0.95	0.78	0.92	0.97	0.85	0.71	0.98	0.85	0.64
	OM-Transport	0.97	0.80	0.89	0.98	0.86	0.69	0.99	0.86	0.62
	EntropyBal	0.95	0.78	0.93	0.97	0.85	0.71	0.98	0.85	0.64

Appendix

TABLE XXII: CATE Calibration under A5 Violations. Slope (ideal=1), R 2 (higher=better), ECE (lower=better).

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