Causal Inference Under Unmeasured Confounding With Negative Controls: A Minimax Learning Approach

Nathan Kallus; Xiaojie Mao; Masatoshi Uehara; Zhiheng Zhang

stat.MLMar 25, 2021

Causal Inference Under Unmeasured Confounding With Negative Controls: A Minimax Learning Approach

Authors: Nathan Kallus, Xiaojie Mao, Masatoshi Uehara, Zhiheng Zhang

Organizations: Cornell University · Tsinghua University

Abstract

We study the estimation of causal parameters when not all confounders are observed and instead negative controls are available. Recent work has shown how these can enable identification and efficient estimation via two so-called bridge functions. In this paper, we tackle the primary challenges in causal inference with negative controls: identifying and estimating these bridge functions. Previous work has typically relied on completeness conditions for identification, imposed uniqueness assumptions for estimation, and focused on parametric models for the bridge functions. Instead, we provide an alternative identification strategy that assumes the existence of two bridge functions rather than completeness conditions. We also provide new estimators for these functions based on minimax learning formulations. These estimators accommodate general function classes such as Reproducing Kernel Hilbert Spaces and neural networks. We study finite-sample convergence results both for estimating bridge functions themselves and for the final estimation of the causal parameter under a variety of combinations of assumptions. We avoid uniqueness conditions on the bridge functions as much as possible.

Figures & tables

Panel of Figure 1 — Figure 1: A typical causal diagram for negative controls. The dashed edges may be absent, and the dashed circle around $U$ indicates that $U$ is unobserved.

Panel of Figure 2 — Figure 2: Different identification strategies. See relevant discussions below Theorem 4 and in Appendices B , C and A .

	Main Assumptions	Est	Convergence Rate	Sta	Uni	Notes
I	$\mathbb{Q}^{\mathrm{obs}}_{0}\cap\mathbb{Q}\neq\emptyset,\mathbb{H}^{\mathrm{obs}}_{0}\cap\mathbb{H}^{\prime}\neq\emptyset$	IPW	$n^{-\min(\frac{1}{2},\beta)}$	N	N	$\hat{h},\hat{q}$ need not converge to any point in $\mathbb{H}^{\mathrm{obs}}_{0},\mathbb{Q}^{\mathrm{obs}}_{0}$
	$\mathbb{H}^{\mathrm{obs}}_{0}\cap\mathbb{H}\neq\emptyset,\pi\mathbb{Q}^{\mathrm{obs}}_{0}\cap\mathbb{Q}^{\prime}\neq\emptyset$	REG
	$\mathbb{Q}^{\mathrm{obs}}_{0}\cap\mathbb{Q}\neq\emptyset,\mathbb{H}^{\mathrm{obs}}_{0}\cap\{h:h-\mathbb{H}\subseteq\mathbb{H}^{\prime}\}\neq\emptyset$	DR
	$\mathbb{H}^{\mathrm{obs}}_{0}\cap\mathbb{H}\neq\emptyset,\mathbb{Q}^{\mathrm{obs}}_{0}\cap\{q:\pi(q-\mathbb{Q})\subseteq\mathbb{Q}^{\prime}\}\neq\emptyset$	DR
II	$\mathbb{Q}^{\mathrm{obs}}_{0}\cap\mathbb{Q}\neq\emptyset,\pi P_{w}(\mathbb{Q}-\mathbb{Q}^{\mathrm{obs}}_{0})\subseteq\mathbb{H}^{\prime}$	IPW / DR	$n^{-\frac{\beta}{2\beta+1}}$	Y	N	$\\|\pi P_{w}(\hat{q}-q_{0})\\|_{2}\to 0$ for any $q_{0}\in\mathbb{Q}^{\mathrm{obs}}_{0}$
			$n^{-\min(\frac{1}{4},\frac{\beta}{2})}$	N	N

Table 1: Convergence rates of different GACE estimators under different assumptions. The identification assumptions in Theorem 4 are always assumed. The sets

\mathbb{H}^{\mathrm{obs}}_{0},\mathbb{Q}^{\mathrm{obs}}_{0}

denote the sets of observed bridge functions (see Lemma 3 ). The function classes

\mathbb{H},\mathbb{Q}^{\prime}

are used to construct the estimator

\hat{h}

and

\mathbb{Q},\mathbb{H}^{\prime}

to construct the estimator

\hat{q}

. We here summarize our conclusions (ignoring polylogs) when all relevant function classes are nonparametric with log covering number at radius

\epsilon

scaling as

\epsilon^{-\beta}

. “Est” refers to which estimating equation is used for the final estimator (see Section 4.4 ). “Sta” refers to whether the bridge function estimators use stabilizers (see Section 4 ). “Uni” refers to whether we assume bridge functions are unique. The projection operators

P_{z},P_{w},P_{u}

are defined in Equations 26 and 33 .

	IPW	IPW(sta)	REG	REG(sta)	DR	DR(sta)	Linear	Plain REG	DR (no W,Z)
MSE	0.0399	0.0353	0.0798	0.0892	0.0771	0.0844	0.1348	0.7718	0.7771
Squared Bias	0.0005	0.0322	0.0693	0.0794	0.0735	0.0800	0.1289	0.7660	0.7732
Variance	0.0394	0.0031	0.0105	0.0098	0.0037	0.0044	0.0059	0.0059	0.0039

Table 2:

g(t)=\sin(t)

n=400

	IPW	IPW(sta)	REG	REG(sta)	DR	DR(sta)	Linear	Plain REG	DR (no W,Z)
MSE	0.0118	0.0153	0.0148	0.0139	0.0083	0.0081	0.0447	0.0183	0.0151
Squared Bias	0.0024	0.0012	0.0006	0.0006	0.0005	0.0005	0.0000	0.0000	0.0000
Variance	0.0094	0.0141	0.0142	0.0133	0.0077	0.0077	0.0447	0.0183	0.0151

Table 6:

g(t)=\sin(t)

n=400

, ablation study-I.

	DR	DR(sta)	Linear(closed-form)	Linear(moment-based)
Treatment effects (SEs)	-0.0528 (0.01194)	-0.0535 (0.01064)	-0.0860 (0.02408)	-0.0828 (0.01977)
$95\%$ CIs	[-0.0762, -0.0294]	[-0.0743, -0.0326]	[-0.1332, -0.0388]	[-0.1216, -0.0441]

Table 8: Treatment effect estimates (standard errors) and

95\%

confidence intervals of the average treatment effect.

Treatment effects (SEs)	-0.0595 (0.00982)	-0.0524 (0.01052)	-0.0436 (0.01009)	-0.0605 (0.01017)
	Half Width		Half Depth
	DR	DR(sta)	DR	DR(sta)
$95\%$ CIs	[-0.0787, -0.0403]	[-0.0730, -0.0318]	[-0.0634, -0.0238]	[-0.0804, -0.0406]

Table 9: Treatment effect estimates (standard errors) and

95\%

confidence intervals for our proposed DR minimax estimators based on neural network bridge function estimation, across two different specifications of the neural networks. The “Half Width” setting reduces the number of hidden neurons in the neural network bridge function estimators by a half, while the “Half Depth” setting reduces the depth of the neural networks by a half.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

A	Z	Y	$P$ (instance 1)	$P$ (instance 2)
0	0	0	0.075	0.025
0	0	1	0.05625	0.04375
0	1	0	0.025	0.075
0	1	1	0.01875	0.13125
1	0	0	0.0375	0.09
1	0	1	0.01875	0.12

Appendix

Table 11: Two instances where the action bridge functions exist and the distributions of the observables

(Z,A,Y)

are identical but the parameter

\mathbb{E}\left[Y(0)\right]

has different values.

Y	A	W	$P$ (instance 1)	$P$ (instance 2)
0	0	0	0.256	0.241
0	0	1	0.064	0.029
0	1	0	0.056	0.023
0	1	1	0.014	0.003
1	0	0	0.064	0.060
1	0	1	0.016	0.020

Appendix

Table 12: Two instances where the outcome bridge functions exist and the distributions of the observables

(W,A,Y)

are identical but the parameter

\mathbb{E}\left[Y(0)\right]

has different values.

Permutation		DR	DR(sta)	Linear(moment-based)
I	Treatment effects (SEs)	-0.0528(0.01194)	-0.0535 (0.01064)	-0.0828 (0.01977)
I	95% CIs	[-0.0762, -0.0294]	[-0.0743, -0.0326]	[-0.1216, -0.0441]
II	Treatment effects (SEs)	-0.0515(0.01175)	-0.0619 (0.01065)	-0.0805 (0.02092)
II	95% CIs	[-0.0745, -0.0284]	[-0.0828, -0.0411]	[-0.1215, -0.0395]
III	Treatment effects (SEs)	-0.0499 (0.01156)	-0.0630 (0.01037)	-0.0710 (0.02148)
III	95% CIs	[-0.0725, -0.0272]	[-0.0833, -0.0427]	[-0.1132, -0.0289]

Appendix

Table 13: The performance of different estimators across

\binom{4}{2}=6

permutations of the proxy variables

Z

and

W

. The linear moment-based baseline estimator is the same as the one in Table 8 . Permutation I: Z=[“pafi1”, “paco21”],W=[“ph1”, “hema1”]; Permutation II: Z = [“pafi1”, “ph1”], W = [“paco21”, “hema1”] ; Permutation III: Z = [“pafi1”, “hema1”], W = [“paco21”, “ph1”]; Permutation IV: Z = [“paco21”, “ph1”], W = [“pafi1”, “hema1”]; Permutation V: Z = [“paco21”, “hema1”], W = [“pafi1”, “ph1”]; Permutation VI: Z = [“ph1”, “hema1”], W = [“pafi1”, “paco21”].

Permutation		DR	DR(sta)	Linear(closed-form)
I	Treatment effects (SEs)	-0.0528 (0.01194)	-0.0535 (0.01064)	-0.0860 (0.02408)
I	95% CIs	[-0.0762, -0.0294]	[-0.0743, -0.0326]	[-0.1332, -0.0388]
II	Treatment effects (SEs)	-0.0515 (0.01175)	-0.0619 (0.01065)	-0.0808 (0.02439)
II	95% CIs	[-0.0745, -0.0284]	[-0.0828, -0.0411]	[-0.1286, -0.0330]
III	Treatment effects (SEs)	-0.0499 (0.01156)	-0.0630 (0.01037)	-0.1588 (1.95893)
III	95% CIs	[-0.0725, -0.0272]	[-0.0833, -0.0427]	[-3.9983, +3.6807]

Appendix

Table 14: The performance of different estimators across

\binom{4}{2}=6

permutations of the proxy variables

Z

and

W

as described in Table 13 . The linear closed-form baseline estimator is the same as the one in Table 8 .

Figure 3: Covariate-omission stress test for DR, DR(sta), and moment-based linear specifications. Points and horizontal bars show treatment-effect estimates and 95% confidence intervals. “A/Y top

k

” and “Proxy top

k

” remove the

k

highest-ranked covariate blocks under the treatment–outcome and proxy-predictive rankings, respectively. The solid vertical line denotes zero; the dashed red line denotes the full-

X

estimate.

Figure 4: Covariate-omission stress test for DR, DR(sta), and closed-form linear specifications. Points and horizontal bars show treatment-effect estimates and 95% confidence intervals. “A/Y top

k

” and “Proxy top

k

” remove the

k

highest-ranked covariate blocks under the treatment–outcome and proxy-predictive rankings, respectively. The solid vertical line denotes zero; the dashed red line denotes the full-

X

estimate.

$X$ omission	DR	DR(sta)	Linear (moment-based)	Linear (Interaction)	Linear (Quadratic)	Linear (Both)
Full $X$	$-0.0528\;(0.01194)$ $[-0.0762,-0.0294]$	$-0.0535\;(0.01064)$ $[-0.0743,-0.0326]$	$-0.0828\;(0.01977)$ $[-0.1216,-0.0441]$	$-0.0372\;(0.07466)$ $[-0.1835,+0.1091]$	$-0.0859\;(0.02427)$ $[-0.1334,-0.0383]$	$+0.1053\;(0.97969)$ $[-1.8149,+2.0255]$
A/Y top $1$	$-0.0577\;(0.01073)$ $[-0.0787,-0.0366]$	$-0.0693\;(0.01091)$ $[-0.0907,-0.0479]$	$-0.2160\;(0.19689)$ $[-0.6019,+0.1699]$	$-0.0237\;(0.06570)$ $[-0.1524,+0.1051]$	$-0.1146\;(0.04802)$ $[-0.2088,-0.0205]$	$+0.1316\;(2.64882)$ $[-5.0601,+5.3233]$
A/Y top $2$	$-0.0472\;(0.01064)$ $[-0.0681,-0.0264]$	$-0.0624\;(0.01096)$ $[-0.0838,-0.0409]$	$-0.2124\;(0.20980)$ $[-0.6236,+0.1989]$	$-0.0122\;(0.06684)$ $[-0.1432,+0.1188]$	$-0.1096\;(0.04773)$ $[-0.2032,-0.0161]$	$+0.0787\;(2.06838)$ $[-3.9753,+4.1328]$
A/Y top $3$	$-0.0774\;(0.01223)$ $[-0.1013,-0.0534]$	$-0.0761\;(0.01065)$ $[-0.0970,-0.0552]$	$-0.7625\;(2.67704)$ $[-6.0095,+4.4845]$	$-0.0007\;(0.10310)$ $[-0.2028,+0.2014]$	$-0.1872\;(0.12910)$ $[-0.4403,+0.0658]$	$-0.0118\;(0.44009)$ $[-0.8743,+0.8508]$
A/Y top $4$	$-0.0731\;(0.01272)$ $[-0.0980,-0.0482]$	$-0.0843\;(0.01243)$ $[-0.1087,-0.0599]$	$+0.0828\;(0.10833)$ $[-0.1296,+0.2951]$	$-0.0271\;(0.04441)$ $[-0.1142,+0.0599]$	$+0.1997\;(0.51015)$ $[-0.8002,+1.1996]$	$-0.1419\;(0.14219)$ $[-0.4206,+0.1368]$
A/Y top $5$	$-0.0658\;(0.01120)$ $[-0.0877,-0.0438]$	$-0.0793\;(0.01155)$ $[-0.1019,-0.0567]$	$+0.0833\;(0.10994)$ $[-0.1322,+0.2988]$	$-0.0270\;(0.04419)$ $[-0.1137,+0.0596]$	$+0.2166\;(0.59490)$ $[-0.9494,+1.3826]$	$-0.1459\;(0.14819)$ $[-0.4363,+0.1446]$

Appendix

Table 15: Covariate-omission analogue. The first line in each cell is the effect estimate (standard error), and the second line is its 95% confidence interval; a bold cell has a 95% confidence interval whose upper endpoint is below zero. Linear interaction, quadratic, and both denote the corresponding moment-based basis specifications.

$X$ omission	DR	DR(sta)	Linear (closed-form)	Linear (Interaction)	Linear (Quadratic)	Linear (Both)
Full $X$	$-0.0528\;(0.01194)$ $[-0.0762,-0.0294]$	$-0.0535\;(0.01064)$ $[-0.0743,-0.0326]$	$-0.0860\;(0.02408)$ $[-0.1332,-0.0388]$	$+0.0017\;(5.32580)$ $[-10.4369,+10.4402]$	$+0.0301\;(0.28169)$ $[-0.5220,+0.5822]$	$-0.2281\;(2.21813)$ $[-4.5756,+4.1194]$
A/Y top $1$	$-0.0577\;(0.01073)$ $[-0.0787,-0.0366]$	$-0.0693\;(0.01091)$ $[-0.0907,-0.0479]$	$-0.0455\;(0.21887)$ $[-0.4745,+0.3835]$	$-0.1369\;(0.06844)$ $[-0.2710,-0.0027]$	$+0.0207\;(0.16058)$ $[-0.2941,+0.3354]$	$-0.3788\;(3.96309)$ $[-8.1464,+7.3889]$
A/Y top $2$	$-0.0472\;(0.01064)$ $[-0.0681,-0.0264]$	$-0.0624\;(0.01096)$ $[-0.0838,-0.0409]$	$-0.0313\;(0.20944)$ $[-0.4418,+0.3792]$	$-0.1212\;(0.06817)$ $[-0.2548,+0.0124]$	$+0.0450\;(0.16952)$ $[-0.2873,+0.3773]$	$-0.2390\;(1.93839)$ $[-4.0382,+3.5602]$
A/Y top $3$	$-0.0774\;(0.01223)$ $[-0.1013,-0.0534]$	$-0.0761\;(0.01065)$ $[-0.0970,-0.0552]$	$-0.1187\;(0.16266)$ $[-0.4375,+0.2001]$	$-0.2382\;(0.29158)$ $[-0.8097,+0.3333]$	$+0.2054\;(0.70717)$ $[-1.1806,+1.5915]$	$+0.2710\;(2.09452)$ $[-3.8342,+4.3763]$
A/Y top $4$	$-0.0731\;(0.01272)$ $[-0.0980,-0.0482]$	$-0.0843\;(0.01243)$ $[-0.1087,-0.0599]$	$+0.3822\;(0.82374)$ $[-1.2323,+1.9967]$	$+0.0320\;(0.05937)$ $[-0.0843,+0.1484]$	$+0.1626\;(0.24639)$ $[-0.3203,+0.6455]$	$+0.0048\;(0.09888)$ $[-0.1890,+0.1986]$
A/Y top $5$	$-0.0658\;(0.01120)$ $[-0.0877,-0.0438]$	$-0.0793\;(0.01155)$ $[-0.1019,-0.0567]$	$+0.3925\;(0.87208)$ $[-1.3167,+2.1018]$	$+0.0325\;(0.05943)$ $[-0.0840,+0.1490]$	$+0.1681\;(0.26488)$ $[-0.3511,+0.6873]$	$+0.0040\;(0.10115)$ $[-0.1942,+0.2023]$

Appendix

Table 16: Covariate-omission analogue. The first line in each cell is the effect estimate (standard error), and the second line is its 95% confidence interval; a bold cell has a 95% confidence interval whose upper endpoint is below zero. Linear interaction, quadratic, and both denote the corresponding closed-form basis specifications.

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