Semiparametric Proximal Causal Inference

Figures & data

Figure 1: DAGs representing treatment and outcome confounding proxies when exchangeability holds.

Figure 2: Coexistence of types (1), (2), and (3) proxies when exchangeability fails.

Figure 3: A causal DAG of proximal causal inference.

Table 1: Simulation results: absolute bias ($\times {10}^{-2}$) and MSE ($\times {10}^{-2}$).

		${\widehat{\psi }}_{\text{DR}}$	${\widehat{\psi }}_{\text{POR}}$	${\widehat{\psi }}_{\text{PIPW}}$	${\widehat{\psi }}_{\text{PDR}}$
Scenario 1	Bias	9.9	0.4	0.5	0.5
	MSE	1.1	0.6	0.7	0.7
Scenario 2	Bias	23.4	64.0	0.5	0.1
	MSE	6.5	47.5	0.7	3.0
Scenario 3	Bias	17.3	0.4	20.4	0.2
	MSE	3.2	0.6	4.5	0.7
Scenario 4	Bias	43.2	30.0	13.7	34.5
	MSE	19.9	19.2	2.3	23.5

${\widehat{\psi }}_{\text{DR}}$ refers to the standard doubly robust estimator; ${\widehat{\psi }}_{\text{POR}}$ refers to the proximal outcome regression approach; ${\widehat{\psi }}_{\text{PIPW}}$ refers to the proximal treatment confounding bridge approach; ${\widehat{\psi }}_{\text{PDR}}$ refers to the proximal doubly robust estimator.

Table 2: Simulation results: coverage ($\%$) and average length ($\times {10}^{-2}$).

		${\widehat{\psi }}_{\text{DR}}$	${\widehat{\psi }}_{\text{POR}}$	${\widehat{\psi }}_{\text{PIPW}}$	${\widehat{\psi }}_{\text{PDR}}$
Scenario 1	Coverage	25.4	95.2	95.8	95.8
	Length	14.4	31.9	33.1	33.1
Scenario 2	Coverage	37.6	14.0	95.8	98.6
	Length	41.2	71.2	33.1	69.5
Scenario 3	Coverage	0.2	95.2	26.0	96.6
	Length	15.6	31.9	33.6	34.2
Scenario 4	Coverage	3.0	83.0	72.2	90.8
	Length	43.8	130.3	37.9	152.1

Table 3: Treatment effect estimates (standard deviations) and 95% confidence intervals of the average treatment effect.

	${\widehat{\psi }}_{\text{DR}}$	${\widehat{\psi }}_{\text{POR}}$	${\widehat{\psi }}_{\text{PIPW}}$	${\widehat{\psi }}_{\text{PDR}}$
Treatment effects (SDs)	$-$1.17 (0.32)	$-$1.80 (0.44)	$-$1.72 (0.30)	$-$1.66 (0.43)
95% CIs	($-$1.79, $-$0.55)	($-$2.65, $-$0.94)	($-$2.30, $-$1.14)	($-$2.50, $-$0.83)