Transporting stochastic direct and indirect effects to new populations

TL;DR

This paper introduces new estimators for transported stochastic direct and indirect effects to understand intervention differences across populations, addressing a gap in existing methods.

Contribution

The paper proposes novel estimators for transported mediation effects, including stabilized weighted, doubly robust, and targeted minimum loss-based estimators.

Findings

The estimators perform well in finite samples in simulations.

The doubly robust estimators provide consistent estimates under model misspecification.

The methods enable analysis of intervention effects in new populations.

Abstract

Transported mediation effects may contribute to understanding how and why interventions may work differently when applied to new populations. However, we are not aware of any estimators for such effects. Thus, we propose several different estimators of transported stochastic direct and indirect effects: an inverse-probability of treatment stabilized weighted estimator, a doubly robust estimator that solves the estimating equation, and a doubly robust substitution estimator in the targeted minimum loss-based framework. We demonstrate their finite sample properties in a simulation study.

Tables

Table 1. Table 1 : Simulation data-generating mechanisms.

Data Generating Mechanism 1
$W_1\sim bernoulli$	$P\left(W_1=1\right)=0.5$
$W_2\sim bernoulli$	$P\left(W_2=1\right)=expit\left(0.4+0.2W_1\right)$
$S\sim bernoulli$	$P\left(S=1\right)=expit\left(3W_2-1\right)$
$A\sim bernoulli$	$P\left(A=1\right)=0.5$
$Z\sim bernoulli$	$P(Z=1)=expit(-0.1A+-0.2S+0.2W_2+5A{W}_2+0.14AS+0.2W_{2}S-0.2A{W}_{2}S-1)$
$M\sim bernoulli$	$P\left(M=1\right)=expit\left(1Z+3Z{W}_2+0.2ZS-0.2W_{2}S+2W_{2}Z+0.2S-0.2Z{W}_{2}S-W_2-2\right)$
$Y\sim bernoulli$	$P\left(Y=1\right)=expit\left(-6Z+0.2Z{W}_2+2ZM+2W_{2}M-2W_2+4M+1Z{W}_{2}M-0.2\right)$
Data Generating Mechanism 2
$W_1\sim bernoulli$	$P\left(W_1=1\right)=0.5$
$W_2\sim bernoulli$	$P\left(W_2=1\right)=expit\left(0.4+0.2W_1\right)$
$S\sim bernoulli$	$P\left(S=1\right)=expit\left(3W_2-1\right)$
$A\sim bernoulli$	$P\left(A=1\right)=0.5$
$Z\sim bernoulli$	$P(Z=1)=expit(-3A+-0.2S+2W_2+0.2A{W}_2-0.2AS+0.2W_{2}S+2A{W}_{2}S-0.2)$
$M\sim bernoulli$	$P\left(M=1\right)=expit\left(1Z+6W_{2}Z-2W_2-2\right)$
$Y\sim bernoulli$	$P\left(Y=1\right)=expit\left(log\left(1.2\right)+log\left(40\right)Z-log\left(30\right)M-log\left(1.2\right)W_2-log\left(40\right)W_{2}Z\right)$

Table 2. Table 2 : Simulation results comparing estimators of the transported stochastic direct effect and transported stochastic indirect effect under DGM 1 and correct model specification for various sample sizes. 1,000 simulations. Estimation methods compared include stabilized inverse probability weighting estimation (IPTW), solving the estimating equation (EE), and targeted minimum loss-based estimation (TMLE). For TMLE and EE, we compare versions of the estimators that incorporate the exclusion restrictions in our statistical model (TMLE efficient, EE efficient) and versions that do not (TMLE general, EE general). Efficiency and 95% CI coverage are calculated separately using 1) the influence curve (IC) and 2) bootstrapping (boot). Bias and RMSE values are averages across the simulations.

Estimator	Bias	Efficiency		95% CI Coverage		RMSE	% Out of Bounds
		IC	Boot	IC	Boot
Transported stochastic direct effect
N=5000
TMLE efficient	0.000	100.17	100.42	0.955	0.957	0.008	0
TMLE general	0.001	321.02	321.92	0.945	0.942	0.027	0
EE efficient	0.000	100.21	100.42	0.955	0.957	0.008	0
EE general	0.001	321.27	321.00	0.946	0.942	0.027	0
IPTW	0.000	412.67	388.11	0.954	0.945	0.033	0
N=500
TMLE efficient	0.000	101.37	105.16	0.946	0.957	0.028	0
TMLE general	-0.004	319.08	327.43	0.929	0.936	0.089	0
EE efficient	0.000	101.40	104.51	0.946	0.957	0.028	0
EE general	-0.004	321.97	316.56	0.947	0.935	0.088	0
IPTW	-0.001	428.43	419.16	0.951	0.942	0.110	0
N=100
TMLE efficient	0.003	102.26	139.52	0.970	0.992	0.064	0
TMLE general	0.007	293.07	375.10	0.852	0.945	0.218	0
EE efficient	0.004	102.13	$118.65$	0.974	0.989	0.060	0
EE general	0.005	316.03	$292.87$	0.957	0.930	0.186	0
IPTW	0.009	442.80	456.03	0.909	0.922	0.267	0
Transported stochastic indirect effect
N=5000
TMLE efficient	0.000	99.98	100.46	0.942	0.942	0.004	0
TMLE general	0.000	101.75	101.99	0.942	0.942	0.004	0
EE efficient	0.000	100.01	100.46	0.942	0.942	0.004	0
EE general	0.000	101.69	101.95	0.942	0.943	0.004	0
IPTW	0.000	166.56	141.45	0.980	0.957	0.005	0
N=500
TMLE efficient	-0.001	99.56	104.59	0.926	0.929	0.012	0
TMLE general	-0.001	101.74	106.99	0.928	0.935	0.012	0
EE efficient	-0.001	99.58	101.11	0.926	0.929	0.012	0
EE general	-0.001	101.94	103.39	0.932	0.930	0.012	0
IPTW	-0.001	177.68	156.98	0.948	0.936	0.018	0
N=100
TMLE efficient	-0.003	93.01	132.95	0.878	0.944	0.033	0
TMLE general	-0.001	99.85	158.57	0.865	0.952	0.041	0
EE efficient	-0.003	93.03	113.13	0.878	0.924	0.033	0
EE general	-0.002	99.09	$117.32$	0.871	0.915	0.036	0
IPTW	0.002	167.51	206.17	0.812	0.867	0.062	0

Table 3. Table 3 : Simulation results comparing estimators of transported stochastic direct effect and transported stochastic indirect effect under various model misspecifications for sample size N=5,000. 1,000 simulations. Estimation methods compared include stabilized inverse probability weighting estimation (IPTW), solving the estimating equation (EE), and targeted minimum loss-based estimation (TMLE). For TMLE and EE, we compare versions of the estimators that incorporate the exclusion restrictions in our statistical model (TMLE efficient, EE efficient) and versions that do not (TMLE general, EE general). Bias and RMSE values are averages across the simulations. Efficiency and 95% CI coverage are calculated separately using 1) the influence curve (IC) and 2) bootstrapping (boot). The DGM used for each misspecification is noted below.

Estimator	Bias	Efficiency		95% CI Coverage		RMSE	% Out of Bounds
		IC	Boot	IC	Boot
Y model misspecified, DGM 2
Transported stochastic direct effect
TMLE efficient	0.000	191.06	118.82	0.999	0.965	0.014	0
TMLE general	0.000	354.81	288.79	0.984	0.954	0.035	0
EE efficient	0.000	240.26	126.67	0.999	0.968	0.015	0
EE general	0.000	374.95	291.49	0.992	0.957	0.035	0
IPTW	0.000	300.52	276.72	0.973	0.957	0.034	0
Transported stochastic indirect effect
TMLE efficient	0.000	186.27	102.20	0.995	0.914	0.002	0
TMLE general	0.000	239.35	186.41	0.986	0.942	0.004	0
EE efficient	0.000	237.87	103.75	0.997	0.919	0.002	0
EE general	0.000	302.49	216.48	0.991	0.945	0.004	0
IPTW	0.000	223.45	174.48	0.988	0.941	0.003	0
Y and Z models misspecified, DGM 1
Transported stochastic direct effect
TMLE efficient	0.032	75.49	77.35	0.000	0.000	0.032	0
TMLE general	0.188	973.02	463.63	0.223	0.003	0.192	0
EE efficient	0.033	80.22	81.92	0.000	0.000	0.034	0
EE general	0.545	1,112.09	1,008.79	0.000	0.000	0.552	0
IPTW	0.108	700.28	456.29	0.574	0.204	0.115	0
Transported stochastic indirect effect
TMLE efficient	-0.014	192.60	92.53	0.380	0.004	0.014	0
TMLE general	-0.040	492.45	143.03	0.185	0.000	0.040	0
EE efficient	0.018	307.60	121.67	0.767	0.029	0.019	0
EE general	0.066	911.65	686.60	0.447	0.231	0.072	0
IPTW	-0.031	439.30	160.87	0.430	0.004	0.031	0
Y and M models misspecified, DGM 2
Transported stochastic direct effect
TMLE efficient	-0.272	280.08	149.17	0.000	0.000	0.273	0
TMLE general	-0.272	698.69	321.77	0.147	0.001	0.276	0
EE efficient	-1.110	852.67	735.38	0.000	0.000	1.118	99.2
EE general	-1.111	1,093.33	877.83	0.000	0.000	1.121	98.4
IPTW	-0.236	589.90	284.02	0.090	0.000	0.240	0
Transported stochastic indirect effect
TMLE efficient	0.104	249.26	173.89	0.000	0.000	0.106	0
TMLE general	0.103	370.51	279.70	0.036	0.016	0.106	0
EE efficient	0.135	336.39	227.03	0.000	0.000	0.137	0
EE general	0.137	535.91	413.33	0.054	0.016	0.141	0
IPTW	0.061	279.01	200.78	0.197	0.090	0.063	0
Y and S models misspecified, DGM 1
Transported stochastic direct effect
TMLE efficient	-0.164	285.91	168.14	0.000	0.000	0.165	0
TMLE general	-0.164	511.22	399.63	0.019	0.010	0.167	0
EE efficient	-0.220	415.64	191.91	0.000	0.000	0.220	0
EE general	-0.218	699.90	586.82	0.029	0.023	0.224	0
IPTW	-0.067	404.82	322.51	0.467	0.319	0.073	0
Transported stochastic indirect effect
TMLE efficient	0.222	507.95	186.55	0.000	0.000	0.223	0
TMLE general	0.222	600.12	343.91	0.000	0.000	0.222	0
EE efficient	0.278	1,033.84	216.14	0.000	0.000	0.279	0
EE general	0.278	1,201.77	675.26	0.000	0.000	0.279	0
IPTW	0.278	520.31	337.87	0.000	0.000	0.278	0
Z, M, and S models misspecified, DGM 1
Transported stochastic direct effect
TMLE efficient	0.000	98.78	100.01	0.956	0.960	0.008	0
TMLE general	0.001	228.37	231.43	0.939	0.941	0.020	0
EE efficient	0.000	98.80	100.03	0.956	0.960	0.008	0
EE general	0.001	228.50	203.24	0.966	0.942	0.018	0
IPTW	0.254	396.11	406.66	0.000	0.000	0.256	0
Transported stochastic indirect effect
TMLE efficient	0.001	54.95	100.51	0.727	0.938	0.010	0
TMLE general	0.001	129.68	136.04	0.879	0.920	0.014	0
EE efficient	0.000	54.95	100.46	0.732	0.939	0.010	0
EE general	0.001	130.08	137.03	0.891	0.930	0.014	0
IPTW	0.014	301.92	123.87	0.976	0.716	0.018	0

Table 4. Table 4 : Simulation data-generating mechanisms.

Data Generating Mechanism 1
$W_1\sim bernoulli$	$P\left(W_1=1\right)=0.5$
$W_2\sim bernoulli$	$P\left(W_2=1\right)=expit\left(0.4+0.2W_1\right)$
$S\sim bernoulli$	$P\left(S=1\right)=expit\left(3W_2-1\right)$
$A\sim bernoulli$	$P\left(A=1\right)=0.5$
$Z\sim bernoulli$	$P(Z=1)=expit(-3A+-0.2S+2W_2+0.2A{W}_2-0.2AS+0.2W_{2}S+2A{W}_{2}S-0.2)$
$M\sim bernoulli$	$P\left(M=1\right)=expit\left(1Z+6W_{2}Z-2W_2-2\right)$
$Y\sim bernoulli$	$P\left(Y=1\right)=expit\left(log\left(1.2\right)+log\left(40\right)Z-log\left(30\right)M-log\left(1.2\right)W_2-log\left(40\right)W_{2}Z\right)$
Data Generating Mechanism 2
$W_1\sim bernoulli$	$P\left(W_1=1\right)=0.5$
$W_2\sim bernoulli$	$P\left(W_2=1\right)=expit\left(0.4+0.2W_1\right)$
$S\sim bernoulli$	$P\left(S=1\right)=expit\left(3W_2-1\right)$
$A\sim bernoulli$	$P\left(A=1\right)=0.5$
$Z\sim bernoulli$	$P(Z=1)=expit(-0.1A+-0.2S+0.2W_2+5A{W}_2+0.14AS+0.2W_{2}S-0.2A{W}_{2}S-1)$
$M\sim bernoulli$	$P\left(M=1\right)=expit\left(1Z+3Z{W}_2+0.2ZS-0.2W_{2}S+2W_{2}Z+0.2S-0.2Z{W}_{2}S-W_2-2\right)$
$Y\sim bernoulli$	$P\left(Y=1\right)=expit\left(-6Z+0.2Z{W}_2+2ZM+2W_{2}M-2W_2+4M+1Z{W}_{2}M-0.2\right)$
Data Generating Mechanism 3
$W_1\sim bernoulli$	$P\left(W_1=1\right)=0.5$
$W_2\sim bernoulli$	$P\left(W_2=1\right)=expit\left(0.4+0.2W_1\right)$
$S\sim bernoulli$	$P\left(S=1\right)=expit\left(3W_2-1\right)$
$A\sim bernoulli$	$P\left(A=1\right)=0.5$
$Z\sim bernoulli$	$P\left(Z=1\right)=expit\left(-3A+2S+2W_2+0.2A{W}_2-0.2AS+0.2W_{2}S+2A{W}_{2}S-0.2\right)$
$M\sim bernoulli$	$P\left(M=1\right)=expit\left(3Z-0.2Z{W}_2+0.2ZS-0.2W_{2}S+2W_{2}Z+0.2S-0.2Z{W}_{2}S-W_2-2\right)$
$Y\sim bernoulli$	$P\left(Y=1\right)=expit\left(-6Z+0.2Z{W}_2+2ZM+2W_{2}M-0.2W_2+4M+1Z{W}_{2}M-0.2\right)$

Table 5. Table 5 : Simulation results comparing estimators of the transported direct and indirect effects for DGP 1 under well-specified models for sample sizes 100, 500 and 5000

Estimator	Bias	Efficiency		95%CI Cov		RMSE	% Out of Bds
		IC	Bootstrapping	IC	Bootstrapping
DGM 1, N=100
Transport stochastic direct effect
SDE_tmle_eff	0.015	86.77	161.49	0.807	0.959	0.114	0
SDE_EE_eff	0.016	86.77	$129.06$	0.812	0.919	0.112	0
SDE_tmle	0.028	201.37	292.39	0.82	0.941	0.252	0
SDE_EE	0.024	216.18	4222.44	0.923	0.924	0.210	0
SDE_iptw	0.039	301.76	305.12	0.906	0.906	0.273	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.005	103.450	52.71	0.833	0.961	0.044	0
SIE_EE_eff	-0.004	102.77	$144.73$	0.843	0.907	0.038	0
SIE_tmle	-0.007	126.57	545.44	0.791	0.948	0.056	0
SIE_EE	-0.004	118.11	158.02	0.829	0.910	0.039	0
SIE_iptw	-0.020	197.71	193.86	0.745	0.741	0.052	0
DGM 1, N=500
Transport stochastic direct effect
SDE_tmle_eff	0.0005	96.77	101.58	0.945	0.950	0.039	0
SDE_EE_eff	0.0005	96.81	101.20	0.946	0.948	0.039	0
SDE_tmle	0.0002	225.77	236.31	0.933	0.941	0.091	0
SDE_EE	0.0003	227.70	226.59	0.948	0.936	0.090	0
SDE_iptw	0.005	300.50	283.20	0.948	0.934	0.111	0
Transport stochastic indirect effect
SIE_tmle_eff	0	101.96	110.93	0.938	0.940	0.008	0
SIE_EE_eff	0	102.35	107.99	0.939	0.939	0.008	0
SIE_tmle	0.0001	125.37	166.92	0.920	0.949	0.011	0
SIE_EE	0.0001	126.16	131.84	0.925	0.940	0.011	0
SIE_iptw	-0.003	243.47	211.11	0.894	0.874	0.015	0
DGM 1, N=5000
Transport stochastic direct effect
SDE_tmle_eff	-0.0001	100.39	100.59	0.949	0.949	0.013	0
SDE_EE_eff	-0.0001	100.39	100.59	0.948	0.949	0.013	0
SDE_tmle	0.001	227.08	227.72	0.96	0.959	0.028	0
SDE_EE	0.001	227.21	227.14	0.96	0.958	0.028	0
SDE_iptw	0.002	301.43	277.55	0.965	0.948	0.033	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.0001	101.12	101.44	0.932	0.924	0.002	0
SIE_EE_eff	-0.0001	101.23	101.43	0.932	0.925	0.002	0
SIE_tmle	-0.0001	130.34	131	0.939	0.940	0.003	0
SIE_EE	-0.0001	130.50	130.94	0.940	0.941	0.003	0
SIE_iptw	-0.0003	222.77	175.23	0.984	0.955	0.003	0

Table 6. Table 6 : Simulation results comparing estimators of the transported direct and indirect effects for DGP 1 under Z 𝑍 Z , M 𝑀 M , and S 𝑆 S models misspecified. For sample sizes 100, 500 and 5000.

Estimator	Bias	Efficiency		95%CI Cov		RMSE	% Out of Bds
		IC	Bootstrapping	IC	Bootstrapping
DGM 1, N=100
Transport stochastic direct effect
SDE_tmle_eff	0.029	60.77	125.16	0.529	0.879	0.139	0
SDE_EE_eff	0.029	60.38	121.08	0.539	0.873	0.134	0
SDE_tmle	0.026	87.39	167.61	0.609	0.922	0.177	0
SDE_EE	0.029	93.83	134.66	0.693	0.878	0.159	0
SDE_iptw	0.027	182.41	172.73	0.867	0.836	0.209	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.009	54.10	200.16	0.616	0.932	0.076	0
SIE_EE_eff	-0.010	56.47	166.06	0.626	0.919	0.075	0
SIE_tmle	-0.012	69.12	244.45	0.572	0.923	0.089	0
SIE_EE	-0.010	77.87	172.43	0.669	0.912	0.079	0
SIE_iptw	-0.007	120.77	112.76	0.833	0.856	0.055	0
DGM 1, N=500
Transport stochastic direct effect
SDE_tmle_eff	-0.005	53.490	100.99	0.643	0.879	0.064	0
SDE_EE_eff	-0.005	53.66	98.63	0.643	0.879	0.063	0
SDE_tmle	-0.007	95.11	141.58	0.747	0.895	0.092	0
SDE_EE	-0.006	97.42	125.71	0.843	0.920	0.078	0
SDE_iptw	0.012	149.01	144.41	0.957	0.935	0.087	0
Transport stochastic indirect effect
SIE_tmle_eff	0.001	64.72	97.48	0.826	0.948	0.019	0
SIE_EE_eff	0.0002	65.52	95.96	0.839	0.952	0.018	0
SIE_tmle	-0.0004	96.56	142.93	0.797	0.932	0.027	0
SIE_EE	-0.001	101.28	118.11	0.889	0.951	0.023	0
SIE_iptw	-0.0002	142.27	134.94	0.910	0.895	0.028	0
DGM 1, N=5000
Transport stochastic direct effect
SDE_tmle_eff	-0.0001	48.68	93.50	0.677	0.942	0.019	0
SDE_EE_eff	-0.0002	48.71	93.31	0.676	0.938	0.019	0
SDE_tmle	-0.0001	94.62	150.35	0.768	0.947	0.030	0
SDE_EE	-0.0002	94.81	124.45	0.854	0.948	0.025	0
SDE_iptw	0.014	138.90	132.79	0.923	0.912	0.031	0
Transport stochastic indirect effect
SIE_tmle_eff	0.0001	61.13	77.84	0.875	0.947	0.006	0
SIE_EE_eff	0.0001	61.18	75.11	0.888	0.945	0.005	0
SIE_tmle	0.0002	94.19	117.94	0.885	0.955	0.008	0
SIE_EE	0.0001	94.53	10	0.941	0.957	0.007	0
SIE_iptw	0.002	128.26	123.34	0.949	0.937	0.009	0

Table 7. Table 7 : Simulation results comparing estimators of the transported direct and indirect effects for DGP 1 under Y 𝑌 Y and S 𝑆 S models misspecified. For sample sizes 100, 500 and 5000.

Estimator	Bias	Efficiency		95%CI Cov		RMSE	% Out of Bds
		IC	Bootstrapping	IC	Bootstrapping
DGM 1, N=100
Transport stochastic direct effect
SDE_tmle_eff	0.196	99.74	130.55	0.348	0.496	0.229	0
SDE_EE_eff	0.23	119.11	$103.82$	0.275	0.230	0.254	0
SDE_tmle	0.195	143.81	234.82	0.556	0.812	0.290	0
SDE_EE	0.226	182.88	$161.11$	0.487	0.494	0.291	0.100
SDE_iptw	0.235	195.46	242.96	0.643	0.767	0.322	0
Transport stochastic indirect effect
SIE_tmle_eff	0.002	221.25	500.77	0.591	0.857	0.070	0
SIE_EE_eff	-0.011	264.85	1,190.67	0.704	0.642	0.061	0
SIE_tmle	-0.005	249.36	509.35	0.583	0.825	0.073	0
SIE_EE	-0.011	324.61	$307.76$	0.700	0.665	0.075	0
SIE_iptw	0.004	387.01	428.64	0.710	0.798	0.071	0
DGM 1, N=500
Transport stochastic direct effect
SDE_tmle_eff	0.11	140.17	175	0.528	0.577	0.149	0
SDE_EE_eff	0.141	294.11	171.95	0.577	0.546	0.172	0
SDE_tmle	0.120	254.54	308.58	0.524	0.571	0.212	0
SDE_EE	0.141	454.66	307.74	0.514	0.475	0.235	0
SDE_iptw	0.173	236.52	255.31	0.477	0.521	0.213	0
Transport stochastic indirect effect
SIE_tmle_eff	0.026	290.33	468.23	0.429	0.673	0.040	0
SIE_EE_eff	0.013	638.03	393.79	0.826	0.664	0.034	0
SIE_tmle	0.022	384.49	542.07	0.606	0.803	0.039	0
SIE_EE	0.013	739.29	525.56	0.810	0.728	0.041	0
SIE_iptw	0.024	409.12	449.20	0.666	0.693	0.036	0
DGM 1, N=5000
Transport stochastic direct effect
SDE_tmle_eff	0.045	226.69	190.97	0.717	0.512	0.054	0
SDE_EE_eff	0.078	519.45	183.26	0.910	0.064	0.083	0
SDE_tmle	0.051	490.16	587.79	0.854	0.924	0.093	0
SDE_EE	0.082	803.98	652.80	0.860	0.834	0.118	0
SDE_iptw	0.124	390.64	378.85	0.244	0.164	0.135	0
Transport stochastic indirect effect
SIE_tmle_eff	0.036	332.17	319.04	0.074	0.106	0.037	0
SIE_EE_eff	0.028	883.10	278.05	0.554	0.129	0.029	0
SIE_tmle	0.035	410.83	458.58	0.136	0.157	0.037	0
SIE_EE	0.027	1,059.64	688.37	0.740	0.391	0.031	0
SIE_iptw	0.028	380.58	339.93	0.173	0.171	0.030	0

Table 8. Table 8 : Simulation results comparing estimators of the transported direct and indirect effects for DGP 1 under well-specified models under the Y 𝑌 Y and Z 𝑍 Z models misspecified. For sample sizes 100, 500 and 5000.

Estimator	Bias	Efficiency		95%CI Cov		RMSE	% Out of Bds
		IC	Bootstrapping	IC	Bootstrapping
DGM 1, N=100
Transport stochastic direct effect
SDE_tmle_eff	0.195	92.71	116.32	0.395	0.524	0.222	0
SDE_EE_eff	0.216	113.89	117.45	0.453	0.469	0.241	0
SDE_tmle	0.076	261.30	292.91	0.824	0.865	0.323	0
SDE_EE	0.082	326.80	293.93	0.921	0.874	0.321	0.100
SDE_iptw	0.040	271.53	271.20	0.892	0.889	0.281	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.025	155.53	155.39	0.740	0.688	0.055	0
SIE_EE_eff	-0.018	181.09	119.44	0.794	0.665	0.046	0
SIE_tmle	-0.023	167.73	219.28	0.664	0.756	0.069	0
SIE_EE	-0.016	219.30	160.33	0.733	0.671	0.060	0
SIE_iptw	-0.017	180.22	176.80	0.818	0.803	0.048	0
DGM 1, N=500
Transport stochastic direct effect
SDE_tmle_eff	0.182	102.42	114.90	0.016	0.032	0.188	0
SDE_EE_eff	0.203	112.52	107.23	0.009	0.006	0.208	0
SDE_tmle	0.041	322.39	327.28	0.930	0.924	0.144	0
SDE_EE	0.039	334.96	319.05	0.942	0.929	0.138	0
SDE_iptw	0.006	278.81	276.85	0.936	0.929	0.115	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.006	200.08	124.34	0.903	0.808	0.011	0
SIE_EE_eff	-0.003	212.20	126.88	0.926	0.843	0.010	0
SIE_tmle	-0.002	241.17	229.07	0.883	0.872	0.016	0
SIE_EE	-0.003	283.03	231.65	0.892	0.862	0.017	0
SIE_iptw	-0.003	216.89	203.55	0.921	0.899	0.014	0
DGM 1, N=5000
Transport stochastic direct effect
SDE_tmle_eff	0.181	104.08	117.69	0	0	0.181	0
SDE_EE_eff	0.202	111.38	105.15	0	0	0.202	0
SDE_tmle	0.050	327.02	332.07	0.782	0.791	0.064	0
SDE_EE	0.047	328.78	317.31	0.805	0.793	0.061	0
SDE_iptw	0.014	275.20	271.82	0.948	0.941	0.036	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.003	179.01	95.78	0.965	0.718	0.004	0
SIE_EE_eff	-0.0004	181.99	101.47	0.994	0.923	0.002	0
SIE_tmle	0.001	222.07	192.19	0.975	0.948	0.004	0
SIE_EE	0	247.54	198.36	0.987	0.946	0.004	0
SIE_iptw	0	195.39	170.37	0.975	0.954	0.004	0

Table 9. Table 9 : Simulation results comparing estimators of the transported direct and indirect effects for DGP 1 under Y 𝑌 Y and M 𝑀 M models misspecified. For sample sizes 100, 500 and 5000.

Estimator	Bias	Efficiency		95%CI Cov		RMSE	% Out of Bds
		IC	Bootstrapping	IC	Bootstrapping
DGM 1, N=100
Transport stochastic direct effect
SDE_tmle_eff	-0.133	245.31	239.69	0.782	0.837	0.283	0
SDE_EE_eff	-1.216	1,423.80	1,532.61	0.895	0.890	1.984	51.100
SDE_tmle	-0.213	429.80	338.75	0.718	0.790	0.418	0
SDE_EE	-1.259	1,611.01	$1632.61$	0.953	0.949	2.078	56.300
SDE_iptw	-0.187	1,098.72	320.43	0.893	0.818	0.382	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.001	233.32	208.17	0.735	0.712	0.141	0
SIE_EE_eff	0.075	394.06	367.59	0.834	0.830	0.215	0.700
SIE_tmle	-0.044	166.40	205.14	0.539	0.575	0.139	0
SIE_EE	0.018	382.64	295.24	0.664	0.627	0.227	1.300
SIE_iptw	-0.038	199.78	157.01	0.653	0.563	0.134	0
DGM 1, N=500
Transport stochastic direct effect
SDE_tmle_eff	-0.260	303.14	180.27	0.482	0.163	0.272	0
SDE_EE_eff	-1.181	1,040.61	1,021.97	0.080	0.066	1.272	76.400
SDE_tmle	-0.309	670.04	318.17	0.823	0.439	0.349	0
SDE_EE	-1.238	1,290.55	1,219.61	0.360	0.305	1.354	77.100
SDE_iptw	-0.267	732.08	298.20	0.916	0.505	0.304	0
Transport stochastic indirect effect
SIE_tmle_eff	0.095	305.14	220.10	0.712	0.467	0.108	0
SIE_EE_eff	0.137	442.19	306.25	0.806	0.424	0.156	0
SIE_tmle	0.060	334.84	297.52	0.892	0.808	0.094	0
SIE_EE	0.112	591.83	473.35	0.965	0.828	0.163	0
SIE_iptw	0.045	316.22	259.63	0.926	0.814	0.075	0
DGM 1, N=5000
Transport stochastic direct effect
SDE_tmle_eff	-0.272	280.08	149.17	0	0	0.273	0
SDE_EE_eff	-1.110	852.67	735.38	0	0	1.118	99.210
SDE_tmle	-0.272	698.69	321.77	0.147	0.001	0.276	0
SDE_EE	-1.111	1,093.33	877.83	0	0	1.121	98.410
SDE_iptw	-0.236	589.90	284.02	0.090	0	0.240	0
Transport stochastic indirect effect
SIE_tmle_eff	0.104	249.26	173.89	0	0	0.106	0
SIE_EE_eff	0.135	336.39	227.03	0	0	0.137	0
SIE_tmle	0.103	370.51	279.70	0.036	0.016	0.106	0
SIE_EE	0.137	535.91	413.33	0.054	0.016	0.141	0
SIE_iptw	0.061	279.01	200.78	0.197	0.090	0.063	0

Table 10. Table 10 : Simulation results comparing estimators of the transported direct and indirect effects for DGP 1 under the Y 𝑌 Y model misspecified. For sample sizes 100, 500 and 5000.

Estimator	Bias	Efficiency		95%CI Cov		RMSE	% Out of Bds
		IC	Bootstrapping	IC	Bootstrapping
DGM 1, N=100
Transport stochastic direct effect
SDE_tmle_eff	0.045	157.49	170.41	0.872	0.893	0.143	0
SDE_EE_eff	0.050	232.07	$170.51$	0.894	0.872	0.149	0
SDE_tmle	0.010	282.54	314.26	0.876	0.912	0.273	0
SDE_EE	0.022	355.18	$289.81$	0.950	0.902	0.258	0
SDE_iptw	0.014	297.66	306.32	0.897	0.909	0.267	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.019	161.13	174.95	0.722	0.777	0.046	0
SIE_EE_eff	-0.019	177.84	$127.50$	0.751	0.710	0.044	0
SIE_tmle	-0.020	175.47	241.36	0.638	0.742	0.055	0
SIE_EE	-0.017	211.04	$162.19$	0.705	0.668	0.051	0
SIE_iptw	-0.018	202.31	200.29	0.764	0.749	0.051	0
DGM 1, N=500
Transport stochastic direct effect
SDE_tmle_eff	0.013	185.58	128.33	0.983	0.918	0.053	0
SDE_EE_eff	0.014	240.63	137.63	0.990	0.915	0.056	0
SDE_tmle	-0.0001	339.55	291.55	0.975	0.939	0.112	0
SDE_EE	0.002	370.42	288.90	0.988	0.939	0.112	0
SDE_iptw	0.005	299.38	283.42	0.957	0.943	0.108	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.002	217.44	131.94	0.930	0.880	0.010	0
SIE_EE_eff	-0.003	251	128.78	0.926	0.842	0.010	0
SIE_tmle	-0.003	255.86	222.71	0.883	0.879	0.015	0
SIE_EE	-0.004	308.32	222.67	0.885	0.833	0.016	0
SIE_iptw	-0.002	249.26	214.45	0.905	0.889	0.015	0
DGM 1, N=5000
Transport stochastic direct effect
SDE_tmle_eff	-0.0002	191.06	118.82	0.999	0.965	0.014	0
SDE_EE_eff	-0.0004	240.26	126.67	0.999	0.968	0.015	0
SDE_tmle	0	354.81	288.79	0.984	0.954	0.035	0
SDE_EE	-0.0002	374.95	291.49	0.992	0.957	0.035	0
SDE_iptw	-0.0001	300.52	276.72	0.973	0.957	0.034	0
Transport stochastic indirect effect
SIE_tmle_eff	0	186.27	102.20	0.995	0.914	0.002	0
SIE_EE_eff	0	237.87	103.75	0.997	0.919	0.002	0
SIE_tmle	0.0001	239.35	186.41	0.986	0.942	0.004	0
SIE_EE	0.0002	302.49	216.48	0.991	0.945	0.004	0
SIE_iptw	0.0001	223.45	174.48	0.988	0.941	0.003	0

Table 11. Table 11 : Simulation results comparing estimators of the transported direct and indirect effects for DGP 2 under well-specified models for sample sizes 100, 500 and 5000.

Estimator	Bias	Efficiency		95%CI Cov		RMSE	% Out of Bds
		IC	Bootstrapping	IC	Bootstrapping
DGM 2, N=100
Transport stochastic direct effect
SDE_tmle_eff	0.003	102.26	139.52	0.970	0.992	0.064	0
SDE_EE_eff	0.004	102.13	$118.65$	0.974	0.989	0.060	0
SDE_tmle	0.007	293.07	375.10	0.852	0.945	0.218	0
SDE_EE	0.005	316.03	292.87	0.957	0.930	0.186	0
SDE_iptw	0.009	442.80	456.03	0.909	0.922	0.267	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.003	93.01	132.95	0.878	0.944	0.033	0
SIE_EE_eff	-0.003	93.03	113.13	0.878	0.924	0.033	0
SIE_tmle	-0.001	99.85	158.57	0.865	0.952	0.041	0
SIE_EE	-0.002	99.09	$117.32$	0.871	0.915	0.036	0
SIE_iptw	0.002	167.51	206.17	0.812	0.867	0.062	0
DGM 2, N=500
Transport stochastic direct effect
SDE_tmle_eff	-0.0003	101.37	105.16	0.946	0.957	0.028	0
SDE_EE_eff	-0.0003	101.40	104.51	0.946	0.957	0.028	0
SDE_tmle	-0.004	319.08	327.43	0.929	0.936	0.089	0
SDE_EE	-0.004	321.97	316.56	0.947	0.935	0.088	0
SDE_iptw	-0.001	428.43	419.16	0.951	0.942	0.110	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.001	99.56	104.59	0.926	0.929	0.012	0
SIE_EE_eff	-0.001	99.58	101.11	0.926	0.929	0.012	0
SIE_tmle	-0.0005	101.74	106.99	0.928	0.935	0.012	0
SIE_EE	-0.0005	101.94	103.39	0.932	0.930	0.012	0
SIE_iptw	-0.0005	177.68	156.98	0.948	0.936	0.018	0
DGM 2, N=5000
Transport stochastic direct effect
SDE_tmle_eff	-0.0003	100.17	100.42	0.955	0.957	0.008	0
SDE_EE_eff	-0.0003	100.21	100.42	0.955	0.957	0.008	0
SDE_tmle	0.001	321.02	321.92	0.945	0.942	0.027	0
SDE_EE	0.001	321.27	321	0.946	0.942	0.027	0
SDE_iptw	0.0002	412.67	388.11	0.954	0.945	0.033	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.0001	99.98	100.46	0.942	0.942	0.004	0
SIE_EE_eff	-0.0001	100.01	100.46	0.942	0.942	0.004	0
SIE_tmle	-0.0001	101.75	101.99	0.942	0.942	0.004	0
SIE_EE	-0.0001	101.69	101.95	0.942	0.943	0.004	0
SIE_iptw	0	166.56	141.45	0.980	0.957	0.005	0

Table 12. Table 12 : Simulation results comparing estimators of the transported direct and indirect effects for DGP 2 under Z 𝑍 Z , M 𝑀 M , and S 𝑆 S models misspecified. For sample sizes 100, 500 and 5000.

Estimator	Bias	Efficiency		95%CI Cov		RMSE	% Out of Bds
		IC	Bootstrapping	IC	Bootstrapping
DGM 2, N=100
Transport stochastic direct effect
SDE_tmle_eff	0.001	90.70	108.20	0.918	0.988	0.062	0
SDE_EE_eff	0.001	90.72	108.25	0.918	0.988	0.062	0
SDE_tmle	0.007	191.08	235.80	0.861	0.956	0.141	0
SDE_EE	0.004	207.43	191.55	0.957	0.951	0.121	0
SDE_iptw	0.206	367.51	375.66	0.729	0.756	0.345	0
Transport stochastic indirect effect
SIE_tmle_eff	0.002	51.42	132.19	0.575	0.927	0.082	0
SIE_EE_eff	0.002	51.31	128.88	0.577	0.920	0.080	0
SIE_tmle	0.002	96.11	158.41	0.721	0.892	0.099	0
SIE_EE	0.002	109.50	151.53	0.779	0.897	0.098	0
SIE_iptw	-0.054	537.90	138.25	0.886	0.846	0.110	0
DGM 2, N=500
Transport stochastic direct effect
SDE_tmle_eff	0.0002	99.14	105.22	0.946	0.969	0.028	0
SDE_EE_eff	0.0002	99.14	105.22	0.946	0.969	0.028	0
SDE_tmle	-0.002	222.27	220.01	0.941	0.947	0.061	0
SDE_EE	-0.002	224.49	199.30	0.971	0.952	0.055	0
SDE_iptw	0.247	396.13	428.63	0.385	0.423	0.273	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.0004	51.46	110.33	0.622	0.934	0.032	0
SIE_EE_eff	-0.0005	51.46	109.03	0.625	0.933	0.032	0
SIE_tmle	0.001	110.16	133.88	0.850	0.921	0.044	0
SIE_EE	0.0003	114.19	136.01	0.871	0.926	0.047	0
SIE_iptw	0.005	389.26	142.21	0.972	0.943	0.044	0
DGM 2, N=5000
Transport stochastic direct effect
SDE_tmle_eff	0	98.78	100.01	0.956	0.960	0.008	0
SDE_EE_eff	0	98.80	100.03	0.956	0.960	0.008	0
SDE_tmle	0.001	228.37	231.43	0.939	0.941	0.020	0
SDE_EE	0.001	228.50	203.24	0.966	0.942	0.018	0
SDE_iptw	0.254	396.11	406.66	0	0	0.256	0
Transport stochastic indirect effect
SIE_tmle_eff	0.0005	54.95	100.51	0.727	0.938	0.010	0
SIE_EE_eff	0.0004	54.95	100.46	0.732	0.939	0.010	0
SIE_tmle	0.001	129.68	136.04	0.879	0.920	0.014	0
SIE_EE	0.001	130.08	137.03	0.891	0.930	0.014	0
SIE_iptw	0.014	301.92	123.87	0.976	0.716	0.018	0

Table 13. Table 13 : Simulation results comparing estimators of the transported direct and indirect effects for DGP 2 under Y 𝑌 Y and S 𝑆 S models misspecified. For sample sizes 100, 500 and 5000.

Estimator	Bias	Efficiency		95%CI Cov		RMSE	% Out of Bds
		IC	Bootstrapping	IC	Bootstrapping
DGM 2, N=100
Transport stochastic direct effect
SDE_tmle_eff	-0.048	198.25	193.84	0.903	0.921	0.146	0
SDE_EE_eff	-0.102	315.28	$200.46$	0.946	0.875	0.181	0
SDE_tmle	-0.057	317.69	365.58	0.819	0.880	0.245	0
SDE_EE	-0.102	554.59	$425.87$	0.960	0.811	0.340	1.100
SDE_iptw	0.054	332.61	378.44	0.844	0.870	0.249	0
Transport stochastic indirect effect
SIE_tmle_eff	0.142	362.76	392.68	0.589	0.652	0.200	0
SIE_EE_eff	0.189	708.70	$403.57$	0.940	0.517	0.246	0
SIE_tmle	0.118	368.02	420.49	0.606	0.748	0.201	0
SIE_EE	0.184	749.61	$475.97$	0.944	0.682	0.267	0
SIE_iptw	0.182	384.16	427.11	0.513	0.627	0.244	0
DGM 2, N=500
Transport stochastic direct effect
SDE_tmle_eff	-0.162	309.07	214.95	0.395	0.121	0.170	0
SDE_EE_eff	-0.219	453.68	234.34	0.553	0.029	0.227	0
SDE_tmle	-0.173	519.71	423.35	0.712	0.599	0.206	0
SDE_EE	-0.226	758.85	619.39	0.817	0.613	0.281	0
SDE_iptw	-0.069	429.11	376.48	0.942	0.794	0.118	0
Transport stochastic indirect effect
SIE_tmle_eff	0.220	529.14	274.22	0.016	0.012	0.222	0
SIE_EE_eff	0.277	1,080.48	309.93	0.203	0.006	0.280	0
SIE_tmle	0.213	600.82	434.83	0.060	0.052	0.219	0
SIE_EE	0.278	1,253.70	726.29	0.442	0.132	0.292	0
SIE_iptw	0.273	544.21	457.14	0.022	0.031	0.278	0
DGM 2, N=5000
Transport stochastic direct effect
SDE_tmle_eff	-0.164	285.91	168.14	0	0	0.165	0
SDE_EE_eff	-0.220	415.64	191.91			0.22	0
SDE_tmle	-0.164	511.22	399.63	0.019	0.010	0.167	0
SDE_EE	-0.218	699.90	586.82	0.029	0.023	0.224	0
SDE_iptw	-0.067	404.82	322.51	0.467	0.319	0.073	0
Transport stochastic indirect effect
SIE_tmle_eff	0.222	507.95	186.55	0	0	0.223	0
SIE_EE_eff	0.278	1,033.84	216.14	0	0	0.279	0
SIE_tmle	0.222	600.12	343.91	0	0	0.222	0
SIE_EE	0.278	1,201.77	675.26	0	0	0.279	0
SIE_iptw	0.278	520.31	337.87	0	0	0.278	0

Table 14. Table 14 : Simulation results comparing estimators of the transported direct and indirect effects for DGP 2 under Y 𝑌 Y and Z 𝑍 Z models misspecified. For sample sizes 100, 500 and 5000.

Estimator	Bias	Efficiency		95%CI Cov		RMSE	% Out of Bds
		IC	Bootstrapping	IC	Bootstrapping
DGM 2, N=100
Transport stochastic direct effect
SDE_tmle_eff	0.028	73.84	84.89	0.881	0.976	0.053	0
SDE_EE_eff	0.031	81.31	95.66	0.884	0.969	0.058	0
SDE_tmle	0.146	892.01	513.89	0.914	0.847	0.360	0
SDE_EE	0.576	1,247.94	1,341.48	0.935	0.923	0.966	24.300
SDE_iptw	0.105	878.55	480.40	0.945	0.856	0.326	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.046	165.05	120.15	0.815	0.728	0.086	0
SIE_EE_eff	-0.029	249.13	155.58	0.819	0.747	0.105	0
SIE_tmle	-0.068	413.29	276.43	0.799	0.695	0.132	0
SIE_EE	-0.003	724.66	686.48	0.825	0.837	0.467	5
SIE_iptw	-0.069	644.23	139.24	0.770	0.529	0.104	0
DGM 2, N=500
Transport stochastic direct effect
SDE_tmle_eff	0.031	74.57	78.71	0.647	0.693	0.038	0
SDE_EE_eff	0.033	79.85	84.01	0.658	0.697	0.040	0
SDE_tmle	0.179	1,002.95	559.87	0.991	0.735	0.224	0
SDE_EE	0.555	1,171.51	1,116.10	0.584	0.545	0.626	6.800
SDE_iptw	0.103	740.10	476.14	0.995	0.851	0.164	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.015	200.43	101.69	0.960	0.728	0.019	0
SIE_EE_eff	0.017	329.17	147.24	0.993	0.808	0.026	0
SIE_tmle	-0.040	469.27	531	0.853	0.667	0.064	0
SIE_EE	0.075	931.81	763.34	0.909	0.817	0.156	0.200
SIE_iptw	-0.033	518.39	166.53	0.729	0.519	0.041	0
DGM 2, N=5000
Transport stochastic direct effect
SDE_tmle_eff	0.032	75.49	77.35	0	0	0.032	0
SDE_EE_eff	0.033	80.22	81.92	0	0	0.034	0
SDE_tmle	0.188	973.02	463.63	0.223	0.003	0.192	0
SDE_EE	0.545	1,112.09	1,008.79	0	0	0.552	0
SDE_iptw	0.108	700.28	456.29	0.574	0.204	0.115	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.014	192.60	92.53	0.380	0.004	0.014	0
SIE_EE_eff	0.018	307.60	121.67	0.767	0.029	0.019	0
SIE_tmle	-0.040	492.45	143.03	0.185	0	0.040	0
SIE_EE	0.066	911.65	686.60	0.447	0.231	0.072	0
SIE_iptw	-0.031	439.30	160.87	0.430	0.004	0.031	0

Table 15. Table 15 : Simulation results comparing estimators of the transported direct and indirect effects for DGP 2 under Y 𝑌 Y and M 𝑀 M models misspecified. For sample sizes 100, 500 and 5000.

Estimator	Bias	Efficiency		95%CI Cov		RMSE	% Out of Bds
		IC	Bootstrapping	IC	Bootstrapping
DGM 2, N=100
Transport stochastic direct effect
SDE_tmle_eff	0.028	122.23	210.47	0.825	0.985	0.114	0
SDE_EE_eff	0.055	561.86	$1131.74$	0.905	0.998	0.724	8
SDE_tmle	0.034	797.09	552.54	0.834	0.825	0.412	0
SDE_EE	0.026	1,816.94	$1828.34$	0.974	0.972	1.374	29.500
SDE_iptw	0.028	639.31	509.92	0.872	0.851	0.364	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.030	283.99	172.61	0.842	0.780	0.092	0
SIE_EE_eff	-0.030	534.83	$549.34$	0.885	0.908	0.345	3.300
SIE_tmle	-0.030	252.15	207.16	0.686	0.734	0.110	0
SIE_EE	0.034	517.82	$503.88$	0.839	0.830	0.415	3.100
SIE_iptw	-0.021	268.93	186.20	0.692	0.769	0.095	0
DGM 2, N=500
Transport stochastic direct effect
SDE_tmle_eff	0.038	79.10	179.89	0.506	0.918	0.061	0
SDE_EE_eff	0.077	208.62	728.54	0.299	0.970	0.195	0
SDE_tmle	0.033	1,209.81	582.41	0.991	0.968	0.138	0
SDE_EE	0.077	1,870.98	1,799.55	0.987	0.990	0.430	3.200
SDE_iptw	0.022	737.59	523.97	0.989	0.963	0.127	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.036	213.86	103.92	0.525	0.230	0.041	0
SIE_EE_eff	-0.018	252.81	228.74	0.395	0.324	0.097	0
SIE_tmle	-0.035	290.63	159.72	0.681	0.399	0.043	0
SIE_EE	-0.017	354.25	346.99	0.588	0.565	0.102	0
SIE_iptw	-0.029	157.96	135.95	0.537	0.425	0.035	0
DGM 2, N=5000
Transport stochastic direct effect
SDE_tmle_eff	0.039	72.52	169.93	0.039	0.211	0.042	0
SDE_EE_eff	0.074	116.09	615.71	0.073	0.715	0.090	0
SDE_tmle	0.040	1,191.15	425.74	1	0.780	0.055	0
SDE_EE	0.077	1,623.22	1,278.49	0.955	0.889	0.135	0
SDE_iptw	0.026	710.95	429.68	0.992	0.874	0.046	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.037	130.59	63.97	0.002	0	0.037	0
SIE_EE_eff	-0.020	126.70	105.49	0.178	0.129	0.034	0
SIE_tmle	-0.036	176.50	85.45	0.005	0	0.037	0
SIE_EE	-0.020	164.23	147.83	0.232	0.206	0.034	0
SIE_iptw	-0.030	134.37	83.04	0	0	0.031	0

Table 16. Table 16 : Simulation results comparing estimators of the transported direct and indirect effects for DGP 2 under Y 𝑌 Y model misspecified. For sample sizes 100, 500 and 5000.

Estimator	Bias	Efficiency		95%CI Cov		RMSE	% Out of Bds
		IC	Bootstrapping	IC	Bootstrapping
DGM 2, N=100
Transport stochastic direct effect
SDE_tmle_eff	0.016	117.52	135.93	0.928	0.991	0.071	0
SDE_EE_eff	0.017	134.03	$127.09$	0.971	0.994	0.069	0
SDE_tmle	-0.005	437.29	464.11	0.856	0.888	0.296	0
SDE_EE	0.003	510.08	$415.06$	0.971	0.887	0.270	0
SDE_iptw	-0.004	437.59	450.42	0.892	0.911	0.279	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.015	154.70	134.11	0.741	0.759	0.061	0
SIE_EE_eff	-0.016	158.75	111.49	0.752	0.711	0.059	0
SIE_tmle	-0.013	170.37	205.62	0.649	0.798	0.076	0
SIE_EE	-0.015	185.12	$139.18$	0.700	0.667	0.072	0
SIE_iptw	-0.004	166.40	199.63	0.795	0.857	0.066	0
DGM 2, N=500
Transport stochastic direct effect
SDE_tmle_eff	-0.001	119.51	107.23	0.981	0.978	0.026	0
SDE_EE_eff	-0.001	122.98	106.84	0.990	0.978	0.026	0
SDE_tmle	-0.002	527.31	463.49	0.974	0.950	0.115	0
SDE_EE	-0.0005	539.36	468.44	0.981	0.957	0.116	0
SDE_iptw	-0.001	429.84	422.32	0.959	0.951	0.107	0
Transport stochastic indirect effect
SIE_tmle_eff	0.0002	207.84	113.29	0.993	0.962	0.012	0
SIE_EE_eff	0.0003	204.81	112.47	0.993	0.962	0.012	0
SIE_tmle	-0.0001	243.79	185.49	0.979	0.951	0.020	0
SIE_EE	-0.0001	240.52	177.94	0.978	0.944	0.021	0
SIE_iptw	0.0003	177.08	158.06	0.969	0.953	0.017	0
DGM 2, N=5000
Transport stochastic direct effect
SDE_tmle_eff	-0.0003	111.40	100.18	0.963	0.936	0.009	0
SDE_EE_eff	-0.0003	111.55	100.18	0.968	0.937	0.009	0
SDE_tmle	0.002	508.10	414.57	0.984	0.947	0.034	0
SDE_EE	0.002	508.50	414.73	0.984	0.947	0.034	0
SDE_iptw	0.002	413.87	389.34	0.958	0.946	0.032	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.0001	195.72	101.34	0.999	0.941	0.004	0
SIE_EE_eff	-0.0001	189.44	101.82	0.999	0.942	0.004	0
SIE_tmle	-0.0002	231.77	163.22	0.992	0.954	0.006	0
SIE_EE	-0.0002	220.13	155.51	0.993	0.958	0.005	0
SIE_iptw	-0.0002	166.19	141.12	0.981	0.951	0.005	0

Table 17. Table 17 : Simulation results comparing estimators of the transported direct and indirect effects for DGP 3 under well-specified models for sample sizes 100, 500 and 5000.

Estimator	Bias	Efficiency		95%CI Cov		RMSE	% Out of Bds
		IC	Bootstrapping	IC	Bootstrapping
DGM 3, N=100
Transport stochastic direct effect
SDE_tmle_eff	-0.020	93.10	158.60	0.823	0.955	0.115	0
SDE_EE_eff	-0.017	93.22	$113.95$	0.828	0.896	0.099	0
SDE_tmle	-0.030	126.80	239.56	0.809	0.951	0.199	0
SDE_EE	-0.020	139.54	$141.15$	0.890	0.875	0.140	0
SDE_iptw	-0.043	346.24	324.14	0.840	0.871	0.327	0
Transport stochastic indirect effect
SIE_tmle_eff	0.011	90.59	159.14	0.803	0.929	0.114	0
SIE_EE_eff	0.008	90.40	$159.14$	0.809	0.905	0.104	0
SIE_tmle	0.018	90.81	173.84	0.742	0.907	0.152	0
SIE_EE	0.009	95.89	$173.84$	0.782	0.886	0.107	0
SIE_iptw	0.085	154.04	133.33	0.634	0.603	0.192	0
DGM 3, N=500
Transport stochastic direct effect
SDE_tmle_eff	-0.002	99.42	106.45	0.931	0.948	0.043	0
SDE_EE_eff	-0.002	99.45	106.04	0.931	0.947	0.043	0
SDE_tmle	-0.003	168.58	169.76	0.933	0.939	0.080	0
SDE_EE	-0.003	177.72	170.88	0.960	0.941	0.085	0
SDE_iptw	0.042	466.12	416.46	0.773	0.780	0.194	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.0005	100.48	107.39	0.928	0.937	0.045	0
SIE_EE_eff	-0.0005	100.48	106.91	0.927	0.937	0.045	0
SIE_tmle	-0.001	107.84	116.48	0.926	0.937	0.050	0
SIE_EE	-0.001	108.51	114.77	0.930	0.938	0.048	0
SIE_iptw	-0.007	174.62	163.60	0.941	0.944	0.062	0
DGM 3, N=5000
Transport stochastic direct effect
SDE_tmle_eff	-0.001	100.04	126.93	0.941	0.962	0.015	0
SDE_EE_eff	-0.0005	100.02	113.84	0.943	0.955	0.014	0
SDE_tmle	0.001	214.33	238.29	0.961	0.965	0.032	0
SDE_EE	0.0003	215.45	222.18	0.964	0.964	0.031	0
SDE_iptw	0.006	475.36	466.38	0.934	0.937	0.059	0
Transport stochastic indirect effect
SIE_tmle_eff	0	100.07	118.92	0.939	0.938	0.015	0
SIE_EE_eff	-0.0002	100.07	100.78	0.940	0.936	0.014	0
SIE_tmle	-0.0004	108.04	125.46	0.938	0.946	0.016	0
SIE_EE	-0.0005	108.09	108.55	0.944	0.944	0.014	0
SIE_iptw	-0.001	147.09	123.72	0.967	0.942	0.016	0

Table 18. Table 18 : Simulation results comparing estimators of the transported direct and indirect effects for DGP 3 under Z 𝑍 Z , M 𝑀 M , and S 𝑆 S models misspecified. For sample sizes 100, 500 and 5000.

Estimator	Bias	Efficiency		95%CI Cov		RMSE	% Out of Bds
		IC	Bootstrapping	IC	Bootstrapping
DGM 3, N=100
Transport stochastic direct effect
SDE_tmle_eff	-0.013	71.12	119.65	0.744	0.896	0.097	0
SDE_EE_eff	-0.013	71.11	119.59	0.745	0.896	0.097	0
SDE_tmle	-0.011	95.20	159.52	0.781	0.942	0.124	0
SDE_EE	-0.012	95.46	131.92	0.844	0.920	0.106	0
SDE_iptw	-0.110	196.15	206.68	0.877	0.889	0.200	0
Transport stochastic indirect effect
SIE_tmle_eff	0.005	29.99	132.19	0.390	0.899	0.050	0
SIE_EE_eff	0.005	30.01	131.03	0.389	0.898	0.050	0
SIE_tmle	0.003	38.27	150.41	0.438	0.915	0.054	0
SIE_EE	0.005	38.70	132.89	0.468	0.901	0.051	0
SIE_iptw	-0.045	85.72	97.43	0.533	0.574	0.060	0
DGM 3, N=500
Transport stochastic direct effect
SDE_tmle_eff	-0.002	69.65	106.13	0.796	0.955	0.042	0
SDE_EE_eff	-0.003	69.61	106.09	0.797	0.954	0.042	0
SDE_tmle	-0.002	95.76	142.10	0.792	0.934	0.057	0
SDE_EE	-0.002	95.89	121.23	0.868	0.946	0.048	0
SDE_iptw	-0.096	198.50	194.22	0.761	0.755	0.120	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.001	27.56	106.69	0.371	0.930	0.023	0
SIE_EE_eff	-0.0004	27.57	106.65	0.368	0.930	0.023	0
SIE_tmle	-0.001	37.31	113.59	0.464	0.930	0.024	0
SIE_EE	-0.0004	37.46	108.94	0.489	0.933	0.023	0
SIE_iptw	-0.055	71.87	65.11	0.036	0.042	0.057	0
DGM 3, N=5000
Transport stochastic direct effect
SDE_tmle_eff	-0.0004	66.47	110.54	0.804	0.962	0.013	0
SDE_EE_eff	-0.0005	66.45	110.18	0.805	0.961	0.013	0
SDE_tmle	-0.0004	93.90	152.31	0.802	0.966	0.018	0
SDE_EE	-0.0005	93.90	125.67	0.890	0.963	0.015	0
SDE_iptw	-0.101	187.34	181.99	0.002	0.001	0.103	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.0002	26.08	109.30	0.382	0.937	0.007	0
SIE_EE_eff		26.090	106.07	0.385	0.940	0.007	0
SIE_tmle	-0.0002	35.98	113.49	0.497	0.929	0.007	0
SIE_EE	0.0001	36	107.81	0.491	0.949	0.007	0
SIE_iptw	-0.057	61.10	53.42	0	0	0.058	0

Table 19. Table 19 : Simulation results comparing estimators of the transported direct and indirect effects for DGP 3 under Y 𝑌 Y and S 𝑆 S models misspecified. For sample sizes 100, 500 and 5000.

Estimator	Bias	Efficiency		95%CI Cov		RMSE	% Out of Bds
		IC	Bootstrapping	IC	Bootstrapping
DGM 3, N=100
Transport stochastic direct effect
SDE_tmle_eff	-0.213	74.55	170.93	0.230	0.654	0.266	0
SDE_EE_eff	-0.254	88.47	$125.74$	0.115	0.274	0.279	0
SDE_tmle	-0.180	119.92	267.80	0.519	0.819	0.297	0
SDE_EE	-0.238	150.22	$177.67$	0.453	0.574	0.284	0
SDE_iptw	-0.210	308.35	290.33	0.788	0.862	0.328	0
Transport stochastic indirect effect
SIE_tmle_eff	0.124	43.16	105.06	0.274	0.492	0.185	0
SIE_EE_eff	0.164	53.82	$41.56$	0.194	0.120	0.193	0
SIE_tmle	0.120	44.87	109.68	0.273	0.478	0.186	0
SIE_EE	0.162	58.43	$54.32$	0.226	0.178	0.192	0
SIE_iptw	0.113	146.94	107.72	0.513	0.500	0.173	0
DGM 3, N=500
Transport stochastic direct effect
SDE_tmle_eff	-0.142	93.50	153.26	0.236	0.334	0.160	0
SDE_EE_eff	-0.197	202.02	148.83	0.287	0.284	0.219	0
SDE_tmle	-0.136	204.33	296.39	0.477	0.717	0.195	0
SDE_EE	-0.191	371.54	301.70	0.301	0.280	0.260	0
SDE_iptw	-0.164	329.74	244.23	0.727	0.501	0.197	0
Transport stochastic indirect effect
SIE_tmle_eff	0.088	60.27	94.55	0.111	0.350	0.101	0
SIE_EE_eff	0.139	73.44	42.17	0.008	0	0.144	0
SIE_tmle	0.087	63.54	112.34	0.169	0.472	0.102	0
SIE_EE	0.139	79.94	53.53	0.020	0	0.144	0
SIE_iptw	0.097	161.70	94.15	0.582	0.268	0.108	0
DGM 3, N=5000
Transport stochastic direct effect
SDE_tmle_eff	-0.056	203.22	244.19	0.477	0.548	0.065	0
SDE_EE_eff	-0.070	875.83	342.74	0.962	0.596	0.082	0
SDE_tmle	-0.062	518.76	616	0.813	0.889	0.111	0
SDE_EE	-0.074	1,340.79	935.49	0.862	0.865	0.162	0
SDE_iptw	-0.099	417.60	429.88	0.620	0.669	0.118	0
Transport stochastic indirect effect
SIE_tmle_eff	0.125	147.45	105.41	0	0	0.126	0
SIE_EE_eff	0.133	67.89	35	0	0	0.134	0
SIE_tmle	0.122	158.62	145.54	0	0	0.124	0
SIE_EE	0.133	74.73	47.12	0	0	0.134	0
SIE_iptw	0.129	157.15	126.67	0	0	0.130	0

Table 20. Table 20 : Simulation results comparing estimators of the transported direct and indirect effects for DGP 3 under Y 𝑌 Y and Z 𝑍 Z models misspecified. For sample sizes 100, 500 and 5000.

Estimator	Bias	Efficiency		95%CI Cov		RMSE	% Out of Bds
		IC	Bootstrapping	IC	Bootstrapping
DGM 3, N=100
Transport stochastic direct effect
SDE_tmle_eff	-0.155	79.58	135.64	0.422	0.658	0.195	0
SDE_EE_eff	-0.150	117.23	136.56	0.580	0.714	0.188	0
SDE_tmle	0.005	264.09	290.02	0.806	0.874	0.287	0
SDE_EE	0.028	380.69	299.92	0.939	0.880	0.292	0.300
SDE_iptw	0.002	280.33	287.94	0.867	0.883	0.272	0
Transport stochastic indirect effect
SIE_tmle_eff	0.047	111.60	126.11	0.640	0.639	0.086	0
SIE_EE_eff	0.038	170.70	110.67	0.863	0.755	0.070	0
SIE_tmle	0.046	120.45	151.05	0.626	0.640	0.094	0
SIE_EE	0.032	195.13	146.58	0.691	0.639	0.088	0
SIE_iptw	0.030	151.98	140.84	0.760	0.809	0.076	0
DGM 3, N=500
Transport stochastic direct effect
SDE_tmle_eff	-0.103	111.77	124.79	0.307	0.383	0.112	0
SDE_EE_eff	-0.106	158.80	114.52	0.526	0.266	0.114	0
SDE_tmle	0.116	397.58	376.82	0.799	0.735	0.192	0
SDE_EE	0.120	481.43	371.48	0.942	0.746	0.191	0
SDE_iptw	0.090	339.25	328.30	0.812	0.776	0.161	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.002	141.83	125.67	0.977	0.954	0.022	0
SIE_EE_eff	0.0002	214.39	101.32	0.993	0.953	0.018	0
SIE_tmle	-0.008	151.37	186.26	0.892	0.923	0.035	0
SIE_EE	-0.004	269.77	193.99	0.973	0.938	0.036	0
SIE_iptw	-0.002	148.16	139.08	0.935	0.921	0.026	0
DGM 3, N=5000
Transport stochastic direct effect
SDE_tmle_eff	-0.098	105.11	103.48	0	0	0.099	0
SDE_EE_eff	-0.104	144.61	97.28	0	0	0.104	0
SDE_tmle	0.128	387.96	351.80	0.210	0.178	0.135	0
SDE_EE	0.127	440.62	351.63	0.282	0.183	0.134	0
SDE_iptw	0.095	334.43	309.47	0.343	0.296	0.102	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.005	122.10	87.11	0.959	0.826	0.007	0
SIE_EE_eff	-0.001	180.44	76.76	1	0.932	0.005	0
SIE_tmle	-0.014	143.41	146.80	0.602	0.626	0.017	0
SIE_EE	-0.005	230.11	164.35	0.992	0.931	0.011	0
SIE_iptw	-0.005	121.01	114.92	0.881	0.859	0.009	0

Table 21. Table 21 : Simulation results comparing estimators of the transported direct and indirect effects for DGP 3 under Y 𝑌 Y and M 𝑀 M models misspecified. For sample sizes 100, 500 and 5000.

Estimator	Bias	Efficiency		95%CI Cov		RMSE	% Out of Bds
		IC	Bootstrapping	IC	Bootstrapping
DGM 3, N=100
Transport stochastic direct effect
SDE_tmle_eff	-0.038	130.75	184.19	0.744	0.913	0.157	0
SDE_EE_eff	-0.035	198.80	$213.89$	0.805	0.873	0.184	0.200
SDE_tmle	0.055	197.50	318.16	0.574	0.859	0.338	0
SDE_EE	-0.006	409.96	$339.35$	0.876	0.876	0.414	2.200
SDE_iptw	0.020	365.53	331.20	0.822	0.860	0.325	0
Transport stochastic indirect effect
SIE_tmle_eff	0.097	108.17	131.11	0.573	0.583	0.196	0
SIE_EE_eff	0.078	203.97	$155.45$	0.656	0.628	0.197	0.100
SIE_tmle	0.109	109.89	121.39	0.533	0.512	0.204	0
SIE_EE	0.082	207.74	$164.23$	0.622	0.575	0.211	0.100
SIE_iptw	0.094	192.23	119.34	0.644	0.562	0.183	0
DGM 3, N=500
Transport stochastic direct effect
SDE_tmle_eff	0.001	170.79	155.19	0.958	0.948	0.058	0
SDE_EE_eff	0.011	252.72	167.82	0.991	0.963	0.059	0
SDE_tmle	0.079	443.35	490.67	0.569	0.642	0.266	0
SDE_EE	-0.010	909.43	847.53	0.984	0.707	0.427	1.800
SDE_iptw	0.056	504.92	437.32	0.664	0.708	0.222	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.017	110.82	158.64	0.871	0.949	0.066	0
SIE_EE_eff	-0.031	308.72	178.99	0.985	0.929	0.081	0
SIE_tmle	-0.010	125.30	181.72	0.876	0.948	0.076	0
SIE_EE	-0.028	328.96	226.06	0.962	0.928	0.098	0
SIE_iptw	-0.009	193.11	160.90	0.933	0.935	0.071	0
DGM 3, N=5000
Transport stochastic direct effect
SDE_tmle_eff	0.004	157.81	131.85	0.975	0.931	0.018	0
SDE_EE_eff	0.013	215.88	125.17	0.994	0.844	0.021	0
SDE_tmle	0.013	628.84	586.62	0.942	0.945	0.076	0
SDE_EE	0.015	777.41	723.73	0.952	0.920	0.091	0
SDE_iptw	0.012	452.62	483.55	0.917	0.949	0.063	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.017	107.69	110.40	0.738	0.761	0.024	0
SIE_EE_eff	-0.029	248.66	133.17	0.992	0.644	0.035	0
SIE_tmle	-0.017	121.09	124.46	0.789	0.806	0.025	0
SIE_EE	-0.029	271.57	175.39	0.983	0.775	0.039	0
SIE_iptw	-0.015	129.58	120.45	0.862	0.825	0.023	0

Table 22. Table 22 : Simulation results comparing estimators of the transported direct and indirect effects for DGP 3 under Y 𝑌 Y model misspecified. For sample sizes 100, 500 and 5000.

Estimator	Bias	Efficiency		95%CI Cov		RMSE	% Out of Bds
		IC	Bootstrapping	IC	Bootstrapping
DGM 3, N=100
Transport stochastic direct effect
SDE_tmle_eff	-0.067	114.03	177.02	0.731	0.893	0.158	0
SDE_EE_eff	-0.095	153.08	$160.44$	0.766	0.817	0.161	0
SDE_tmle	0.019	189.07	312.17	0.598	0.876	0.314	0
SDE_EE	-0.063	335.43	$258.40$	0.857	0.854	0.335	1.600
SDE_iptw	-0.015	349.83	326.79	0.868	0.883	0.306	0
Transport stochastic indirect effect
SIE_tmle_eff	0.093	107.56	142.73	0.533	0.582	0.200	0
SIE_EE_eff	0.094	172.64	$120.52$	0.612	0.562	0.185	0
SIE_tmle	0.101	109.75	138.89	0.516	0.532	0.203	0
SIE_EE	0.096	180.93	$132.24$	0.604	0.545	0.193	0
SIE_iptw	0.083	159.86	136.27	0.648	0.589	0.183	0
DGM 3, N=500
Transport stochastic direct effect
SDE_tmle_eff	-0.007	147.33	140.13	0.954	0.949	0.054	0
SDE_EE_eff	-0.025	215.05	133.32	0.971	0.923	0.056	0
SDE_tmle	0.066	411.77	462.31	0.607	0.674	0.238	0
SDE_EE	-0.025	729.40	644.29	0.992	0.752	0.314	1
SDE_iptw	0.042	463.89	418.71	0.768	0.756	0.200	0
Transport stochastic indirect effect
SIE_tmle_eff	-0.009	113.07	157.65	0.898	0.955	0.060	0
SIE_EE_eff	0.004	270.99	147.05	0.985	0.959	0.058	0
SIE_tmle	-0.005	126.18	184.44	0.896	0.958	0.068	0
SIE_EE	0.003	293.04	190.52	0.970	0.950	0.074	0
SIE_iptw	-0.005	176.15	164.44	0.937	0.944	0.066	0
DGM 3, N=5000
Transport stochastic direct effect
SDE_tmle_eff	-0.0002	147.13	118.60	0.984	0.947	0.015	0
SDE_EE_eff	-0.001	250.94	124.08	0.992	0.950	0.014	0
SDE_tmle	0.006	574.32	554.59	0.925	0.910	0.076	0
SDE_EE	-0.002	716.64	643.04	0.947	0.910	0.087	0
SDE_iptw	0.004	470.22	463.33	0.903	0.915	0.063	0
Transport stochastic indirect effect
SIE_tmle_eff	0.0001	130.82	108.84	0.984	0.952	0.014	0
SIE_EE_eff	0.001	220.90	108.07	1	0.939	0.015	0
SIE_tmle	-0.0001	146.59	126.90	0.955	0.939	0.017	0
SIE_EE	0.001	240.50	147.62	0.999	0.938	0.020	0
SIE_iptw	-0.0001	149.24	124.84	0.964	0.940	0.017	0

Equations

$$S$$

$$W$$

$$A$$

$$Z$$

$$M$$

$$Y$$

$$g_{M\mid a^{\ast },W}^{\ast }(M\mid W)=\sum _{z}Pr(M\mid Z=z,W)Pr(Z=z\mid A=a^{\ast },W).$$

$${\Psi }^F(P_{UO})=\mathbb{E}\left[Y_{a,g_{M\mid a^{\ast },W,s}^{\ast }}\mid S=0\right]$$

$$g_{M\mid a^{\ast },W,s}^{\ast }(M\mid W)=\sum _{z}Pr(M\mid Z=z,W,S=s)Pr(Z=z\mid A=a^{\ast },W,S=s),$$

$$\mathbb{E}\left[Y_{1,g_{M\mid 0,W,s}^{\ast }}-Y_{0,g_{M\mid 0,W,s}^{\ast }}\mid S=0\right]$$

$$\mathbb{E}\left[Y_{1,g_{M\mid 1,W,s}^{\ast }}-Y_{1,g_{M\mid 0,W,s}^{\ast }}\mid S=0\right].$$

$$\displaystyle\Psi(P)=\Psi^{F}(P_{UX})=\mathbb{E}\biggr{[}\mathbb{E}\biggr{[}\mathbb{E}_{\hat{g}_{M\mid a^{*},W,s}}\biggr{[}\mathbb{E}\biggr{[}Y|W,Z,M,S=1\biggr{]}|W,Z\biggr{]}|W,a,S=0\biggr{]}|S=0\biggr{]}$$

$$\displaystyle=\mathbb{E}\biggr{[}\mathbb{E}\biggr{[}\sum_{m}\biggr{[}\mathbb{E}Y\hat{g}_{M\mid a^{*},W,s}(m\mid W)\mid M=m,W,Z,A=a,S=1\biggr{]}\mid A=a,W,S\biggr{]}\mid S=0\biggr{]}$$

$$\begin{gathered} H(M,Z,A,W,S)= \\ \frac{{\hat{g}}_{M\mid a^{\ast },W,s}(M\mid W)p_Z\left(Z\mid A=a,W,S=0\right)p_W\left(S=0\mid W\right)I(S=1,A=a)}{p_M\left(M\mid Z,W,S=1\right)p_Z\left(Z\mid A=a,W,S=1\right)p_A\left(a\mid W,S=1\right)p_W\left(S=1\mid W\right)P_S(S=0)}. \end{gathered}$$

$${\hat{\Psi }}_n^{IPTW}=\frac{\frac{1}{n}{\sum }_i{\hat{H}}_n(M_i,Z_i,A_i,W_i,S_i)Y_i}{\frac{1}{n}{\sum }_i{\hat{H}}_n(M_i,Z_i,A_i,W_i,S_i)},$$

$$IC(P)=\frac{\hat{H}_{n}(M_{i},Z_{i},A_{i},W_{i},S_{i})}{\frac{1}{n}\sum_{i}\hat{H}_{n}(M_{i},Z_{i},A_{i},W_{i},S_{i})}\bigg{(}Y_{i}-\frac{1}{n}\sum_{i}\hat{H}_{n}(M_{i},Z_{i},A_{i},W_{i},S_{i})Y_{i}\bigg{)}.$$

$$\begin{array}{rl}D(P)& =D_Y(P)+D_Z(P)+D_W(P),\text{ where }\\ D_Y(P)& =\left(Y-{\bar{Q}}_Y(M,Z,W)\right)\\ & \times \frac{{\hat{g}}_{M\mid a^{\ast },W,s}(M|W)p_Z\left(Z\mid A=a,W,S=0\right)p_W\left(S=0\mid W\right)I(S=1)}{g_M\left(M\mid Z,W,S=1\right)p_Z\left(Z\mid W,S=1\right)p_W\left(S=1\mid W\right)P_S(S=0)},\\ D_Z(P)& =\left({\bar{Q}}_M(Z,W,S)-{\bar{Q}}_Z(a,W,S)\right)\frac{I(S=0,A=a)}{p_A\left(a\mid W,S\right)p_S(S=0)},\text{ and }\\ D_W(P)& =\left({\bar{Q}}_Z(a,W,S)-\Psi (P)\right)\frac{I(S=0)}{p_S(S=0)}\text{ (notation is explained further below}).\end{array}$$

$$\begin{array}{rl}D(P)& =D_Y(P)+D_Z(P)+D_W(P),\text{ where }\\ D_Y(P)& =\left(Y-{\bar{Q}}_Y(M,Z,A,W)\right)\\ & \times \frac{{\hat{g}}_{M\mid a^{\ast },W,s}(M|W)p_Z\left(Z\mid A=a,W,S=0\right)p_W\left(S=0\mid W\right)I(S=1,A=a)}{g_M\left(M\mid Z,A,W,S=1\right)p_Z\left(Z\mid A=a,W,S=1\right)p_A\left(a\mid W,S=1\right)p_W\left(S=1\mid W\right)P_S(S=0)},\\ D_Z(P)& =\left({\bar{Q}}_M(Z,A,W,S)-{\bar{Q}}_Z(a,W,S)\right)\frac{I(S=0,A=a)}{p_A\left(a\mid W,S\right)p_S(S=0)},\text{ and }\\ D_W(P)& =\left({\bar{Q}}_Z(a,W,S)-\Psi (P)\right)\frac{I(S=0)}{p_S(S=0)}\text{ (notation is explained further below}).\end{array}$$

$$H(M,Z,W,S)=\frac{{\hat{g}}_{M\mid a^{\ast },W,s}(M|W)p_Z\left(Z\mid A=a,W,S=0\right)p_W\left(S=0\mid W\right)I(S=1)}{g_M\left(M\mid Z,W,S=1\right)p_Z\left(Z\mid W,S=1\right)p_W\left(S=1\mid W\right)P_S(S=0)},$$

$$H_a(a,W,S)=\frac{I(S=0,A=a)}{p_A\left(A\mid W,S\right)p_S(S=0)},$$

$$\displaystyle\Psi(P)=\Psi^{F}(P_{UX})=\mathbb{E}\biggr{[}\mathbb{E}\biggr{[}\mathbb{E}_{\hat{g}_{M\mid a^{*},W,s}}\biggr{[}\mathbb{E}\biggr{[}Y|M,Z,W,S=1\biggr{]}|W,Z\biggr{]}|W,a,S=0\biggr{]}|S=0\biggr{]}$$

$$\displaystyle=\mathbb{E}\biggr{[}\mathbb{E}\biggr{[}\sum_{m}\biggr{[}\mathbb{E}Y\hat{g}_{M\mid a^{*},W,s}(m\mid W)\mid M=m,Z,A=a,W,S=1\biggr{]}\mid A=a,W,S\biggr{]}\mid S=0\biggr{]}$$

$${\Psi }^F(P_{UX})$$

$$\displaystyle\overset{\text{tower law}}{=}\mathbb{E}\biggr{[}\mathbb{E}\biggr{[}Y_{a,\hat{g}_{M\mid a^{*},W,s}}\mid Z,A,W\biggr{]}\mid S=0\biggr{]}$$

$$\displaystyle\overset{\text{assumption 2}}{=}\mathbb{E}\biggr{[}\sum_{m}\biggr{[}\mathbb{E}Y_{am}\hat{g}_{M\mid a^{*},W,s}(m\mid W)\mid M=m,Z,A,W,S=1\biggr{]}\mid S=0\biggr{]}$$

$$\displaystyle\overset{\text{tower law}}{=}\mathbb{E}\biggr{[}\mathbb{E}\biggr{[}\sum_{m}\biggr{[}\mathbb{E}Y_{am}\hat{g}_{M\mid a^{*},W,s}(m\mid W)\mid M=m,Z,A,W,S=1\biggr{]}$$

$$\displaystyle\hskip 56.9055pt\mid A,W,S=0\biggr{]}\mid S=0\biggr{]}$$

$$\displaystyle\overset{\text{assumption 3}}{=}\mathbb{E}\biggr{[}\mathbb{E}\biggr{[}\sum_{m}\biggr{[}\mathbb{E}Y\hat{g}_{M\mid a^{*},W,s}(m\mid W)\mid M=m,Z,A=a,W,S=1\biggr{]}$$

$$\displaystyle\hskip 56.9055pt\mid A=a,W,S=0\biggr{]}\mid S=0\biggr{]}$$

$$H(M,Z,A,W,S)=\frac{{\hat{g}}_{M\mid a^{\ast },W,s}\left(M\mid W\right)p_Z\left(Z\mid A,W,S=0\right)p_{S\mid W}\left(S=0\mid W\right)I(S=1,A=a)}{p_M\left(M\mid Z,W,S\right)p_Z\left(Z\mid A,W,S\right)p_A\left(A\mid W,S\right)p_{S\mid W}\left(S\mid W\right)P_S(S=0)}.$$

$${\Psi }_n^1-{\Psi }_n^0=-{\left(\frac{d}{d\Psi }{P}_n{\hat{H}}_n(Y-{\Psi }_n^0)\right)}^{-1}{P}_n{\hat{H}}_n(Y-{\Psi }_n^0),$$

$${\Psi }_n^{IPTW}-\Psi (P_0)$$

$$={\Psi }_n^0+(P_n-P_0){\hat{D}}_n+P_0{\hat{D}}_n-\Psi (P_0)$$

$${\Psi }_n^0+P_0{\hat{D}}_n-\Psi (P_0)$$

$$=\frac{(P_n-P_0){\hat{H}}_n{\Psi }_n^0}{P_n{\hat{H}}_n}+P_0({\hat{H}}_n-H)Y+\frac{P_0{\hat{H}}_{n}Y}{P_n{\hat{H}}_n}\left((P_0-P_n){\hat{H}}_n+P_0(H-{\hat{H}}_n)\right)$$

$$=P_0({\hat{H}}_n-H)Y+\frac{(P_0-P_n){\hat{H}}_{n}Y(P_0-P_n){\hat{H}}_n}{P_n{\hat{H}}_n}+\frac{P_0{\hat{H}}_{n}Y}{P_n{\hat{H}}_n}{P}_0(H-{\hat{H}}_n)$$

$$=P_0({\hat{H}}_n-H)Y+\frac{(P_0-P_n){\hat{H}}_{n}Y((P_0-P_n)({\hat{H}}_n-H)+H)}{P_n{\hat{H}}_n}+\frac{P_0{\hat{H}}_{n}Y}{P_n{\hat{H}}_n}{P}_0(H-{\hat{H}}_n)$$

$$\prod \left(I{C}_{IPTW}\Vert T_{Y\times S}\right)=I{C}_{IPTW}(O)-\mathbb{E}\left[I{C}_{IPTW}(O)\mid M,Z,A,W,S\right]=D_Y^{\ast }(O)$$

$$\Psi(P)=\mathbb{E}\biggr{[}\mathbb{E}\biggr{[}\sum_{m}\biggr{[}\mathbb{E}Y\hat{g}_{M\mid a^{*},W,s}(m\mid W)\mid M=m,W,Z,A=a,S=1\biggr{]}\mid A=a,W,S\biggr{]}\mid S=0\biggr{]}$$

Full text

Keywords: Causal inference; Mediation; Transportability; Generalizability; External validity; Instrumental variables; Targeted maximum likelihood estimation

1 Introduction

Often, an intervention, program, or policy that works in one place or population fails to replicate in another place or population (21) or can even have unintended harmful effects (6). This is problematic from a public policy or public health perspective in that the goals of such interventions are to help—not harm, and problematic from a financial perspective in that limited resources may be not be spent optimally.

When such initiatives fail to replicate or have unintended effects in new populations, transportability theory and methods offer a chance to understand why. Transportability is the ability (based on identifying assumptions) to transport a causal effect from a source population to a new, target population, accounting for differences between the two populations (e.g., differences in compositional factors, treatment adherence, etc.) (16). Previous work developed estimators to transport total effects from a source to target population (20) or, similarly, to generalize effects from a sample to the population (13, 23, 4, 5).

In some cases, examining transportability of the total effect may shed light on reasons for lack of replication. However, in other cases, transporting the total effect may not identify the relevant differences and it may be beneficial to go further and examine transportability of the underlying mediation mechanisms. Although there has been work on the identification on transported indirect effects (2, 16), we are not aware of any previous work developing estimators for transporting mediation effects (direct and indirect effects) from a source to target population. Thus, we address this research gap by proposing several different estimators of stochastic direct and indirect effects: an inverse-probability of treatment stabilized weighted estimator, a doubly robust estimator that solves the estimating equation, and a doubly robust substitution estimator in the targeted minimum loss-based framework.

The paper is organized as follows. In Section 2, we introduce notation and the structural equations model generating our data. In Section 3, we define the parameters of interest, the transported stochastic direct and indirect effects, and give identification results. Sections 4, 5, and 6, detail an inverse-probability of treatment weighted estimator, an estimating equation estimator, and a targeted minimum loss-based estimator for estimating these transported stochastic direct and indirect effects. In Section 7, we present the results of simulation studies that demonstrates the relative performance of the aforementioned estimators in finite samples. Section 8 concludes.

2 Notation and structural equations model

The full data is generated by a structural equations model (SEM) (26, 17), which consists of the data generating process to which we would like to have access. The SEM first generates a random draw of a vector $U$ of unknown measurements (15), where $U=(U_{S},U_{W},U_{A},U_{Z},U_{M},U_{Y})\sim P_{U}$. Then our variables are generated in the following time ordering:

[TABLE]

where $S$ is a binary indicator of site, $W$ is a vector of covariates, $A$ is a binary treatment, $Z$ is a binary intermediate variable, $M$ is a binary mediator, and $Y$ is a binary or continuous outcome. The SEM generates the full data as $(U,O)\sim P_{UO}\in\cal{M}^{F}$, our full-data statistical model. If we had access to the SEM, we could generate potential (i.e., counterfactual) outcomes (14, 18), which define our causal parameters of interest. We observe data $O=(S,W,A,Z,M,S\times Y)$ for $n$ participants, with the true distribution $O_{1},...,O_{n}\overset{iid}{\sim}P_{O}\in\cal{M}$, our observed data statistical model. Note, we will only observe the outcome, $Y$, for site $S=1$.

We consider a structural causal model for these data aligned with our motivating example in which $A$ is an instrumental variable (IV) (1), which puts several restrictions on the statistical model, $\cal{M}$, where $\cal{M}$ is the collection of probability distributions under which the causal quantities of interest are identified—we define and identify such quantities in the following section. The restrictions are: 1) $A$ is randomly assigned (possibly conditional on $(W,S)$), and 2) there is no direct effect of $A$ on $M$ or of $A$ on $Y$—downstream effects of $A$ only operate through $Z$ (22). However, the methods we propose can be used in the more general case where $A$ directly affects $M$ and/or $Y$; we discuss such an extension in the appendix.

In our notation, we use lowercase letters to denote fixed, assignment values of variables and uppercase letters to denote observed values. We use subscripts for descriptive purposes—subscripts are not to be considered a variable. For instance, we use a capital letter in $p_{Y}$, the conditional density of $Y$, because it is a density of the random variable $Y$.

3 Parameters of interest

We consider two causal quantities of interest that we call transported stochastic direct and indirect effects. These causal quantities represent stochastic direct and indirect effects (22, 25) transported from a source population to a new, target population. Stochastic direct and indirect effects, also called randomized interventional direct and indirect effects (25), represent the 1) direct effect of $A$ on $Y$ not through $M$ and the 2) indirect effect of $A$ on $Y$ through $M$. As has been described previously (19), one can consider versions of these effects that condition on $Z$ and thus estimate the indirect pathway of $A$ to $M$ to $Y$, not through $Z$ (27), or versions that marginalize over $Z$ and thus estimate the combined indirect pathways of 1) $A$ to $Z$ to $M$ to $Y$ plus 2) $A$ to $M$ to $Y$ (22, 25). Adhering to the IV constraints on our statistical model, no effect operates through pathway $A$ to $M$ to $Y$, so we focus herein on the versions of these effects that marginalize over $Z$. Previously, such a stochastic intervention on $M$ has been defined (22, 25)

[TABLE]

The subscript for $\hat{g}_{M\mid a^{*},W}$ specifies that it is a conditional density of random variable $M$ given random variable $W$, and value $a^{*}$ for which a lower case letter indicates they are fixed and the same for all participants. Previously, the parameter of interest was defined as $\Psi^{F}(P_{UO})=\mathbb{E}\left[Y_{a,g^{*}_{M\mid a^{*},W}}\right]$ where the expectation is taken over the full data model and $Y_{a,g^{*}_{M\mid a^{*},W}}$ is a potential outcome intervening on $A$ to set it to $a$, and then downstream, intervening on $M$ to set it to a random (i.e., stochastic) draw from the distribution of $M$ defined by $g^{*}_{M\mid a^{*},W}(M|W)$ (22). We wish to transport this parameter to a new site where the outcome was not observed $(S=0)$, and thus make the following modification:

[TABLE]

where

[TABLE]

and where we impose a certain $a^{*}$ and a certain $s$ in both the $Z$ and $M$ models. We note that $g^{*}_{M\mid a^{*},W,s}(M|W)$ represents any stochastic intervention, which can include stochastic draws from the true models, but can also include a data-dependent version, estimated from observed data distributions, which we denote $\hat{g}_{M\mid a^{*},W,s}(M|W)$.

The transported stochastic direct effect entails setting $a^{*}$ to 0 and taking the difference in mean outcome between setting $a$ to 1 and setting $a$ to 0, denoted

[TABLE]

and the transported stochastic indirect effect entails setting $a=1$ and then taking the difference in mean outcome between setting $a^{*}=1$ and $a^{*}=0$, denoted

[TABLE]

4 Identifiability

To identify the stochastic direct effect and stochastic indirect effect we will need to impose additional assumptions on $\mathcal{M}^{F}$ and $\mathcal{M}$, listed below.

Positivity: For all $S$ and $W$ we need a positive probability of assigning any level of $A$. For all combinations of $S,W,$ and $A=a$, we have a positive probability of any level of $Z.$ For $S=1$ and all combinations of $Z$ and $W$ we need a positive probability of any level of the mediator, $M$. 2. 2.

Common outcome model across sites: $\mathbb{E}\left[Y\mid M,Z,W,S=1\right]=\mathbb{E}\left[Y\mid M,Z,W,S=0\right]$. The null hypothesis of a common outcome model may be tested nonparametrically (12). 3. 3.

Sequential Randomization: $Y_{am}\perp A\mid W,S$ and $Y_{am}\perp M\mid W,Z,S$. This is akin to a two-time point longitudinal intervention where at the first time point, we statically intervene to set the treatment, $A=a$, and at the second time point, we stochastically intervene on the mediator, $M$.

Theorem 4.1**.**

Given the above assumptions, we can establish the following identification:

[TABLE]

The proof is in the appendix.

5 Stabilized Inverse probability of treatment weighted estimator

First, we describe a stabilized inverse probability of treatment weighted (IPTW) estimator of $\Psi(P)$. The R code to implement this estimator is provided in the appendix. We describe model fitting using regression language for simplicity but note that machine learning can be used instead. We will use the knowledge of our smaller model, $\cal{M}$, with restrictions detailed in Section 2, so do not include $A$ in the regression model for $M$.

We use the weights

[TABLE]

$\hat{g}_{M|a^{*},W,s}(M|W)$ is a data-dependent stochastic intervention on $M$, which can be estimated $\hat{g}_{M|a^{*},W,s}(M|W)=\sum^{1}_{z=0}P(M=1|Z=z,W,S=s)P(Z=z|A=a^{*},W,S=s).$ $P(M=1|Z=z,W,S=s)$ can be estimated using a logistic regression estimating the probability of $M=1$ given $Z$, $W$, and $S$ and thereby getting predicted probabilities for $M=1$ setting $S=s$ and separately setting $Z=1$ and $Z=0$. $P(Z=z|A=a^{*},W,S=s)$ can be estimated using a logistic regression estimating the probability of $Z=1$ given $A$, $W$, and $S$ and thereby getting predicted probabilities for $Z=1$ and for $Z=0$, setting $A=a^{*}$ and $S=s$ and using observed values for $W$. We note that marginalizing out $Z$ introduces dependence on $A$.

The stabilized IPTW estimate of $\Psi(P)$ is the empirical mean of $Y$ weighted by $\hat{H}_{n}(M,Z,A,W,S)$, stabilized by the empirical mean of $\hat{H}(M,Z,A,W,S)$:

[TABLE]

where $\hat{H}_{n}$ is an approximation of $H(M,Z,A,W,S)$.

The standard error of the stabilized IPTW estimator is calculated as the sample standard deviation of the approximation of the IPTW influence curve of $\Psi(P)$, given by

[TABLE]

The stochastic direct effect (SDE) entails setting $a^{*}$ to 0 and taking the difference in estimates between setting $a$ to 1 and setting $a$ to 0. The corresponding influence curve approximation is a difference of the influence curve approximation for the parameter defined by setting $a^{*}=0,a=1$ and the influence curve approximation for the parameter defined by setting $a^{*}=0,a=0$. The stochastic indirect effect (SIE) entails setting $a=1$ and then taking the difference in estimates between setting $a^{*}=1$ and $a^{*}=0$. The corresponding influence curve approximation is again a difference of the two influence curve approximations (one for parameter defined by setting $a^{*}=1,a=1$ and the other for $a^{*}=0,a=1$). For each estimator, we used the sample standard deviation of their respective influence curve approximations, divided by $\sqrt{n}$ for the standard error estimate.

6 Estimating equation estimator

Next, we describe two estimating equation (EE) estimators of $\Psi(P)$: 1) one that incorporates the exclusion restrictions on our statistical model, $\cal{M}$, that there is no direct effect of $A$ on $M$ or of $A$ on $Y$, and 2) another that does not impose those restrictions. The R code to implement these estimators is provided in the appendix. We describe model fitting using regression language for simplicity but note that machine learning can be used instead.

6.1 Estimator incorporating exclusion restrictions

This EE estimator solves the efficient influence curve (EIC) equation of the parameter $\Psi(P)$ for the restricted model where $M$ and $Y$ do not depend directly on $A$. This EIC is given by

[TABLE]

We reiterate that this EIC is only efficient for model with the exclusion restrictions. We approximate $D_{Y}(P)$ as follows. First, we estimate $\mathbb{E}\left[Y\mid M,Z,W\right]$ as $\bar{Q}_{Y,n}(M,Z,W),$ which can be calculated using predicted values from a regression of $Y$ on $M,Z,W$ among those with $S=1$. $p_{Z}(Z|W,S=1)$ can be calculated $p_{Z}(Z|W,S=1)=\sum_{a=0}^{1}P(Z=1|A=a,W,S=1)P(A=a|W,S=1)$. Each of the remaining probabilities in $D_{Y}(P)$ can be calculated using predicted probabilities from a logistic regression of the relevant conditional mean outcome models for M, Z and S, as described in the stabilized IPTW estimator section. Estimation of $\hat{g}_{M|a^{*},W,s}(M|W)$ is also described in the stabilized IPTW estimator section.

We next approximate $D_{Z}(P)$. To do so, we apply the stochastic intervention on $\bar{Q}_{Y,n}(M,Z,W)$ via the computation $\bar{Q}_{M,n}(Z,W,S)=\mathbb{E}_{\hat{g}_{M\mid a^{*},W,s}}[\bar{Q}_{Y,n}(M,Z,W)\mid Z,W,S]$. Specifically, we can generate predicted values of $\bar{Q}_{Y,n}(1,Z,W)$ and $\bar{Q}_{Y,n}(0,Z,W)$ and then marginalize over $\hat{g}_{m\mid a^{*},W,s}(M|W)$: $\sum^{1}_{m=0}\bar{Q}_{Y,n}(m,Z,W)\hat{g}_{m\mid a^{*},W,s}(m|W)$. Next, we estimate $\bar{Q}_{Z,n}(a,W,S)$ by regressing $\bar{Q}_{M,n}(Z,W,S)$ on $A,W,$ and $S$, and getting predicted values setting $A=a$.

The estimate of $\Psi(P)$ is obtained by solving $\frac{1}{n}\sum_{i}\hat{D}_{n}(O_{i})=0$, where $\hat{D}_{\Psi,n}$ is the EIC approximation as computed above. The variance of the EE estimate is the sample variance of the EIC approximation. The SDE, SIE, and their corresponding ICs and standard errors can be calculated as described in the stabilized IPTW estimator section. The conditions for consistency and asymptotic efficiency of this estimator are discussed in the appendix.

6.2 Estimator not imposing exclusion restrictions

This EE estimator solves the efficient influence curve (EIC) equation of the parameter $\Psi(P)$ for the unrestricted model where $M$ and $Y$ may depend directly on $A$. It will be inefficient under the restricted model where $M$ and $Y$ do not depend directly on $A$. This EIC is given by

[TABLE]

We approximate $D_{Y}(P)$ as follows. First, we estimate $\mathbb{E}\left[Y\mid M,Z,A,W\right]$ as $\bar{Q}_{Y,n}(M,Z,A,W),$ which can be calculated using predicted values from a regression of $Y$ on $M,Z,A,W$ among those with $S=1$. Each of the probabilities in $D_{Y}(P)$ can be calculated using predicted probabilities from a logistic regression of the relevant conditional mean outcome models for M, Z, A, and S. $\hat{g}_{M|a^{*},W,s}(M|W)$ can be estimated $\hat{g}_{M|a^{*},W,s}(M|W)=\sum^{1}_{z=0}P(M=1|Z=z,A=a^{*},W,S=s)P(Z=z|A=a^{*},W,S=s).$

We next approximate $D_{Z}(P)$. To do so, we apply the stochastic intervention on $\bar{Q}_{Y,n}(M,Z,A,W)$ via the computation $\bar{Q}_{M,n}(Z,A,W,S)=\mathbb{E}_{\hat{g}_{M\mid a^{*},W,s}}[\bar{Q}_{Y,n}(M,Z,A,W)\mid Z,W,S]$. Specifically, we can generate predicted values of $\bar{Q}_{Y,n}(1,Z,A,W)$ and $\bar{Q}_{Y,n}(0,Z,A,W)$ and then marginalize over $\hat{g}_{m\mid a^{*},W,s}(M|W)$: $\sum^{1}_{m=0}\bar{Q}_{Y,n}(m,Z,A,W)\hat{g}_{m\mid a^{*},W,s}(m|W)$. Next, we estimate $\bar{Q}_{Z,n}(a,W,S)$ by regressing $\bar{Q}_{M,n}(Z,A,W,S)$ on $A,W,$ and $S$, and getting predicted values setting $A=a$.

The estimate of $\Psi(P)$ is obtained by solving $\frac{1}{n}\sum_{i}\hat{D}_{n}(O_{i})=0$, where $\hat{D}_{\Psi,n}$ is the EIC approximation as computed above. The variance of the EE estimate is the sample variance of the EIC approximation. The SDE, SIE, and their corresponding ICs and standard errors can be calculated as described in the stabilized IPTW estimator section. The conditions for consistency and asymptotic efficiency of this estimator are discussed in the appendix.

7 Targeted minimum loss-based estimator

We now describe how to estimate $\Psi(P)$ using targeted minimum loss-based estimation (TMLE). This estimation approach, which is just one of several TMLE approaches that could be used, uses sequential regression, updating the conditional outcome model at each stage to both solve the EIC equation while also lowering the empirical negative log-likelihood loss. The process is identical to a two time-point longitudinal intervention (10). Similar to the EE estimator section, we describe two TMLE estimators of $\Psi(P)$: 1) one that incorporates the exclusion restrictions on our statistical model, $\cal{M}$, that there is no direct effect of $A$ on $M$ or of $A$ on $Y$, and 2) another that does not impose those restrictions. We include R code for these estimators in the appendix. As in the previous sections, we describe model fitting using regression language for simplicity but note that machine learning can be used instead.

7.1 Estimator incorporating exclusion restrictions

This TMLE estimator solves the EIC equation of the parameter $\Psi(P)$ for the restricted model where $M$ and $Y$ do not depend directly on $A$.

Let $\bar{Q}_{Y,n}^{0}(M,Z,W)$ be an initial estimate of $\mathbb{E}\left[Y\mid M,Z,W\right]$. $\bar{Q}_{Y,n}^{0}(M,Z,W)$ can be estimated by predicted values from a regression of $Y$ on $M,Z,W$ among those with $S=1$.

Next, we update that initial estimate using the weights

[TABLE]

which are approximated with $\hat{H}_{n}(M,Z,W,S).$ $\bar{Q}_{Y,n}^{0}(M,Z,W)$ is updated by performing a weighted parametric logistic regression of $Y$ with $logit(\bar{Q}_{Y,n}^{0}(M,Z,W))$ as an offset, intercept $\epsilon_{Y}$, and weights $\hat{H}_{n}(M,Z,W,S)$. $\epsilon_{Y,n}$ is the MLE fit of intercept $\epsilon_{Y}$. The update is given by $\bar{Q}_{Y,n}^{*}(M,Z,W)=\bar{Q}_{Y,n}^{0}(\epsilon_{Y,n})(M,Z,W)$.

We then perform the stochastic intervention on $\bar{Q}_{Y,n}^{*}(M,Z,W)$ via the computation $\bar{Q}_{M,n}^{*}(Z,W,S)=\mathbb{E}_{\hat{g}_{M\mid a^{*},W,s}}[\bar{Q}_{Y,n}^{*}(M,Z,W)\mid Z,W,S]$. This can be done by generating predicted values of $\bar{Q}_{Y,n}^{*}(1,Z,W)$ and $\bar{Q}_{Y,n}^{*}(0,Z,W)$ and then marginalizing over $\hat{g}_{m\mid a^{*},W,s}(M|W)$: $\sum^{1}_{m=0}\bar{Q}_{Y,n}^{*}(m,Z,W)\hat{g}_{m\mid a^{*},W,s}(m|W)$.

Next, we estimate $\bar{Q}_{Z,n}^{0}(a,W,S)$ by regressing $\bar{Q}_{M,n}^{*}(Z,W,S)$ on $A,W,$ and $S$, and getting predicted values setting $A=a$. We then update this initial estimate using a second set of weights,

[TABLE]

in a weighted logistic regression of $logit(\bar{Q}_{M,n}^{*}(Z,W,S))$ with $logit(\bar{Q}_{Z,n}^{0}(a,W,S))$ as an offset, intercept $\epsilon_{Z}$. $\epsilon_{Z,n}$ is the MLE fit of intercept $\epsilon_{Z}$. The updated estimate will be notated $\bar{Q}_{Z,n}^{*}(a,W,S)=\bar{Q}_{Z,n}^{0}(\epsilon_{Z,n})(a,W,S)$.

The empirical mean of $\bar{Q}_{Z,n}^{*}(a,W,S)$ among those for whom $S=0$ is the TMLE estimate of $\Psi(P)$. The influence curve approximation obtained by incorporating our updated model fits solves the empirical mean of the EIC equation (see Equation 4 in Section 6), replacing $P$ with $P^{*}_{n}$ defined by $\bar{Q}_{Y,n}^{*}(M,Z,W))$, $\bar{Q}_{M,n}^{*}(Z,W,S)$, $\bar{Q}_{Z,n}^{*}(a,W,S)$, our known $p_{A}(A|W,S)$ and $p_{S}(S=0)$, our regression fit for $p_{M}(M|Z,W,S),p_{Z}(Z|A,W,S)$ and $p_{S|W}(S|W)$, and the empirical distribution as an estimate of $p_{W}(W)$. We denote this EIC $D(P^{*}_{n})$. The TMLE updating step also decreases the empirical loss of the model fits. The variance of the TMLE estimate is the sample variance of the EIC approximation. The SDE, SIE, and their corresponding ICs and standard errors can be calculated as described in the stabilized IPTW estimator section. The conditions for consistency and asymptotic efficiency of this estimator are discussed in the appendix.

7.2 Estimator not imposing exclusion restrictions

This TMLE estimator solves the EIC equation of the parameter $\Psi(P)$ for the unrestricted model where $M$ and $Y$ may depend directly on $A$. It will be inefficient under the restricted model where $M$ and $Y$ do not depend directly on $A$.

Let $\bar{Q}_{Y,n}^{0}(M,Z,A,W)$ be an initial estimate of $\mathbb{E}\left[Y\mid M,Z,A,W\right]$. $\bar{Q}_{Y,n}^{0}(M,Z,A,W)$ can be estimated by predicted values from a regression of $Y$ on $M,Z,A,W$ among those with $S=1$.

Next, we update that initial estimate using the weights, $H(M,Z,A,W,S)$, given in Equation 2, which are approximated with $\hat{H}_{n}(M,Z,A,W,S).$ $\bar{Q}_{Y,n}^{0}(M,Z,A,W)$ is updated by performing a weighted parametric logistic regression of $Y$ with $logit(\bar{Q}_{Y,n}^{0}(M,Z,A,W))$ as an offset, intercept $\epsilon_{Y}$, and weights $\hat{H}_{n}(M,Z,A,W,S)$. $\epsilon_{Y,n}$ is the MLE fit of intercept $\epsilon_{Y}$. The update is given by $\bar{Q}_{Y,n}^{*}(M,Z,A,W)=\bar{Q}_{Y,n}^{0}(\epsilon_{Y,n})(M,Z,A,W)$.

We then perform the stochastic intervention on $\bar{Q}_{Y,n}^{*}(M,Z,A,W)$ via the computation $\bar{Q}_{M,n}^{*}(Z,A,W,S)=\mathbb{E}_{\hat{g}_{M\mid a^{*},W,s}}[\bar{Q}_{Y,n}^{*}(M,Z,A,W)\mid Z,A,W,S]$. This can be done by generating predicted values of $\bar{Q}_{Y,n}^{*}(1,Z,A,W)$ and $\bar{Q}_{Y,n}^{*}(0,Z,A,W)$ and then marginalizing over $\hat{g}_{m\mid a^{*},W,s}(M|W)$: $\sum^{1}_{m=0}\bar{Q}_{Y,n}^{*}(m,Z,A,W)\hat{g}_{m\mid a^{*},W,s}(m|W)$.

Next, we estimate $\bar{Q}_{Z,n}^{0}(a,W,S)$ by regressing $\bar{Q}_{M,n}^{*}(Z,A,W,S)$ on $A,W,$ and $S$, and getting predicted values setting $A=a$. The remainder of the steps for this TMLE are identical to those for the restricted TMLE in the above subsection.

8 Simulation

8.1 Overview

We compare finite sample performance of our three types estimators in estimating the transported SDE and transported SIE using simulation. For both TMLE and EE, we include the 1) versions that are efficient under the restricted model in which $M$ and $Y$ do not depend directly on $A$ (henceforth TMLE efficient and EE efficient) and 2) versions that are more general in that they allow for the exclusion restriction not to hold but are inefficient under resetricted model $\cal{M}$ (henceforth TMLE general and EE general). We show estimator performance in terms of absolute bias, efficiency, 95% confidence interval (CI) coverage, root mean squared error (RMSE), and percent of estimates lying outside the bounds of the parameter space across 1,000 simulations. For calculating the efficiency and the 95% CI coverage, we use both the IC and 500 bootstrap replicates.

We consider two data-generating mechanisms (DGMs) within the structural causal model described in Section 2. The DGMs are detailed in Table 1 using the same notation as in Section 2.

8.2 Results

First, in Table LABEL:restabcorrect, we show results under correct specification of all models for sample sizes of N=5,000, N=500, and N=100 using DGM 1. Results using alternative DGMs were similar and are shown in the appendix.

Table LABEL:restabcorrect shows that under correct parametric model specifications, all estimators result in consistent estimates with 95% CI coverage close to 95% for sample sizes of N=5,000 and N=500. Influence curve-based efficiency is close to 100% of the efficiency bound for both the efficient TMLE and EE estimators for all sample sizes. For the transported direct effect, efficiency is about 3 times the efficiency bound for the general TMLE and EE estimators and about 4 times the efficiency bound for the stabilized IPTW estimator. For the transported indirect effect, IC- and bootstrapped-based efficiencies are similar and close to the efficiency bound for the TMLE and EE estimators. The stabilized IPTW estimator has efficiencies about 70% larger than the efficiency bound. In the appendix, we review the theory underlying the empirical finding that the stabilized IPTW estimator is least efficient.

Performance of all estimators degrades with a smaller sample size of N=100. For this sample size, bias slightly increases, the estimates stray from the efficiency bound, and IC-based coverage generally decreases (though bootstrapping can recover some of this lost coverage by approximating the true variance of the estimator).

Next, we show results under various model misspecifications in Table LABEL:restabincorrect for sample size N=5,000. We consider: 1) misspecification of the $Y$ model, 2) misspecification of the $Y$ and $Z$ models, 3) misspecification of the $Y$ and $M$ models, 4) misspecification of the $Y$ and $S$ models, and 5) misspecification of the $Z$, $M$, and $S$ models. We use DGM 2 for misspecification of the $Y$ model and misspecification of the $Y$ and $M$ models. We use DGM 1 for misspecification of the $Y$ and $Z$ models, misspecification of the $Y$ and $S$ models, and misspecification of the $Z,$ $M,$ and $S$ models. Full results for misspecified models under more DGMs and various sample sizes are shown in the appendix.

As derived from the robustness properties proven in the appendix, we expect the TMLE and EE estimators to be consistent if 1) the $Y$ model is correctly specified or if 2) the $Z$, $M$, and $S$ models are correctly specified. The stabilized IPTW estimator will be consistent only in the latter case: where the $Z$, $M$, and $S$ models are correctly specified. Note that we assume $A$ is randomly assigned, so we don’t examine misspecification of the $A$ model. However, in cases where $A$ is nonrandom, condition 2 above would change to require correct specification of the $A$, $Z$, $M$, and $S$ models.

We see in Table LABEL:restabincorrect that, as expected, all estimators remain unbiased when the $Y$ model is misspecified. Compared to the correctly specified case (Table LABEL:restabcorrect, there is a slight reduction in efficiency, evidenced by efficiencies greater than 100% of the efficiency bound and slight overcoverage when using the IC as opposed to bootstrapping (Table LABEL:restabincorrect).

When the $Z,M,$ and $S$ models are misspecified, the TMLE and EE estimators also remain unbiased, as expected, but the stabilized IPTW estimator is no longer consistent, resulting in 0% coverage of the transported SDE (Table LABEL:restabincorrect). However, in this scenario, coverage of the TMLE and EE efficient estimators is anticonservative when using the IC, evidenced by efficiencies as low as 55% of the efficiency bound in for the transported SIE and coverage as low as 73%. This result is not unexpected—the EIC may no longer provide accurate inference under these misspecifications, because the misspecified models may bias the influence curve approximation. In this particular misspecified scenario, our influence curve approximation becomes a less variant mean-zero function of the observed data than our efficiency bound given by the true influence curve, causing poor coverage. In such instances of model misspecification, using the bootstrap can recover coverage (as long the fitting methods applied yield valid standard errors under the non-parametric bootstrap. For linear regressions, this is the case, but for data adaptive machine learning algorithms, such might not be the case (24)).

In the remaining model misspecifications shown in Table LABEL:restabincorrect, none of the estimators are guaranteed to be unbiased. In the case where the $Y$ and $M$ models are misspecified, all three estimators perform poorly in estimating both the transported direct and indirect effects. Performances of the EE estimators are particularly poor in terms of bias and inefficiency. The marked inefficiency may be due, in part, to estimates lying outsides the bounds of the parameter space. Indeed, we see that nearly 100% of the EE transported SDE estimates lie outside the parameter space. In contrast, even though the TMLE estimators solve the same influence curve, none of the TMLE estimates lie outside the parameter space, demonstrating its advantage as a substitution estimator.

9 Conclusion

In this paper, we defined and identified parameters that transport stochastic direct and indirect mediating effects from a source population ($S=1$) to a new, target population ($S=0$). Identification of such parameters rely on the typical sequential randomization and positivity assumptions of other stochastic mediation effects (22, 27, 25) as well as a common outcome model assumption, described previously for transport estimators (20), which can be tested nonparametrically (12). Such parameters enable the prediction of mediating effects in new populations based on data about the mediation mechanism in a source population and the differing distributions of compositional characteristics between the two populations. Thus, transport SDE and SIE parameters contribute to understanding the how and why interventions may work differently and/or have differing effects when applied to new populations.

We proposed three estimation approaches for such effects: a stabilized IPTW estimator, an estimating equation approach, and a TMLE approach. For the EE and TMLE approaches, we describe a version of each estimator that is efficient under a statistical model with exclusion restrictions such that $A$ does not directly affect $M$ or $Y$, and another version that is efficient under a statistical model without those restrictions such that $A$ can directly affect $M$ and $Y$. The EE and TMLE estimators solve the EIC for a particular statistical model, which results in double robustness, meaning that they are unbiased if one either consistently estimates the $Y$ model or $Z,M,S$ models. The TMLE estimator has the additional advantage of staying within the bounds of the parameter space by virtue of being a substitution estimator. We demonstrated the finite sample advantage of staying within the parameter bounds in a simulation study; in a particularly challenging scenario, nearly all of the EE estimates were outside of the parameter space but the TMLE estimates stayed within bounds. The simulation study also demonstrated that, as expected, the stabilized IPTW estimator is less efficient than the EE and TMLE estimators under correct model specification and in many scenarios where models are misspecified.

We also saw empirical evidence that, even in cases where robustness properties guarantee estimator consistency, the EIC may no longer provide accurate inference when models are misspecified. If parametric models are used, as in our simulation, the bootstrap can be used to recover appropriate coverage in such scenarios. However, the bootstrap is not an appropriate strategy if data-adaptive methods are used in model fitting. An alternative approach is to target the nuisance parameters of the IC (e.g, $g_{M}$, $g_{Z}$, $g_{S}$, etc.) in addition to the parameter of interest in a TMLE (3). Applying such extra targeting the to TMLE estimators we describe here is an area for future work.

To facilitate use of these methods, we include commented R code for implementing them in the appendix.

The estimators we propose are limited in that they consider a stochastic intervention on mediator, $M$, that is assumed known and estimated from observed data. However, we plan to extend them to a true, unknown stochastic intervention in the future. Another limitation is that the parameters are only identified if one assumes a common outcome model across the source and target populations. There will be some research questions for which it is not possible to establish evidence for or against this assumption, as in questions about predicting a long-term outcome in a new population. However, when the research question instead focuses on establishing the extent to which mechanisms are shared across populations, and the full set of data $O=(S,W,A,Z,M,Y)$ is observed for both populations, one can empirically test whether there is evidence against such a shared outcome model (12).

In the main text, we focused on transporting mediation estimates where an instrument, $A$, was statically intervened on and mediator $M$ was stochastically intervened on. Moreover, we were primarily concerned with a statistical model that imposed instrumental variable assumptions such as the exclusion restriction assumption. However, we describe how each estimator can be easily modified to accommodate statistical models that do not impose instrumental variable assumptions, allowing for a direct effect of $A$ on $M$ and of $A$ on $Y$. Extending our proposed estimators for data generating mechanisms that do not include an intermediate variable, $Z$ is straightforward. Thus, our transport mediation estimators can be applied to a wide-range of common data generating mechanisms.

Appendix A Identifiability

To identify the stochastic direct effect and stochastic indirect effect we will need to impose additional assumptions on $\mathcal{M}^{F}$ and $\mathcal{M}$, listed below.

Positivity: For all $S$ and $W$ we need a positive probability of assigning any level of $A$. For all combinations of $S,W,$ and $A=a$, we have a positive probability of any level of $Z.$ For $S=1$ and all combinations of $Z$ and $W$ we need a positive probability of any level of the mediator, $M$. 2. 2.

Common outcome model across sites: $\mathbb{E}\left[Y\mid M,Z,W,S=1\right]=\mathbb{E}\left[Y\mid M,Z,W,S=0\right]$. The null hypothesis of a common outcome model may be tested nonparametrically (12). 3. 3.

Sequential Randomization: $Y_{am}\perp A\mid W,S$ and $Y_{am}\perp M\mid W,Z,S$. This is akin to a two-time point longitudinal intervention where at the first time point, we statically intervene to set the treatment, $A=a$, and at the second time point, we stochastically intervene on the mediator, $M$.

Theorem A.1**.**

[TABLE]

Proof:

[TABLE]

The conditional expectations are well-defined due to assumption 1 above.

Appendix B Estimator derivations

B.1 Stabilized inverse probability of treatment weighted estimator

This estimator solves the estimating equation $P_{n}\hat{H}_{n}(Y-\Psi_{n})=0$, where $\hat{H}_{n}(M,Z,A,W,S)$ is an estimate of the true $H$ given by

[TABLE]

To solve the estimating equation, we find an initial estimate, $\Psi_{n}^{0}=P_{n}\hat{H}_{n}Y$, i.e., the standard unstabilized IPTW estimator.

[TABLE]

where our stablilized estimator is then $\Psi_{n}^{IPTW}=\Psi_{n}^{1}=P_{n}\hat{H}_{n}Y/P_{n}\hat{H}_{n}$.

B.1.1 Efficiency

Theorem B.1**.**

Assume

A1: $\|H-\hat{H}_{n}\|_{L^{2}(P_{0})}=o_{P}(n^{-0.5})$ 2. 2.

A2: $\hat{H}_{n}$ and $\hat{D}_{n}$ are in a $P_{0}$ Donsker class.

Let $IC_{IPTW}$ be the true influence curve or $H(Y-\Psi(P_{0}))/P_{0}H=H(Y-\Psi(P_{0}))$, noting $P_{0}H=1$ for the true $H$. Then $\sqrt{n}(\Psi^{IPTW}_{n}-\Psi_{0}(P))$ converges in distribution to $\sqrt{n}$ times the empirical mean of the true influence curve. In this case, our standard error times $\sqrt{n}$ will converge to the standard deviation of $IC_{IPTW}$.

Proof:

[TABLE]

By A2 $\sqrt{n}(P_{n}-P_{0})\hat{D}_{n}$ converges in distribution to a normal distribution of mean 0 and variance equal to the true variance of $IC_{IPTW}$. Thus, we must show $\Psi_{n}^{0}+P_{0}\hat{D}_{n}-\Psi(P_{0})=o_{P}(1/\sqrt{n})$ under A1.

[TABLE]

We can notice in Equation 9 that the first term is bounded above by $C\|\hat{H}_{n}-H\|_{L^{2}(P_{0})}$ for some constant $C$. By A2, the second term is $o_{P}(1/\sqrt{n})$. By A1 the third term is also $o_{P}(1/\sqrt{n})$.

Theorem B.2**.**

The IPTW estimator’s limiting variance under the assumptions of theorem B.1 is larger than that of the TMLE or EE under well-specified models, if the model is non-parametric. If we use the TMLE and EE incorporating the efficient influence curve for the restricted model, then these are even more efficient.

Proof:

Let’s assume the unrestricted model, $\mathcal{M}^{np}$. We project the IPTW influence curve onto the tangent space subspace, $T_{Y\times S}$ constisting of $\{\gamma(O)\mid\mathbb{E}[\gamma(O)\mid M,Z,A,W,S]=0$ via the formula

[TABLE]

Likewise we have $\prod\left(IC_{IPTW}(O)\|T_{Z}\right)=\mathbb{E}\left[IC_{IPTW}(O)\mid Z,A,W,S\right]-\mathbb{E}\left[IC_{IPTW}(O)\mid A,W,S\right]=D_{Z}^{*}(O)$, where $T_{Z}$ is the tangent space consisting of $\{\gamma(O)\mid\mathbb{E}[\gamma(O)\mid A,W,S]=0$. The empirical mean of $\prod\left(IC_{IPTW}(O)\|T_{S}\right)=\mathbb{E}\left[IC_{IPTW}(O)\mid W\right]-\mathbb{E}\left[IC_{IPTW}(O)\mid W\right]$ = the empirical mean of $D_{W}^{*}(O)$, the $T_{S}$ component of the efficient influence curve for $\mathcal{M}^{np}$. We have non-trivial projections of $IC_{IPTW}$ onto $T_{M}$ and $T_{A}$. Since the asymptotic variance of $IC_{IPTW}(O)$ is a sum of the variances of its orthogonal components, it must exceed that of the EE and TMLE estimators, and the proof is complete.

B.1.2 Robustness

The IPTW estimator requires all models for $M$, $Z$, $A$ and $S$ to be estimated at parametric rates in order to be consistent.

B.2 Estimating equation estimator

B.2.1 Derivation of efficient influence curve for model without exclusion restrictions

First we derive the efficient influence curve (EIC) for the less restricted case where $A$ may directly affect $M$ and $Y$. For more information on deriving an efficient influence curve in general, we refer the reader to a tutorial by Levy, 2019 (11). We will also allow our treatment mechanism not to be known, which will have no effect on the derivation but means that the EIC is applicable in situations when treatment is not randomly assigned.

Theorem B.3**.**

Consider a non-parametric model or semiparametric model with one or both the treatment and mediator mechanisms known (mechanisms for $A$ and $M$). Consider the parameter defined by

[TABLE]

where the expectations are taken with respect to $P$. Then the efficient influence curve is given by

[TABLE]

where

[TABLE]

Proof: The density factorizes as follows:

$p(O)=p_{Y\times S}(Y\times\gamma\mid M,Z,A,W,S)p_{M}(M\mid Z,A,W,S)p_{Z}(Z\mid A,Z,W,S)p_{A}(A\mid W,S)p_{W\mid S}(W\mid S)p_{S}(S)$

This means

$p_{\epsilon}(O)=p_{Y\times S,\epsilon}(Y\times S\mid M,Z,A,W,S)p_{M,\epsilon}(M\mid Z,A,W,S)p_{Z,\epsilon}(Z\mid A,W,S)p_{A,\epsilon}(A\mid W,S)p_{W\mid S,\epsilon}(W\mid S)p_{S,\epsilon}(S)$

[TABLE]

Our estimand, which identifies our parameter of interest (see Section 3 of the main text) is given by

[TABLE]

We then take the pathwise derivative for a path along score, $\gamma$. We note that this derivative is unaffected by knowledge of the treatment mechanism, $E[A\mid S,W]$, or the mediator mechansim, $E[M\mid Z,A,W,S]$, due to the estimand not depending on these models as well as the fact that scores, $\gamma_{A}$ and $\gamma_{M}$ are orthogonal (have 0 covariance) to $\gamma_{Y},\gamma_{Z},\gamma_{W})$ in the Hilbert Space $L^{2}(P)$. This is why for a semi-parametric model where the treatment mechanism is known, the efficient influence curve will be the same as that for $\mathcal{M}^{np}$.

[TABLE]

The first term in 13:

[TABLE]

where

[TABLE]

The reader may notice $D^{*}_{Y}(P)(O)$ is not a mean 0 function of $Y\mid M,Z,W$ because it also depends on the variable, $A$. Hence, it is not an element of the tangent space under the restricted model, $\mathcal{M}$, that incorporates the instrumental variable exclusion restrictions as described in Section 2 of the main text, which is necessarily a proper subspace of $L^{2}(P)$ for $P\in\mathcal{M}^{np}$. Therefore, $D^{*}(P)(O)$ has an extra orthogonal component in addition to the efficient influence curve under $\mathcal{M}$, making its variance necessarily bigger than the lower bound under model $\mathcal{M}$. This is why the estimators using $D^{*}(P)(O)$ (the EE and TMLE estimators) are not asymptotically efficient for model, $\mathcal{M}$ but are so for $\mathcal{M}^{np}$.

The second term in 13:

[TABLE]

We substitute

[TABLE]

and since $x$ represents the treatment, $A$, in the integrals above, we get

$D_{Z}^{*}(P)(O)=\left(\bar{Q}_{M}(Z,A,W)-\bar{Q}_{Z}(A,W,S)\right)\frac{I(S=0,A=a)}{p_{A}\left(A\mid W,S\right)p_{S}(S=0)}.$

The third term in 13:

[TABLE]

where $D_{W}^{*}(P)(O)=\left(\bar{Q}_{Z}(A=a,W,S)-\Psi(P)\right)\frac{I(S=0)}{p_{S}(S=0)}$

Thus the efficient influence curve is the sum of its orthogonal components:

[TABLE]

B.2.2 Derivation of efficient influence curve for model with exclusion restrictions

Next, we derive the EIC for the model that restricts $A$ to affect $M$ and $Y$ only through $Z$. Similar to the above subsection, we allow our treatment mechanism not to be known.

Theorem B.4**.**

The efficient influence curve for our restricted model, where $M$ and $Y$ do not depend directly on $A$, is given by

[TABLE]

where

[TABLE]

Proof:

We can note that our only task here is to project $D^{*}_{Y}(P)$, our component of the influence curve in $T_{Y}$, onto the subspace of $T_{Y}$ given by

$T_{Y,r}=\overline{\{\gamma:\mathbb{E}(\gamma(O)\mid YS,M,Z,W,S)=0,\mathbb{E}\gamma(O)^{2}<\infty\}}$.

$p_{YS,r}$ is the conditional density of $ys$ given $m,z,w$ and $p_{M,r}$ is the conditional density of $m$ given $z,w,s$ in the restricted model, i.e. we don’t put the instrument, $a$, in those conditional statements as that is the model assumption. We remind the reader that a "bar" signifies the variable and all past variables as in, $\bar{M}=m,z,x,w,s$.

Notice the following:

[TABLE]

[TABLE]

Thus from 14 and 15

[TABLE]

And the proof is complete since the other components of the unrestricted model’s influence curve will remain the same. The reader may note that $p_{Z\mid W,S}(z\mid w,s)=p_{Z}(z\mid 1,w,s)g_{A}(1\mid w,s)+p_{Z}(z\mid 0,w,s)g_{A}(0\mid w,s)$, so we need not perform any additional regressions for this restricted model.

B.2.3 Robustness

Here we derive the remainder term for the EE estimator. We first form an initial estimate

[TABLE]

.

Then we update this estimate by adding the empirical mean of the approximated influence curve. If we call our approximated influence curve, $\hat{D}_{n}^{EE}$ our estimating equation (EE) estimate is given by

[TABLE]

This then leads to a second order expansion

[TABLE]

where $R_{2}(P_{n},P_{0})=\hat{\Psi}_{n}^{0}-\Psi(P_{0})+P_{0}\hat{D}_{n}^{EE}(O)$.

The behavior of $R_{2}$ determines the robustness of our estimator and conditions under which we can guarantee consistency and efficiency (9, 8).

Theorem B.5**.**

[TABLE]

Where we substituted the following: $f_{1,0}(o)=g_{M,0}(m\mid m,z,w)$, $f_{2,0}=p_{Z,0}(m\mid a,w,1)$, $f_{3,0}=p_{Z,0}(m\mid a,w,0)$, $f_{4,0}=p_{A,0}(x=a\mid w,1)$, $f_{5,0}=p_{A,0}(x=a\mid w,0)$, $f_{6,0}=p_{S,0}(s=1\mid w)$ and $f_{7,0}=p_{S,0}(s=0\mid w)$ and dropping the subscript, 0, indicates the estimated counterpart. $h_{i}$ is a bounded function by the positivity assumption (see section 4) and thus the last inequality holds with a sufficiently large $k$.

[TABLE]

The treatment mechanism (model for A) being well-specified makes term 4 disappear above. Clearly terms 3 and 5 only cancel if the models for M, Z and S also are well-specified.

Integrating terms 3 and 5, we get

[TABLE]

If we add fractions and put everything in one integral we have:

[TABLE]

$h_{i}$ is a bounded function by the positivity assumption (see section 4) and thus the last inequality holds with a sufficiently large $k$, completing the proof

We conclude that a product of $L^{2}(P_{0})$ norms between the bias in estimating the outcome model and the bias of the regression models for $M\mid Z,W,S$, $Z\mid A,W,S$, $A\mid W,S$ and $S\mid W$ must be such that the product converges to 0 in probability when multiplied by $\sqrt{n}$. Also, $\sqrt{n}\|\bar{Q}_{L_{1}}-\bar{Q}_{L_{1},0}\|_{L^{2}{P_{0}}}\|p_{A}-p_{A,0}\|_{L^{2}{P_{0}}}$ must also converge to 0 in probability for the TMLE to be consistent and asymptotically linear with limiting variance that of the TMLE influence curve (efficient for the broader model including dependence of $Y$ and $M$ on $A$). Such conditions are guaranteed asymptotically when using the highly adaptive lasso to fit the regressions if the true regressions are of finite sectional variation norm and are left-hand continuous with right-hand limits (7).

To summarize, if the outcome model is misspecified then we need to correctly specify all the models $M\mid Z,W,S$, $Z\mid A,W,S$, $A\mid W,S$ and $S\mid W$ for the EE estimator to be consistent and asymptotically efficient. If we specify the outcome model correctly so that $\sqrt{n}\|\bar{Q}_{Y}-\bar{Q}_{Y,0}\|_{L^{2}(P_{0})}$ converges to 0 in probability, then clearly term 1 disappears by the Cauchy-Schwarz inequality, and we have to consider term 2, which disappears if $\sqrt{n}\|\bar{Q}_{L_{1}}-\bar{Q}_{L_{1},0}\|_{L^{2}{P_{0}}}\|p_{A}-p_{A,0}\|_{L^{2}{P_{0}}}$ converges to 0 in probability. Thus if we correctly specify the regression, $\bar{Q}_{L_{1}}(Z,W)\mid A,W,S$, along with the regression for the outcome, then we need not specify any other model correctly to obtain a consistent estimator. If we correctly specify the outcome and misspecify the second regression, $\bar{Q}_{L_{1}}\mid A,W,S$, then we need to correctly specify the treatment mechanism in order to be consistent.

B.3 Targeted minimum loss-based estimator

B.3.1 Derivation of efficient influence curve

The derivations of the EICs for the less restricted and more restricted models are given in subsections B.2.1 and B.2.2.

B.3.2 Robustness

For our TMLE estimator: $\Psi(P_{n}^{*})-\Psi(P_{0})=(P_{n}-P_{0})D^{*}(P_{n}^{*})+R_{2}(P_{n}^{*},P_{0})$ where $P_{0}$ is the true observed data generating distribution and $P_{n}^{*}$ is the TMLE updated estimation of $P_{0}$. Since the empirical mean of $D^{*}(P_{n}^{*})=0$ by virtue of the TMLE algorithm, we have $R_{2}(P_{n}^{*},P_{0})=\Psi(P_{n}^{*})-\Psi(P_{0})+P_{0}D^{*}(P_{n}^{*})$. The robustness properties of the TMLE estimator are identical to the EE estimator, see Subsection B.2.3.

Appendix C Simulation

DGM 1 is intended to break when Y and M models are misspecified, especially. DGM 2 is intended to break when Y and Z models are misspecified, especially. DGM 3 is intended to break when Y and S models are misspecified, especially.