Nonparametric estimation of natural direct and indirect effects based on inverse probability weighting

Yu-Chin Hsu; Martin Huber; Tsung-Chih Lai

doi:10.1515/jem-2017-0016

Published by De Gruyter June 13, 2018

Nonparametric estimation of natural direct and indirect effects based on inverse probability weighting

Yu-Chin Hsu , Martin Huber and Tsung-Chih Lai

From the journal Journal of Econometric Methods

https://doi.org/10.1515/jem-2017-0016

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Abstract

Using a sequential conditional independence assumption, this paper discusses fully nonparametric estimation of natural direct and indirect causal effects in causal mediation analysis based on inverse probability weighting. We propose estimators of the average indirect effect of a binary treatment, which operates through intermediate variables (or mediators) on the causal path between the treatment and the outcome, as well as the unmediated direct effect. In a first step, treatment propensity scores given the mediator and observed covariates or given covariates alone are estimated by nonparametric series logit estimation. In a second step, they are used to reweigh observations in order to estimate the effects of interest. We establish root-n consistency and asymptotic normality of this approach as well as a weighted version thereof. The latter allows evaluating effects on specific subgroups like the treated, for which we derive the asymptotic properties under estimated propensity scores. We also provide a simulation study and an application to an information intervention about male circumcisions.

Keywords: causal channels; causal mechanisms; causal pathways; direct effects; indirect effects; inverse probability weighting; mediation analysis; nonparametric estimation; propensity score; series logit estimation

JEL Classification: C21

A Appendix

A.1 Proof of Theorem 1

Define Y_dd′ = Y(d, M(d′)) and μ_dd′ = E[Y_dd′] for d, d′ ∈ {0, 1}. By Assumption 1 and Assumption 2,

μ 11 = E [ D Y p ( X ) ] , μ 00 = E [ ( 1 − D ) Y 1 − p ( X ) ] , μ 10 = E [ D Y p ( M , X ) 1 − p ( M , X ) 1 − p ( X ) ] , μ 01 = E [ ( 1 − D ) Y 1 − p ( M , X ) p ( M , X ) p ( X ) ] ,

see equations (4) and (5) of Huber (2014). This allows defining the direct and indirect effects of interest in terms of μ_dd′, e.g. θ(1) = μ₁₁ − μ₀₁. We estimate μ_dd′ for d, d′ ∈ {0, 1} by the normalized sample analogs

μ ^ 11 = 1 n ∑ i = 1 n D i Y i p ^ ( X i ) / 1 n ∑ i = 1 n D i p ^ ( X i ) , μ ^ 00 = 1 n ∑ i = 1 n ( 1 − D i ) Y i 1 − p ^ ( X i ) / 1 n ∑ i = 1 n ( 1 − D i ) 1 − p ^ ( X i ) , μ ^ 10 = 1 n ∑ i = 1 n D i Y i p ^ ( M i , X i ) 1 − p ^ ( M i , X i ) 1 − p ^ ( X i ) / 1 n ∑ i = 1 n D i p ^ ( M i , X i ) 1 − p ^ ( M i , X i ) 1 − p ^ ( X i ) , μ ^ 01 = 1 n ∑ i = 1 n ( 1 − D i ) Y i 1 − p ^ ( M i , X i ) p ^ ( M i , X i ) p ^ ( X i ) / 1 n ∑ i = 1 n ( 1 − D i ) 1 − p ^ ( M i , X i ) p ^ ( M i , X i ) p ^ ( X i ) .

To prove Theorem 1, it is thus sufficient to show that for d, d′ ∈ {0, 1},

n ( μ ^ d d ′ − μ d d ′ ) = 1 n ∑ i = 1 n ψ d d ′ ( Y i , M i , D i , X i ) + o p ( 1 ) .

As the results regarding μ₁₁ and μ₀₀ have been established by HIR, we subsequently focus on the proof for μ₁₀ and note that the derivations for μ₀₁ proceed in an analogous way. Let μ~10 be the numerator of μ^10 and ω~10 be the denominator of μ^10. We first show that

n ( μ ~ 10 − μ 10 ) = 1 n ∑ i = 1 n ψ 10 ( Y i , M i , D i , X i ) + o p ( 1 ) ,

using a similar approach as HIR. Note that

n ( μ ~ 10 − μ 10 ) = 1 n ∑ i = 1 n { D i Y i p ^ ( M i , X i ) 1 − p ^ ( M i , X i ) 1 − p ^ ( X i ) − μ 10 } = 1 n ∑ i = 1 n { D i Y i p ( M i , X i ) 1 − p ( M i , X i ) 1 − p ( X i ) − μ 10 } + 1 n ∑ i = 1 n { D i Y i p ( M i , X i ) 1 − p ( M i , X i ) ( 1 − p ( X i ) ) 2 ( p ^ ( X i ) − p ( X i ) ) } + 1 n ∑ i = 1 n { − D i Y i p 2 ( M i , X i ) 1 1 − p ( X i ) ( p ^ ( M i , X i ) − p ( M i , X i ) ) } + o p ( 1 )

where the second equality holds by a second order mean valued expansion around p(X_i) and p(M_i, X_i) and the fact that supx∈X|p^(x)−p(x)|=op(n−1/4) and supm∈M,x∈X|p^(m,x)−p(m,x)|=op(n−1/4). The converge rate of supx∈X|p^(x)−p(x)|=Op(Kx(Kx/n+K−p¯x/2dx)) is derived in Lemma A3 of Hsu (2017) and the conditions given in this paper are sufficient for supx∈X|p^(x)−p(x)|=op(n−1/4). By the same argument as in Theorem 1 of HIR, we have

1 n ∑ i = 1 n { D i Y i p ( M i , X i ) 1 − p ( M i , X i ) ( 1 − p ( X i ) ) 2 ( p ^ ( X i ) − p ( X i ) ) } = 1 n ∑ i = 1 n { E [ D i Y i p ( M i , X i ) 1 − p ( M i , X i ) ( 1 − p ( X i ) ) 2 | X i ] ( D i − p ( X i ) ) } + o p ( 1 ) = 1 n ∑ i = 1 n { ρ 10 ( X i ) ( 1 − p ( X i ) ) ( D i − p ( X i ) ) } + o p ( 1 ) .

Similarly, it holds that

1 n ∑ i = 1 n { − D i Y i p 2 ( M i , X i ) 1 1 − p ( X i ) ( p ^ ( M i , X i ) − p ( M i , X i ) ) } = 1 n ∑ i = 1 n { E [ − D i Y i p 2 ( M i , X i ) 1 1 − p ( X i ) | M i , X i ] ( D i − p ( M i , X i ) ) } + o p ( 1 ) = 1 n ∑ i = 1 n { − ζ 1 ( M i , X i ) p ( M i , X i ) 1 1 − p ( X i ) ( D i − p ( M i , X i ) ) } + o p ( 1 ) .

Next, one can easily show that

n ( w ~ 10 − 1 ) = o p ( 1 ) ,

by replacing Y_i’s with 1’s in the proof for μ~₁₀ and acknowledging that the influence functions for w~₁₀ are all zero. Finally, it follows that

n ( μ ^ 10 − μ 10 ) = n ( μ ~ 10 w ~ 10 − μ 10 ) = n ( μ ~ 10 − μ 10 ) − μ 10 n ( w ~ 10 − 1 ) + o p ( 1 ) = n ( μ ~ 10 − μ 10 ) + o p ( 1 ) ,

where the second equality follows by a mean-value expansion and the last equality holds by the fact that μ₁₀ is bounded and that n(w~10−1)=op(1). This completes the proof. □

A.2 Proof of Theorem 2

Define μ_dd′,g = E[g(X)Y_dd′]/E[g(X)] for d, d′ ∈ {0, 1}. Similarly,

μ 11 , g = E [ g ( X ) D Y p ( X ) ] E [ g ( X ) ] , μ 00 , g = E [ g ( X ) ( 1 − D ) Y 1 − p ( X ) ] E [ g ( X ) ] , μ 10 , g = E [ g ( X ) D Y p ( M , X ) 1 − p ( M , X ) 1 − p ( X ) ] E [ g ( X ) ] , μ 01 , g = E [ g ( X ) ( 1 − D ) Y 1 − p ( M , X ) p ( M , X ) p ( X ) ] E [ g ( X ) ] .

We estimate μ_dd′,g for d, d′ ∈ {0, 1} by the normalized sample analogs

μ ^ 11 , g = 1 n ∑ i = 1 n g ( X i ) D i Y i p ^ ( X i ) / 1 n ∑ i = 1 n g ( X i ) D i p ^ ( X i ) , μ ^ 00 , g = 1 n ∑ i = 1 n g ( X i ) ( 1 − D i ) Y i 1 − p ^ ( X i ) / 1 n ∑ i = 1 n g ( X i ) ( 1 − D i ) 1 − p ^ ( X i ) , μ ^ 10 , g = 1 n ∑ i = 1 n g ( X i ) D i Y i p ^ ( M i , X i ) 1 − p ^ ( M i , X i ) 1 − p ^ ( X i ) / 1 n ∑ i = 1 n g ( X i ) D i p ^ ( M i , X i ) 1 − p ^ ( M i , X i ) 1 − p ^ ( X i ) , μ ^ 01 , g = 1 n ∑ i = 1 n g ( X i ) ( 1 − D i ) Y i 1 − p ^ ( M i , X i ) p ^ ( M i , X i ) p ^ ( X i ) / 1 n ∑ i = 1 n g ( X i ) ( 1 − D i ) 1 − p ^ ( M i , X i ) p ^ ( M i , X i ) p ^ ( X i ) .

Define P_g = E[g(X)]. We would like to show that n(μ^11,g−μ11,g)=n−1/2∑i=1nψμ11,g(Yi,Mi,Di,Xi)+op(1). First, we can be demonstrated that

n ( 1 n ∑ i = 1 n g ( X i ) D i Y i p ^ ( X i ) − μ 11 , g ⋅ P g ) = 1 n ∑ i = 1 n ( g ( X i ) D i Y i p ( X i ) − g ( X i ) ρ 11 ( X i ) p ( X i ) ( D i − p ( X i ) ) − μ 11 , g ⋅ P g ) + o p ( 1 ) , and n ( 1 n ∑ i = 1 n g ( X i ) D i p ^ ( X i ) − μ 11 , g ⋅ P g ) = 1 n ∑ i = 1 n ( g ( X i ) D i p ( X i ) − g ( X i ) p ( X i ) ( D i − p ( X i ) ) − P g ) + o p ( 1 ) = 1 n ∑ i = 1 n ( g ( X i ) − P g ) + o p ( 1 ) .

Then by delta method, we have

n ( μ ^ 11 , g − μ 11 , g ) = 1 P g 1 n ∑ i = 1 n ( g ( X i ) D i Y i p ( X i ) − g ( X i ) μ 1 ( X i ) p ( X i ) ( D i − p ( X i ) ) − μ 11 , g ⋅ P g ) − μ 11 , g P g ( g ( X i ) − P g ) + o p ( 1 ) = 1 n ∑ i = 1 n g ( X i ) E [ g ( X ) ] ( D i Y i p ( X i ) − ρ 11 ( X i ) p ( X i ) ( D i − p ( X i ) ) − μ 11 , g ) + o p ( 1 ) .

This gives the result for the μ_11,g case. Using the same arguments, we can derive the results for μ_00,g, μ_01,g and μ_10,g, too, which is sufficient to prove Theorem 2. □

A.3 Proof of Theorem 3

Define μ_dd′,t = E[p(X)Y_dd′]/E[p(X)] for d, d′ ∈ {0, 1}. Similarly,

μ 11 , t = E [ p ( X ) D Y p ( X ) ] E [ p ( X ) ] , μ 00 , t = E [ p ( X ) ( 1 − D ) Y 1 − p ( X ) ] E [ p ( X ) ] , μ 10 , t = E [ p ( X ) D Y p ( M , X ) 1 − p ( M , X ) 1 − p ( X ) ] E [ p ( X ) ] , μ 01 , t = E [ p ( X ) ( 1 − D ) Y 1 − p ( M , X ) p ( M , X ) p ( X ) ] E [ p ( X ) ] .

We estimate μ_dd′,t for d, d′ ∈ {0, 1} by the normalized sample analogs

μ ^ 11 , t = 1 n ∑ i = 1 n D i Y i / 1 n ∑ i = 1 n D i , μ ^ 00 , t = 1 n ∑ i = 1 n p ^ ( X i ) ( 1 − D i ) Y i 1 − p ^ ( X i ) / 1 n ∑ i = 1 n p ^ ( X i ) ( 1 − D i ) 1 − p ^ ( X i ) , μ ^ 10 , t = 1 n ∑ i = 1 n p ^ ( X i ) D i Y i p ^ ( M i , X i ) 1 − p ^ ( M i , X i ) 1 − p ^ ( X i ) / 1 n ∑ i = 1 n p ^ ( X i ) D i p ^ ( M i , X i ) 1 − p ^ ( M i , X i ) 1 − p ^ ( X i ) , μ ^ 01 , t = 1 n ∑ i = 1 n ( 1 − D i ) Y i 1 − p ^ ( M i , X i ) p ^ ( M i , X i ) / 1 n ∑ i = 1 n ( 1 − D i ) 1 − p ^ ( M i , X i ) p ^ ( M i , X i ) .

Define P_t = E[p(X)]. Note that we have

n ( 1 n ∑ i = 1 n D i Y i − μ 11 , t P t ) = 1 n ∑ i = 1 n ( D i Y i − μ 11 , t P t ) , n ( 1 n ∑ i = 1 n p ^ ( X i ) ( 1 − D i ) Y i 1 − p ^ ( X i ) − μ 00 , t P t ) = 1 n ∑ i = 1 n ( p ( X i ) ( 1 − D i ) Y i 1 − p ( X i ) + μ 0 ( X i ) 1 − p ( X i ) ( D i − p ( X i ) ) − μ 00 , t P t ) + o p ( 1 ) , n ( 1 n ∑ i = 1 n p ^ ( X i ) D i Y i p ^ ( M i , X i ) 1 − p ^ ( M i , X i ) 1 − p ^ ( X i ) − μ 10 , t P t ) = 1 n ∑ i = 1 n ( p ( X i ) D i Y i p ( M i , X i ) 1 − p ( M i , X i ) 1 − p ( X i ) + ρ 10 ( X i ) 1 − p ( X i ) ( D i − p ( X i ) ) − p ( X i ) ζ 1 ( M i , X i ) p ( M i , X i ) ( 1 − p ( X i ) ) ( D i − p ( M i , X i ) ) − μ 10 , t P t ) + o p ( 1 ) , and n ( 1 n ∑ i = 1 n ( 1 − D i ) Y i 1 − p ^ ( M i , X i ) p ^ ( M i , X i ) − μ 01 , t P t ) = 1 n ∑ i = 1 n ( ( 1 − D i ) Y i 1 − p ( M i , X i ) p ( M i , X i ) + ζ 0 ( M i , X i ) ( 1 − p ( M i , X i ) ) ( D i − p ( M i , X i ) ) − μ 01 , t P t ) + o p ( 1 ) .

Furthermore,

n ( 1 n ∑ i = 1 n D i − P t ) = 1 n ∑ i = 1 n ( D i − P t ) + o p ( 1 ) , n ( 1 n ∑ i = 1 n p ^ ( X i ) ( 1 − D i ) Y i 1 − p ^ ( X i ) − P t ) = 1 n ∑ i = 1 n ( D i − P t ) + o p ( 1 ) n ( 1 n ∑ i = 1 n p ^ ( X i ) D i p ^ ( M i , X i ) 1 − p ^ ( M i , X i ) 1 − p ^ ( X i ) − P t ) = 1 n ∑ i = 1 n ( D i − P t ) + o p ( 1 ) , and n ( 1 n ∑ i = 1 n ( 1 − D i ) 1 − p ^ ( M i , X i ) p ^ ( M i , X i ) − P t ) = 1 n ∑ i = 1 n ( D i − P t ) + o p ( 1 ) .

By delta method, we have

n ( μ ^ 11 , t − μ 11 , t ) = 1 n ∑ i = 1 n 1 P t ( D i Y i − μ 11 , t P t − μ 11 , t ( D i − P t ) ) + o p ( 1 ) = 1 n ∑ i = 1 n 1 P t ( D i ( Y i − μ 11 , t ) ) + o p ( 1 ) .

Similarly,

n ( μ ^ 00 , t − μ 00 , t ) = 1 n ∑ i = 1 n 1 P t ( p ( X i ) ( 1 − D i ) Y i 1 − p ( X i ) + ρ 00 ( X i ) 1 − p ( X i ) ( D i − p ( X i ) ) − μ 00 , t P t − μ 00 , t ( D i − P t ) ) + o p ( 1 ) = 1 n ∑ i = 1 n 1 P t ( p ( X i ) ( 1 − D i ) ( Y i − ρ 00 ( X i ) ) 1 − p ( X i ) + ( ρ 00 ( X i ) − μ 00 , t ) D i ) + o p ( 1 ) .

Next,

n ( μ ^ 10 , t − μ 10 , t ) = 1 n ∑ i = 1 n 1 P t ( p ( X i ) D i Y i p ( M i , X i ) 1 − p ( M i , X i ) 1 − p ( X i ) + ρ 10 ( X i ) ( 1 − p ( X i ) ) ( D i − p ( X i ) ) − p ( X i ) ζ 1 ( M i , X i ) p ( M i , X i ) ( 1 − p ( X i ) ) ( D i − p ( M i , X i ) ) − μ 10 , t D i ) + o p ( 1 ) , n ( μ ^ 01 , t − μ 01 , t ) = 1 n ∑ i = 1 n 1 P t ( ( 1 − D i ) Y i 1 − p ( M i , X i ) p ( M i , X i ) + ζ 0 ( M i , X i ) ( 1 − p ( M i , X i ) ) ( D i − p ( M i , X i ) ) − μ 01 , t D i ) + o p ( 1 ) .

This completes the proof. □

A.4 Effects of Circumcision on the Treated

Table 4:

Direct and indirect effects among treated.

	θ(1)			θ(0)			δ(1)			δ(0)
	est	se	pval	est	se	pval	est	se	pval	est	se	pval
ipw cv	0.04	0.02	0.13	0.03	0.03	0.19	0.01	0.01	0.24	0.01	0.01	0.29
ipw ofit	0.03	0.03	0.34	0.01	0.03	0.74	0.02	0.01	0.04	0.00	0.01	0.78
semi ipw	0.04	0.02	0.13	0.03	0.03	0.19	0.01	0.01	0.24	0.01	0.01	0.28

‘est’, ‘se’, and ‘pval’ denote the effect estimate, the standard error, and the p-value, respectively. ‘ipw cv’, ‘ipw ofit’, and ‘semi ipw’ denote IPW using SLE based on cross-validation, IPW using SLE based on overfitting, and semiparametric IPW using probit, respectively. Standard errors and/or p-values are based on 1999 bootstrap replications.

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Published Online: 2018-06-13

Nonparametric estimation of natural direct and indirect effects based on inverse probability weighting

Abstract

A Appendix

A.1 Proof of Theorem 1

A.2 Proof of Theorem 2

A.3 Proof of Theorem 3

A.4 Effects of Circumcision on the Treated

References

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