A semiparametric multiply robust multiple imputation method for causal inference

Abstract

Evaluating the impact of non-randomized treatment on various health outcomes is difficult in observational studies because of the presence of covariates that may affect both the treatment or exposure received and the outcome of interest. In the present study, we develop a semiparametric multiply robust multiple imputation method for estimating average treatment effects in such studies. Our method combines information from multiple propensity score models and outcome regression models, and is multiply robust in that it produces consistent estimators for the average causal effects if at least one of the models is correctly specified. Our proposed estimators show promising performances even with incorrect models. Compared with existing fully parametric approaches, our proposed method is more robust against model misspecifications. Compared with fully non-parametric approaches, our proposed method does not have the problem of curse of dimensionality and achieves dimension reduction by combining information from multiple models. In addition, it is less sensitive to the extreme propensity score estimates compared with inverse propensity score weighted estimators and augmented estimators. The asymptotic properties of our method are developed and the simulation study shows the advantages of our proposed method compared with some existing methods in terms of balancing efficiency, bias, and coverage probability. Rubin’s variance estimation formula can be used for estimating the variance of our proposed estimators. Finally, we apply our method to 2009–2010 National Health Nutrition and Examination Survey to examine the effect of exposure to perfluoroalkyl acids on kidney function.

Appendix

1.1 A: Regularity conditions

Before providing sketched proofs of Theorems 1 and 2, we will first provide some necessary regularity conditions.

Let $\hat{Z}_{i}^{(g)}=\left( \hat{Z}_{1i}^{(g)},\hat{Z}_{2i}^{(g)}\right)$ where $\hat{Z}_{1i}^{(g)}=\hat{m}_{g,i}$ and $\hat{Z}_{2i}^{(g)}=\hat{p}_{g,i}$. Next, let $\beta _{0g}^{*(k)}$,$\alpha ^{*(j)}$, and $\eta _{m_g}^{*}$ be the probability limits for the estimators $\hat{\beta }_{0g}^{(k)}$, $\;\hat{\alpha }^{(j)}$, and $\;\hat{\eta }_{m_g}$, respectively.

Denote

$$\begin{aligned} Z_i^{*(g)}=(Z_{1i}^{*(g)},Z_{2i}^{*(g)}) \end{aligned}$$

with

$$\begin{aligned} Z_{1i}^{*(g)}=\hat{U}_{m_{g},i}^{*\top }\; \frac{\eta _{m_g}^{*2}}{\eta ^{*T}_{p}\eta ^{*}_{m_1}} ,\;\; Z_{2i}^{*(1)}=\hat{U}_{p,i}^{*\top }\; \frac{\eta _{p}^{*2}}{\eta ^{*T}_{p}\eta ^{*}_{p}},\;\text {and}\;\; Z_{2i}^{*(0)}=1-\left( \hat{U}_{p,i}^{*\top }\; \frac{\eta _{p}^{*2}}{\eta ^{*T}_{p}\eta ^{*}_{p}}\right) \end{aligned}$$

where

$$\begin{aligned} \hat{U}_{m_{g},i}^{*}=\{m_g^{(1)}(X_i;\beta _{0g}^{*(1)}),...,m_g^{(K)}(X_i;\beta _{0g}^{*(K)})\}^\top \end{aligned}$$

and

$$\begin{aligned} \hat{U}_{p,i}^*=\{p^{(1)}(X_i;\alpha ^{*(1)}),...,p^{(J)}(X_i;\alpha ^{*(J)})\}^\top . \end{aligned}$$

Finally, let $f_g(Z^{*(g)})$ be the density function of $Z^{*(g)}$. We assume the following regularity conditions necessary for proving Theorem 1 and Theorem 2. Conditions (C1) and (C2) apply to each propensity score model, $j=1,...,J$, and each pair of outcome regression models, $k=1,...,K$.

• (C1) $\hat{\alpha }^{(j)}$ is the unique solution of $S_p^{(j)}(\alpha ^{(j)})=0$ and $\;\hat{\beta }_{0g}^{(k)}$ is the unique solution of $S_{m_g}^{(k)}(\beta _{0g}^{(k)})=0$ where $S_p^{(j)}(\alpha ^{(j)})$ and $S_{m_g}^{(k)}$ are as defined in Sect. 3.

• (C2) $S_p^{(j)}(\alpha ^{(j)})$ converges almost surely to $S_p^{*(j)}(\alpha ^{(j)})=E\{S_p^{(j)}(\alpha ^{(j)})\}$, uniformly in $\alpha ^{(j)}$, and $S_p^{*(j)}(\alpha ^{(j)})=0$ has a unique solution $\alpha ^{*(j)}$. Also, $S_{m_g}^{(k)}(\beta _{0g}^{(k)})$ converges almost surely to $S_{m_g}^{*(k)}(\beta _{0g}^{(k)})=E\{S_{m_g}^{(k)}(\beta _{0g}^{(k)})\}$, uniformly in $\beta _{0g}^{(k)}$, and $S_{m_g}^{*(k)}(\beta _{0g}^{(k)})=0$ has a unique solution in $\beta _{0g}^{*(k)}$.

• (C3) $E(Y_g^2)<\infty$ and $E\{\mu _g^{2}(Z^{*(g)})\}<\infty$, where $\mu _g^{2}(Z^{*(g)})=E(Y_g\vert Z^{*(g)})$.

• (C4) $H/n=o(1)$ and $\text {log}(n)/H=o(1)$.

• (C5) $f_g(Z^{*(g)})$ and $\pi _g(Z^{*(g)})$ are continuous and bounded away from 0 in the compact support of $Z^{*(g)}$.

The consistency of $\hat{\alpha }^{(j)}$ and $\hat{\beta }_{0g}^{(k)}$ is ensured by Conditions (C1) and (C2). These conditions are satisfied for most linear (and generalized linear) models. Condition (C3) is useful for deriving the asymptotic expansion and normality of $\hat{\tau }_{MRMI}$. Condition (C4) is used to control the asymptotic order of H. Condition (C5), a common condition in nonparametric statistics, helps avoid extreme values of the propensity and density scores.

1.2 B: Sketched proof of Theorem 1

Let $\hat{\alpha }=(\hat{\alpha }^{(1)},...,\hat{\alpha }^{(J)})$, $\hat{\beta }_{0g}=(\hat{\beta }_{0g}^{(1)},...,\hat{\beta }_{0g}^{(K)})$, $\hat{\alpha }^*=(\hat{\alpha }^{*(1)},...,\hat{\alpha }^{*(J)})$, and $\hat{\beta }^*_{0g}=(\hat{\beta }_{0g}^{*(1)},...,\hat{\beta }_{0g}^{*(K)})$. According to (C1), (C2), and Van der Vaart (2000), it can be shown that $\hat{\alpha }\rightarrow ^{p}\alpha ^*$ and $\hat{\beta }_{0g}\rightarrow ^{p}\beta _{0g}^{*}$.

Assume that one of the pairs of outcome regression models is correct, say $m_1^{(1)}(X_i;\beta _{01}^{(1)})$ and $m_0^{(1)}(X_i;\beta _{00}^{(0)})$. Then, we have $\beta _{0g}^{*(1)}=\beta _{0g}$ and $\;\eta _{m_g}^{*}=(1,0,...,0)^{T}$, which implies that $Z_{1i}^{*(g)}=m_g(X_i;\beta _{0g})$.

It follows that

$$\begin{aligned} \begin{aligned} E\left( Y_1\vert T,Z^{*(1)}\right)&= E\left\{ E\left( Y_1\vert T,Z^{*(1)},X\right) \vert T,Z^{*(1)}\right\} \\&=E\left\{ E\left( Y_1\vert Z^{*(1)},X\right) \vert T,Z^{*(1)}\right\} \\&=E\left( Z_1^{*(1)}\vert T,Z^{*(1)}\right) =Z_1^{*(1)}=E\left( Y_1\vert Z^{*(1)}\right) \end{aligned} \end{aligned}$$

If one of the propensity score models is correctly specified, say $p_1^{(1)}(X_i;\alpha ^{(1)})$, then $\alpha ^{*(1)}=\alpha$,$\;\eta _p^*=(1,0,...,0)^\top$ and $\;Z_{2i}^{*(1)}=p_1(X_i;\alpha )$. Therefore,

$$\begin{aligned} E\left( Y_1 \vert T,Z^{*(1)}\right) =E\left( Y_1\vert Z^{*(1)}\right) \end{aligned}$$

due to the fact that $Y_1$ is independent of T given $Z_2^{*(1)}$.

According to Devroye and Wagner (1977) and Silverman (1978), it can be shown that

$$\begin{aligned} \sum _{j \in R_H^{(1)}(i)} \frac{1}{H}Y_{1j}\rightarrow ^{p}E\left( Y_{1i} \vert Z_i^{*(1)},T_i=1\right) \; \; \; \; \; \end{aligned}$$

(B.1)

and

$$\begin{aligned} \frac{1}{H}\sum _{j \in R_H^{(1)}(i)}(Y_{1j}-\bar{Y}_{1R_H^{(1)}(i)})^2\rightarrow ^{p}V(Y_{1i} \vert Z_i^{*(1)},T_i=1) \; \; \; \; \; \end{aligned}$$

(B.2)

uniformly for $i\in s$ as $n \rightarrow \infty$, $L \rightarrow \infty$, $H \rightarrow \infty$, and conditions (C4) and (C5). In addition, according to Chebyshev’s inequality, (B.1), and (B.2) we have

$$\begin{aligned}&Pr\left( \left| \frac{1}{n} \sum _{i=1}^n(1-T_i)\frac{1}{L}\sum _{l=1}^LY_{1i}^{*(l)}-E\left\{ (1-T_i)E(Y_{1i}^{*(l)}\vert Y_1,X,T)\right\} \right| > \epsilon \right) \nonumber \\&\quad \le \epsilon ^{-2}V\left\{ \frac{1}{n}\sum _{i=1}^n(1-T_i)\frac{1}{L}\sum _{l=1}^L Y_{1i}^{*(l)}\right\} \nonumber \\&\quad =\epsilon ^{-2}\left[ V \left\{ \frac{1}{n}\sum _{i=1}^n(1-T_i)E\left( Y_{1i}^{*(l)}\vert Y_1,X,T\right) \right\} \right. \nonumber \\&\qquad \left. +E \left\{ \frac{1}{n^2}\sum _{i=1}^n\left( 1-T_i\right) \frac{1}{L}V\left( Y_{1i}^{*(l)}\vert Y_1,X,T\right) \right\} \right] \nonumber \\&\quad =\epsilon ^{-2} \left[ V\left\{ \frac{1}{n}\sum _{i=1}^n \left( 1-T_i\right) \sum _{j \in R_H^{(1)}(i)}\frac{1}{H}Y_{1j}\right\} \right. \nonumber \\&\quad \left. + E\left\{ \frac{1}{n^2}\sum _{i=1}^n(1-T_i) \frac{1}{L} \frac{1}{H}\sum _{j \in R_H^{(1)}(i)}\left( Y_{1j}-\bar{Y}_{1R_H^{(1)}(i)}\right) ^2\right\} \right] \nonumber \\&=\epsilon ^{-2}\left[ V\left\{ \frac{1}{n}\sum _{i=1}^n \left( 1-T_i\right) E(Y_{1i}\vert Z_i^{*(1)},T_i=1)\right\} \right. \nonumber \\&\quad \left. + E\left\{ \frac{1}{n^2}\sum _{i=1}^n(1-T_i)\frac{1}{L}V(Y_{1i} \vert Z_i^{*(1)},T_i=1)\right\} \right] \nonumber \\&\quad +o(n^{-1})+o(n^{-1}L^{-1})\nonumber \\&=o(n^{-1})+o(n^{-1}L^{-1}) \; \; \; \; \; \end{aligned}$$

(B.3)

According to (B.3), we have

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^n \left( 1-T_i\right) \frac{1}{L}\sum _{l=1}^LY_{1i}^{*(l)} \rightarrow ^p E\{(1-T_i)E(Y_{1i}^{*(l)}\vert Y_1,X,T)\}\; \; \; \; \end{aligned}$$

(B.4)

as $n \rightarrow \infty$, $L \rightarrow \infty$, and $H \rightarrow \infty$. Therefore, if at least one of the propensity score models or one of the pairs of regression models is correctly specified, according to (B.1) and (B.4), we have

$$\begin{aligned} \begin{aligned} \hat{\mu }_{1MRMI}&=\frac{1}{n}\sum _{i=1}^n \left\{ T_iY_{1i}+(1-T_i)\frac{1}{L}\sum _{l=1}^LY_{1i}^{*(l)}\right\} \\&\rightarrow ^p E(T_iY_{1i})+E \left\{ (1-T_i)E(Y_{1i}^{*(l)}\vert Y_1,X,T)\right\} \\&=E(T_iY_{1i})+E\left\{ (1-T_i)E \left( \sum _{j \in R_H^{(l)(1)}(i)}\frac{1}{H}Y_{1i}^{*(l)}\vert X,Y_1,T \right) \right\} \\&=E(T_iY_{1i})+E\left\{ (1-T_i)\sum _{j \in R_H^{(1)}(i)}\frac{1}{H}Y_{1j}\right\} \rightarrow E(T_iY_{1i})\\&+E\left\{ (1-T_i)E\left( Y_{1i}\vert Z_i^{*(1)},T_i=1\right) \right\} \\&=E(T_iY_{1i})+E\left\{ (1-T_i)Y_{1i}\right\} \\&=E(Y_1) \end{aligned} \end{aligned}$$

as $n \rightarrow \infty$, $L \rightarrow \infty$, and $H \rightarrow \infty$, where we used the consistent probability weights argument from Stone (1977) in the derivation.

It follows by similar argument that $E(\hat{\mu }_{0MRMI}) \rightarrow ^p E(Y_0)$ as $n \rightarrow \infty$, $L \rightarrow \infty$, and $H \rightarrow \infty$.

Then

$$\begin{aligned} E(\hat{\tau })=E(\hat{\mu }_{1MRMI})-E(\hat{\mu }_{0MRMI})\rightarrow ^p E(Y_1)-E(Y_0) \end{aligned}$$

as $n \rightarrow \infty$, $L \rightarrow \infty$, and $H \rightarrow \infty$. Therefore, Theorem 1 is proven.

1.3 C: Sketched proof of Theorem 2

We can write $\hat{\mu }_{1MRMI}$ as

$$\begin{aligned} \hat{\mu }_{1MRMI}=\hat{\mu }_1+\hat{\mu }_{1MRMI}-\hat{\mu }_1 \end{aligned}$$

where

$$\begin{aligned} \hat{\mu }_1=\frac{1}{n}\sum _{i=1}^n\left\{ T_iY_{1i}+(1-T_i)\sum _{j\in R_H^{(1)}(i)}\frac{1}{H}Y_{1j}\right\} \; \; \; \; \; \end{aligned}$$

(C.1)

and

$$\begin{aligned} \hat{\mu }_{1MRMI}-\hat{\mu }_1=\frac{1}{n}\sum _{i=1}^n\left\{ (1-T_i)\left( \frac{1}{L}\sum _{l=1}^L Y_{1i}^{*(1)(l)}-\frac{1}{H}\sum _{j \in R_H^{(1)}(i)}Y_{1j}\right) \right\} \; \; \; \; \; \end{aligned}$$

(C.2)

Then, using a first-order Taylor expansion of $\hat{\mu }_1$ about $\theta _1=\theta ^*_1$, we obtain

$$\begin{aligned} \hat{\mu }_1=\hat{\mu }_1^*+E\left( \frac{\partial \hat{\mu }_1^*}{\partial \theta _1}\right) (\hat{\theta _1}-\theta ^*_1)+o_p(n^{-\frac{1}{2}}) \; \; \; \; \; \end{aligned}$$

(C.3)

where $\partial \hat{\mu }_1^*/\partial \theta _1$ is $\partial \hat{\mu }_1/\partial \theta _1$ evaluated at $\theta ^*_1$. Because $\theta _1$ is the solution of the estimating equation $S_{m_1p}(\theta _1)=0$, one can show that

$$\begin{aligned} \hat{\theta _1}-\theta ^*_1=-\left[ E\left\{ \frac{\partial S_{m_1p}(\theta ^*_1)}{\partial \theta _1}\right\} \right] ^{-1}S_{m_1p}(\theta ^*_1)+o_p(n^{-\frac{1}{2}}) \; \; \; \; \; \end{aligned}$$

(C.4)

Plugging (C.4) into (C.3) yields

$$\begin{aligned} \hat{\mu }_1=\hat{\mu }_1^*+A_1S_{m_1p}(\theta ^*_1) \; \; \; \; \; \end{aligned}$$

(C.5)

where $A_1$ is defined in Theorem 2 of Sect. 4. Following a similar argument to that of Long et al. (2012), given regularity conditions (C1)–(C5), one can show that

$$\begin{aligned} \hat{\mu }_1^*-\mu _1=Q_1+Q_2+Q_3+o_p(n^{-\frac{1}{2}}) \; \; \; \; \; \end{aligned}$$

(C.6)

where

$$\begin{aligned}&Q_1=\frac{1}{n}\sum _{i=1}^n\left\{ \psi _1\left( Z_i^{*(1)}\right) -\mu _1\right\} \;,\; Q_2=\frac{1}{n}\sum _{i=1}^n T_i\left\{ Y_{1i}-\psi _1\left( Z_i^{*(1)}\right) \right\} ,\; \text {and} \\&Q_3=\frac{1}{n}\sum _{i=1}^n \frac{1-\pi _1\left( Z_i^{*(1)}\right) }{\pi _1\left( Z_i^{*(1)}\right) }T_i\left\{ Y_{1i}-\psi _1\left( Z_i^{*(1)}\right) \right\} . \end{aligned}$$

Because $\hat{\mu }_{1MRMI}-\hat{\mu }_1$ and $\hat{\mu }_1$ are asymptotically independent, we have

$$\begin{aligned} \begin{aligned} V(\hat{\mu }_{1MRMI}-\hat{\mu }_1)&=V\left\{ E(\hat{\mu }_{1MRMI}-\hat{\mu }_1\vert X, Y_1,T)\right\} \\&=E\left\{ V(\hat{\mu }_{1MRMI}-\hat{\mu }_1\vert X, Y_1,T)\right\} \\&=\frac{1}{nL}E\left\{ \frac{1}{n}\sum _{i=1}^n(1-T_i)V(Y_{1i}^{*(l)} \vert Y_1,X,T)\right\} \\&=\frac{1}{nL}E\left\{ \frac{1}{n}\sum _{i=1}^n(1-T_i)\frac{1}{H}\sum _{j \in R_H^{(1)}(i)}(Y_{1j}-\bar{Y}_{1R_{H^{(1)}(i)}})^2 \right\} \\&=O\left( \frac{1}{nL}\right) \; \; \; \; \; \end{aligned} \end{aligned}$$

(C.7)

where $\bar{Y}_{1R_{H^{(1)}(i)}}=H^{-1}\sum _{j \in R_H^{(1)}(i)}Y_{1j}$. According to the asymptotic independence between $\hat{\mu }_{1MRMI}-\hat{\mu }_1$ and $\hat{\mu }_1$, $Q_1$ and $Q_2$, $Q_1$ and $Q_3$, Eqs. (C.1), (C.2), (C.5)–(C.7), and regularity condition (C3), we have

$$\begin{aligned} \begin{aligned} \hat{\mu }_{1MRMI}&=\frac{1}{n}\sum _{i=1}^n \left[ \frac{T_i}{\pi _1\left( Z_i^{*(1)}\right) }Y_{1i}+\left\{ 1-\frac{T_i}{\pi _1\left( Z_i^{*(1)}\right) }\right\} \right. \\&\left. \psi _1\left( Z_i^{*(1)}\right) +A_1s_1(X_i;T_i;\theta ^*_1)\right] \\&\quad +o_p(n^{-\frac{1}{2}}) \end{aligned} \end{aligned}$$

as $n \rightarrow \infty$ and $L \rightarrow \infty$.

We may apply a similar argument for $\hat{\mu }_{0MRMI}$ to obtain

$$\begin{aligned} \begin{aligned} \hat{\mu }_{0MRMI}&=\frac{1}{n}\sum _{i=1}^n \left[ \frac{1-T_i}{\pi _0\left( Z_i^{*(0)}\right) }Y_{0i}+\left\{ 1-\frac{1-T_i}{\pi _0\left( Z_i^{*(0)}\right) }\right\} \right. \\&\quad \left. \psi _0\left( Z_i^{*(0)}\right) +A_0s_0(X_i;T_i;\theta ^*_0)\right] \\&\quad +o_p(n^{-\frac{1}{2}}) \end{aligned} \end{aligned}$$

as $n \rightarrow \infty$ and $L \rightarrow \infty$.

Then, because $\hat{\tau }_{MRMI}=\hat{\mu }_{1MRMI}-\hat{\mu }_{0MRMI}$, we have

$$\begin{aligned} \begin{aligned} \hat{\tau }_{MRMI}&=\frac{1}{n}\sum _{i=1}^n \Bigg [\left\{ \frac{T_i}{\pi _1\left( Z_i^{*(1)}\right) }Y_{1i}+\left\{ 1-\frac{T_i}{\pi _1\left( Z_i^{*(1)}\right) }\right\} \psi _1\left( Z_i^{*(1)}\right) +A_1s_1(X_i;T_i;\theta ^*_1)\right\} \\&-\left\{ \frac{1-T_i}{\pi _0\left( Z_i^{*(0)}\right) }Y_{0i}+\left\{ 1-\frac{1-T_i}{\pi _0\left( Z_i^{*(0)}\right) }\right\} \psi _0\left( Z_i^{*(0)}\right) +A_0s_0(X_i;T_i;\theta ^*_0)\right\} \Bigg ]+o_p(n^{-\frac{1}{2}}) \end{aligned} \end{aligned}$$

Then, by the Central Limit Theorem, we obtain

$$\begin{aligned} n^{\frac{1}{2}}(\hat{\tau }_{MRMI}-\tau )\rightarrow ^dN(0,\sigma ^2_{MRMI}) \end{aligned}$$

with $\sigma ^2_{MRMI}$ defined in Theorem 2 of Sect. 4.