An improved multiply robust estimator for the average treatment effect

Let \({{\varvec{X}}}_{i}\) be a \(p\)-dimensional vector of covariates, \({Y}_{i}\) be the observed outcome, and \({Z}_{i}\) be the treatment status taking value 1 if treated or 0 if untreated. Let (\({Y}^{1}, {Y}^{0}\)) be the two potential outcomes in the treatment and control groups, respectively, and the ATE is defined as

And to draw a correct causal inference in the study, exchangeability, consistency, and positivity assumptions hold [24].

The previous MR approach proposed by Han [9] provides multiple protection to the model misspecification. Specifically, specifying two sets of parametric models, \(\mathcal{P}=\left\{{\pi }^{l}\left({\varvec{X}}\right),l=1, 2, 3,\dots ,L\right\}\) for propensity score and \(\mathcal{M}=\left\{{m}_{z}^{k}\left({\varvec{X}}\right),k=\mathrm{1,2},3,\dots ,K\right\}\) for outcome, where \({m}_{z}^{k}\left({\varvec{X}}\right)={m}^{k}\left({\varvec{X}},Z\right)\). Without loss of generality, let \({\mathbb{I}}=1,\dots ,{n}_{1}\) and \({\mathbb{J}}=1,\dots ,{n}_{0}\) be the indexes for treated and untreated subjects, respectively. Let \({n}_{1}\) and \({n}_{0}=n-{n}_{1}\) represent the size of treatment and control groups, respectively.

To recover the treated population average from subjects in the treatment group, the empirical likelihood weights \({w}_{i}\left(i\in {\mathbb{I}}\right)\) for the outcome \({Y}_{i}\left(i\in {\mathbb{I}}\right)\) in the treatment group are estimated by maximizing \(\prod_{i\in {\mathbb{I}}}{w}_{i}\) subject to the following constraints:where \({\widehat{\theta }}_{1}^{l}={n}^{-1}{\sum }_{i=1}^{n}{\pi }^{l}\left({{\varvec{X}}}_{i}\right)\) and \({\widehat{\eta }}_{1}^{k}={n}^{-1}{\sum }_{i=1}^{n}{m}_{1}^{k}\left({{\varvec{X}}}_{i}\right)\). By symmetry, the weights \({w}_{j}\left(j\in {\mathbb{J}}\right)\) for the control group are given by maximizing \(\prod_{j\in {\mathbb{J}}}{w}_{j}\) according to the following constraints: where \({\widehat{\theta }}_{0}^{l}={n}^{-1}{\sum }_{i=1}^{n}(1-{\pi }^{l}\left({{\varvec{X}}}_{i}\right))\) and \({\widehat{\eta }}_{0}^{k}={n}^{-1}{\sum }_{i=1}^{n}{m}_{0}^{k}\left({{\varvec{X}}}_{i}\right)\). The \({w}_{i}\) and \({w}_{j}\) can be given with Lagrange multiplier method as follows:

where

\({\widehat{{\varvec{\rho}}}}_{1}^{T}\) and \({\widehat{{\varvec{\rho}}}}_{0}^{T}\) are the \((J+K\))-dimensional Lagrange multipliers solving

\({\widehat{{\varvec{\rho}}}}_{1}^{T}\) and \({\widehat{{\varvec{\rho}}}}_{0}^{T}\) must satisfy \(1+{\widehat{\rho }}_{1}^{T}{\widehat{g}}_{1}\left({{\varvec{X}}}_{i}\right)>0\) and \(1+{\widehat{\rho }}_{0}^{T}{\widehat{g}}_{0}\left({{\varvec{X}}}_{j}\right)>0\) due to the non-negativity of \({w}_{i}\) and \({w}_{j}\), respectively. The estimation of \({\widehat{w}}_{i}\) and \({\widehat{w}}_{j}\) can be solved by the Newton–Raphson algorithm [9].

In summary, the ATE estimated by MR method [9] is defined as

The previous MR method [9] allows multiple parametric models, increasing the likelihood of including the correct model; However, there is still a biased estimates of ATE when all parametric models are misspecified. Based on the previous MR method, the proposed MR method allows multiple parametric models, and also includes a nonparametric PS model and a nonparametric OR model.

We select the neural network (NNET) as nonparametric models in the proposed MR method. NNET, one machine learning algorithm, has been used to estimate the ATE [15, 16]. We specified three-layer (input layer, one hidden layer, output layer) NNET, which may be practical [12], and the hidden layer consists of 4 nodes. We performed the NNET using the nnet R package with default parameters. Finally, the proposed MR method added a NNET-based outcome regression model (NN-OR) and a NNET-based propensity score model (NN-PS) in base of previous MR method.

Similar to the previous MR method [9], we specify two sets of models, \(\mathcal{P}=\left\{{\pi }^{l}\left({\varvec{X}}\right),l=1, 2, 3,\dots ,L,L+1\right\}\) for propensity score and \(\mathcal{M}=\left\{{m}_{z}^{k}\left({\varvec{X}}\right),k=\mathrm{1,2},3,\dots ,K,K+1\right\}\) for outcome. Assume \({\pi }^{L+1}\left({\varvec{X}}\right)\) and \({m}_{z}^{K+1}({\varvec{X}})\) are the NN-PS and NN-OR models, respectively, and the other are parametric models. The empirical likelihood weights \({w}_{i}\left(i\in {\mathbb{I}}\right)\) for the outcome \({Y}_{i}\left(i\in {\mathbb{I}}\right)\) in the treatment group are estimated by maximizing \(\prod_{i\in {\mathbb{I}}}{w}_{i}\) subject to the following constraints:where \({\widehat{\theta }}_{1}^{l}={n}^{-1}{\sum }_{i=1}^{n}{\pi }^{l}\left({{\varvec{X}}}_{{\varvec{i}}}\right)\) and \({\widehat{\eta }}_{1}^{k}={n}^{-1}{\sum }_{i=1}^{n}{m}_{1}^{k}\left({{\varvec{X}}}_{i}\right)\). By symmetry, the weights \({w}_{j}\left(j\in {\mathbb{J}}\right)\) for the control group are given by maximizing \(\prod_{j\in {\mathbb{J}}}{w}_{j}\) according to the following constraints:where \({\widehat{\theta }}_{0}^{l}={n}^{-1}{\sum }_{i=1}^{n}(1-{\pi }^{l}\left({{\varvec{X}}}_{i}\right))\) and \({\widehat{\eta }}_{0}^{k}={n}^{-1}{\sum }_{i=1}^{n}{m}_{0}^{k}\left({{\varvec{X}}}_{i}\right)\). The \({w}_{i}\) and \({w}_{j}\) can be given with Lagrange multiplier method as follows:

where

\({\widehat{{\varvec{\rho}}}}_{1}^{T}\) and \({\widehat{{\varvec{\rho}}}}_{0}^{T}\) are the \((J+K+2\))-dimensional Lagrange multipliers solving

\({\widehat{{\varvec{\rho}}}}_{1}^{T}\) and \({\widehat{{\varvec{\rho}}}}_{0}^{T}\) must satisfy \(1+{\widehat{{\varvec{\rho}}}}_{1}^{{\varvec{T}}}{\widehat{{\varvec{g}}}}_{1}\left({{\varvec{X}}}_{i}\right)>0\) and \(1+{\widehat{{\varvec{\rho}}}}_{0}^{T}{\widehat{{\varvec{g}}}}_{0}\left({{\varvec{X}}}_{j}\right)>0\) due to the non-negativity of \({w}_{i}\) and \({w}_{j}\), respectively. The estimation of \({\widehat{w}}_{i}\) and \({\widehat{w}}_{j}\) can be solved by the Newton–Raphson algorithm [9].

The ATE estimated by the proposed method is defined as

The confidence interval of the estimators \(\Delta\) could be obtained by the bootstrap method, where \(\Delta\) maybe IPW, OR, or MR estimator. Specifically, \(n\) individuals first are resampled with replacement from the original data for \(B\) times to obtain \(B\) bootstrap sample, where \(B\) is the pre-specified number. For \(b=1,\dots ,B\), let \({\widehat{\Delta }}^{b}\) be the estimates of the estimator from the \(b\)-th bootstrap sample. Then the bootstrap variance estimator for \({\widehat{\Delta }}^{b}\) is given by

A normality-based 95% confidence interval for \(\Delta\) is \({\widehat{\Delta }}^{b}\pm 1.96\sqrt{\widehat{var}\left({\widehat{\Delta }}^{b}\right)}\)

Table 1 and Figure S1 showed the simulation results of estimating ATE where the proposed MR method included the correct parametric models. And Table 2 and Figure S2 showed the simulation results when there were no correct parametric models in MR method. We could get a few conclusions from the simulation studies.

According to Table 1 and Figure S1, biases of IPW, OR, or MR estimators were ignorable when the parametric models were correctly specified or NNET models were included. The RMSEs of estimators with correct PS or NN-PS models were larger than those with correct OR or NN-OR, respectively. And the RMSEs of the estimators with the correct parametric OR model (OR.model2, MR000010) had the smallest RMSEs among all estimators, and were significantly less than those with NN-OR model (OR.model1, MR000100). However, it can easily be seen that the biases and RMSEs of the estimator with a wrong parametric model are much larger (OR.model3, MR000001).

Together with Table 2 and Figure S2, the proposed MR estimators improved the robustness to the model misspecification even if the parametric models were all incorrectly modeled (MR111000, MR000111, MR111011, MR011111, MR111111 in Table 2 and Figure S2). Although the biases of the six estimators are negligible, the MR000111 had the smallest RMSE among the six estimators. Further, MR000111 with a correct OR model (MR000111 in Table 1 and Figure S1) had a smaller RMSE than the MR estimators with both parametric PS and OR models; and the RMSE of MR000111 was small as that of OR estimator with the correct parametric model (OR.model2 in Table 1 and Figure S1). Even if the two parametric OR models are incorrectly specified, the RMSE of MR000111 is similar to that of NN-OR model.

In addition, Table 3 and Figure S3 showed the simulation results when NNET models included all covariates where parametric models included a correct model; and Table 4 and Figure S4 showed the simulation results when NNET models included all covariates where parametric models did not include a correct model. Similar results were observed in Tables 3 and 4 and Figures S3 and S4.

In conclusion, the simulation results showed that the proposed MR estimators were robust to model misspecification even if all parametric models were incorrectly specified. Further, considering the robustness to model misspecification and RMSE, the MR estimators with only OR models, where one of the models was a NNET model, was the most recommended. The recommended estimators were robust to model misspecification and tended to have the smallest RMSE when the estimators included a correct OR model; and the performance of the recommended estimators was comparative even if all parametric models were misspecified.

The China Health and Retirement Longitudinal Study (CHARLS) is a large-scale, nationally representative longitudinal survey of people aged 45 or older and their spouses in China, including assessments of the social, economic, and health status of community residents [25]. The study aimed to estimate the treatment effect of social activity on the depression level in the real-world data from CHARLS. The depression level was evaluated by CES-D Depression Scale, and the total score was between 0 and 30: a higher total score denotes a higher depression level, while a lower total score denotes a lower depression level. The self-reported social activity includes 11 categories based on individual responses to the question, “Have you done any of these activities in the last month”. The value of the variable is 1 if the participant takes part in any activities; otherwise, the value is 0. The group with a value 1 was defined as a social activity group, and the group with a value 0 was defined as a non-social activity group. Baseline information included age, marital status, sex, region, smoke status, self-reported hypertension, self-reported diabetes, self-reported heart disease, and self-reported stroke. Inclusion criteria are: (1) participants who took part in the survey in 2011–2012 (2) complete baseline. A total of 10,119 participants were included in the analysis. The baseline information was summarized in Table S1. We found that the non-social activity group had a higher level of depression.

In the study, we specified three PS models (including one NN-PS model and two logistic models) and three OR models (including one NN-OR model and two linear regression) in the MR method. For the NNET models, we included all covariates. For the parametric models, we explored the association of social activity group/non-social activity group with all covariates via a logistic model, and the association of depression with all covariates through a linear model; and we identified candidate models with a significant level at 0.05 and 0.01. Hence, three set of covariates in [\({\pi }^{1}\left({\varvec{X}}\right),{\pi }^{2}\left({\varvec{X}}\right),{\pi }^{3}({\varvec{X}})\)] are: (i) age, marital status, sex, region, smoke status, self-reported hypertension, self-reported diabetes, self-reported heart disease and self-reported stroke; (ii) age, marital status, sex, region, smoke status, and self-reported diabetes; (iii) age, region and smoke status. Three sets of covariates in [\({m}^{1}({\varvec{X}},Z),{m}^{2}({\varvec{X}},Z),{m}^{3}({\varvec{X}},Z)\)] are: (i) age, marital status, sex, region, smoke status, self-reported hypertension, self-reported diabetes, self-reported heart disease, self-reported stroke and activity group; (ii) marital status, sex, region, self-reported diabetes, self-reported heart disease, self-reported stroke, and social activity; (iii) marital status, sex, region, self-reported heart disease, self-reported stroke and activity group. We applied the MR methods to estimate the effect. The results were shown in Table 5 and Figure S5.

From Table 5, all estimates showed that the social activity group had lower depression scores than the non-social activity group. However, when only specifying NN-PS model and NN-OR model (MR100000, MR000100, MR100100), the estimators tended to have higher estimated values and higher standard errors. The other estimators had similar estimates, and MR000111 tended to have the smallest standard errors.