Estimation of average treatment effect based on a multi-index propensity score

Suppose that \({\mathbf{Z}}_{i}={\left({Y}_{i},{A}_{i},{\mathbf{X}}_{i}^{{\top }}\right)}^{{\top }}, i=1,\dots ,n\) be the observed data for \({i}^{\mathrm{th}}\) subject from independent and identically distributed copies of \(\mathbf{Z}={\left(Y,A,{\mathbf{X}}^{{\top }}\right)}^{{\top }}\), where \(Y\) is the outcome, \(A\) is the binary indicator of treatment (\(A=1\) if treated and \(A=0\) if controlled), and \(\mathbf{X}\) is the p-dimensional vector of pretreatment covariates. Let \({Y}^{1}\) and \({Y}^{0}\) represent the potential outcomes if a subject was assigned to treated or controlled group, respectively. The formula for average treatment effect (ATE) is

Under causal inference framework, the identifiability assumptions are usually assumed, that is [6],

The IPW estimator is usually used for correcting confounding bias. The propensity score (PS) \(\pi \left(\mathbf{X}\right)=P\left(A=1 \right| \mathbf{X})\) can be modeled as \(\pi \left(\mathbf{X};\boldsymbol{\alpha }\right)={g}_{\pi }\left({\alpha }_{0}+{\boldsymbol{\alpha }}_{1}^{\mathrm{T}}\mathbf{X}\right)\), where \({g}_{\pi }\left(\cdot \right)\) is a specified link function, for example, the inverse of the logit function for the logistic regression, and \(\boldsymbol{\alpha }={\left({\alpha }_{0},{\boldsymbol{\alpha }}_{1}^{\mathrm{T}}\right)}^{\mathrm{T}}\) are the unknown parameters and can be estimated from maximum likelihood estimation. Under causal inference assumptions, the ATE can be estimated by the IPW estimator

where \(\widehat{\boldsymbol{\alpha }}\) is the estimated value of \(\boldsymbol{\alpha }\). If \(\pi \left(\mathbf{X};\boldsymbol{\alpha }\right)\) is correctly specified, \({\widehat{\Delta }}_{IPW}\) is a consistent estimator of \(\Delta\).

The OR estimator is another commonly used approach for correcting confounding bias. Let \({\mu }_{A}\left(\mathbf{X}\right)=E\left(Y \right| \mathbf{X},A)\) denote outcome regression (OR), where \(A\in \{\mathrm{0,1}\}\). It can be modeled as \({\mu }_{A}\left(\mathbf{X};{\varvec{\beta}}\right)={g}_{\mu }\left({\beta }_{0}+{{\varvec{\beta}}}_{1}^{T}\mathbf{X}+{\beta }_{2}A\right)\), where \({g}_{\mu }(\cdot )\) is a specified link function, for example, the identity function for the linear regression, \({\varvec{\beta}}={\left({\beta }_{0},{{\varvec{\beta}}}_{1}^{{\top }},{\beta }_{2}\right)}^{{\top }}\) are the unknown parameters and can be estimated from maximum likelihood estimation. Interactions between \(A\) and \(\mathbf{X}\) in OR model can also be accommodated by estimating the OR separately by treated and controlled groups [19]. Under causal inference assumptions, the ATE also can be estimated by the OR estimator

where \(\widehat{{\varvec{\beta}}}\) is the estimated value of \({\varvec{\beta}}\). If \(\mu \left(\mathbf{X},A;{\varvec{\beta}}\right)\) is correctly specified, \({\widehat{\Delta }}_{OR}\) is a consistent estimator of \(\Delta\).

If the PS model for IPW estimator or the OR model for OR estimator is incorrectly specified, the estimation consistency of \({\widehat{\Delta }}_{IPW}\) or \({\widehat{\Delta }}_{OR}\) with \(\Delta\) can not be guaranteed. To provide protection against model misspecification, Cheng et al. [19] considered integrating the information of PS \(\pi \left(\mathbf{X};\boldsymbol{\alpha }\right)\) and OR \({\mu }_{a}\left(\mathbf{X};{\varvec{\beta}}\right)\) to construct double-index propensity score (DiPS), which is denoted by \(\pi \left(\mathbf{X};{\boldsymbol{\alpha }}_{1},{{\varvec{\beta}}}_{1}\right)=E\left[A | {\boldsymbol{\alpha }}_{1}^{\mathrm{T}}\mathbf{X},{{\varvec{\beta}}}_{1}^{\mathrm{T}}\mathbf{X}\right]\). In order to estimate this conditional expectation, Cheng et al. [19] firstly got the estimated value \({\widehat{\boldsymbol{\alpha }}}_{1}\) of PS model and the estimated value \({\widehat{{\varvec{\beta}}}}_{1}\) of OR model, then used the Nadaraya-Watson kernel estimator [34] to conduct nonparametric regression of \(A\) on \({\widehat{\boldsymbol{\alpha }}}_{1}^{\mathrm{T}}\mathbf{X}\) and \({\widehat{{\varvec{\beta}}}}_{1}^{\mathrm{T}}\mathbf{X}\), to get the estimated value of DiPS as

where \({\widehat{\mathbf{S}}}_{i}=\left({\widehat{\boldsymbol{\alpha }}}_{1}^{\mathrm{T}}{\mathbf{X}}_{i},{\widehat{{\varvec{\beta}}}}_{1}^{\mathrm{T}}{\mathbf{X}}_{i}\right)\) and \(\widehat{\mathbf{S}}=\left({\widehat{\boldsymbol{\alpha }}}_{1}^{\mathrm{T}}\mathbf{X},{\widehat{{\varvec{\beta}}}}_{1}^{\mathrm{T}}\mathbf{X}\right)\) are bivariate regressors, which is named double-index. \({\mathcal{K}}_{\mathbf{H}}\left(\bullet \right)\) is a kernel function with a bandwidth \(\mathbf{H}\) of \(2\times 2\) matrix. Using the estimated DiPS \(\widehat{\pi }\left(\mathbf{X};{\widehat{\boldsymbol{\alpha }}}_{1},{\widehat{{\varvec{\beta}}}}_{1}\right)\), the ATE can be estimated by

Cheng et al. [19] demonstrated that \({\widehat{\Delta }}_{DiPS}\) is a doubly robust estimator: it is consistent when \(\pi \left(\mathbf{X};\boldsymbol{\alpha }\right)\) is correctly specified, or \({\mu }_{A}\left(\mathbf{X};{\varvec{\beta}}\right)\) is correctly specified, but not necessarily both.

Although \({\widehat{\Delta }}_{DiPS}\) in (3) can achieve doubly robust estimation for ATE, the DiPS estimated by the Nadaraya-Watson kernel estimator in (2), which may make the estimated probability outside the range of 0 to1, then the above Assumption 3 is violated. Furthermore, \({\widehat{\Delta }}_{DiPS}\) in (3) only allows a single model for PS and a single model for OR, the estimation consistency cannot be guaranteed when both models are incorrect. To provide more protection on estimation consistency, we would like to develop an approach that allows multiple candidate models for PS and/or OR, to achieve multiple robustness: the estimator is consistent when any model for PS or any model for OR is correctly specified.

Specifically, we consider multiple candidate models for PS \(\{{\pi }^{k}\left(\mathbf{X};{\boldsymbol{\alpha }}^{k}\right)={g}_{\pi }\left({\alpha }_{0}^{k}+{\boldsymbol{\alpha }}_{1}^{k\mathrm{T}}\mathbf{X}\right),k=1,\dots ,K\}\) and multiple candidate models for OR \(\left\{{\mu }_{A}^{l}\left(\mathbf{X};{{\varvec{\beta}}}^{l}\right)={g}_{\mu }\left({\beta }_{1}^{l}+{{\varvec{\beta}}}_{1}^{l\mathrm{T}}\mathbf{X}+{\beta }_{2}^{l}A\right),l=1,\dots ,L\right\}\), probably with different choices or functional forms of covariates. Then we integrate the information from multiple PS models and multiple OR models to construct multi-index propensity score (MiPS), which is denoted by \(\pi \left(\mathbf{X};{\boldsymbol{\alpha }}_{1}^{1},...,{\boldsymbol{\alpha }}_{1}^{K},{{\varvec{\beta}}}_{1}^{1},...,{{\varvec{\beta}}}_{1}^{L}\right)=E\left[A | {\boldsymbol{\alpha }}_{1}^{1\mathrm{T}}\mathbf{X},...{\boldsymbol{\alpha }}_{1}^{K\mathrm{T}}\mathbf{X},{{\varvec{\beta}}}_{1}^{1\mathrm{T}}\mathbf{X},...,{{\varvec{\beta}}}_{1}^{L\mathrm{T}}\mathbf{X}\right]\). In order to estimate this conditional expectation, we firstly get the estimated values \({\widehat{\boldsymbol{\alpha }}}_{1}^{1}\),…, \({\widehat{\boldsymbol{\alpha }}}_{1}^{K}\) of multiple PS models and the estimated values \({\widehat{{\varvec{\beta}}}}_{1}^{1}\),…, \({\widehat{{\varvec{\beta}}}}_{1}^{L}\) of multiple OR models, then a naive idea is to use the multivariate Nadaraya-Watson kernel estimator to conduct nonparametric regression of \(A\) on \({\widehat{\boldsymbol{\alpha }}}_{1}^{1\mathrm{T}}\mathbf{X}\),…, \({\widehat{\boldsymbol{\alpha }}}_{1}^{K\mathrm{T}}\mathbf{X}\) and \({\widehat{{\varvec{\beta}}}}_{1}^{1\mathrm{T}}\mathbf{X}\),…, \({\widehat{{\varvec{\beta}}}}_{1}^{L\mathrm{T}}\mathbf{X}\) to get the estimated value of MiPS as

where \({\widehat{\mathbf{S}}}_{j}=\left({\widehat{\boldsymbol{\alpha }}}_{1}^{1\mathrm{T}}{\mathbf{X}}_{j},\dots , {\widehat{\boldsymbol{\alpha }}}_{1}^{K\mathrm{T}}{\mathbf{X}}_{j},{\widehat{{\varvec{\beta}}}}_{1}^{1\mathrm{T}}{\mathbf{X}}_{j},\dots , {\widehat{{\varvec{\beta}}}}_{1}^{L\mathrm{T}}{\mathbf{X}}_{j}\right)\) and \(\widehat{\mathbf{S}}=\left({\widehat{\boldsymbol{\alpha }}}_{1}^{1\mathrm{T}}\mathbf{X},\dots , {\widehat{\boldsymbol{\alpha }}}_{1}^{K\mathrm{T}}\mathbf{X},{\widehat{{\varvec{\beta}}}}_{1}^{1\mathrm{T}}\mathbf{X},\dots , {\widehat{{\varvec{\beta}}}}_{1}^{L\mathrm{T}}\mathbf{X}\right)\) are multivariate regressors, which is named multi-index. \({\mathcal{K}}_{\mathbf{H}}\left(\bullet \right)\) is a kernel function with a bandwidth \(\mathbf{H}\) of \(\left(K+L\right)\times \left(K+L\right)\) matrix. Using the estimated kernel-based MiPS \({\widehat{\pi }}^{Ker}\left(\mathbf{X};{\widehat{\boldsymbol{\alpha }}}_{1}^{1},...,{\widehat{\boldsymbol{\alpha }}}_{1}^{K},{\widehat{{\varvec{\beta}}}}_{1}^{1},...,{\widehat{{\varvec{\beta}}}}_{1}^{L}\right)\), the ATE can be estimated by

However, if there are no additional assumptions about the regression structure, the performance of Nadaraya-Watson kernel estimator in (5) degrades as the number of regressors increases. This degradation in performance is often referred to as the “curse of dimensionality” [22‐24]. Our following simulation results also show that \({\widehat{\Delta }}_{MiPS}^{Ker}\) has obvious bias when multiple candidate models are included in \({\widehat{\pi }}^{Ker}\left(\mathbf{X};{\widehat{\boldsymbol{\alpha }}}_{1}^{1},...,{\widehat{\boldsymbol{\alpha }}}_{1}^{K},{\widehat{{\varvec{\beta}}}}_{1}^{1},...,{\widehat{{\varvec{\beta}}}}_{1}^{L}\right)\), even if the correct PS and/or OR model is covered.

With the development of scalable computing and optimization techniques [25, 26], the use of machine learning has been one of the most promising approaches in connection with applications related to approximation and estimation of multivariate functions [27, 28]. Artificial neural network (ANN) is one of machine learning approaches. Benefiting from its flexible structure, the ANN becomes a universal approximator of a variety of functions [31‐33]. The ANN comprises an input layer, a researcher-specified number of hidden layer(s), and an output layer. The hidden layer(s) and output layer consist of a number of neurons (also specified by researchers) with activation functions [35]. The operation of ANN includes following steps: 1) Information is input from the input layer, which passes it to the hidden layer; 2) In the hidden layer(s), the information is multiplied by the weight and a bias is added, and then passed to the next layer after transforming by the activation function; 3) The information is passed layer by layer until the last layer, where it is multiplied by the weight and then transformed by the activation function to provide the output; and 4) Calculate the error between the output and the actual value, and minimize the error by optimizing the weight parameters and bias parameters through the backpropagation algorithm [36]. In addition to having the potential of overcoming the “curse of dimensionality” [29, 30], the ANN is capable of automatically capturing complex relationships between variables [27]. It may be suited for modeling the relationship between treatment and multi-index because interactions commonly exist between indexes due to shared covariates in candidate PS and/or OR models. Therefore, we replaced the kernel function by ANN and proposed our ANN-based MiPS (ANN.MiPS) estimator.

Now we propose the ANN-based MiPS. We firstly get the estimated values \({\widehat{\boldsymbol{\alpha }}}_{1}^{1}\),…, \({\widehat{\boldsymbol{\alpha }}}_{1}^{K}\) of multiple PS models and the estimated values \({\widehat{{\varvec{\beta}}}}_{1}^{1}\),…, \({\widehat{{\varvec{\beta}}}}_{1}^{L}\) of multiple OR models, then use the ANN to conduct nonparametric regression of \(A\) on multiple indexes \({\widehat{\boldsymbol{\alpha }}}_{1}^{1\mathrm{T}}\mathbf{X}\),…, \({\widehat{\boldsymbol{\alpha }}}_{1}^{K\mathrm{T}}\mathbf{X}\) and \({\widehat{{\varvec{\beta}}}}_{1}^{1\mathrm{T}}\mathbf{X}\),…, \({\widehat{{\varvec{\beta}}}}_{1}^{L\mathrm{T}}\mathbf{X}\) to get the estimated value of MiPS as \({\widehat{\pi }}^{Ann}\left(\mathbf{X};{\widehat{\boldsymbol{\alpha }}}_{1}^{1},...,{\widehat{\boldsymbol{\alpha }}}_{1}^{K},{\widehat{{\varvec{\beta}}}}_{1}^{1},...,{\widehat{{\varvec{\beta}}}}_{1}^{L}\right)\). Then the ATE can be estimated by

Our following simulations indicate the multiple robustness of \({\widehat{\Delta }}_{MiPS}^{Ann}\): its bias is ignorable when any model for PS or any model for OR is correctly specified.

We implemented the ANN that contains 2 hidden layers with 4 neurons in each hidden layer using AMORE package [37] for ANN.MiPS estimator. Therefore, the total number of parameters to be estimated in the ANN is \(4*(K+L)+32\), including \(4*(K+L)+24\) weight parameters and 8 bias parameters. The learning rate is set as 0.001 [10, 12]. The momentum is set as 0.5, the default value in the AMORE package. The hyperbolic tangent function was specified as the activation function for hidden layer. The sigmoid function was specified as the activation function for output layer to ensure the estimated ANN-based MiPS is between 0 to 1 [38]. To examine the performance stability of the estimator, we performed a sensitivity analysis using different hyperparameter selections. The simulations, real data analysis, and all statistical tests were conducted using R software (Version 4.1.0) [39]. A zip file of AMORE package and an example code for implementing the ANN.MiPS approach can be found in the attachment.

We conducted simulation studies to evaluate the performance of (i) single model-based estimators: IPW estimator in (1) and OR estimator in (2); (ii) doubly robust estimators: augmented inverse probability weighting (AIPW) [17] and target maximum likelihood estimator (TMLE) [18], which allows a single model for PS and a single model for OR; (iii) multiple models-based estimators: kernel-based estimator in (6) and ANN-based estimator in (7), which allows multiple candidate models for PS and/or OR.

Ten covariates \({X}_{1}-{X}_{10}\) were generated from standard normal distribution, and the correlation between them are shown in Fig. 1. The binary treatment indicator \(A\) was generated from a Bernoulli distribution according to the following propensity score

\({\alpha }_{0}\) was set to be 0 or -1.1 to make approximately 50% or 25% subjects entering the treatment group. The continuous outcome \(Y\) was generated from

where \(\varepsilon\) follows the standard normal distribution. The true ATE was \(\Delta =E\left({Y}^{1}\right)-E\left({Y}^{0}\right)=-0.4\).

In the estimation, two estimation models were specified

for propensity score, and two estimation models were specified

for outcome regression. According to the data-generating mechanism, \({\pi }^{1}\left(\mathbf{X};{\boldsymbol{\alpha }}^{1}\right)\) and \({{\mu }_{A}}^{1}\left(\mathbf{X};{{\varvec{\beta}}}^{1}\right)\) were correct PS and correct OR models, whereas \({\pi }^{2}\left(\mathbf{X};{\boldsymbol{\alpha }}^{2}\right)\) and \({{\mu }_{A}}^{2}\left(\mathbf{X};{{\varvec{\beta}}}^{2}\right)\) were incorrect PS and incorrect OR models, due to the mis-specified functional forms of covariates. To distinguish these estimation methods, each estimator is denoted as "method-0000". Each of the four numbers, from left to right, represents if \({\pi }^{1}\left(\mathbf{X};{\boldsymbol{\alpha }}^{1}\right)\), \({\pi }^{2}\left(\mathbf{X};{\boldsymbol{\alpha }}^{2}\right)\), \({{\mu }_{A}}^{1}\left(\mathbf{X};{{\varvec{\beta}}}^{1}\right)\) or \({{\mu }_{A}}^{2}\left(\mathbf{X};{{\varvec{\beta}}}^{2}\right)\) is included in the estimator, where “1” indicates yes and “0” indicates no.

We investigated sample sizes of \(n=300\) and \(n=1000\) with 1000 replications in all settings. Tables 1 and 2 show the estimation results of all estimators, along with five evaluation measures including percentage of bias (BIAS, in percentage), root mean square error (RMSE), Monte Carlo standard error (MC-SE), bootstrapping standard error (BS-SE) based on 100 resamples, and coverage rate of 95% Wald confidence interval (CI-Cov). Our bootstrapping procedure resamples from the original sample set with replacement until the bootstrapping sample size reaches the original sample size. Fig. S1 shows the distribution of the estimated ATEs of Ker.MiPS and ANN.MiPS estimators. The following conclusions can be obtained. For estimation bias,

For estimation efficiency,

To evaluate the performance of the MiPS estimator when the number of specified models increases, we have considered three additional estimators: MiPS-1111-2PS, adding two additional incorrect PS models \(\left\{\begin{array}{c}logit\left[{\pi }^{3}\left(\mathbf{X};{\boldsymbol{\alpha }}^{3}\right)\right]=\left(1,{X}_{1},{X}_{2},{X}_{3}\right){\boldsymbol{\alpha }}^{3}\\ logit\left[{\pi }^{4}\left(\mathbf{X};{\boldsymbol{\alpha }}^{4}\right)\right]=\left(1,{X}_{1}^{2},{X}_{2}^{2},{X}_{3}^{2}\right){\boldsymbol{\alpha }}^{4}\end{array}\right\}\) on the basis of the MiPS-1111; MiPS-1111-2OR, adding two additional incorrect OR models \(\left\{\begin{array}{c}{\mu }_{A}^{3}\left(\mathbf{X};{{\varvec{\beta}}}^{3}\right)=\left(1,{X}_{1},{X}_{2},{X}_{3},A\right){{\varvec{\beta}}}^{3}\\ {\mu }_{A}^{4}\left(\mathbf{X};{{\varvec{\beta}}}^{4}\right)=\left(1,{X}_{1}^{2},{X}_{2}^{2},{X}_{3}^{2},A\right){{\varvec{\beta}}}^{4}\end{array}\right\}\) on the basis of the MiPS-1111; MiPS-1111-2PS-2OR, adding two additional incorrect PS models \({\pi }^{3}\left(\mathbf{X};{\boldsymbol{\alpha }}^{3}\right)\) and \({\pi }^{4}\left(\mathbf{X};{\boldsymbol{\alpha }}^{4}\right)\) and two additional incorrect OR models \({\mu }_{A}^{3}\left(\mathbf{X};{{\varvec{\beta}}}^{3}\right)\) and \({\mu }_{A}^{4}\left(\mathbf{X};{{\varvec{\beta}}}^{4}\right)\) on the basis of the MiPS-1111. Table 3 shows the estimation results. The following conclusions can be obtained.

We also evaluated the performance of ANN.MiPS estimator under the simulation scenario with both continuous and discrete covariates. The simulation setting was described in Supplementary Document. Similar conclusions can be obtained as the above scenario with all continuous covariates (Table S1, S2). The sensitivity analysis of hyperparameters selection in ANN revealed the performance stability of ANN.MiPS estimator (Table S3).

To illustrate our proposed method, we analyzed a subset of real data from the National Health and Nutrition Examination Survey Data | Epidemiologic Follow-up Study (NHEFS) (wwwn.cdc.gov/nchs/nhanes/nhefs/). The dataset consists of 1,507 participants aged 25–74 who smoked at the first survey and were followed for approximately 10 years. The empirical study aimed to estimate the ATE of smoking cessation (coded as quitting and non-quitting, with non-quitting as the reference group) on weight gain. Participants were categorized as treated if they quit smoking during follow-up, otherwise controlled. Weight gain for each individual was measured as weight at the end of follow-up minus weight at baseline survey (in kilograms). During the 10-year follow-up, 379 (25.15%) participants quit smoking. The average weight gain was greater for those who quit smoking with an unadjusted difference of 2.4 kg.

Table 4 summarized the baseline characteristics, including age, gender, race, baseline weight, active life level, education level, exercise, smoking intensity, smoking years, and ever use of weight loss medication between the smoking quitters and non-quitters. As shown in the table, the distribution of age, gender, race, education level, smoking intensity, and smoking years was different between quitters and non-quitters. When estimating the ATE of smoking cessation on weight gain, these factors should be adjusted for if they are confounders.

To identify candidate models for ANN.MiPS estimator, we explored the association of smoking cessation with all potential risk factors by logistic regression, and explored the association of weight gain with all potential risk factors by linear regression. The covariates in model 1 and model 2 for both PS and OR models were identified at significant levels of 0.05 and 0.1, respectively. The covariates in PS model 1 and model 2 were (i) age, gender, race, smoking intensity, and smoking years; (ii) age, gender, race, smoking intensity, smoking years, education level, and exercise situation. The covariates in OR model 1 and model 2 were (i) age, weight at baseline, smoking intensity, education level, and active life level; (ii) age, weight at baseline, smoking intensity, education level, active life level, and family income level. We applied the single model-based IPW estimator, single model-based OR estimator, and our proposed ANN.MiPS estimator to estimate the ATE. The four numbers in the ANN.MiPS estimator, from left to right, represents if PS model 1, PS model 2, OR model 1, or OR model 2 is included in the estimator, where “1” indicates yes and “0” indicates no. For example, “ANN.MiPS-1010” represents that the PS model 1 and OR model 1 are included in the estimator. The standard error of estimation was estimated based on 500 resampled bootstrapping.

The estimation results in Table 5 indicated that all estimators suggested quitting smoking significantly increased participants' weight gain. Most of the estimated adjusted effects based on these estimators were greater than the estimated unadjusted effects of 2.4, which seems more precise and reliable. The point estimation and its bootstrap standard error for ATE of the ANN.MiPS estimator was stable under different model specifications.