Highly robust causal semiparametric U-statistic with applications in biomedical studies

Abstract

With our increased ability to capture large data, causal inference has received renewed attention and is playing an ever-important role in biomedicine and economics. However, one major methodological hurdle is that existing methods rely on many unverifiable model assumptions. Thus robust modeling is a critically important approach complementary to sensitivity analysis, where it compares results under various model assumptions. The more robust a method is with respect to model assumptions, the more worthy it is. The doubly robust estimator (DRE) is a significant advance in this direction. However, in practice, many outcome measures are functionals of multiple distributions, and so are the associated estimands, which can only be estimated via U-statistics. Thus most existing DREs do not apply. This article proposes a broad class of highly robust U-statistic estimators (HREs), which use semiparametric specifications for both the propensity score and outcome models in constructing the U-statistic. Thus, the HRE is more robust than the existing DREs. We derive comprehensive asymptotic properties of the proposed estimators and perform extensive simulation studies to evaluate their finite sample performance and compare them with the corresponding parametric U-statistics and the naive estimators, which show significant advantages. Then we apply the method to analyze a clinical trial from the AIDS Clinical Trials Group.

1. Introduction

In observational investigations such as case-control studies, the treatment assignment is often not entirely random, or the covariates are not balanced among treatment groups. It is known that the treatment effects estimates based on group sample means are biased [1]. proposed the inverse propensity weighting estimator (IPWE) to correct the bias. However, the IPWE is not robust in that when the propensity model is incorrectly specified, or the propensity score has some values close to zero or one, it will still be biased or even aberrant [2–4]. Such a problem belongs to the general field of causal inference [5, 6]. Thus, robustness with respect to model assumptions is vital in causal inference. The more robust the method is, the more worthy it is. The doubly robust estimator (DRE [7–11]), is a significant improvement over the IPWE. The DRE is constructed using the propensity score and the outcome model. It is unbiased as long as the propensity score, or the outcome model is correctly specified, hence the name DRE. Due to this remarkable property, there has been considerable effort in applying the DRE in biomedical studies and applications. However, these existing methods still rely on unverifiable model assumptions. As pointed out in [12, 13]; among others, the DRE is highly sensitive to mild model misspecifications; namely, when both models deviate even slightly from the true ones, the DRE may give a misleading result. To overcome this problem and enhance the robustness of DRE, [14]; and [15] proposed semiparametric DREs, where both propensity score and outcome models are semiparametric models.

In practice, many outcome measures of interest are functionals of multiple distributions, and so are the associated estimands, which can only be estimated via U-statistics, and the existing DRE does not apply in these cases [16–19]. Again if the treatment assignment is not entirely random, the naive U-statistic would be a biased estimator of the estimand of interest, and the DRE for U-statistic is called for. The U-statistic [20, 21] is a well-studied estimation tool. It produces the minimal variance unbiased estimator ([22]; p. 176) and is used in a variety of fields, including biomedicine, business, economics, and marketing [23–27]. To our knowledge, progress towards a highly robust approach has been slow and incremental [28]. proposed IPW adjusted estimator of the mean of a K-sample U-statistic when data on the outcomes of interest are missing and illustrated the methods with the estimation of the area under a receiver operating characteristic curve (AUC) [29]. proposed an outcome regression adjusted Mann–Whitney U test. Still, none of them are DREs, while [30] developed a DRE U-statistic but only focused on the estimation of the AUC, and studied it in the parametric specifications. More recently [31], proposed the DRE U-statistic with several illustrative parametric examples, while [32] proposed a Mann–Whitney-Type DRE. Still, both methods are based on parametric models.

This article proposes a general methodology for developing a highly robust estimator (HRE) for U- statistic with semiparametric specifications for both the propensity score and outcome models via single index models. Without shape restriction, often the link function is constructed with kernel or smoothing methods [33]. investigated a single index model that employed least squares estimation for both the link function and the regression parameters. A kernel smoothing method is used to construct the link function. Since minimizing the least squares estimation depends on an optimization problem in a high-dimensional space, its implementation is challenging [34]. proposed a single index model that minimizes the average conditional variance and also used a kernel smoothing method to estimate the link function. In addition [35], investigated such a model using a penalized spline estimation for the link function [36]. Fitted the polynomial spline smoothing method for the unknown link function in the single index model. In most of these methods, the unknown link function is estimated using the kernel or spline method. As a result, the estimates largely depend on the choice of the kernel, bandwidth, or the spline knots, degree, and basis functions, which are subjective and do not always guarantee improved model specification.

In contrast, using semiparametric MLE to estimate single index model parameters is fully objective without any subjective inputs. However, the MLE of the nonparametric link function exists only under certain shape constraints (e.g. monotonicity, convexity, and concavity). Among them, the monotonicity assumption is widely used in biomedical studies for both practicality and ease of interpretation. For example, in biology, monotone regression approaches have been used to search for gene-gene interactions [37]. In medicine, they have been used to investigate the expression of a leukemia antigen as a function of white blood cell count and DNA index [38]. Several researchers have studied the semiparametric monotone single index model and used it to solve a variety of real problems; see, for instance, [39–42]; and [43]. The monotone constraint on the link function in the single index model provides a ready interpretation of the effect of the covariate on the response in comparison to the unconstrained single index model. For example, with monotone increasing link function, the response will increase or remain the same if the linear transformation of covariates increases. This property can be useful in biomedical research. Another advantage of using the shape-constrained single index model is that it can be estimated objectively without subjective choices of (say) kernel and/or spline features, and tuning parameters required of the unconstrained single index model. Consequently, there is a trade-off between objectivity and model constraints. We have chosen to use the constrained model. In fact, some commonly used parametric models are special cases of monotone single index models, such as the parametric logistic model (with the logistic link), and the linear model (with identity link).

We also derive asymptotic properties of the highly robust causal U-statistic under semiparametric specifications. The statistical implications are significant because of the inherent robustness with semiparametric specifications and broad application areas. The semiparametric model performs well even with moderate sample size and can model the covariate effects in addition to the treatment effects. In contrast, the nonparametric methods require a large sample size to achieve the desired accuracy due because of their low efficiency. They often perform poorly with small to moderate sample sizes or multiple covariates.

The rest of the article is organized below. Section 2 introduces the semiparametric construction of the proposed highly robust U-statistic estimator (HRUE) and gives some practical examples. In Section 3, we derive the asymptotic properties of HRUE. In Section 4, we conduct extensive simulation studies to evaluate the performance of the proposed estimator and compare it with the parametric DR U-statistic estimator (DRUE) and the naive U-statistic estimator. Section 5 applies the proposed method to analyze an actual clinical trial. We leave the proofs of theorems and the regularity conditions in the Supplementary Material.

2. The proposed method

We first give a brief review of the existing DRE construction, then describe the proposed HRUE, specify the semiparametric models and estimate the parameters, and give some illustrative examples.

2.1. Brief review of DRE

Consider observed data $D_n=\left\{\left(y_i,t_i,x_i\right):i=1,\dots ,n\right\}\text{iid}(Y,T,X)$, for subject $i$, where $y_i$ is the response, $t_i$ is the treatment indicator, $t_i=1/0$ for treatment/control, $x_i$ is the covariate. Let $y_{i1}$ and $y_{i0}$ be the potential outcomes under the treatment and the control group, respectively. Then the response can be written as $y_i=t_i{y}_{i1}+\left(1-t_i\right)y_{i0}$. Assume that conditioning on $x_i$, $t_i$ and $\left(y_{i0},y_{i1}\right)$ are independent. The estimand of interest is the effects difference $\text{Δ}=E\left(Y_1\right)-E\left(Y_0\right):={\mu }_1-{\mu }_0$. If the treatment assignment $T$ is independent of the response $Y$, it is known that the naive sample difference ${\overline{Y}}_1-{\overline{Y}}_0$ is an unbiased estimator of $\text{Δ}$, where ${\overline{Y}}_1={\sum }_{i=1}^n{t}_i{y}_i/{\sum }_{i=1}^n{t}_i$ and ${\overline{Y}}_0={\sum }_{i=1}^n\left(1-t_i\right)y_i/{\sum }_{i=1}^n\left(1-t_i\right)$. However, in observational studies, often the treatment assignment is not entirely random, and it is known that ${\overline{Y}}_1-{\overline{Y}}_0$ is biased for $\text{Δ}$. To overcome this problem [1], proposed the inverse probability weighted estimator (IPWE) $n^{-1}{\sum }_{i=1}^n\left(2t_i-1\right)y_i/g\left(x_i\right)$, and it is unbiased for $\text{Δ}$ if the propensity score model $g(x)=P(T=1\mid X=x)$ is correctly specified. However, it is still biased if $g(\cdot )$ is misspecified. To improve its robustness against model specification [7, 9], and others proposed and studied the DRE

$$\widehat{\text{Δ}}=\frac{1}{n}\sum _{i=1}^n\left(\frac{t_i\left(y_i-m_1\left(x_i\right)\right)}{g\left(x_i\right)}+m_1\left(x_i\right)-\frac{\left(1-t_i\right)\left(y_i-m_0\left(x_i^{\top }\right)\right)}{1-g\left(x_i\right)}-m_0\left(x_i^{\top }\right)\right),$$

where $m_j(x)=E(Y\mid X=x,T=j)$ is the outcome model. The DRE is unbiased for $\text{Δ}$ if either $g(\cdot )$ or the $m_j(\cdot )\text{’s}$ is correctly specified, thus providing double protection against model specification and hence the name DRE.

2.2. The proposed HRUE

In biomedical studies, often the interest is to compare the efficacy of a new treatment versus that of an existing one, or more generally, compare one group versus another. So we focus on the U-statistic with a two-variate kernel, the case of the general multi-variate kernel can be dealt with similarly. Given a kernel function $h(\cdot ,\cdot )$, the goal is to find an unbiased estimate of

$$\theta =E\left[h\left(Y_i,Y_j\right)\mid T_i=1,T_j=0\right].$$

Generally $h\left(y_i,y_j\right)$ is not linear in $\left(y_i,y_j\right)$ so existing DREs do not apply.

Often in observational studies or nonrandomized trials, the treatment assignments are not completely random and the covariates are not well balanced, so, as in the classical situation, the naive U-statistic is a biased estimator for the object of interest $\theta$. To construct the HRE for U-statistic, define the true propensity score and outcome model $g_{•}\left(x_1,x_2\right)$ and $m_{•}\left(x_1,x_2\right)$ below

$$g_{•}\left(x_1,x_2\right)=E\left[T_1\left(1-T_2\right)\mid X_1=x_1,X_2=x_2\right],$$

and

$$m_{•}\left(x_1,x_2\right)=E\left[h\left(Y_1,Y_2\right)\mid T_1=1,T_2=0;X_1=x_1,X_2=x_2\right].$$

In practice $g_{•}\left(x_1,x_2\right)$ and $m_{•}\left(x_1,x_2\right)$ are unknown, and need to be estimated. Assume $\widehat{g}\left(x_1,x_2\right)$ and $\widehat{m}\left(x_1,x_2\right)$ are their estimates. Denote $z_i=\left(y_i,t_i,x_i\right)$. We define the two-sample HRUE as

$$\widehat{\theta }=\frac{1}{n(n-1)}\sum _{i\ne j}h\left(z_i,z_j\mid \widehat{g},\widehat{m}\right).$$ (1)

where

$$h\left(z_i,z_j\mid \widehat{g},\widehat{m}\right)=\frac{t_i\left(1-t_j\right)\left\{h\left(y_i,y_j\right)-\widehat{m}\left(x_i,x_j\right)\right\}}{\widehat{g}\left(x_i,x_j\right)}+\widehat{m}\left(x_i,x_j\right).$$

To check the asymptotic double robustness of $\widehat{\theta }$, note if g is correctly specified, then $\widehat{g}$ is consistently estimated of $g_{•}(\cdot ,\cdot )$, i.e., $\widehat{g}(\cdot ,\cdot )=g_{•}(\cdot ,\cdot )+o(1)$, then

$$E\left[h\left(z_i,z_j\mid \widehat{g},\widehat{m}\right)\right]=E\left\{E\left[h\left(z_i,z_j\mid \widehat{g},\widehat{m}\right)\mid x_i,x_j\right]\right\}$$

$$=E\left\{\frac{E\left[t_i\left(1-t_j\right)\mid x_i,x_j\right]\left\{E\left[h\left(y_i,y_j\right)\mid x_i,x_j\right]-\widehat{m}\left(x_i,x_j\right)\right\}}{\widehat{g}\left(x_i,x_j\right)}+\widehat{m}\left(x_i,x_j\right)\right\}$$

$$=E\left\{\frac{g_{•}\left(x_i,x_j\right)\left\{m_{•}\left(x_i,x_j\right)-\widehat{m}\left(x_i,x_j\right)\right\}}{g_{•}\left(x_i,x_j\right)+o(1)}+\widehat{m}\left(x_i,x_j\right)\right\}$$

$$=E\left\{o(1)+m_{•}\left(x_i,x_j\right)-\widehat{m}\left(x_i,x_j\right)+\widehat{m}\left(x_i,x_j\right)\right\}=\theta +o(1);$$

If m is correctly specified, then $\widehat{m}$ is asymptotically unbiased, i.e., $\widehat{m}(\cdot ,\cdot )=m_{•}(\cdot ,\cdot )+o(1)$, then

$$E\left[h\left(z_i,z_j\mid \widehat{g},\widehat{m}\right)\right]=E\left\{E\left[h\left(z_i,z_j\mid \widehat{g},\widehat{m}\right)\mid x_i,x_j\right]\right\}$$

$$=E\left\{\frac{E\left[t_i\left(1-t_j\right)\mid x_i,x_j\right]\left\{E\left[h\left(y_i,y_j\right)\mid x_i,x_j\right]-\widehat{m}\left(x_i,x_j\right)\right\}}{\widehat{g}\left(x_i,x_j\right)}+\widehat{m}\left(x_i,x_j\right)\right\}$$

$$=E\left\{\frac{g_{•}\left(x_i,x_j\right)\left\{m_{•}\left(x_i,x_j\right)-\widehat{m}\left(x_i,x_j\right)\right\}}{\widehat{g}\left(x_i,x_j\right)}+\widehat{m}\left(x_i,x_j\right)\right\}$$

$$=E\left\{\widehat{m}\left(x_i,x_j\right)\right\}=E\left\{m_{•}\left(x_i,x_j\right)+o(1)\right\}=\theta +o(1);$$

Therefore, if either $g(\cdot )$ or $m(\cdot )$ is correctly specified, $E\left[h\left(z_i,z_j\mid \widehat{g},\widehat{m}\right)\right]=\theta +o(1)$, thus $\widehat{\theta }$ is a HRUE of $\theta$. To see the specific forms of $h(\cdot ,\cdot )$, $g(\cdot ,\cdot )$ and $m(\cdot ,\cdot )$, we give a range of illustrative examples in the next subsection.

2.3. Practical examples

Let $\widehat{g}$ and $\widehat{m}$ be the estimates of g and m, which are obtained with estimation algorithms described later in Section 2.4.

Example 1.

(Two-sample mean comparison). Most literature in causal inference focus on the average treatment effect, then the kernel is $h\left(y_1,y_0\right)=y_1-y_0$, in this case $m\left(x_1,x_0\right)=E\left[h\left(Y_1,Y_0\right)\mid t,0,x_1,x_0\right]=m_1\left(x_1^{\top }{\beta }_1\right)-m_0\left(x_0^{\top }{\beta }_0\right)$, and $\theta =E\left[m\left(x_1,x_0\right)\right]=E\left[m_1\left(x_1\right)\right]-E\left[m_0\left(x_0\right)\right]={\mu }_1-{\mu }_0=\text{Δ}$. This is the classical DRE of $\text{Δ}$ given by $\widehat{\text{Δ}}={\widehat{\mu }}_1-{\widehat{\mu }}_0$ described at the beginning of this Section. The U-statistic version of the HRE is

$$\widehat{\theta }=\frac{1}{n(n-1)}\sum _{i\ne j}\left(\frac{t_i\left(1-t_j\right)\left\{h\left(y_i,y_j\right)-\widehat{m}\left(x_i,x_j\right)\right\}}{\widehat{g}\left(x_i\right)\left(1-\widehat{g}\left(x_j\right)\right)}+\widehat{m}\left(x_i,x_j\right)\right)$$

$$=\frac{1}{n(n-1)}\sum _{i\ne j}\left(\frac{t_i\left(1-t_j\right)\left\{y_i-y_j-{\widehat{m}}_1\left(x_i\right)+{\widehat{m}}_0\left(x_j\right)\right\}}{\widehat{g}\left(x_i\right)\left(1-\widehat{g}\left(x_j\right)\right)}+{\widehat{m}}_1\left(x_i\right)-{\widehat{m}}_0\left(x_j\right)\right).$$

Example 2.

(Gini’s mean difference). Gini’s mean difference is the expected absolute difference between two random variables. It is an alternative measure of dispersion in non-normal distributions. It has the kernel $h\left(y_1,y_0\right)=\left|y_1-y_0\right|$. Let $m\left(x_1,x_0\right)=E\left[\left|Y_1-Y_0\right||1,0,x_1,x_0\right]$. Assume ${ϵ}_1\sim N\left(0,{\sigma }_1^2\right)$, ${ϵ}_0\sim N\left(0,{\sigma }_0^2\right)$, then $(y_1-y_0)\mid \left(1,0,x_1,x_0\right)\sim N\left(\mu ,{\sigma }^2\right)$, with $\mu =m_1\left(x_1^{\top }{\beta }_1\right)-m_0\left(x_0^{\top }{\beta }_0\right)$, ${\sigma }^2={\sigma }_1^2+{\sigma }_0^2$, and

$$\left|y_1-y_0\right|\mid \left(1,0,x_1,x_0\right)\sim N_{(0,\infty )}\left(\mu ,{\sigma }^2\right){\mathrm{𝟙}}_{y_1>y_0}-N_{(-\infty ,0)}\left(\mu ,{\sigma }^2\right){\mathrm{𝟙}}_{y_1<y_0},$$

where $N_{(0,\infty )}\left(\mu ,{\sigma }^2\right)$ is the truncated normal on $(0,\infty )$, similarly for $N_{(-\infty ,0)}\left(\mu ,{\sigma }^2\right)$. By the formula for the mean of truncated normal, we get

$$m\left(x_1,x_0\right)=\left(\mu +\frac{\varphi \left(-\frac{\mu }{\sigma }\right)}{1-\text{Φ}\left(-\frac{\mu }{\sigma }\right)}\sigma \right)\left(1-\text{Φ}\left(-\frac{\mu }{\sigma }\right)\right)-\left(\mu -\frac{\varphi \left(-\frac{\mu }{\sigma }\right)}{\text{Φ}\left(-\frac{\mu }{\sigma }\right)}\sigma \right)\text{Φ}\left(-\frac{\mu }{\sigma }\right)$$

where $\mu =m_1\left(x_1^{\top }{\beta }_1\right)-m_0\left(x_0^{\top }{\beta }_0\right)$, and ${\sigma }^2={\sigma }_1^2+{\sigma }_0^2$.

Example 3.

(Sample variance). Let $h\left(y_1,y_0\right)={\left(y_1-y_0\right)}^2$, and $m\left(x_1,x_0\right)=E\left[{\left(Y_1-Y_0\right)}^2\mid 1,0;x_1,x_0\right]=E\left[\left(Y_1^2+Y_0^2-2Y_1{Y}_0\right)\mid 1,0;x_1,x_0\right]=E\left[\left(Y_1^2+Y_0^2\right)\mid 1,0;x_1,x_0\right]-2m_1\left(x_1^{\top }{\beta }_1\right)m_0\left(x_0^{\top }{\beta }_0\right)$. Note that $E[Y_1^2\mid 1,0;x_1,x_0]=E\left[Y_1^2\mid 1,x_1\right]=\text{Var}\left[Y_1\mid 1,x_1\right]+E^2\left[Y_1\mid 1,x_1\right]={\sigma }_1^2+m_1^2\left(x_1^{\top }{\beta }_1\right)$. Similarly, $E\left[Y_0^2\mid 1,0;x_1,x_0\right]={\sigma }_0^2+m_0^2\left(x_0^{\top }{\beta }_0\right)$. Thus,

$$m\left(x_1,x_0\right)={\left(m_1\left(x_1^{\top }{\beta }_1\right)-m_0\left(x_0^{\top }{\beta }_0\right)\right)}^2+{\sigma }_1^2+{\sigma }_0^2.$$

In the above Examples, $m(\cdot ,\cdot )$ is linear in $m_1$ and $m_0$, which can estimated separately. In the following examples, $m(\cdot ,\cdot )$ is not linear, it has the form $m\left(x_0,x_1\right)=G\left(m_0\left(x_0^{\top }{\beta }_0\right),m_1\left(x_1^{\top }{\beta }_1\right)\right)$ for some known $G(\cdot )$, and the estimation procedure is different.

Example 4.

(Wilcoxon two-sample test, Kendal’s $\tau$). The kernel is $h\left(y_1,y_0\right)=I\left(y_1>y_0\right)$, and $y\mid 1=x_1^{\top }{\beta }_1+{ϵ}_1,y\mid 0=x_0^{\top }{\beta }_0+{ϵ}_0,{ϵ}_0-{ϵ}_1\sim F$. Wilcoxon’s test is $H_0:P\left(y_1>y_0\right)=1/2$. Kendal’s $\tau =4E\left[h\left(Y_1,Y_0\right)\right]-1$. In this case,

$$m\left(x_1,x_0\right)=E\left[h\left(Y_1,Y_0\right)\mid 1,0;x_1,x_0\right]=P\left(y_1>y_0\mid 1,0;x_1,x_0\right)$$

$$=P\left({ϵ}_0-{ϵ}_1<x_1^{\top }{\beta }_1-x_0^{\top }{\beta }_0\mid 1,0;x_1,x_0\right)=F\left(x_1^{\top }{\beta }_1-x_0^{\top }{\beta }_0\right)$$

It is worth noting that as we only observe either $y_1$ or $y_0$ for each individual, not both. Thus the joint distribution of $\left(y_1,y_0\right)$ is not estimable, so is the integral $E\left[h\left(Y_1,Y_0\right)\right]$. So the estimand in the causal framework is in fact defined by $\theta =E[h\left(Y_1,Y_0\mid T_1=1,T_0=0\right]=\int h\left(y_1>y_0\right)F_1\left(\text{d}{y}_1\right)F_0\left(\text{d}{y}_0\right)$, where $F_1(\cdot )$ and $F_0(\cdot )$ are the marginal distribution functions of $y_1$ and $y_0$. This subtle difference has caused confusion on how the relevant estimand is defined and thus sometimes their misinterpretation (see more discussions in [44, 45]. The same cautionary note applies to Example 5 below.

Example 5.

[16, 18] and others used the following kernel clinical trial study. The estimand of interest is

$$\theta (F)=\sum _{j=1}^d{\alpha }_j\left(P\left(Y_{1j}<Y_{0j}\right)-P\left(Y_{1j}>Y_{0j}\right)\right)$$

Here the kernel is $h\left(t_1,y_1;t_2,y_2\right)={\sum }_{j=1}^d{\alpha }_j\left(I\left(y_{1j}<y_{0j}\right)I\left(t_{1j}=1,t_{0j}=0\right)-I\left(y_{1j}>y_{0j}\right)I\left(t_{1j}=1,t_{0j}=0\right)\right)$, $m(\cdot ,\cdot )$ has a form similar to that in Example 4.

Example 6.

[18] considered two test kits and want to compare their AUCs. The data is $\{z_i:=\left(y_i,t_i,{\delta }_i,x_i\right):i=1,\dots ,n\}$, ${\delta }_i$ is the test indicator, ${\delta }_i=1/0$ denote the first/second test. Define the kernel $h\left(z_{1i},z_{0j}\right)=I(y_{0j}\le y_{1i})I\left(t_{1i}=1,t_{0j}=0\right)\left\{I\left({\delta }_{1i}=1,{\delta }_{0j}=1\right)-I\left({\delta }_{1i}=0,{\delta }_{0j}=0\right)\right\}$, and $m(\cdot ,\cdot )$ has a form similar to that in Example 4.

Further examples are given in the Supplementary Material.

2.4. Model specification and estimation

To construct the HRUE, we will formulate a robust propensity score model $g(\cdot ,\cdot )$ and an outcome model $m(\cdot ,\cdot )$ that can be estimated efficiently. The parametric model is simple to use and efficient (if correctly specified). However, any subjectively selected model more or less deviates from the true one, and for DRE such deviation may cause serious bias. Nonparametric models are most robust and flexible, but resulting covariates’ effects are challenging to explain. In addition, they may have deplorable finite sample properties for high dimensional data $\left(x_i,x_j\right)$, which is often the case in real applications. Thus, we adopt the semiparametric approach. We will utilize a semiparametric single index model, which is more flexible and robust than the parametric model, and the coefficients of the covariates provide an easy-to-understand interpretation of the relationship between covariates and response.

2.4.1. The propensity score model

Utilizing the single index model, we formulate

$$P(T=1\mid \chi )=g\left(x^{\top }\gamma \right),g(\cdot )\in 𝓖$$ (2)

where $𝓖$ is a collection of non-negative monotone increasing functions upper bounded by 1, in particular the collection of all distribution functions, $\gamma$ is the coefficients satisfying $\gamma \in \text{Γ}=\{\gamma :\parallel \gamma \parallel =1\}$ and at least one continuous covariate with a non-zero coefficient for the identifiability. For model identifiability, a general single index model also requires ${\gamma }_1>0$ (or $<0$, see for example [43], however, since we have assumed $g(\cdot )$ to be monotone increasing, this additional requirement is not needed. In addition, we assume $\dim (x)>1$ otherwise $\gamma$ is absorbed into $g(\cdot )$.

Estimation of $(\gamma ,g)$. The log-likelihood for the treatment (or group) assignment is

$${\ell }_T\left(\gamma ,g\mid D_n\right)=\sum _{i=1}^n\left(t_i\log g\left(x_i^{\top }\gamma \right)+\left(1-t_i\right)\log \left(1-g\left(x_i^{\top }\gamma \right)\right)\right).$$ (3)

Thus, the MLE, $(\widehat{\gamma },\widehat{g})$, of the model parameter $(\gamma ,g)$ is given by

$$(\widehat{\gamma },\widehat{g})=\arg \underset{(\gamma ,g)\in (\text{Γ},𝓖)}{\max }{\ell }_T\left(\gamma ,g\mid D_n\right)$$

Since g is nonparametric, and the joint optimization is non-trivial, we describe the iterative algorithm to obtain $(\widehat{\gamma },\widehat{g})$ step-by-step below.

Step 1. Given an initial estimate ${\gamma }^{(0)}$, obtain ${\widehat{g}}^{(0)}$, then

$${\gamma }^{(1)}=\arg \underset{\gamma \in \text{Γ}}{\max }{\ell }_T\left(\gamma ,g^{(0)}\mid D_n\right).$$

Step 2. Given ${\gamma }^{(r)}$, $r=1,2,\dots ,$ we compute $g^{(r)}$ in the following way.

Arrange the ${\chi }_i^{\top }{\gamma }^{(r)}\text{'s}$ in increasing order, let the corresponding $t_i\text{'s}$ be their concomitant, we still denote them as $\left\{\left(t_i,{\chi }_i^{\top }{\gamma }^{(r)}\right):i=1,\dots ,n\right\}$ without causing confusion. Denote $g_i=g\left(x_i^{\top }{\gamma }^{(r)}\right)$. Then

$$g^{(r)}=\arg \underset{g\in 𝓖}{\max }\ell \left({\gamma }^{(r)},g\mid D_n\right)$$

$$=\arg \underset{g\in 𝓖}{\max }\sum _{i=1}^n\left(t_i\log g\left(x_i^{\top }{\gamma }^{(r)}\right)+\left(1-t_i\right)\log \left(1-g\left(x_i^{\top }{\gamma }^{(r)}\right)\right)\right).$$

By Theorem 1.5.1 in Robertson et al. [46] (see also Example 3.2.15 in [47]; p. 298), the above can be re-written as

$$g^{(r)}=\arg \underset{g\in 𝓖}{\min }\sum _{i=1}^n{\left(g\left(x_i^{\top }{\gamma }^{(r)}\right)-t_i\right)}^2.$$

Consequently, $g^{(r)}$ is equal to the solution of an isotonic regression, and $g^{(r)}$ can be obtained using (e.g.) the pool adjacent violators algorithm (PAVA), a useful one to perform order restricted maximization or minimization, and can be readily implemented with the R-package isoreg or ufit.

It can be shown readily that the sequence $\left\{\left({\gamma }^{(r)},g^{(r)}\right)\right\}$ increases the likelihood at each iteration, and will converge to at least some local maxima of $\ell \left(\gamma ,g\mid D_n\right)$. To locate the global maxima, we should try multiple starting values. If more than one stationary point is found, their log-likelihoods achieved by different starting values will be compared to determine the global maxima, which is then the MLE $(\widehat{\gamma },\widehat{g})$. Starting with the MLE of the parametric model is a reasonable choice.

If we plug in $\widehat{g}$ into the previous expression and estimate $\widehat{\gamma }$ by,

$$\widehat{\gamma }=\arg \underset{\gamma \in \text{Γ}}{\min }\sum _{i=1}^n{\left(t_i-\widehat{g}\left(x_i^{\top }\gamma \right)\right)}^2=:\arg \underset{\gamma \in \text{Γ}}{\min }{P}_n\ell (\gamma ,\widehat{g}\mid D)$$

where $P_n$ is the empirical CDF of $D_n$ and $D=(Y,T,X)$, then the $\sqrt{n}$-consistency of $\widehat{\gamma }$ is an open question [48]; Section 3.2.2; [49]; Sections 2 and 3; and [50]. The reason is that $\widehat{g}$ is a discontinuous step function, and $\gamma$ is bundled inside it [43]. proposed $\sqrt{n}$-consistent and asymptotic normal estimator of $\gamma$. In their method, they still used a discontinuous function to estimate $\gamma$, and therefore, the estimator $\gamma$ is in the sense of zero-crossing rather than an exact solution. Motivated by their method, we propose to replace $\widehat{g}$ with its smoothed version $\tilde{g}$, then we can get an exact solution instead of zero-crossing in the score equations. So we have the following step 3.

Step 3. Given a kernel $K(\cdot )$ (a density function), denote $K_h(\cdot )=h^{-1}K(\cdot /h)$ and ${\mathbb{K}}_h(\cdot )=\mathbb{K}(\cdot /h)$, with $\mathbb{K}(\cdot )=\int K(y)\text{d}y$. Then a smoothed version $\tilde{g}(\cdot )$ is

$$\tilde{g}(t)=\int {\mathbb{K}}_h(t-s)d\widehat{g}(s)=\sum _{j=1}^m{\mathbb{K}}_h\left(t-s_j\right)\left(\widehat{g}\left(s_j\right)-\widehat{g}\left(s_j-0\right)\right),$$ (4)

where $s_1,\dots ,s_m(m<n)$ are the jump points of $\widehat{g}(\cdot )$.

Then we estimate $\dot{\tilde{g}}$ by,

$$\dot{\tilde{g}}(t)=\tilde{g}(t)/\text{d}t=\int K_h(t-s)d\widehat{g}(s)=\sum _{j=1}^m{K}_h\left(t-s_j\right)\left(\widehat{g}\left(s_j\right)-\widehat{g}\left(s_j-0\right)\right)$$

and the estimator $\tilde{\gamma }$ is,

$$\tilde{\gamma }=\underset{\gamma \in \text{Γ}}{\arg }{P}_n\left\{\dot{\tilde{g}}\left(x^T\gamma \right)\left(t-\tilde{g}\left(x^{\top }\gamma \right)\right)\right\}=\mathbf{0},$$

or equivalently, without computation of $\dot{\tilde{g}}(\cdot )$,

$$\tilde{\gamma }=\arg \underset{\gamma \in \text{Γ}}{\min }{P}_n{\left(t-\tilde{g}\left(x^{\top }\gamma \right)\right)}^2.$$ (5)

2.4.2. The outcome model

Suppose $\left(y_1,x_1\right)$ is from the treatment group, and $\left(y_0,x_0\right)$ from the control group. We consider two scenarios that often occur in practice:

1. Instead of one joint model $m\left(x_1,x_0\right)$, we specify and estimate $\left(m_1,{\beta }_1\right)$ and $\left(m_0,{\beta }_0\right)$ for the two groups separately as in the typical outcome model in [15]. So we specify

   $$y_1=m_1\left(x_1^{\top }{\beta }_1\right)+{ϵ}_1,y_0=m_0\left(x_0^{\top }{\beta }_0\right)+{ϵ}_0,m_j\in 𝓜.$$ (6)

   $E\left({ϵ}_1\right)=E\left({ϵ}_0\right)=0$, and (m₁, m₀, ${\beta }_1$, ${\beta }_0$) are be estimated as in [15].
2. There are other cases in which unknown parameters can not be estimated using separate samples. Then we specify

   $$h\left(y_1,y_0\right)=m\left(x_1^{\top }{\beta }_1,x_0^{\top }{\beta }_0\right)+ϵ,m\in 𝓜.$$ (7)

Now for given $h(\cdot ,\cdot )$ we need to compute $m\left(x_1,x_0\right)=E\left[h\left(Y_1,Y_0\right)\mid x_1,x_0\right]$ in terms of $\left(m_1,m_0,{\beta }_1,{\beta }_0\right)$. Note that scenario 1 corresponds to Examples 1–3 in Section 2.3, and scenario 2 corresponds to Examples 4–6 therein. In addition, the causal estimands with $h(\cdot ,\cdot )$ given in Examples 2–6 can only be constructed using U-statistics.

Estimation of $(\beta ,m)$. We need to consider the following 2 scenarios:

1. $y_1=m_1\left(x_1^{\top }{\beta }_1\right)+{ϵ}_1,y_0=m_0\left(x_0^{\top }{\beta }_0\right)+{ϵ}_0,m_j\in 𝓜.$

   Let $a_{ij}=I\left(t_i=j\right)$, we estimate $\left({\beta }_j,m_j\right)$ by the least squares estimator (LSE),

   $$\begin{array}{c}\left({\widehat{\beta }}_j,{\widehat{m}}_j\right)=\arg \underset{\left({\beta }_j,m_j\right)\in (𝓑,𝓜)}{\min }\sum _i{a}_{ij}{\left(y_i-m_j\left(x_i^{\top }{\beta }_j\right)\right)}^2\\ :=\arg \underset{\left({\beta }_j,m_j\right)\in (𝓑,𝓜)}{\min }{P}_n\ell \left({\beta }_j,m_j\mid D\right),(j=0,1).\end{array}$$ (8)

   In particular, $\left({\widehat{\beta }}_j,{\widehat{m}}_j\right)$ can be obtained using the iterative isotonic regression method as described in [15].

   Similarly, the $\sqrt{n}$-consistency of $\widehat{\beta }$ is an open question. Therefore, we use similar smoothing techniques that are used on link function g in the propensity score model, and take an extra step and replace ${\widehat{m}}_j$ by its smoothed version ${\tilde{m}}_j$, then derived the estimator ${\tilde{\beta }}_j$ as follows.

   $${\tilde{m}}_j(t)=\int {\mathbb{K}}_h(t-s)d{\widehat{m}}_j(s)=\sum _{i=1}^m{\mathbb{K}}_h\left(t-s_i\right)\left({\widehat{m}}_j\left(s_i\right)-{\widehat{m}}_j\left(s_i-0\right)\right),(j=0,1).$$

   and the estimator ${\tilde{\beta }}_j$ is,

   $${\tilde{\beta }}_j=\arg \underset{{\beta }_j\in 𝓑}{\min }{P}_n{\left(y_j-{\tilde{m}}_j\left(x_i^{\top }{\beta }_j\right)\right)}^2,(j=0,1).$$ (9)
2. In the second scenario, we consider $m(\cdot )$ of the form

   $$h\left(y_1,y_0\right)=m\left(x_1^{\top }{\beta }_1-x_0^{\top }{\beta }_0\right)+ϵ,m\in 𝓜.$$ (10)
   Denote $\beta =\left({\beta }_1,{\beta }_0\right),x_{ij}={\left(x_{1i}^{\top },-x_{0j}^{\top }\right)}^{\top }$, then $x_{1i}^{\top }{\beta }_1-x_{0j}^{\top }{\beta }_0=x_{ij}^{\top }\beta$. Let $n_1={\sum }_{i=1}^n{t}_i$, and $n_0={\sum }_{i=1}^n\left(1-t_i\right)$, we use U-statistics to estimate $(\beta ,m)$ by the following least squares estimate.

   $$(\widehat{\beta },\widehat{m})=\arg \underset{(\beta ,m)\in \left({𝓑}^2,𝓜\right)}{\min }\frac{1}{n_1{n}_0}\sum _{i=1}^{n_1}\sum _{j=1}^{n_0}{\left(h\left(y_{1i},y_{0j}\right)-m\left(x_{ij}^{\top }\beta \right)\right)}^2.$$ (11)

In particular, $(\widehat{\beta },\widehat{m})$ can be obtained using the following iterative isotonic regression method and the steps below.

Step 1. Given an initial estimate ${\beta }^{(0)}$, get ${\widehat{m}}^{(0)},\dots$, in the rth iteration, given ${\beta }^{(r)}$, arrange the $\left(x_{1i}^{\top }{\beta }_1-x_{0j}^{\top }{\beta }_0\right)$’s in increasing order. Then

$$m^{(r)}=\arg \underset{m\in 𝓜}{\min }\frac{1}{n_1{n}_0}\sum _{i=1}^{n_1}\sum _{j=1}^{n_0}{\left(h\left(y_{1i},y_{0j}\right)-m\left(x_{ij}^{\top }{\beta }^{(r-1)}\right)\right)}^2.$$

Step 2. Plugging $m^{(r)}(\cdot )$ into (11), we update the estimate of ${\beta }_0$ to ${\beta }^{(r+1)}$ as below,

$${\beta }^{(r+1)}=\arg \underset{\beta \in B^2}{\min }\frac{1}{n_1{n}_0}\sum _{i=1}^{n_1}\sum _{j=1}^{n_0}{\left(h\left(y_{1i},y_{0j}\right)-m^{(r)}\left(x_{ij}^{\top }\beta \right)\right)}^2.$$

Let the above process iterates until convergence of $\left\{\left({\beta }^{(r)},m^{(r)}\right)\right\}$ to $(\widehat{\beta },\widehat{m})$, or just the convergence of ${\beta }^{(r)}$, since this in turn implies convergence of $m^{(r)}$. Here $(\widehat{\beta },\widehat{m})$ is computed via U-statistic, by a property of U-statistic as shown in the Supplementary Material, the $\sqrt{n}$-consistency of $\widehat{\beta }$ is guaranteed, and so the smoothing of $\widehat{m}(\cdot )$ by $\tilde{m}()$ and computation of the corresponding $\tilde{\beta }$ is not needed.

Recall that in Example 1 of Section 2.3,

$$\widehat{m}\left(x_i^{\top }{\tilde{\beta }}_1,x_j^{\top }{\tilde{\beta }}_0\right)={\widehat{m}}_1\left(x_i^{\top }{\tilde{\beta }}_1\right)-{\widehat{m}}_0\left(x_j^{\top }{\tilde{\beta }}_0\right).$$ (12)

In Example 2,

$$\widehat{m}\left(x_i^{\top }{\tilde{\beta }}_1,x_j^{\top }{\tilde{\beta }}_0\right)=\left(\mu +\frac{\varphi \left(-\frac{\mu }{\sigma }\right)}{1-\text{Φ}\left(-\frac{\mu }{\sigma }\right)}\sigma \right)\left(1-\text{Φ}\left(-\frac{\mu }{\sigma }\right)\right)-\left(\mu -\frac{\varphi \left(-\frac{\mu }{\sigma }\right)}{\text{Φ}\left(-\frac{\mu }{\sigma }\right)}\sigma \right)\text{Φ}\left(-\frac{\mu }{\sigma }\right),$$ (13)

where $\mu ={\widehat{m}}_1\left(x_i^{\top }{\tilde{\beta }}_1\right)-{\widehat{m}}_0\left(x_j^{\top }{\tilde{\beta }}_0\right)$, ${\sigma }^2={\sigma }_1^2+{\sigma }_0^2$.

In Example 3,

$$\widehat{m}\left(x_i^{\top }{\tilde{\beta }}_1,x_j^{\top }{\tilde{\beta }}_0\right)={\left({\widehat{m}}_1\left(x_i^{\top }{\tilde{\beta }}_1\right)-{\widehat{m}}_0\left(x_j^{\top }{\tilde{\beta }}_0\right)\right)}^2+{\sigma }_1^2+{\sigma }_0^2.$$ (14)

In the above, we can estimate ${\widehat{m}}_1\left(x_i^{\top }{\tilde{\beta }}_1\right)$ and ${\widehat{m}}_0\left(x_j^{\top }{\tilde{\beta }}_0\right)$ respectively by (8).

2.5. Construction of the HRUE

After the model parameters are estimated, we can construct the HRUE according to the forms of outcome models. We consider the following two main forms

1. With $\widehat{m}$ given in (12)–(14) and $\tilde{\beta }$ given in (10),

   $$\widehat{\theta }=\frac{1}{n(n-1)}\sum _{i\ne j}h\left(z_i,z_j\mid \widehat{g},\widehat{m},\tilde{\beta },\tilde{\gamma }\right),$$ (15)

   where

   $$h\left(z_i,z_j\mid \widehat{g},\widehat{m},\tilde{\beta },\tilde{\gamma }\right)=\frac{t_i\left(1-t_j\right)\left\{h\left(y_i,y_j\right)-\widehat{m}\left(x_i^{\top }{\tilde{\beta }}_1,x_j^{\top }{\tilde{\beta }}_0\right)\right\}}{\widehat{g}\left(x_i^{\top }\tilde{\gamma }\right)\left(1-\widehat{g}\left(x_j^{\top }\tilde{\gamma }\right)\right)}+\widehat{m}\left(x_i^{\top }{\tilde{\beta }}_1,x_j^{\top }{\tilde{\beta }}_0\right).$$
2. With $\widehat{m}$ and $\widehat{\beta }$ given in (11),

   $$\widehat{\theta }=\frac{1}{n(n-1)}\sum _{i\ne j}h\left(z_i,z_j\mid \widehat{g},\widehat{m},\widehat{\beta },\tilde{\gamma }\right),$$ (16)

   where

   $$h\left(z_i,z_j\mid \widehat{g},\widehat{m},\widehat{\beta },\tilde{\gamma }\right)=\frac{t_i\left(1-t_j\right)\left\{h\left(y_i,y_j\right)-\widehat{m}\left(x_i^{\top }{\widehat{\beta }}_1-x_j^{\top }{\widehat{\beta }}_0\right)\right\}}{\widehat{g}\left(x_i^{\top }\tilde{\gamma }\right)\left(1-\widehat{g}\left(x_j^{\top }\tilde{\gamma }\right)\right)}+\widehat{m}\left(x_i^{\top }{\widehat{\beta }}_1-x_j^{\top }{\widehat{\beta }}_0\right).$$

3. Asymptotic theory

Since the HRE U-statistic involves numerous estimators under different scenarios, we first study the asymptotic properties of these estimators, then that of the HRE U-statistic. In the case of separated outcome model, asymptotic properties of $\left({\widehat{\beta }}_j,{\widehat{m}}_j\right)$ and $(\widehat{\gamma },\widehat{g})$ are given in [15]. As mentioned in Section 2.4 that the rate $\sqrt{n}$-convergence and asymptotic distribution of ${\widehat{\beta }}_j$ is an open problem, below we will show $\sqrt{n}$-convergence and asymptotic distribution of ${\tilde{\beta }}_j$.

Since we consider two different scenarios for the outcome model: (1) $y=m_j\left(x^{\top }{\beta }_j\right)+{ϵ}_j$, $m_j\in 𝓜$, $(j=0,1)$; (2) $h\left(y_1,y_0\right)=m\left(x_1^{\top }{\beta }_1-x_0^{\top }{\beta }_0\right)+ϵ,(m\in 𝓜)$, we need to study the asymptotic properties of the smoothed version $\left({\tilde{\beta }}_j,{\tilde{m}}_j\right)$ and U-statistic version $(\widehat{\beta },\widehat{m})$ (Theorems 1–3) respectively, obtain those of $(\tilde{\gamma },\tilde{g})$ as their corollaries (Theorems 4–6), and those of the HR-U estimator $\widehat{\theta }$ the last (Theorems 7–8).

Let $\left({\gamma }_0,g_0\right)$ and $\left({\beta }_{j,0},m_{j,0}\right)$ be the pseudo true parameters $(j=0,1)$ as defined below,

$$\left({\gamma }_0,g_0\right)=\arg \underset{(\gamma ,g)\in (\text{Γ},𝓖)}{\max }E\left[T\log g\left(X^{\top }\gamma \right)+(1-T)\log \left(1-g\left(X^{\top }\gamma \right)\right)\right]$$

Denote $\ell \left({\beta }_0,m_0\mid t_i,y_i,x_i\right)=\left(1-t_i\right){\left(y_i-m_0\left(x_i^{\top }{\beta }_0\right)\right)}^2$, and $\ell \left({\beta }_1,m_1\mid t_i,y_i,x_i\right)=t_i{\left(y_i-m_1\left(x_i^{\top }{\beta }_1\right)\right)}^2$,

$$\left({\beta }_{j,0},m_{j,0}\right)=\arg \underset{\left({\beta }_j,m_j\right)\in (\beta ,𝓜)}{\min }E\left[\ell \left({\beta }_j,m_j\mid T,Y,X\right)\right],(j=0,1)$$

where E implies expectation with respect to the true distribution of (T, Y, X). We call $g_0\left(\cdot {\gamma }_0\right)$, $m_{j,0}\left(\cdot {\beta }_{j,0}\right)(j=0,1)$ the Kullback-Leibler projections of the true propensity score model $g_{•}(\cdot )$ and outcome model $m_{j,•}(\cdot )$ onto the specified model spaces. These projections are the true parameters only if the models are correctly specified, and then $g_0\left(x^{\top }{\gamma }_0\right)=g_{.}(x)$, $m_{j,0}\left(x^{\top }{\beta }_{j,0}\right)=m_{j,•}(x),\forall x\in 𝓧,(j=0,1)$.

Below we present the theorems under regularity conditions (C1)–(C25) given in the Supplementary Material. The proofs of theorems and the conditions are given in the Supplementary Material. Let $D_j$ be the support of $\left\{X^{\top }{\beta }_j:{\beta }_j\in 𝓑\right\}(j=0,1)$. Theorem 1 below gives strong consistency of $\left({\tilde{\beta }}_j,{\tilde{m}}_j\right)(j=0,1)$ and $\left({\widehat{\beta }}_j,\widehat{m}\right)$ to their pseudo true values.

Theorem 1.

1. With ${\tilde{m}}_j(s)$ given in (9) and ${\tilde{\beta }}_j$ given in (10), under conditions (C1) and (C2), (C4) and (C13),

   $${\tilde{\beta }}_j\stackrel{\mathit{\text{a.s.}}}{\to }{\beta }_{j,0},\underset{s\in D_j}{\sup }\left|{\tilde{m}}_j(s)-m_{j,0}(s)\right|\stackrel{\mathit{\text{a.s.}}}{\to }0,(j=0,1).$$
2. With $\widehat{m}(s)$ and $\widehat{\beta }$ given in (11), assume conditions (C2), (C16)–(C19),

   $$\widehat{\beta }\stackrel{\mathit{\text{a.s.}}}{\to }{\beta }_0,\underset{s\in D}{\sup }\left|\widehat{m}(s)-m_0(s)\right|\stackrel{\mathit{\text{a.s.}}}{\to }0.$$

To study the asymptotic distribution, and use the notations given in the Supplementary Material, we have

Theorem 2.

1. With ${\tilde{\beta }}_j$ given in (10), assume conditions of Theorem 1(i), (C5) and (C15),

   $$\sqrt{n}\left({\tilde{\beta }}_j-{\beta }_{j,0}\right)\stackrel{D}{\to }N\left(0,{\text{Ω}}_j\right),(j=0,1)$$
2. With $\widehat{\beta }$ given in (11), assume conditions of Theorem 2(i), (C18) and (C25),

   1. If $0<{\text{Φ}}_2<\infty$, then as $\min \left\{n_1,n_0\right\}\to \infty$,

      $$\sqrt{\frac{n_1{n}_0}{n_1+n_0}}\left(\widehat{\beta }-{\beta }_0\right)\stackrel{D}{\to }N\left(0,{\text{Ω}}_2\right)$$
   2. If ${\text{Φ}}_2=0$ then as $\min \left\{n_1,n_0\right\}\to \infty$,

      $$\frac{n_1{n}_0}{n_1+n_0}\left(\widehat{\beta }-{\beta }_0\right)\stackrel{D}{\to }-\sqrt{\lambda (1-\lambda )}{V}^{-}{Z}_{(1,1)}$$

Remark 1.

In the second scenario, we do not need to replace $\widehat{\beta }$ with its smoothed version $\tilde{\beta }$ since by the property of U-statistics, although the kernel function is discontinuous in β, its projection is continuous in β, and the asymptotic property is determined by the projection.

Theorem 3.

1. With ${\tilde{m}}_j(t)$ given in (9), assume (C2), (C4) and (C5), (C13), (C20)–(C23) and conditions of Theorem 2(i),

   $$n^{2/5}\left({\tilde{m}}_j(t)-m_{j,0}(t)+n^{-2/5}{b}_j\right)\stackrel{D}{\to }N\left(0,{\sigma }_j^2(t)\right),(j=0,1)$$

   where $b_j=2^{-1}{\ddot{m}}_{j,0}(t)\int \parallel x{\parallel }^{2}K(x)\text{d}x$, and ${\sigma }_j^2(t)=\int K^2(z)\text{d}z\int \frac{{(y-t)}^2}{f_i(t)}{f}_j(y\mid t)\text{d}y$ and $f_j(\cdot )$ is the density of $X^{\top }{\beta }_{j,0},f_j(y\mid t)$ is the density of $Y\mid X^{\top }{\beta }_{j,0}=t$.
2. With $\widehat{m}(s)$ given in (11), assume (C1)–(C7), (C18), (C25) and $f(s)>0$, then

   $${\left(\frac{n_1{n}_0}{n_1+n_0}\right)}^{1/3}\left(\widehat{m}(s)-m_0(s)\right)\stackrel{D}{\to }{\left(\frac{4{\sigma }^2{\dot{m}}_0(s)}{f(s)}\right)}^{1/3}\arg \underset{h\in R}{\min }\left\{\mathbb{B}(q)+q^2\right\}$$

   where ${\sigma }^2=E\left[{\left(h\left(Y_1,Y_0\right)-m_0\left(X^{\top }{\beta }_0\right)\right)}^2\right]$, and $\mathbb{B}(\cdot )$ is the two-sided Brownian motion with $\mathbb{B}(0)=0$, which is a mean zero Gaussian process on R with $E{(\mathbb{B}(r)-\mathbb{B}(q))}^2=|r-q|$ for all $r,q\in R$.

Theorem 4.

With $\tilde{g}(z)$ given in (4) and $\tilde{\gamma }$ given in (5), assume conditions of Theorem 1(i), with $\left({\beta }_{j,0},m_{j,0}\right)$ replaced by $\left({\gamma }_0,g_0\right)$, and (C14),

$$\tilde{\gamma }\stackrel{\mathit{\text{a.s.}}}{\to }{\gamma }_0,\underset{z\in [a,b]}{\sup }\left|\tilde{g}(z)-g_0(z)\right|\stackrel{\mathit{\text{a.s.}}}{\to }0.$$

Theorem 5.

With $\tilde{\gamma }$ given in (5), assume conditions of Theorem 2(i) and 4, with $\left({\beta }_{j,0},m_{j,0}\right)$ replaced by $\left({\gamma }_0,g_0\right)$, and (C10),

$$\sqrt{n}\left(\tilde{\gamma }-{\gamma }_0\right)\stackrel{D}{\to }N\left(0,{\text{Ω}}_3\right)$$

where ${\text{Ω}}_3=A_2^{-}{\text{Φ}}_3{A}_2^{-\top }$, $A_2=-E\left[{\dot{g}}_0^2\left({\gamma }_0^{\top }X\right)\left(X-{\gamma }_0^{\top }X{\gamma }_0\right)X^{\top }\right]$, and ${\text{Φ}}_3=E[{\dot{g}}_0^2\left({\gamma }_0^{\top }X\right){\left(T-g_0\left({\gamma }_0^{\top }X\right)\right)}^2\left(X-{\gamma }_0^{\top }X{\gamma }_0\right)(X-{{\gamma }_0^{\top }X{\gamma }_0)}^{\top }]$, and $A_2^{-}$ is the Moore-Penrose generalized inverse of $A_2$.

Theorem 6.

With $\tilde{g}(s)$ given in (4), assume conditions of Theorems 3(i) and 5, with $\left({\beta }_{j,0},m_{j,0}\right)$ replaced by $\left({\gamma }_0,g_0\right)$, and (C10), (C14), (C24),

$$n^{2/5}\left(\tilde{g}(s)-g_0(s)+n^{-2/5}{b}_2\right)\stackrel{D}{\to }N\left(0,{\sigma }^2(s)\right)$$

where $b_2=2^{-1}{\ddot{g}}_0(s)\int \parallel x{\parallel }^{2}K(x)\text{d}x$, and ${\sigma }^2(s)=\int K^2(z)dz\int \frac{{(t-s)}^2}{f(s)}f(t\mid s)\text{d}t$, and $f(\cdot )$ is the density of $X^{\top }{\gamma }_0,f(t\mid s)$ is the density of $T\mid X^{\top }{\gamma }_0=s$.

Finally, we study the asymptotic properties of the HRUE $\widehat{\theta }$ given in (15) and (16).

Theorem 7.

Assume that at least one of models g and m are correctly specified, and conditions of (C1)–(C3), (C8)–(C9), and conditions of Theorems 1 and 4, thenwith $\widehat{\theta }$ given in (15) or (16), $\widehat{\theta }\to {\theta }_0$ (a.s.).

To get the asymptotic distribution of $\widehat{\theta }$, the notations for Theorem 8 are given in the Supplementary Material.

Theorem 8.

Assume that at least one of models g and m are correctly specified. Assume conditions of (C1)–(C5), (C8)–(C9) and (C12), and conditions of Theorems 2 and 5 in [15]; and $0<\lambda ={\lim }_n{n}_0/n<1$.

1. With $\widehat{\theta }$ given in (15),

   $$\sqrt{n}\left(\widehat{\theta }-{\theta }_0\right)\stackrel{D}{\to }N\left(0,{\tau }_1^2\right)$$

   where ${\tau }_1^2=\text{Var}[2{\tilde{h}}_1\left(z_1\right)+S\left(Z\mid {\xi }_0\right)+{\lambda }^{-1/2}{S}_0\left(Z\mid {\xi }_0\right)+{(1-\lambda )}^{-1/2}{S}_1\left(Z\mid {\xi }_0\right)$

   $$\begin{array}{c}+A_0^{-}E\left(A_4\left(X\mid g_0,m_{0,0},m_{1,0},{\xi }_0\right)\right)S_2\left(Z\mid {\xi }_0\right)+A_1^{-}E\left(A_5\left(X\mid g_0,m_{0,0},m_{1,0},{\xi }_0\right)\right)S_3\left(Z\mid {\xi }_0\right)\\ +A_2^{-}E\left(A_6\left(X\mid g_0,m_{0,0},m_{1,0},{\xi }_0\right)\right)S_4\left(Z\mid {\xi }_0\right)]\\ {\tilde{h}}_1\left(z_1\right)={\tilde{h}}_1\left(z_1\mid g_0,m_{0,0},m_{1,0},{\xi }_0\right)\end{array}$$

   $=E\left[h\left(z_1,Z_j\mid g_0,m_{0,0},m_{1,0},{\xi }_0\right)\right]-h\left(g_0,m_{0,0},m_{1,0},{\xi }_0\right)$ with $A_0$ and $A_1$ given in Theorem 2(i), and $A_2$ given in Theorem 5.
2. With $\widehat{\theta }$ given in (16),

   $$\sqrt{n}\left(\widehat{\theta }-{\theta }_0\right)\stackrel{D}{\to }N\left(0,{\tau }_2^2\right)$$

   where ${\tau }_2^2=\text{Var}[2{\tilde{h}}_1\left(z_1\right)+S\left(Z\mid {\beta }_0,{\gamma }_0\right)+{[\lambda (1-\lambda )]}^{-1/2}{S}_{2,1}\left(Z\mid {\beta }_0,{\gamma }_0\right)$

   $$+V^{-}E\left(A_3\left(X\mid g_0,m_0,{\beta }_0,{\gamma }_0\right)\right)\mathbb{\text{H}}\left(V_1,V_2\right)+A_2^{-}E\left(A_{4,2}\left(X\mid g_0,m_0,{\beta }_0,{\gamma }_0\right)\right)S_{4,2}\left(Z\mid {\beta }_0,{\gamma }_0\right)]$$

   ${\tilde{h}}_1\left(z_1\right)={\tilde{h}}_1\left(z_1\mid g_0,m_0,{\beta }_0,{\gamma }_0\right)=E\left[h\left(z_1,Z_j\mid g_0,m_0,{\beta }_0,{\gamma }_0\right)\right]-h\left(g_0,m_0,{\beta }_0,{\gamma }_0\right)$, with V and $\mathbb{H}\left(V_1,V_2\right)$ given in Theorem 2(ii), $A_2$ given in Theorem 5.

This theorem establishes the asymptotic normality of $\widehat{\theta }$ (centered at ${\theta }_0$). The asymptotic variance ${\tau }_1^2$ or ${\tau }_2^2$ can be used as an approximate to the asymptotic variance of $\widehat{\theta }$ (centered at ${\theta }_0$). In application, these variances can be computed using re-sampling methods.

Remark 2.

1. In Theorem 8(i), replace $\tilde{\xi }$ in $A_{j,n}$ by ${\xi }_0(j=1,2,3,4,5,6)$.

   1. If $\left(m_{j,0},{\beta }_{j,0}\right)$ is correctly specified, then $m_{j,0}\left(x_i^{\top }{\beta }_{j,0}\right)=m_{j,•}\left(x_i\right)(j=0,1)$, and we have

      $$\sqrt{n}\left(\widehat{\theta }-{\theta }_0\right)\stackrel{D}{\to }N\left(0,{\tau }_{11}^2\right),$$

      where ${\tau }_{11}^2=\text{Var}\left[2{\tilde{h}}_1\left(z_1\right)+{\lambda }^{-1/2}{S}_0\left(Z\mid {\xi }_0\right)+{(1-\lambda )}^{-1/2}{S}_1\left(Z\mid {\xi }_0\right)\right]$.
   2. If $\left(g_0,{\gamma }_0\right)$ is correctly specified, then $g_0\left(x_i^{\top }{\gamma }_0\right)=g_{•}\left(x_i\right)$,

      $$\sqrt{n}\left(\widehat{\theta }-{\theta }_0\right)\stackrel{D}{\to }N\left(0,{\tau }_{12}^2\right),$$

      where ${\tau }_{12}^2=\text{Var}\left[2{\tilde{h}}_1\left(z_1\right)+S\left(Z\mid {\xi }_0\right)\right].$
   3. If all the models are correctly specified, then $m_{j,0}\left(x_i^{\top }{\beta }_{j,0}\right)=m_{j,•}\left(x_i\right)(j=0,1)$ and $g_0\left(x_i^{\top }{\gamma }_0\right)=g_{•}\left(x_i\right)$,

      $$\sqrt{n}\left(\widehat{\theta }-{\theta }_0\right)\stackrel{D}{\to }N\left(0,{\tau }_{13}^2\right),$$

      where ${\tau }_{13}^2=\text{Var}\left[2{\tilde{h}}_1\left(z_1\right)\right].$
2. In Theorem 8(ii), replace $(\widehat{\beta },\tilde{\gamma })$ in ${\tilde{A}}_{j,n}$ by $\left({\beta }_0,{\gamma }_0\right)(j=1,2,3,4)$.

   1. If $\left(m_0,{\beta }_0\right)$ is correctly specified, then $m_0\left(x_{ij}^{\top }{\beta }_0\right)=m_{•}\left(x_{ij}\right)$ and we have

      $$\sqrt{n}\left(\widehat{\theta }-{\theta }_0\right)\stackrel{D}{\to }N\left(0,{\tau }_{21}^2\right),$$

      where ${\tau }_{21}^2=\text{Var}\left[2{\tilde{h}}_1\left(z_1\right)+{(\lambda (1-\lambda ))}^{-1/2}{S}_{2,1}\left(Z\mid {\beta }_0,{\gamma }_0\right)\right]$.
   2. If $\left(g_0,{\gamma }_0\right)$ is correctly specified, then $g_0\left(x_i^{\top }{\gamma }_0\right)=g_{•}\left(x_i\right)$,

      $$\sqrt{n}\left(\widehat{\theta }-{\theta }_0\right)\stackrel{D}{\to }N\left(0,{\tau }_{22}^2\right),$$

      where ${\tau }_{22}^2=\text{Var}\left[2{\tilde{h}}_1\left(z_1\right)+S\left(Z\mid {\beta }_0,{\gamma }_0\right)\right]$.
   3. If all the models are correctly specified, then $m_0\left(x_{ij}^{\top }{\beta }_0\right)=m_{•}\left(x_{ij}\right)$ and $g_0\left(x_i^{\top }{\gamma }_0\right)=g_{•}\left(x_i\right)$,

      $$\sqrt{n}\left(\widehat{\theta }-{\theta }_0\right)\stackrel{D}{\to }N\left(0,{\tau }_{23}^2\right),$$

      where ${\tau }_{23}^2=\text{Var}\left[2{\tilde{h}}_1\left(z_1\right)\right]$.

Therefore, by the theory in [8]; $\widehat{\theta }$ is locally efficient in that when both $g\left(x^{\top }\gamma \right)$ and $m\left(x^{\top }\beta \right)$ are correctly specified, the asymptotic variance of $\widehat{\theta }$ achieves the asymptotic information lower bound.

Significance Test.

Typically we are interested in testing the null hypothesis $H_0:{\theta }_0=0$ versus $H_1:{\theta }_0>0$. Then the Wald test statistic can be constructed below. When at least one of models (2) and (6) (or (7)) is correctly specified, under $H_0$, Theorem 8 suggests that

$$T_n:=\frac{\widehat{\theta }}{\sqrt{\text{Var}(\widehat{\theta })}}\stackrel{D}{\to }N(0,1).$$

The analytic form of the asymptotic variance of $\widehat{\theta }$ given in Theorem 8 characterizes the innate relationships among the variables and the model components. In principle, we can obtain the standard error estimates using the derived analytic form. However, the analytic form involves a large number of parameters to be estimated and of very complicated functional forms. Direct computation of them is not efficient and may be infeasible, as in many papers (e.g. [51]; Theorem 1.1 in [52]; Theorem 5 in ä [53]. Since we have proved the asymptotic normality, to construct the statistical test of relevant parameters we only need to estimate the variance. Therefore, it is quite efficient to use the Bootstrap or re-sampling method to compute $\text{Var}(\widehat{\theta })$ in the above. For a given significance level $\alpha$, $H_0$ is rejected if $T_n>{\text{Φ}}^{-1}(1-\alpha )$, where ${\text{Φ}}^{-1}(\cdot )$ is the quantile function of $N(0,1)$.

4. Simulation study

We consider 4 examples that correspond to Examples 1–4 in Section 2.3. In Examples 1–3, we simulate 1000 datasets for each scenario, each with 300, 500, 1000, and 1500 subjects. In Example 4, we simulate 100 datasets for each scenario with 300 subjects. We will compare the performance of the proposed estimator with the corresponding parametric DRE U-statistic and the naive U-statistic estimator. The data generating processes are described below.

For each scenario, we first generate covariates $x_i=\left(x_{1i},x_{0i}\right)$ from 3, 5 to 7-dimensional multivariate normal distribution respectively, with different mean vectors ${\mu }_x$ and some given covariance matrix $\text{Ω}$. For the propensity score model, the treatment assignment indicator $t_i^{\prime }\text{s}$ are generated from model (2), with three different distribution functions for $g(\cdot )$: (1) $N(2,1)$; (2) logistic(2, 1); (3) logistic(0, 1). For the outcome model, in Examples 1–3, the response data $y_1$ and $y_0$ are generated from model (6), and $m_1(\cdot )$ and $m_0(\cdot )$ are generated with different distribution functions. In Example 4, $y_1$ and $y_0$ are generated as follows. The error terms ${ϵ}_1$ and ${ϵ}_0$ are $N\left(\mu ,{\sigma }^2\right)$ with different ${\mu }^{\prime }\text{s}$ and ${\sigma }^{\prime }\text{s}$.

$$y_1=x_1^{\top }{\beta }_1+{ϵ}_1,y_0=x_0^{\top }{\beta }_0+{ϵ}_0,{ϵ}_0-{ϵ}_1\sim F.$$

Then we estimate the HRUE $\widehat{\theta }$ from model (1). Also, the propensity score estimator may yield extreme weights, therefore we truncated the weights to be between 0.2 and 0.8. Given the page limit, some of the tables and figures are left in the Supplementary Material, which are referred to as, e.g., Table S6, Figure S1, with the letter S before the figure and table numbers.

Table 1 shows estimation of ${\gamma }_0$ and its standard error under three different $g^{\prime }\text{s}$: smoothed function $\tilde{g}$, step function $\widehat{g}$ and standard logistic model logistic(0, 1). The true distribution $g_0$ are N(2, 1), logistic(2, 1) and logistic(0, 1) respectively. In each scenario, ${\gamma }_0,\tilde{\gamma },\widehat{\gamma }$ and $\check{\gamma }$ represent true parameter, parameter under smoothed function $\tilde{g}$, step function $\widehat{g}$ and standard logistic model. The estimates from smoothed function and step function are similar and close to the true value. In scenarios 1 and 2, the true model deviates from the logistic model, so the parameter $\check{\gamma }$ from the standard logistic model has a large bias. However, when the true model is the logistic model in scenario 3, the parameter estimate $\check{\gamma }$ improved. Even in this scenario $\tilde{\gamma }$ and $\widehat{\gamma }$ from our proposed method are very close to $\check{\gamma }$ of the parametric model.

Table 1:

Parameter estimation of ${\gamma }_0$ under three $g^{\prime }\text{s}(n=500)$.

	${\gamma }_1$	${\gamma }_2$	${\gamma }_3$	${\gamma }_4$	${\gamma }_5$	${\gamma }_6$	${\gamma }_7$
(1) ${\gamma }_0$	0.669	−0.579	0.467
$\tilde{\gamma }$	0.666	−0.579	0.468
(SE)	0.020	0.018	0.028
$\widehat{\gamma }$	0.666	−0.579	0.468
(SE)	0.017	0.016	0.025
$\check{\gamma }$	0.413	−0.658	0.629
(SE)	0.018	0.017	0.026
(2) ${\gamma }_0$	0.403	0.522	0.568	0.304	−0.387
$\tilde{\gamma }$	0.383	0.500	0.533	0.292	−0.367
(SE)	0.126	0.134	0.181	0.155	0.089
$\widehat{\gamma }$	0.379	0.500	0.539	0.299	−0.374
(SE)	0.118	0.126	0.169	0.146	0.084
$\check{\gamma }$	0.199	0.423	0.574	0.408	−0.445
(SE)	0.126	0.115	0.182	0.140	0.076
(3) ${\gamma }_0$	−0.308	0.515	0.431	0.431	−0.380	−0.265	−0.232
$\tilde{\gamma }$	−0.280	0.482	0.419	0.402	−0.359	−0.250	−0.223
(SE)	0.133	0.117	0.181	0.137	0.071	0.105	0.111
$\widehat{\gamma }$	−0.280	0.487	0.424	0.404	−0.361	−0.251	−0.228
(SE)	0.125	0.105	0.170	0.126	0.067	0.097	0.104
$\check{\gamma }$	−0.285	0.481	0.439	0.391	−0.363	−0.266	−0.218
(SE)	0.119	0.106	0.164	0.120	0.064	0.100	0.108

$g(\cdot )$, propensity score model; ${\gamma }_0$, true parameter; $\tilde{\gamma }$, estimate under smoothed function $\tilde{g}$; $\widehat{\gamma }$, estimate under step function $\widehat{g}$; $\check{\gamma }$, estimater under standard logistic model logistic(0, 1).

The plots for function g are displayed in Figure 1 and Figures S1 and S2 (see Supplementary Material). Again, the true functions are $N(2,1)$, logistic(2, 1) and logistic(0, 1) respectively. In each figure, we present function g under different sample size: $n=300$, $n=500$, $n=1000$ and $n=3000$. As the sample size increase, the estimated step and smoothed function are closer to the true function. From Figures 1 and S1, both the step function and smoothed function are more accurate, whereas the standard logistic model deviates a lot.

Figure 1:

Propensity score model. True function N(2, 1), (solid black line), estimated step function $\widehat{g}$ (solid blue line), estimated smoothed function $\tilde{g}$ (solid red line), Logistic(0, 1, dotted line); (a): sample size n = 300, (b): n = 500, (c): n = 1000, (d): n = 3000.

For the proposed method, we used parameters $\tilde{\gamma }$ and $\tilde{\beta }$ from the smoothed function, and the step function $\widehat{g}$ and $\widehat{m}$ to estimate the treatment effect $\theta$. Note that in Example 4, we used $\tilde{\beta }$ instead of $\widehat{\beta }$ to estimate $\theta$ because the estimated $\tilde{\beta }$ and $\widehat{\beta }$ are quite similar.

Example 1.

Tables S1 and S2 (see Supplementary Material) shows regression model under treatment group $m_1$ and control group $m_0$, respectively. For details, we refer to the Supplementary Material. In each scenario, ${\beta }_0,\tilde{\beta },\widehat{\beta }$ and $\check{\beta }$ represent true parameter, parameter under smoothed function ${\tilde{m}}_j$, step function ${\widehat{m}}_j$ and linear regression model. When the true model is non-linear, the parameters from the smoothed function and the step functions are relatively close and have a smaller bias, compared to the parameters from the linear regression model.

In addition, Table 2 provides the estimate of treatment effect θ from four methods: the proposed highly robust U-statistic method (HR-U), parametric DR-U, doubly robust semiparametric method [14] and Naive method, with sample size n = 300, 500, 1000 and 1500. Our proposed HR-U method remains relatively stable and accurate. However, the parametric DR-U method has a large bias, even though in scenario 3, the logistic model is the true model and only the regression model is misspecified. Although the semiparametric doubly robust method (DR [15]), is also close to the true value, it has a larger standard error compared to our HR-U. The naive method always has a large bias and overestimates the true value. Moreover, we used the re-sampling method to examine the performance of the standard error (SE) estimates of the proposed HRUE. Table 3 shows the SE estimates of different methods under 1000 and 2000 times repetition, and the SE estimates of our proposed HR-U are relatively stable. Table 4 displays the coverage probability (CP) estimates from the four methods with Example 7, with total sample sizes of n = 50, 100, 150, 300, 500, and 1000 for both groups. The CPs of the proposed HRUE and DR are reasonably good, but those of the parametric DR-U and Naïve estimator are way off, due to their significant bias.

Table 2:

Example 1: Estimate of ${\theta }_0$ from four estimators with sample size n = 300, 500, 1000, 1500.

Scenario		$\theta$
		n = 300	n = 500	n = 1000	n = 1500
(1)	True	3.659	3.659	3.659	3.659
	HRUE	3.948	3.799	3.789	3.802
	(SE)	0.173	0.124	0.083	0.076
	Parametric DRUE	2.545	2.527	2.551	2.547
	(SE)	0.730	0.558	0.374	0.320
	DR	3.972	3.789	3.812	3.833
	(SE)	0.234	0.170	0.122	0.104
	Naïve	7.006	7.018	7.023	6.999
	(SE)	0.492	0.392	0.273	0.226
(2)	True	2.781	2.781	2.781	2.781
	HRUE	2.890	2.786	2.716	2.710
	(SE)	0.439	0.392	0.307	0.312
	Parametric DRUE	3.859	3.865	3.841	3.857
	(SE)	0.417	0.324	0.242	0.244
	DR	2.809	2.729	2.681	2.669
	(SE)	0.452	0.400	0.313	0.317
	Naïve	7.776	7.786	7.783	7.784
	(SE)	0.310	0.240	0.177	0.172
(3)	True	1.753	1.753	1.753	1.753
	HRUE	1.791	1.765	1.673	1.701
	(SE)	0.161	0.123	0.089	0.071
	Parametric DRUE	1.929	1.887	1.891	1.900
	(SE)	0.574	0.435	0.306	0.256
	DR	1.662	1.715	1.751	1.734
	(SE)	0.280	0.215	0.152	0.127
	Naïve	4.650	4.645	4.641	4.631
	(SE)	0.618	0.487	0.323	0.267

$\theta$, treatment effect; HRUE, the proposed semiparametric HRE U-statistic; Parametric DRUE, parametric DRE U-statistic; DR, doubly robust semiparametric method [14]; Naïve, naïve U-statistic; SE, estimated standard error of the corresponding estimator; The 3 scenarios correspond to scenarios 1–3 in Tables 1–3.

Table 3:

Example 1: Estimate of standard error (SE) under 1000 and 2000 repetitions from four estimators with sample size n = 300, 500, 1000, 1500.

Number of repetitions		SE
		n = 300	n = 500	n = 1000	n = 1500
1000	HRUE	0.173	0.124	0.083	0.076
	Parametric DR-U	0.730	0.558	0.374	0.320
	DR	0.234	0.170	0.122	0.104
	Naïve	0.492	0.392	0.273	0.226
2000	HRUE	0.296	0.213	0.140	0.122
	Parametric DR-U	0.712	0.551	0.372	0.316
	DR	0.324	0.233	0.160	0.144
	Naïve	0.493	0.375	0.271	0.224

HRUE, the proposed semiparametric HRE U-statistic; Parametric DRUE, parametric DRE U-statistic; DR, doubly robust semiparametric method [13]; Naïve, naïve U-statistic; SE, estimated standard error of the corresponding estimator; Number of repetitions, used the re-sampling method under 1000 and 2000 repetitions to derive the standard error estimates.

Table 4:

Example 1: Estimate of coverage probability (CP) from four estimators with different total sample size.

	CP
	n = 50	n = 100	n = 150	n = 300	n = 500	n = 1000
HRUE	0.646	0.741	0.771	0.891	0.938	0.983
Parametric DR-U	0.637	0.608	0.516	0.359	0.205	0.047
DR	0.73	0.791	0.823	0.906	0.949	0.986
Naïve	0.235	0.056	0.007	0	0	0

HRUE, the proposed semiparametric HRE U-statistic; Parametric DRUE, parametric DRE U-statistic; DR, doubly robust semiparametric method [15]; Naïve, naïve U-statistic; CP, coverage probability of the corresponding estimator.

Example 2.

Table 5 presents the parameter estimation of regression model under treatment group m₁. The three scenarios are (1): 3 covariates and $m_1:15\times \exp (1.5)$; (2): 5 covariates and $m_1:12\times N(1.5,1)$; (3): 7 covariates and $m_1:13\times \text{Beta}(0.5,0.5)$. Table 6 shows parameters from $m_0$: (1): 3 covariates and $m_0$: 13× Gamma(0.5, 1); (2) 5 covariates and $m_0$: 19× Gamma(1, 1); (3) 7 covariates and $m_0$: 8× Beta(2, 2). Table S3 shows the estimate of treatment effect $\theta$ from three methods: the proposed HRE U-statistic, parametric DRE U-statistic, and Naive method, with different sample sizes. The Naive estimator in this case is given by $\widehat{\theta }={\sum }_{i=1}^{n_1}{\sum }_{j=1}^{n_0}\left|y_{1i}-y_{0j}\right|/\left(n_1{n}_0\right)$. The proposed HRE U-statistic is still superior to the other two methods.

Table 5:

Example 2: Parameter estimation of ${\beta }_{1,0}$ under three $m_1(\cdot {)}^{\prime }\text{s}$ (n = 500).

	$\sigma$	${\beta }_1$	${\beta }_2$	${\beta }_3$	${\beta }_4$	${\beta }_5$	${\beta }_6$	${\beta }_7$
(1) True	1.150	0.689	0.552	0.469
Smoothed	1.097	0.801	0.574	0.472
(SE)	0.240	0.166	0.199	0.178
Step function	1.097	0.739	0.599	0.498
(SE)	0.240	0.149	0.198	0.176
Parametric	5.000	3.804	0.113	−0.471
(SE)	0.004	0.160	0.259	0.221
(2) True	0.840	0.528	0.322	0.299	0.465	0.558
Smoothed	0.804	0.541	0.356	0.328	0.483	0.560
(SE)	0.090	0.069	0.073	0.119	0.080	0.055
Step function	0.804	0.533	0.349	0.344	0.476	0.548
(SE)	0.090	0.035	0.058	0.071	0.050	0.043
Parametric	4.315	2.285	0.901	−0.295	−1.036	1.310
(SE)	0.456	0.232	0.202	0.334	0.246	0.145
(3) True	0.650	0.312	0.234	0.308	0.430	0.516	0.430	0.340
Smoothed	0.673	0.406	0.304	0.414	0.562	0.651	0.578	0.441
(SE)	0.150	0.126	0.135	0.118	0.156	0.106	0.112	0.118
Step function	0.673	0.355	0.266	0.368	0.489	0.570	0.502	0.396
(SE)	0.150	0.091	0.085	0.096	0.108	0.087	0.091	0.096
Parametric	5.000	0.215	1.020	−0.504	0.527	1.247	−0.628	0.824
(SE)	0.000	0.598	0.626	0.471	0.666	0.328	0.350	0.423

$m_1(\cdot )$, regression model under treatment group; True, true parameter; Smoothed, estimate under smoothed function ${\tilde{m}}_1$; Step function, estimate under step function ${\widehat{m}}_1$; Parametric, estimate under linear regression model.

Table 6:

Example 2: estimate of ${\beta }_{0,0}$ under three $m_0(\cdot {)}^{\prime }\text{s}$ (n = 500).

	$\sigma$	${\beta }_1$	${\beta }_2$	${\beta }_3$	${\beta }_4$	${\beta }_5$	${\beta }_6$	${\beta }_7$
(1) True	1.080	0.663	0.310	0.682
Smoothed	1.043	0.769	0.367	0.804
(SE)	0.232	0.058	0.038	0.058
Step function	1.043	0.657	0.313	0.678
(SE)	0.232	0.047	0.026	0.045
Parametric	5.000	3.078	0.796	0.810
(SE)	0.000	0.165	0.175	0.116
(2) True	1.030	0.322	0.477	0.466	0.310	0.596
Smoothed	0.851	0.345	0.521	0.508	0.349	0.645
(SE)	0.149	0.040	0.039	0.060	0.048	0.031
Step function	0.851	0.324	0.484	0.468	0.319	0.598
(SE)	0.149	0.020	0.023	0.027	0.028	0.019
Parametric	5.000	2.490	1.928	0.881	−0.211	2.658
(SE)	0.000	0.622	0.565	0.919	0.667	0.391
(3) True	0.740	0.496	0.266	0.283	0.318	0.414	0.318	0.479
Smoothed	0.788	0.748	0.447	0.450	0.557	0.614	0.544	0.734
(SE)	0.162	0.200	0.177	0.151	0.205	0.149	0.150	0.192
Step function	0.788	0.606	0.402	0.353	0.487	0.513	0.451	0.604
(SE)	0.162	0.121	0.117	0.097	0.144	0.104	0.104	0.147
Parametric	5.000	0.215	1.020	−0.504	0.527	1.247	−0.628	0.824
(SE)	0.000	0.598	0.626	0.471	0.666	0.328	0.350	0.423

$m_0(\cdot )$, regression model under control group; True, true parameter; Smoothed, estimate under smoothed function ${\tilde{m}}_0$; Step function, estimate under step function ${\widehat{m}}_0$; Parametric, estimate under linear regression model.

Example 3.

Tables S4 and S5 provide the parameter estimation of $m_1$ and $m_0$ respectively (see details in the Supplementary Material). In this example, the Naive estimator is $\widehat{\theta }={\sum }_{i=1}^{n_1}{\sum }_{j=1}^{n_0}{\left(y_{1i}-y_{0j}\right)}^2/\left(n_1{n}_0\right)$. The estimate of treatment effect $\theta$ from three methods in Table S6 also shows the superiority of the proposed HRE U-statistic method.

The plots for function $m_1$ and $m_0$ are provided in Figures 2 and 3 respectively. The sample sizes for both are $n=500$. The estimated smoothed and step functions are quite close to the respective true functions

Figure 2:

Regression model $m_1$ (sample size n = 500). True function (black line), estimated step function ${\widehat{m}}_1$ (blue line), estimated smoothed function ${\widehat{m}}_1$ (red line); (a): $m_1$: 21× N(0, 1.5), (b): 12× Gamma(1, 2), (c): 15× exp(1.5), (d): 13× Beta(0.5, 0.5).

Figure 3:

Regression model $m_0$ (sample size n = 500). True function (black line), estimated step function ${\widehat{m}}_0$ (blue line), estimated smoothed function ${\tilde{m}}_0$ (red line); (a): $m_0$: 13× Gamma(0.5, 1), (b): 8× Beta(2, 2), (c): 11× N(0, 4), (d): 8× N(1, 3).

Example 4.

Here, we simulate 100 datasets with 300 subjects. Table S7 presents the parameter estimation of the regression model $m\left(x_1,x_0\right)$ (see details in the Supplementary Material). The parameter estimates of the step function and the smoothed function are more accurate than the parametric model with a smaller bias. In addition, the parameter estimation of the propensity score model is given in scenario 1 of Table 1, with $g(\cdot )=N(2,1)$

Table S8 presents the estimate of treatment effect $\theta$ from three methods: the proposed HRE U-statistic, parametric DRE U-statistic, and Naive method $\left(\widehat{\theta }={\sum }_{i=1}^{n_1}{\sum }_{j=1}^{n_0}I\left(y_{1i}>y_{0j}\right)/\left(n_1{n}_0\right)\right)$, with sample size $n=300$. Our proposed HRE U-statistic is quite close to the true effect whereas the Naive method deviates from the true effect a lot. The parametric DRE U-statistic performs well since the true distribution functions are normal distributions, and we assume the parametric model also has a normal distribution, even in this scenario, our proposed HRE U-statistic performs better than the parametric model.

Further simulation results support the same conclusion but are given in the Supplementary Material.

5. Real data analysis

Now we analyze the data ACTG 175 (https://actgnetwork.org/) using the proposed method and compare the results with those from the parametric DRU and naive U-statistic. ACTG 175 is a randomized, double-blind, placebo-controlled trial for studying different treatments in adults infected with HIV (type I) with CD4 cell counts between 200 and 500/mm³. Patients from 43 AIDS Clinical Trials Units and 9 US and Puerto Rico National Hemophilia Foundation sites are randomly assigned to one of four treatment groups: (1) zidovudine and didanosine, (2) zidovudine and zalcitabine, (3) zidovudine alone, and (4) didanosine alone.

Our analysis will compare CD4 counts between treatment 1 (n = 522): zidovudine and didanosine and treatment 2 (n = 524): zidovudine and zalcitabine [14]. utilized a robust semiparametric version of the doubly robust method to estimate $h\left(y_1,y_0\right)=y_1-y_0$ and the estimated treatment difference is 30.681. However, CD4 cell count is not normally distributed and a doubly robust estimator with Wilcoxon rank-sum-type statistics in comparing two treatments may be more appropriate. Here we extend this problem to estimating $h\left(y_1,y_0\right)=I\left(y_1>y_0\right)$, which are illustrated as Example 10 in Section 2.3.

In this analysis, the CD4 cell counts were treated as responses, and four variables such as age (years), homosexual activity (0 = no, 1 = yes), Karnofsky score (on a scale of 0–100) and number of days of previously received antiretroviral therapy, were covariates of interest. The parameter estimates from the regression model $m\left(x_1,x_0\right)$ is $\tilde{\beta }$ = (−0.274, 0.257, 1.830, 0.176, −0.297, 0.146, 1.669, −0.049).

Table 7 shows the U-statistic estimate of $\theta$ from our proposed HR-U method, parametric DR-U method and naive method under $h\left(y_1,y_0\right)=I\left(y_1>y_0\right)$. In addition, $\widehat{\theta }$ and $SE(\widehat{\theta })$ are obtained by bootstrap. Since it is a randomized trial and the covariates are balanced between two groups, the treatment effect θ for our proposed HRUE, the parametric DRUE and the Naïve method are relatively close but the parametric DRUE is not as efficient with a larger SE. The treatment effects are statistically significant (P-value < 0.001) for both the proposed DRUE and the Naïve estimate, whereas the effect is not significant under parametric DRUE. Figure 4 shows the estimated propensity score and regression functions.

Table 7:

Real data analysis: estimation of ${\theta }_0$ under three methods.

Method	$\theta$	SE ($\theta$)	P-value
HRUE	0.513	0.074	<0.001
Parametric DRUE	0.533	0.320	0.095
Naïve	0.554	0.017	<0.001

$\theta$, treatment effect; HRUE, the proposed semiparametric HRE U-statistic; Parametric DRUE, parametric DR U-statistic method; Naïve, naïve U-statistic; SE, estimated standard error of the corresponding estimator.

Figure 4:

Propensity score model and regression models (real data). (a) And (b): g and $m\left(x_1,x_0\right)$. Estimated step function (black line), estimated smoothed function (red line).

6. Conclusion and discussion

We have proposed a novel and broadly applicable methodology for a general class of causal estimands based on U-statistic and enhanced doubly robustness. The method utilizes semiparametric specifications for the propensity score and outcome models, further improving robustness in DRE. We have derived comprehensive asymptotic properties of U-statistic HRE. Simulation studies have shown that the new U-statistic HREs are more accurate and stable than the corresponding parametric U-statistics and the naive or the average treatment effect estimators. We have shown that the proposed highly robust U-statistic estimate addressed a fatal issue with DRE [12, 13]; namely, it is too sensitive to mild model misspecifications.

Our simulations considered 4 examples that correspond to different causal estimands. Based on the simulation results, the proposed model outperforms the parametric U-statistics and the naive estimators. However, when the dimension of the covariate is high, e.g., greater than 10, it will require considerable computational time to achieve convergence. This phenomenon is quite common for most models. In this case, we can use dimension reduction methods such as LASSO to select relevant covariates.

We have considered various problems where the outcome measures and the estimands of interest are functionals of multiple distributions estimated using U-statistics and where the existing DRE does not apply. It has relevance to economics to biomedicine. The Gini’s mean difference is widely used in economics and is a measure of dispersion to represent income inequality or wealth inequality among individuals. It can also be a dispersion measure for different kinds of data. In addition, U-statistics are used in studies with multiple primary endpoints that compare the new treatment against existing ones, AUCs of two diagnostic tests, and non-normal and non-numeric data. The proposed approach provides a causal estimate for these problems for the first time. Indeed the robust causal U-statistic estimate framework is general and can be applied to other real-world issues. Therefore, the significance of the proposed is not only because of its inherent robustness with semiparametric specifications but also its broad applicability. It would be of interest to extend the methods and apply them to new applications in the future.