Average treatment effect on the treated, under lack of positivity

Consider a non-randomized study, with a treatment assignment indicator $Z\in\{0,1\}$ that determines whether a participant received the treatment ($Z=1$) or a control ($Z=0$). Let $Y$ denote the observed continuous outcome, and ${X}=(X_{1},\ldots,X_{p})^{\prime}$ denote the vector of measured covariates. The observed data $\mathcal{O}=\{(Z_{i},X_{i},Y_{i}):i=1,\dots,N\}$ represent an independently identically distributed (i.i.d.) sample of $N$ participants from a large population.

We follow the potential outcome framework of Neyman-Rubin (neyman1923applications, ; imbens2015causal, ) and assume that each participant has two (ex-ante fixed, but a priori unknown) potential outcomes associated with treatment assignment, denoted by $Y_{i}(z)$ for $z=0,1$, where $Y_{i}=Z_{i}Y_{i}(1)+(1-Z_{i})Y_{i}(0)$. The potential outcome $Y_{i}(z)$ represents the outcome a participant $i$ will experience if, possibly contrary to fact, assigned to treatment $Z_{i}=z_{i}$.

Although a participant’s causal treatment effect is defined as $Y_{i}(1)-Y_{i}(0)$, we cannot observed both $Y_{i}(1)$ and $Y_{i}(0)$ simultaneously. Instead, we must rely on a sample of participants to estimate average treatment effects. For this purpose, we assume the stable-unit treatment value assumption (SUTVA), i.e., there is only one version of the treatment, and the potential outcome $Y_{i}(z)$ of an individual does not depend on and does not impact another individual’s received treatment. (rosenbaum1983central, )

The PS defined by a subject-specific probability of receiving the treatment given the subject’s covariates, i.e., $e_{i}({x})=P(Z_{i}=1\mid{X_{i}=x_{i}})$, will play a major role in this paper. The PS is unknown for a non-randomized study; we usually postulate a (parametric) model (e.g., using a generalized linear model (GLM)) to estimate it. Hereafter, we assume the PS model is correctly specified by a GLM and use the notation $e_{i}(X_{i};\beta)$ where $\beta$ is the vector of regression coefficients for the covariates in the model. The correct model specification means that there exists $\beta^{*}$ in the space of parameters $\beta$ such that $e_{i}(X_{i};\beta^{*})$ is indeed the true PS. The estimated PS is then given by $\widehat{e}_{i}(X_{i})=e(X_{i};\widehat{\beta}_{N})$ where $\widehat{\beta}_{N}$ is a consistent estimator for $\beta$.

Our main estimand of interest, the average treatment effect on the treated (ATT), is defined by

The first term $\mathbb{E}\{Y\mid Z=1\}$ is potentially observable and can be estimated from the data. However, the second term, $\mathbb{E}\{Y(0)\mid Z=1\}$—the average outcome among treated participants had they not been treated—is not observable since for any particular treated participant, we cannot observe (i.e. measure) $Y(0)$. The main idea of most causal inference methods in estimating ATT is to leverage measured data from control participants to impute missing the counterfactuals $Y_{i}(0)$ in treated participants (i.e., as good proxies) to provide a consistent estimate of $\mathbb{E}\{Y(0)\mid Z=1\}$ by making some specific, often untestable, assumptions.

To estimate ATT using PS weighting methods, there are two important assumptions to be made.

1. 1.

   Unconfoundness: $\mathbb{E}\{Y(0)\mid Z=0,X\}=\mathbb{E}\{Y(0)\mid Z=1,X\}$

2. 2.

   Positivity: (a) $P(Z=1)>0$, and (b) $P(e(X)<1)=1$ on control participants.

Remarks. Although the Assumption 1 is weaker than the uncounfoundness assumption often used in the literature, it however suffices (along with Assumption 2) to identify ATT. Assumption 2(a), $P(Z=1)>0$, is trivial as we need a fraction of the population to receive the treatment to estimate ATT. The assumption 2(b), $e(X)<1$ with probability 1 on control participants, implies that the support $Supp(X\mid Z=0)$ of covariates for the participants under control contains the support $Supp(X\mid Z=1)$ of treated participants (see Abadie (abadie2005semiparametric, ) and Heckman et al. (heckman1998matching, ) for important insights on these conditions). Note that the above Assumption 2(b) is an identification, which is not needed in other estimation methods of ATT, such as matching where the focus is on the common support $Supp(X\mid Z=1)\cap Supp(X\mid Z=0)$. For matching, the impetus is to find matches that are fairly similar; thus, we emphasize a good overlap of the PS distributions between the treatment groups.stuart2010matching

Under these two assumptions, it can be shown that (see Appendix A.1)

where $w_{0}(X)={e(X)}\{1-e(X)\}^{-1}$. These conventional ATT weights $w_{0}(X)$ generate a pseudo-population among the control participants by re-weighting their contributions, hirano2003efficient ; li2018balancing which helps the covariate distribution of these control group participants look as similar as possible to that of the treatment participants. Thus, by Assumption 1 (unconfoundness), the (weighted) observed outcomes of the control participants can serve as counterfactual alternatives of $Y_{i}(0)$ for treated participants $i$ to finally estimate $\tau_{att}.$ In a sense, the weights explicitly indicate the contribution of each control participant in approximating such counterfactuals, based on their covariate distributions. Thus, using the group of the treated participants as reference, control participants are either up-weighted or down-weighted, depending on whether their PSs are similar to those of treated participants: the higher the PS, the higher the weight.

An estimator of $\tau_{att}$ is given by

where $\widehat{w}_{0}(X_{i})={\widehat{e}(X_{i})}\{{1-\widehat{e}(X_{i})}\}^{-1}$. Such a normalized estimator (i.e., where the weights within each treatment group sum up to 1) such as (2) is often preferred in finite sample as it is more stable and more efficient. busso2009finite ; matsouaka2020framework ; matsouaka2023variance

When the PS model is correctly specified, the above estimator is consistent for ATT and has an asymptotic normal distribution. It can be seen that the positivity assumption is extremely important since large PSs (close to 1) for the control participants will result in extreme values of $w_{0}(x)$. Under finite-sample, extreme weights may incur an inefficient estimate. In addition, it has been shown that the estimator (2) is sensitive to extreme weights and model misspecifications. (matsouaka2024overlap, ; matsouaka2023variance, )

Two widely-used strategies for dealing with the lack of positivity (i.e., when extreme weights exist) are PS trimming and truncation. ATT trimming drops all control participants whose estimated PSs are outside of the interval $[0,1-\alpha]$ from the study sample, with $\alpha\in(0,0.5)$ a user-specified threshold. Because trimming is not smooth, one of its key drawbacks is that the use of bootstrap to estimate the variance is not always valid, as the condition needed to adequately apply the bootstrap technique does not hold.yang2018asymptotic Instead of abruptly discarding control participants whose PS is above $1-\alpha$, Yang and Ding yang2018asymptotic propose to reduce their weights gradually and smoothly toward 0 (but still approximate the above trimmed sample) and derive an estimator that has better asymptotic properties. With truncation, instead of reducing the weights $w_{0}(x)={e(x)}\{{1-e(x)}\}^{-1}$ of control participants whose PSs fall outside of the interval $[0,1-\alpha]$ to 0, they are replaced by a constant value $w_{0}(x)=({1-\alpha})\alpha^{-1}$.

There have been two approaches for ATT trimming (resp. ATT truncation) participants in the literature. One is to trim (resp. truncate) both the treated and control participants, the other is to only trim (resp. truncate) the control group. Heckman et al.heckman1998matching ; heckman1998characterizing and Smith and Toddsmith2005does excluded all observations in both the treated and control groups with estimated PS of 1 or near 1. Yang and Ding argued to trim both the treated and control groups, because the control units with estimated PS close to 1 lead to large weights while the treated units with large estimated PS have few counterparts in the control group to make inference on.yang2018asymptotic Austin excluded any participant with an estimated PS outside the range of [$\alpha$, $1-\alpha$].austin2022bootstrap

In contrast, Lee et al. set all weights in the control group above a certain threshold equal to the threshold. lee2011weight Dehejia and Wahba discarded the units in the control group whose estimated PS was smaller than the smallest estimated PS in the treated group. dehejia1999causal Crump et al. followed Dehejia and Wahba by dropping participants in the control group only, but those with estimated PS greater than a certain threshold.crump2006moving

In this paper, we trim the control participants whose estimated PS falls outside the range $[0,1-\alpha]$, but leave the treated group intact. In addition, after trimming, we considered two specific strategies: (1) not to re-estimate the PS (for instance, as adopted by Sturmer et al. sturmer2010treatment ) and (2) to re-run the PS model on the trimmed sample and re-estimate PSs, following the recommendation from Li et al.li2019addressing

The above ATT trimming and ATT truncation estimators (in Section 2.2) proposed to deal with issues related to violations of the positivity Assumption 2 are all scaled version of the ATT, in the sense that the weights $w_{0}(X)$ for control participants are scaled by a multiplying factor $h(X)$ (a function of the PS $e(X)$) to circumvent such violations. Indeed, these methods change the weights on the control in the estimand (1), going from $w_{0}(X)$ to $w_{0}(X)h(X)$. In this regard, they estimate the treatment effect on the treated over a subset $\mathcal{H}$ of the support of $X$.

First, for the standard (non-smooth) ATT trimming with threshold $\alpha\in(0,0.5)$, we can either use (i) $\mathcal{H}=\{(X,Z):0\leq Ze(X)\leq 1,0\leq(1-Z)e(X)\leq 1-\alpha\}$ and $h(X)=1$, or (ii) $\mathcal{H}=\{X:0\leq e(X)\leq 1\}$ and ${h(X)=\boldsymbol{1}\{0\leq e(X)\leq 1-\alpha\}}$ to define its target estimand. Illustration (i) shows that trimming is applying ATT to the subpopulation defined by $\mathcal{H}$ , whereas (ii) indicates that trimming can also be viewed as using a non-smooth function to weight on the control group.

For smooth ATT trimming,yang2018asymptotic with a threshold $\alpha$, we have $\mathcal{H}=\{X:0\leq e(X)\leq 1\}$ and $h(X)=\Phi_{\varepsilon}(1-e(X)-\alpha)$ where, for some $\varepsilon>0$, $\Phi_{\varepsilon}$ is the cumulative distribution function (CDF) of a normal distribution, with mean zero and variance $\varepsilon^{2}.$ Whenever $\varepsilon\to 0$, $h(X)\to{\boldsymbol{1}\{0\leq e(X)\leq 1-\alpha\}}$, i.e., for some small $\varepsilon$, this method approximates standard ATT trimming (based on the above illustration (ii)), but using a smooth weight function (everywhere) on the control. In other words, smooth ATT trimming cannot be illustrated using (i) above, because it does not really trim samples with $e(X)>1-\alpha$, but applies on them a weight that is very close to 0.

Finally, for ATT truncation, with a threshold $\alpha,$ we have $\mathcal{H}=\{X:0\leq e(X)\leq 1\}$ and $h(X)=\boldsymbol{1}(0\leq e(X)<1-\alpha)+(1-\alpha)\alpha^{-1}{w_{0}(X)}^{-1}\boldsymbol{1}\{e(X)\geq 1-\alpha\}$.

Overall, following the weighted average treatment effects (WATE) framework,li2018balancing ; matsouaka2020framework all the above estimands can be wrapped up into a class of generalized versions of ATT, called weighted average treatment effect on the treated (WATT), and defined by

The function $h(x)$, hereafter the tilting function, specifies the weighted target population $\mathcal{H}$ for control participants, which is a subset $\mathcal{H}$ of the support $Supp(X\mid Z=0)$ of the covariates $X$. By abuse of language, the estimand $\tau_{watt}^{h}(\mathcal{H})$ can also be referred to as the treatment effect on the treated over the subset $\mathcal{H}$, whenever the context allows.

To identify WATT from the observed data, we propose the following simple weighted estimators:

where $\widehat{\omega}_{0h}(X_{i})=\widehat{w}_{0}(X_{i})\widehat{h}(X_{i})$ and $\widehat{h}(X_{i})$ is calculated by plugging $\widehat{e}(X_{i})$ into the function $h(X_{i})$. Evidently, when $h(X)\propto 1$, $\tau_{watt}^{h}(\mathcal{H})$ and $\widehat{\tau}_{watt}^{h}(\mathcal{H})$ simplify to $\tau_{att}$ and $\widehat{\tau}_{att}$, respectively. Furthermore, by classic asymptotic theory, we can show that when the PS model is correctly specified, $\widehat{\tau}_{watt}^{h}(\mathcal{H})$ is a consistent estimator of $\tau_{watt}^{h}(\mathcal{H})$.

By using either ATT trimming or truncation, we implicitly make the decision to change the target of interest, crump2006moving as shown by the above formulas (3) and (4). ATT trimming (resp. truncation) excludes (resp. stabilizes) the weights of control participants whose PS are near or equal to 1. Unfortunately, these methods are ad-hoc and thus subjective; they depend on the choice of the user-specified threshold $\alpha$, which can have a substantial impact on the estimation results. Furthermore, for the smooth ATT trimming,yang2018asymptotic we also need to select an appropriate value for $\varepsilon$.

Although Yang and Dingyang2018asymptotic suggested using a small $\varepsilon$, such as $\varepsilon=10^{-4}$ or $10^{-5}$, there is actually a bias-variance trade-off that needs to be made. While choosing smaller $\varepsilon$ helps achieve a better estimate of $\tau_{watt}^{h}(\mathcal{H})$, when the PSs are correctly estimated, it also leads to weights that are closer or similar to those from the standard ATT trimming and increases the variance of $\widehat{\tau}_{watt}^{h}(\mathcal{H})$. Thus, it is often recommended to run a sensitivity analysis by varying $\varepsilon$ over a grid of possible values.

For both ATT trimming and truncation, there is not a general consensus on which value of $\alpha$ (and $\varepsilon$) achieves the optimal bias-variance trade-off. (crump2009dealing, ; yang2018asymptotic, ) Similar concerns have been voiced in the estimation of the average treatment effect.zhou2020propensity ; matsouaka2020framework Therefore, given the broad discretion to subjectively choose a threshold $\alpha$ (and the smoothing parameter $\varepsilon$), there is a higher risk to cherry-pick and report results that suit us, hence falling into the trap we wanted to avoid by using a PS method in the first place.(rubin2007design, ; rubin2008objective, )

What should we do then? We propose the use of a method that is robust to and independent of such user-specified threshold and parameters.

We consider $\mathcal{H}=\{X:0\leq e(X)\leq 1\}$ and $h(X)=e(X)\{1-e(X)\}$, i.e., the tilting function of the overlap weight estimator. li2018balancing We refer to $\tau_{watt}^{h}(\mathcal{H})$ as the overlap weight average treatment effect on the treated (OWATT). It circumvents the need to specify a threshold $\alpha$ (and $\varepsilon$), but still provides smooth weighting (without extreme weights or the need for ATT trimming), and has better asymptotic properties. (hirano2003efficient, ; li2018balancing, )

The function $h(X)=e(X)\{1-e(X)\}$ targets the population of participants for whom there is a clinical equipoise, i.e., those who have high probabilities of receiving either treatment option. Since $h(X)$ reaches its maximum value at $e(X)=0.5$, it weighs the most participants at “equipoise”(matsouaka2020framework, ; matsouaka2024overlap, ) (mimicking 1:1 allocation in a randomized trial). The weights $\omega_{0h}(X)$ simplifies to $e(X)^{2}$ and do not involve $(1-e(X))^{-1}$, which circumvent the issue of large weights altogether.

As function of $e(X)$, the weights $\omega_{0h}(X)=e(X)^{2}$ are monotone increasing throughout the whole PS domain $\{e(X):0\leq e(X)\leq 1\}$. They reach their highest value at $e(X)=1$, which contrast drastically with the behaviors of ATT truncation and ATT trimming weights, in the region $\mathcal{R}=\{e(X):e(X)\geq 1-\alpha\}$. For ATT truncation $\omega_{0h}(X)=(1-\alpha)\alpha^{-1}$ is constant on $\mathcal{R}$, regardless of the individual PSs, while the ATT trimming weights are either 0 (standard trimming) or slowly decrease to 0 (smooth ATT trimming). Furthermore, when $e(X)$ approaches 1, oddly enough, the weights for smooth ATT trimming go back up to infinity, defeating the purpose of why ATT trimming was used in the first place (see Figure 1 below).

Why are these properties of $\omega_{0h}(X)=e(X)^{2}$ important? The goal of any PS weighting to estimate ATT-like estimands is to judiciously use the contributions of control participants and, ultimately, impute missing counterfactuals outcomes $Y(0)$ for treated participants. Therefore, the rationale behind PS weighting when estimating ATT (or ATT-like estimand) is to use the group of treated participants as the reference group from which the controls cases are standardized to, based on their PSs: control participants whose covariates are most similar to the reference group are up-weighted while those with dissimilar covariates are down-weighted. Hence, of all the weights we consider in this paper, only the overlap weights follow this rationale and continue to up-weight the contribution of control participants whose characteristics, encapsulated by the PSs, are similar to those of treated participants. When $e(X)=0,$ all the $\omega_{0h}(X)$’s are equal to 0, including $\omega_{0h}(X)=e(X)^{2}=0$ for OWATT. However, when $e(X)=1$, $\omega_{0h}(X)$ is still equal to 0 for standard ATT trimming and remains constant (and equal to $(1-\alpha)/\alpha$) for ATT truncation, but $\omega_{0h}(X)$ goes to infinity when we do not trim or when we use smooth ATT trimming ATT estimators, for some values of $\varepsilon$.

As noted from Figure 1, the smooth ATT trimming with $\alpha=0.1$ and $\varepsilon=0.05$ has weights that decrease in the region $\{e(X):e(X)\in[1-\alpha,1)\}$, but go sharply back to infinity as soon as $e(x)$ is fairly close or equal to 1. The reason being that when $e(x)$ is close or equal to 1, $\Phi_{\varepsilon}(1-e(X)-\alpha)=\Phi_{0.05}(0.90-e(x))$ does not equal or approximate 0. Thus, the weights are still dominated by ${e(x)}\{1-e(x)\}^{-1}$ which remains large when $e(x)\approx 1$. Thus, oddly enough, the smooth trimming method sometimes may still suffer from the extreme weights when the duo $(\alpha,\varepsilon)$ is not selected appropriately. However, this behavior is not observed for instance when $\varepsilon=0.01$.

Using asymptotic theory, we can show that when the PS model is correctly specified, $\widehat{\tau}_{watt}(\mathcal{H})$ is consistent to $\tau_{watt}^{h}(\mathcal{H})$. Furthermore, we have the following asymptotic result:

Proof. See Appendix A.2.

It ensues from Theorem 1 the following remarks:

1. 1.

   The use of the standard bootstrap for variance estimation is valid, because $\widehat{\tau}_{watt}(\mathcal{H})$ is asymptotic linear. shao2012jackknife The empirical performance of bootstrap variance estimation is examined in the simulation study of Section 4.

2. 2.

   The above asymptotic variance of $\widehat{\tau}_{watt}^{h}(\mathcal{H})$ reveals that the uncertainty to estimate $\tau_{watt}^{h}(\mathcal{H})$ comes from multiple sources of variability, including the uncertainty to estimate $\beta$ and the variability in observed outcome. The choice of $h(x)$ also affects the variance of $\widehat{\tau}_{watt}^{h}(\mathcal{H})$. For example, when $h(x)=1$, $\omega_{0h}(x)=w_{0}(x)=e(x)/\{1-e(x)\}$ and $\widehat{\tau}_{watt}^{h}(\mathcal{H})$ estimates ATT. If some $e(x)$’s in control group are close to 1, $\eta_{0}(x)\propto e(x)^{2}/\{1-e(x)\}$ will be large, thus will also incur large variance. When choosing the overlap function $h(x)=e(x)\{1-e(x)\}$, $\eta_{0}(x)\propto e(x)^{4}\{1-e(x)\}$, which is then bounded and less variable.

3. 3.

   We have shown (see Appendix A.3) that, when the PS model is possibly misspecified and equal to $\widetilde{e}(X)$, the asymptotic bias of $\widehat{\tau}_{watt}^{h}(\mathcal{H})$ is

   $\displaystyle\text{ABias}(\widehat{\tau}_{watt}^{h}(\mathcal{H}))$

   $\displaystyle=\frac{\mathbb{E}\{\omega_{0h}(X)\{1-e(X)\}m_{0}(X)\}}{\mathbb{E}\{\omega_{0h}(X)\{1-e(X)\}\}}-\frac{\mathbb{E}\{\widetilde{\omega}_{0h}(X)\{1-e(X)\}m_{0}(X)\}}{\mathbb{E}\{\widetilde{\omega}_{0h}(X)\{1-e(X)\}\}},$

   where $m_{0}(X)=\mathbb{E}\{Y(0)\mid X\}$, and $\widetilde{\omega}_{0h}(X)$ denotes the limit of $\widehat{\omega}_{0h}(X)$ using misspecified PS model $\widetilde{e}(X)$ in lieu of the true PS $e(X)$. Clearly, whether we choose $h(x)=1$ vs. $h(x)=e(x)\{1-e(x)\}$, the asymptotic bias is, respectively,

   	$\displaystyle\frac{\mathbb{E}\{e(X)m_{0}(X)\}}{\mathbb{E}\{e(X)\}}-\frac{\mathbb{E}\left\{\dfrac{\widetilde{e}(X)}{1-\widetilde{e}(X)}\{1-e(X)\}m_{0}(X)\right\}}{\mathbb{E}\left\{\dfrac{\widetilde{e}(X)}{1-\widetilde{e}(X)}\{1-e(X)\}\right\}}$
   	$\displaystyle\text{ or }\quad\frac{\mathbb{E}\{e(X)^{2}\{1-e(X)\}m_{0}(X)\}}{\mathbb{E}\{e(X)^{2}\{1-e(X)\}\}}-\dfrac{\mathbb{E}\left\{\widetilde{e}(X)^{2}\{1-e(X)\}m_{0}(X)\right\}}{\mathbb{E}\left\{\widetilde{e}(X)^{2}\{1-e(X)\}\right\}},$

   when we use $\widetilde{e}(X)$ in lieu of the true PS $e(X)$. This shows that the latter bias is less sensitive and less extreme to large values of $\widetilde{e}(x)$, i.e., when $\widetilde{e}(x)\to 1$.

We provide more details on the asymptotic bias when the PS model is (possibly) misspecified in the Appendix A.3. In the simulation study in Section 4, we also compare the performance of the OWATT estimator against the trimming and truncation estimators, when the PS model is misspecified.

We considered two different DGPs, one similar to the DGP used by Li and Lili2021propensity (DGP I) and the other following the classic DGP of Kang and Schaferkang2007demystifying (DGP II).

To evaluate and compare the performance of the estimators considered (and their sensitivity to model misspecifications), we used 3 specific measures: the absolute relative percent bias (ARBias%) $=100\%*\left|\text{mean of }\left(\widehat{\tau}_{watt}^{h}-\tau_{watt}^{h}\right)/\tau_{watt}^{h}\right|$, the root mean square error (RMSE) $=\left\{\text{mean of }\left(\widehat{\tau}_{watt}^{h}-\tau_{watt}^{h}\right)^{2}\right\}^{\frac{1}{2}}$, and the 95% coverage probability (CP), i.e., and the proportion of times $\tau_{watt}^{h}$ falls inside of its estimated confidence interval, over the $1000$ Monte Carlo replicates.

We used $R=200$ standard bootstrap samples, in each simulated data replicate, and calculated the empirical variance of the 200 bootstrap point estimates as the variance estimate. We use ARBias to meaningfully compare different estimators, via tables or graphs, since the underlying causal estimands are expected to be different across methods; the smaller ARBias%, the measure, the better. Also, since we used $M=1000$ simulated data, a 95% CP is considered significantly different from the nominal 95% level if it is outside of the interval $[0.937,0.963].$

In the Appendix B, we also report the relative root mean square error (RRMSE) $=\left\{\text{mean of }\left[({\widehat{\tau}_{watt}^{h}-\tau_{watt}^{h}})/{\tau_{watt}^{h}}\right]^{2}\right\}^{\frac{1}{2}}$ and the relative efficiency (RE), i.e., the mean of the ratios of the empirical variance over the bootstrap variance estimate of $\widehat{\tau}_{watt}^{h}$ over the $1000$ Monte Carlo replicates.

In this section, to be succinct and concise, we only present the results under the cases of (i) poor PS overlap and the higher adjusted $R^{2}$ of DGP I, and (ii) $N=1000$ of DGP II. Results from additional simulations (good and moderate overlaps, smaller adjusted $R^{2}$ of DGP I, as well as $N=200$ of DGP II) are reported in Appendix B. The main conclusion by simulation remains the same across different cases.

Figure 2 shows the PS distributions by treatment groups under the true DGP I (the left panel) of the “poor overlap” PS model and under the true DGP II (the right panel). Clearly, in the left panel, the tails of the PS distributions of two treatment groups have very limited overlap and there is a number of observations in the control group with $e(X)\approx 1$, which can incur extreme weights when estimating ATT. In the right panel, we see moderate overlap between PS distributions and there are a smaller number of extreme weights than the left panel.