Counterfactual Prediction Under Outcome Measurement Error

Prior work has conducted detailed assessments of specific modeling issues (Lakkaraju et al., 2017; De-Arteaga et al., 2021; Kleinberg et al., 2018; Coston et al., 2020b; Kallus and Zhou, 2018; Wang et al., 2021), which have been synthesized into broader critiques of AI validity and functionality (Coston et al., 2022; Raji et al., 2022; Wang et al., 2022). Raji et al. (2022) surface AI functionality harms in which models fail to achieve their purported goal due to systematic design, engineering, deployment, and communication failures. Coston et al. (2022) highlight challenges related to value alignment, reliability, and validity that may draw the justifiability of RAIs into question in some contexts. We build upon this literature by studying intersectional threats to model reliability arising from outcome measurement error (Jacobs and Wallach, 2021; Wang et al., 2021), treatment effects (Coston et al., 2020b; Perdomo et al., 2020), and selection bias (Kallus and Zhou, 2018) in parallel.

Modeling outcome measurement error is challenging because it introduces two sources of uncertainty: which error model is reasonable for a given proxy, and which specific error parameters govern the relationship between target and proxy outcomes under the assumed measurement model (Jacobs and Wallach, 2021). Popular error models studied in the machine learning literature include uniform (Angluin and Laird, 1988; Van Rooyen et al., 2015b), class-conditional (Menon et al., 2015; Scott et al., 2013), and instance-dependent (Xia et al., 2020; Chen et al., 2021) structures of outcome misclassification. Work in algorithmic fairness has also studied settings in which measurement error varies across levels of a protected attribute (Wang et al., 2021), and proposed sensitivity analysis frameworks that are model agnostic(Fogliato et al., 2020).

Numerous statistical approaches have been developed for measurement error parameter estimation in the quantitative social sciences literature (Roberts, 1985; Bishop, 1998). Application of these approaches is tightly coupled with domain knowledge of the phenomena under study, as in biostatistics (Hui and Walter, 1980) or psychometrics (Shrout and Lane, 2012). To date, data-driven techniques for error parameter estimation have primarily been applied in the machine learning literature, which rely on key assumptions relating the target outcome of interest and its proxy (Xia et al., 2019; Liu and Tao, 2015; Scott, 2015; Scott et al., 2013; Menon et al., 2015; Northcutt et al., 2021). In this work, we build upon an existing “anchor assumptions” framework that estimates error parameters by linking the proxy and target outcome probabilities at specific instances (Xia et al., 2019). In contrast to prior work, we provide a range of anchoring assumptions, which can be flexibly combined depending on which are reasonable in a specific algorithmic decision support (ADS) domain.

Natarajan et al. (2013) propose a widely-adopted unbiased risk minimization approach for learning under noisy labels given knowledge of measurement error parameters (Patrini et al., 2017; Chou et al., 2020; Van Rooyen et al., 2015a). This method constructs a surrogate loss $\tilde{\ell}$ such that the $\tilde{\ell}$-risk over proxy outcomes is equivalent to the $\ell$-risk over target outcomes in expectation. Additionally, Natarajan et al. (2013) show that the minimizer of $\tilde{\ell}$-risk over proxy outcomes is optimal with respect to target outcomes if $\ell$ is symmetric (e.g., Huber, logistic, and squared losses). In this work, we develop a novel variant of this unbiased risk minimization approach designed for settings with treatment-conditional OME.

Recent work has shown that counterfactual modeling is necessary when the decision informed by a predictive model serves as a risk-mitigating intervention (Coston et al., 2020b). Building off of this result, we argue that it is necessary to account for treatment effects on target outcomes of interest and their observed proxy while modeling OME. Our methods build upon conditional average treatment effect (CATE) estimation techniques from the causal inference literature (Shalit et al., 2017; Johansson et al., 2020; Abrevaya et al., 2015). Subject to identification conditions (Pearl, 2009; Rubin, 2005), these approaches predict the difference between the expected outcome under treatment (e.g., high-risk program enrollment) versus control (e.g., no program enrollment) conditional on covariates. One family of outcome regression estimators predicts the CATE by directly estimating the expected outcome under treatment or control conditional on covariates (Hill, 2011; Künzel et al., 2019; Chernozhukov et al., 2017). However, these methods suffer from statistical bias when prior decisions were non-randomized (i.e., due to distribution shift induced by selection bias) (Shimodaira, 2000; Angluin and Laird, 1988). Therefore, we leverage a re-weighting strategy proposed by (Johansson et al., 2020) to correct for this selection bias during risk minimization. Our re-weighting method performs a similar bias correction as inverse probability weighting (IPW) methods (Shimodaira, 2000, 2000; Rosenbaum and Rubin, 1983).

Outcome measurement error has also been studied in causal inference literature. Finkelstein et al. (2021) bound the average treatment effect (ATE) under multiple plausible OME models. Shu and Yi (2019) propose a doubly robust method which accounts for measurement error during ATE estimation, while Díaz and van der Laan (2013) provide a sensitivity analysis framework for examining robustness of ATE estimates to OME. This work is primarily concerned with estimating population statistics rather than predicting outcomes conditional on measured covariates (i.e., the CATE).

In this section, we develop an approach for estimating $\eta^{*}_{t}$ given observational data drawn from $p(X,T,Y)$ and measurement error parameters $\alpha_{t}$, $\beta_{t}$. Let $f_{t}\in\mathcal{H}$ for $\mathcal{H}\subset\{f_{t}:\mathcal{X}\rightarrow[0,1]\}$ be a probabilistic decision function targeting $Y^{*}_{t}$ and let $\ell:\mathcal{Y}\times[0,1]\rightarrow\mathbb{R}_{+}$ be a loss function. If we observed target potential outcomes $Y^{*}_{t}\sim p^{*}$, we could directly apply supervised learning techniques to minimize the expected $\ell$-risk of $f_{t}$ over target potential outcomes

and learn an estimator for $\eta^{*}_{t}$ via standard empirical risk minimization approaches. Given a strongly proper composite loss such that $\operatorname*{arg\,min}_{f_{t}}R_{\ell}^{*}(f_{t})$ is a monotone transform $\psi$ of $\eta^{*}_{t}$ (e.g., the logistic and exponential loss), this would enable recovering class probabilities from the optimal prediction via the link function $\psi$ (Agarwal, 2014; Menon et al., 2015). However, directly minimizing (2) is not possible in our setting because we sample observational proxies instead of target potential outcomes. We address this challenge by constructing a re-weighted surrogate risk $R_{t,\tilde{\ell}}^{w}$ such that evaluating this risk over observed proxy outcomes is equivalent to $R_{\ell}^{*}$ in expectation.

In particular, let $w:\mathcal{X}\rightarrow\mathbb{R}_{+}$ be a weighting function satisfying $\mathbb{E}_{X}[w(X)|T=t]=1$ and let $\ell:\mathcal{Y}\times[0,1]\rightarrow\mathbb{R}_{+}$ be a surrogate loss function. We construct a re-weighted surrogate risk

such that $R_{\ell}^{*}(f_{t})=R_{t,\tilde{\ell}}(f_{t})$ in expectation. Theorem 4.1 shows that we can recover a surrogate risk satisfying this property by constructing $w(x)$ as in (4) and $\tilde{\ell}$ as in (5). Note that this surrogate risk requires knowledge of $\alpha_{t}$, $\beta_{t}$.

We prove Theorem 4.1 in Appendix A.2. Intuitively, $R_{t,\tilde{\ell}}^{w}\left(f_{t}\right)$ applies a joint bias correction for OME and distribution shift introduced by historical decision-making policies (i.e., selection bias). The unbiased risk minimization framework dating back to Natarajan et al. (2013) corrects for OME by minimizing a surrogate loss $\tilde{\ell}$ on proxies $Y$ observed over the full population unconditional on treatment. Yet this approach is untenable when decisions impact outcomes ($T\centernot{\mathrel{\perp\mspace{-10.0mu}\perp}}\{Y^{*},Y\}$) and error rates differ across treatments. One possible extension of unbiased risk minimizers to the treatment-conditional setting involves minimizing $\tilde{\ell}$ over the treatment population $p(X|T=t)$

However, $R_{t,\tilde{\ell}}\neq R_{\ell}^{*}$ in observational settings because the treatment population $p(X|T=t)$ can differ from the marginal population $p(X)$ under historical selection policies when $X\centernot{\mathrel{\perp\mspace{-10.0mu}\perp}}T$. Therefore, our re-weighting procedure applies a second bias correction that adjusts $p(X|T=t)$ to resemble $p(X)$.

Learning algorithm. As a result of Theorem 4.1, we can learn a predictor $\hat{\eta}^{*}_{t}$ by minimizing the re-weighted surrogate risk over observed samples $(X_{1},T_{1},Y_{1}),...,(X_{n},T_{n},Y_{n})\sim p$. First, we estimate the weighting function $\hat{w}(x)$ through a finite sample, which boils down to learning propensity scores $\hat{\pi}(x)$ (as shown in (4)). Estimating the propensity scores can be done by applying supervised learning algorithms to learn a predictor from $X$ to $T$. Then for any treatment $t$, weighting function $\hat{w}$, and predictor $f_{t}$, we can approximate $R_{t,\tilde{\ell}}^{w}\left(f_{t}\right)$ by taking the sample average over the treatment population

for $n_{t}=\sum_{i=1}^{n}\mathbbm{1}[T_{i}=t]$. Therefore, given $\hat{w}$ we can learn a predictor from observational data by minimizing the empirical risk

We refer to solving (8) as re-weighted risk minimization with a surrogate loss (Algorithm 1).

Because our risk minimization approach requires knowledge of OME parameters, we develop a method for estimating $\alpha_{t}$, $\beta_{t}$ from observational data. Error parameter estimation is challenging in decision support applications because target outcomes often result from nuanced social and organizational processes. Understanding the measurement error properties of proxies targeted in criminal justice, medicine, and hiring domains remains an ongoing focus of domain-specific research (Fogliato et al., 2021; Akpinar et al., 2021; Mullainathan and Obermeyer, 2021; Obermeyer et al., 2019; Zwaan and Singh, 2015; Chalfin et al., 2016). Therefore, we develop an approach compatible with multiple sources of domain knowledge about proxies, which can be flexibly combined depending on which assumptions are deemed reasonable in a specific context.

Error parameters are identifiable if they can be uniquely computed from observational data. Because our error model (e.q. 1) expresses the proxy class probability as a linear equation with two unknowns, $\alpha_{t}$, $\beta_{t}$ are identifiable if the target class probability $c^{*}_{t,i}=\eta^{*}_{t}(x_{i})$ and proxy class probability $c_{t,i}=\eta_{t}(x_{i})$ are known at two distinct points $(c^{*}_{t,i},c_{t,i})$ and $(c^{*}_{t,j},c_{t,j})$ such that $c^{*}_{t,i}\neq c^{*}_{t,j}$. Following prior literature (Han et al., 2020), we refer to knowledge of $(c^{*}_{t,i},c_{t,i})$ as an anchor assumption because it requires knowledge of the unobserved quantity $\eta^{*}_{t}$. We now introduce several anchor assumptions that are practical in ADS, before showing that these can be flexibly combined to identify $\alpha_{t}$, $\beta_{t}$ in Theorem 4.2.

Min anchor. A min anchor assumption holds if there is an instance at no risk of the target potential outcome under intervention $t$: $c_{t,i}^{*}=\inf_{x_{i}\in\mathcal{X}}{\{\eta^{*}_{t}(x_{i})\}}=0$. Because $\eta_{t}$ is a strictly monotone increasing transform of $\eta^{*}_{t}$, the corresponding value of $\eta_{t}$ can be recovered via $c_{t,i}=\inf_{x_{i}\in\mathcal{X}}\{\eta_{t}(x_{i})\}$ (Menon et al., 2015). Min anchors are reasonable when there are cases that are confirmed to be at no risk based on domain knowledge of the data generating process. For example, a min anchor may be reasonable in diagnostic testing if a patient is confirmed to be negative for a medical condition based on a high-precision gold standard medical test (Enøe et al., 2000).

Max anchor. A max anchor assumption holds if there is an instance at certain risk of the target outcome under intervention $t$: $c_{t,i}^{*}=\sup_{x_{i}\in\mathcal{X}}\{\eta^{*}_{t}(x_{i})\}=1$. The corresponding value of $\eta_{t}$ can be recovered via $c_{t,i}=\sup_{x_{i}\in\mathcal{X}}\{\eta_{t}(x_{i})\}$ because $\eta_{t}$ is a strictly monotone increasing transform of $\eta^{*}_{t}$. Max anchors are reasonable when there are confirmed instances of a positive target potential outcome based on domain knowledge of the data generating process. For example, a max anchor may be justified in a medical setting if a subset of patients have confirmed disease diagnoses based on biopsy results (Begg and Greenes, 1983), or if a disease prognosis (and resulting health outcomes) are known from pathology.

Base rate anchor. A base rate anchor assumption holds if the expected value of $\eta^{*}_{t}$ is known under intervention $t$: $c_{t,i}^{*}=\mathbb{E}[\eta^{*}_{t}(X)]$. The corresponding value of $\eta_{t}$ can be recovered by taking the expectation over the proxy class probability $c_{t,i}=\mathbb{E}[\eta_{t}(X)]$. Base rate anchors are practical because the prevalence of unobservable target outcomes (e.g., medical conditions (Walter and Irwig, 1988), crime (Kruttschnitt et al., 2014; Lohr, 2019), student performance (Schouwenburg, 2004; Di Folco et al., 2022)) is routinely estimated via domain-specific analyses of measurement error. For instance, studies have been conducted to estimate the base rate of undiagnosed heart attacks (i.e., accounting for measurement error in diagnosis proxy outcomes) (Orso et al., 2017). Additionally, the conditional average treatment effect $\mathbb{E}[\eta^{*}_{1}(X)]-\mathbb{E}[\eta^{*}_{0}(X)]$ is commonly estimated in randomized controlled trials (RCTs) while assessing treatment effect heterogeneity (Hill, 2011). While the conditional average treatment effect is normally estimated via proxies $Y_{0}$ and $Y_{1}$, measurement error analysis is a routine component of RCT design and evaluation (Gamerman et al., 2019).

Anchor assumptions can be flexibly combined to identify error parameters based on which set of assumptions are reasonable in a given ADS domain. In particular, Theorem 4.2 shows that combinations of anchor assumptions listed in Table 1 are sufficient for identifying error parameters under our causal assumptions.

We prove Theorem 4.2 in Appendix A.2. In practice, we estimate the error rates on finite samples $(X_{i},T_{i},Y_{i})\sim p$, which gives an approximation $\hat{\eta}_{t}$. Therefore, we propose a conditional class probability estimation (CCPE) method for parameter estimation which estimates $\hat{\alpha}_{t}$, $\hat{\beta}_{t}$ by fitting $\hat{\eta}_{t}$ on observational data then applying the relevant pair of anchor assumptions to estimate error rates. Algorithm 2 provides pseudocode for this approach with min and max anchors, which can easily be extended to other pairs of identifying assumptions shown in Table 1. The combination of min and max anchors is known as weak separability (Menon et al., 2015) or mutual irreducibility (Scott et al., 2013; Scott, 2015) in the observational label noise literature. Prior results in the observational setting show that unconditional class probability estimation (i.e., fitting $\hat{\eta}(x)=p(Y=1|X=x$)) yields a consistent estimator for observational error rates under weak seperability (Scott et al., 2013; Reeve et al., 2019). Statistical consistency results extend to the treatment-conditional setting under positivity (4) because $p(T=t|X=x)>0,\;\forall t\in\{0,1\},\;x\in\mathcal{X}$. However, asymptotic convergence rates may be slower under strong selection bias if $p(T=t|X=x)$ is near 0.

We compare several modeling approaches in our evaluation to examine how existing modeling practices are impacted by treatment-conditional outcome measurement error:

• •

  Unconditional proxy (UP). Predict the observed outcome unconditional on treatment: $X\rightarrow Y$. This model does not adjust for OME or treatment effects., and reflects model performance in a scenario in which practitioners overlook all challenges examined in this work. ³³3This baseline is also called an observational risk assessment in experiments reported by Coston et al. (2020b).

• •

  Unconditional target (UT). Predict the target outcome unconditional on treatment: $X\rightarrow Y^{*}$. Here, we determine $Y^{*}$ by applying consistency: $Y^{*}=(1-T)\cdot Y^{*}_{0}+T\cdot Y^{*}_{1}$. This method reflects a setting in which no OME is present but modeling does not account for treatment effects (Wang et al., 2021; Natarajan et al., 2013; Menon et al., 2015; Obermeyer et al., 2019).

• •

  Conditional proxy (CP). Predict the proxy outcome conditional on treatment: $X,T\rightarrow Y$. This is a counterfactual model that estimates a conditional expectation without correcting for OME (Coston et al., 2020b; Shalit et al., 2017; Künzel et al., 2019).⁴⁴4This model is known by different names in the causal inference literature, including the backdoor adjustment (G-computation) formula (Pearl, 2009; Robins, 1986), T-learner (Künzel et al., 2019), and plug-in estimator (Kennedy, 2022).

• •

  Re-weighted surrogate loss (RW-SL). Our proposed risk minimization approach, as defined in equation (8). This method corrects for both OME and treatment effects in parallel. Additionally, this method corrects for distribution shift due to selection bias in the prior decision-making policy via re-weighting.

• •

  Target Potential Outcome (TPO). Directly predict the target potential outcome: $X\rightarrow Y^{*}_{t}$. This model is an oracle that provides an upper-bound on model performance under no OME or treatment effects.

We also perform an ablation of our proposed RW-SL method by including a model that applies a surrogate loss correction $\tilde{\ell}$ over the treatment population without re-weighting (SL).

We begin by experimentally manipulating treatment effects and measurement error via a synthetic evaluation. Because this provides full control over the data generating process, we can evaluate methods against target potential outcomes. This evaluation would not possible with real-world data because counterfactual target outcomes are unobserved. Our experiment design is consistent with prior synthetic evaluations of counterfactual risk assessments (Coston et al., 2020b) and causal inference methods (Shi et al., 2019; Nie et al., 2021). In our evaluation, we sample outcomes via the following data generating process:

1. (1)

   $Y^{*}_{t}:=\;\;\sim\text{Bern}(\eta_{t}^{*}(X)),\;\forall t\in\{0,1\}$

2. (2)

   $Y_{t}:=\begin{cases}1-\epsilon_{+}&\text{if $Y^{*}_{t}=1$, where $\epsilon_{+}\sim\text{Bern}(\beta_{t})$ }\\ \epsilon^{-}&\text{if $Y^{*}_{t}=0$, where $\epsilon_{-}\sim\text{Bern}(\alpha_{t})$ }\\ \end{cases},\forall t\in\{0,1\}\;$

3. (3)

   $T:=\;\;\sim\text{Bern}(\pi(X))$

4. (4)

   $Y^{*}:=(1-T)\cdot Y^{*}_{0}+T\cdot Y^{*}_{1};\;Y:=(1-T)\cdot Y_{0}+T\cdot Y_{1}$

As shown in Figure 4, we draw $X\sim U(-1,1)$ and sample target potential outcomes from sinusoidal class probability functions (see Appendix A.4 for details). Note that our choice of $\eta^{*}_{0}(x)$, $\eta^{*}_{1}(x)$ satisfies min and max anchor assumptions. Because $\eta^{*}_{0}(x)$ and $\eta^{*}_{1}(x)$ differ, models that do not condition on treatment (i.e., UP, UT) will learn an average of the two class probability functions. Under our choice of $\pi(x)$, fewer samples are drawn from $\eta^{*}_{1}(x)$ in the region where $\pi(x)$ is small (near $x=-1$), and fewer samples are drawn from $\eta^{*}_{0}(x)$ in the region where $1-\pi(x)$ is small (near $x=1$). This introduces selection bias when sampling from $\pi(x)$.

In addition to our synthetic evaluation, we conduct experiments using real-world data collected as part of randomized controlled trials (RCTs) in healthcare and employment domains. While this evaluation affords less control over the data generating process, it provides a more realistic sample of data likely to be encountered in real-world model deployments. Evaluation via data from randomized or partially randomized experimental studies is useful for validating counterfactual prediction approaches because random assignment ensures that causal assumptions are satisfied (Shalit et al., 2017; Johansson et al., 2020; Coston et al., 2020a).

Our work speaks to key complexities faced by domain experts, model developers, and other stakeholders while examining proxies in ADS. One decision surfaced by our work entails which measurement error model best describes the relationship between the unobserved outcome of policy interest and its recorded proxy. We open this work by highlighting a measurement model decision made by Obermeyer et al. (2019) during their audit of a clinical risk assessment: that error rates are fixed across treatments. Our work suggests that failing to account for treatment-conditional error in OME models may exacerbate reliability concerns. However, at the same time, the error model we adopt in this work intentionally abstracts over other factors known to impact proxies in decision support tasks, including error rates that vary across covariates. Although this simplifying assumption can be unreasonable in some settings (Fogliato et al., 2021; Akpinar et al., 2021), including the one studied by Obermeyer et al. (2019), it is helpful in foregrounding previously-overlooked challenges involving treatment effects and selection bias. In practice, model developers correcting for measurement error may wish to combine our methods with existing unbiased risk minimization approaches designed for group-dependent error where appropriate (Wang et al., 2021). Further, analyses of measurement error should not overlook more fundamental conceptual differences between target outcomes and proxies readily available for modeling (e.g., when long-term child welfare related outcomes targeted by a risk assessment differ from immediate threats to child safety weighted by social workers (Kawakami et al., 2022, 2022)). This underscores the need to carefully weigh the validity of proxies in consultation with multiple stakeholders (e.g., domain experts, data scientists, and decision-makers) while deciding whether OME correction is warranted.

A second decision point highlighted in this work entails the specific measurement error parameters that govern the relationship between target and proxy outcomes. In particular, our work underscores the need for a tighter coupling between domain expertise and data-driven approaches for error parameter estimation. Current techniques designed to address OME in the machine learning literature – which typically examine settings with “label noise” – rely heavily upon data-driven approaches without close consideration of whether the underlying measurement assumptions hold (Wang et al., 2021; Northcutt et al., 2021; Menon et al., 2015; Xia et al., 2019). While application of these assumptions may be practical for methodological evaluations and theoretical analysis (Scott et al., 2013; Scott, 2015; Reeve et al., 2019), these assumptions should be carefully vetted when applying OME correction to real-world proxies. This is supported by our findings in Figure 5, which show that RW-SL performs poorly when the anchor assumptions used for error parameter estimation are violated. Our flexible set of anchor assumptions provides a step towards a tighter coupling between domain expertise and data-driven approaches in measurement parameter estimation.

Our counterfactual modeling approach requires several causal identifiability assumptions (Pearl, 2009), which may not be satisfied in all decision support contexts. Of our assumptions, the most stringent is likely ignorability, which requires that no unobserved confounders influenced past decisions and outcomes. While recent modeling developments may ease ignorability-related concerns in some cases (Coston et al., 2020b; Rambachan et al., 2022), model developers should carefully evaluate whether confounders are likely to impact a model in a given deployment context. At the same time, our results show that formulating algorithmic decision support as a “pure prediction problem” that optimizes predictive performance without estimating causal effects (Kleinberg et al., 2015) imposes equally serious limitations. If the policy-relevant target outcome of interest is risk conditional on intervention (as is often the case in decision support applications), an observational model will generate invalid predictions for cases that historically responded most to treatment (Coston et al., 2020b). Our results, which empirically demonstrate poor performance of observational PU and TU models that overlook treatment-effects, corroborate prior findings indicating that counterfactual modeling is required to ensure the reliability of RAIs in decision support settings (Coston et al., 2020b). Taken together, our work suggests that domain experts and model developers should exercise considerable caution while mapping the causal estimand of policy interest to the statistical estimand targeted by a predictive model (Lundberg et al., 2021).