Conformalized Lee Inference: Distribution-Free Individual Treatment Effect Intervals under Monotone Sample Selection

Conformalized Lee Inference: Distribution-Free Individual Treatment Effect Intervals under Monotone Sample Selection

JEL Classification: C14, C21, C24, C53.

Keywords: Conformal prediction; Lee bounds; sample selection; individual treatment effects; counterfactual prediction; partial identification; distribution-free inference.

Introduction

Introduction

Empirical studies often observe outcomes only for units who are selected into the sample. This creates a fundamental problem when treatment itself affects selection. In a randomized treatment setting, selected treated and selected control units need not represent the same latent population: comparisons among observed outcomes may combine treatment effects with changes in who is observed. The monotone-selection framework of lee2009training addresses this problem by assuming that treatment can only increase selection. Under this restriction, selected controls are always-selected units, while treated-selected units are a mixture of always-selected units and marginal-in units induced into selection by treatment.

This paper develops conformalized Lee inference, a distribution-free method for constructing prediction intervals under monotone sample selection. The primary target is a prediction set for the treated potential outcome of a future draw from the always-selected population. For selected controls, the untreated outcome is observed and monotonicity implies membership in the always-selected stratum. Hence, a prediction set for the missing treated potential outcome can be translated into a prediction interval for the realized individual treatment effect by subtracting the observed untreated outcome. The resulting guarantee is marginal over the selected-control population, not conditional on a fixed individual's covariates and untreated outcome.

The central difficulty is that the calibration data are not drawn from the target law. Treated-selected observations come from a mixture of always-selected and marginal-in units, whereas the target law contains only always-selected units. Let \(P\) denote the observed treated-selected law of \((X,Y(1))\), and let \(Q_0\) denote the corresponding always-selected target law. If the selection rate is \(p_d=\Pr(S(d)=1)\) for $d \in \{0,1\}$, then monotonicity identifies \[ \pi=\frac{p_0}{p_1}, \] the fraction of treated-selected units who are always-selected. The identity of those units is not observed, so \(Q_0\) is not point identified.

The first contribution is to characterize the resulting distributional identification problem. The always-selected target law must be dominated by the treated-selected law and cannot overweight any event by more than \(1/\pi\): \[ Q_0 \ll P, \qquad 0 \leq \frac{dQ_0}{dP} \leq \frac{1}{\pi}. \] This is the distributional analogue of Lee's trimming argument. Lee bounds trim the treated-selected outcome distribution because the always-selected treated units are unidentified. Here, the same nesting logic is applied to the joint law of covariates and treated potential outcomes. The paper shows that this ambiguity set is sharp relative to the reduced observables \((P,p_0,p_1)\): every law satisfying the likelihood-ratio restriction can arise under random assignment, monotone selection, and the same observed selection rates.

The second contribution is to turn this identification result into a conformal prediction procedure. Ordinary split conformal prediction would calibrate treated-selected scores as if the future target unit were drawn from the same law as the calibration sample. That is generally invalid here. The Lee ambiguity set gives the necessary correction. Since the target law may place up to \(1/\pi\) times more probability on high-score events than the treated-selected law, the method calibrates at the higher treated-selected score quantile \(1-\alpha\pi\), rather than at the usual \(1-\alpha\) quantile. Equivalently, the treated-selected score tail is made small enough that even the worst admissible always-selected target law has tail probability at most \(\alpha\).

The resulting algorithm is simple. It trains any prediction rule using treated-selected observations, computes held-out nonconformity scores from treated-selected calibration observations, and chooses the Lee-adjusted conformal cutoff. The coverage result is finite-sample and distribution-free for arbitrary prediction rules and score functions. When the always-selected share \(\pi\) is unknown, the paper distinguishes plug-in estimation from lower-bound implementation. If the value used in calibration is no larger than the true \(\pi\), the procedure remains conservative; one-sided lower confidence bounds therefore provide formal high-probability coverage guarantees, while plug-in estimates offer a less conservative empirical implementation.

The third contribution is a minimax interpretation of the cutoff. Over the sharp Lee ambiguity set, the worst-case target law places as much mass as allowed on the upper tail of the nonconformity score. Therefore, the \(1-\alpha\pi\) population score quantile is not merely a sufficient conservative adjustment. It is the smallest score threshold that guarantees uniform coverage over all target laws consistent with the reduced monotone-selection information. Any smaller threshold fails for some observationally equivalent data-generating process.

The simulations compare naive conformal prediction, oracle Lee adjustment, plug-in Lee adjustment, and lower-bound Lee procedures across benign, tail-selection, and smooth-selection designs. Naive conformal prediction is valid when the calibration and target laws coincide, but it under-covers when always-selected units are more likely to have large nonconformity scores. The Lee-adjusted procedures restore coverage close to the target in the shifted designs, while Clopper--Pearson and Hoeffding lower-bound versions are conservative, as expected.

The rest of the paper proceeds as follows. Section 2 introduces the setup, principal strata, target population, and Lee ambiguity set, and establishes sharp identification under reduced observables. Section 3 develops the split-conformal Lee procedure, proves finite-sample coverage, derives individual treatment-effect intervals for selected controls, and establishes minimax optimality of the cutoff. Section 4 reports simulation evidence. The appendix gives proofs, practical lower-bound constructions for \(\pi\), and the full observed law identification result.

Related Work

This paper connects four literatures: treatment-effect inference under sample selection, principal stratification and partial identification, contamination models, and conformal prediction under distribution shift and causal inference.

\paragraph{Sample selection and Lee bounds.} This paper contributes to the literature on treatment-effect inference under sample selection (see, e.g., lee2009training, honore2020selection, honore2024sample). The closest starting point is lee2009training, who shows that monotone selection can deliver sharp bounds on average treatment effects by trimming the treated-selected outcome distribution. The present paper keeps the same monotone-selection logic but changes the inferential target. Rather than bounding only an average treatment effect, it constructs distribution-free prediction sets for counterfactual treated outcomes and individual treatment effects in the always-selected population.

Recent work extends Lee-type selection correction in richer settings, including covariate-rich selection models, generalized Lee bounds, continuous treatments, random objects, and treatment endogeneity (see, e.g., heiler2024heterogeneous, semenova2025generalized, lee2024lee, kurisu2026lee, dong2026sharp). These papers focus primarily on bounding average or distributional causal parameters. The present paper is complementary: it uses Lee-type nesting to derive a sharp ambiguity set for the target counterfactual law and then converts that ambiguity set into a conformal prediction rule. The appendix's full observed law result is especially related to covariate-rich Lee methods, since it shows that the selected-control covariate distribution identifies the target covariate marginal and that the conditional treated-outcome law satisfies the local restriction \[ Q_x \ll P_x, \qquad 0 \leq \frac{dQ_x}{dP_x} \leq \frac{1}{\pi(x)}. \] The main procedure, however, deliberately works with the reduced information \((P,p_0,p_1)\), yielding a simple and distribution-free robust calibration rule.

\paragraph{Principal stratification and partial identification.} The paper is also related to principal stratification. frangakis2002principal introduce principal stratification as a framework for defining causal effects within latent subpopulations determined by post-treatment variables, and zhang2003estimation apply this logic to outcomes truncated by death. The present setting has the same structure, with sample selection replacing survival. Under monotone selection, selected controls reveal the always-selected stratum, while treated-selected observations combine always-selected and marginal-in units. The contribution here is to use this principal-strata structure not only to define the causal target, but also to derive the distributional restriction required for valid conformal calibration.

The identification result is also connected to partial identification and contamination models (see, e.g., huber1981robust, horowitz1995identification, manski2003partial). The observed treated-selected law can be written as \[ P=\pi Q+(1-\pi)R, \] where \(Q\) is the always-selected target law and \(R\) is an unrestricted marginal-in law. Because the identity of always-selected treated units is unobserved, \(Q\) is not point identified. Monotone selection and random assignment identify the mixture share \(\pi=p_0/p_1\), which yields the sharp likelihood-ratio ambiguity set. This is the distributional analogue of Lee's trimming argument.

\paragraph{Conformal prediction under distribution shift.} On the prediction side, the paper builds on conformal prediction under exchangeability and split conformal prediction for regression (see, vovk2005algorithmic, lei2018distribution). Standard split conformal prediction is not directly valid here because calibration scores are computed from treated-selected observations drawn from \(P\), while the target counterfactual law is an unknown \(Q\) in the Lee ambiguity set.

This connects the paper to conformal prediction under distribution shift, including weighted conformal prediction under covariate shift and conformal prediction beyond exchangeability (see, tibshirani2019conformal, barber2023conformal). The distinction is that, in the present setting, the pointwise density ratio is not known. Instead, Lee monotonicity identifies a sharp upper bound on the joint density ratio, \(dQ/dP\leq 1/\pi\). Moreover, the shift is not only a change in the covariate distribution: principal-stratum membership may be related to both \(X\) and \(Y(1)\), so the target law may differ from the calibration law in both its covariate marginal and its conditional treated-outcome distribution. The conformalized Lee cutoff is therefore a robust conformal threshold for a partially identified joint distribution shift.

\paragraph{Conformal causal inference and sensitivity analysis.} The paper also contributes to conformal causal inference. lei2021conformal develop conformal methods for counterfactuals and individual treatment effects, and chernozhukov2021exact use conformal ideas for counterfactual and synthetic control inference. The closest comparison is robust conformal sensitivity analysis for individual treatment effects (see, e.g., jin2023sensitivity, yin2024conformal). Related sensitivity-analysis approaches study robustness to deviations such as unobserved confounding or analyst-specified departures from identifying assumptions (see, e.g., rosenbaum1987sensitivity, yadlowsky2022bounds, dorn2023sharp, dorn2025doubly).

The present paper differs because the robustness class is not chosen by the analyst as a sensitivity model. It is identified by Lee monotonicity and the observed treatment-control selection rates. This delivers a closed-form adjustment from the usual \((1-\alpha)\) conformal cutoff to the \((1-\alpha\pi)\) treated-selected score quantile. The paper further shows that this cutoff is minimax optimal over the reduced-information Lee ambiguity set: any smaller population threshold fails for some target law that is observationally equivalent under the maintained monotone-selection assumptions.

Setup, Principal Strata, and Predictive Target

We consider a randomized treatment setting with sample selection. For each unit $i=1,...,n$, let $D_i \in \{0,1 \}$ denote treatment assignment, let $X_i \in \mathcal{X} \subseteq \mathbb{R}^d$ denote predetermined covariates, and let $Y_i(1), Y_i(0)$ be the treated and untreated potential outcomes. Outcomes are observed only when the unit is selected into the sample. Let $S_i(1), S_i(0) \in \{0,1\}$ denote the potential selection indicators under treatment and control. The realized selection indicator is $S_i = D_i S_i(1) + (1 - D_i)S_i(0)$, and the observed outcome is $Y_i = D_i Y_i(1) + (1 - D_i) Y_i(0)$ when $S_i = 1$. Thus, the observed data are $\{(D_i,X_i,S_i, S_i Y_i)\}_{i=1}^n$.

The central challenge is that the treated-selected sample is not necessarily the same population as the always-selected target population. Under monotone selection, treatment may induce additional units to become selected. Therefore, treated-selected observations contain both always-selected units and marginal-in units, while selected-controls contain only always-selected units.

We impose the following assumptions.

Assumption (ref) identifies potential-outcome and potential-selection distributions from treatment arms. Assumption (ref) rules out units who would be selected under control but not under treatment. Hence, the population can be partitioned into three principal strata:$$ \mathcal{A}=\{S(0)=S(1)=1\}, \qquad \mathcal{M}=\{S(0)=0,S(1)=1\}, \qquad \mathcal{N}=\{S(0)=S(1)=0\}. $$ where $\mathcal{A}$ is the always-selected stratum, $\mathcal{M}$ is the marginal-in stratum induced into selection by treatment, and $\mathcal{N}$ is the never-selected stratum. Because of monotonicity, every selected-control unit belongs to $\mathcal{A}$, whereas a treated-selected unit may belong either to $\mathcal{A}$ or to $\mathcal{M}$.

The object of interest in this paper is the distribution-free prediction problem for individual treatment effects in the always-selected stratum. For an always-selected unit, the individual treatment effect is $$\tau_i = Y_i(1) - Y_i(0), \quad i \in \mathcal{A}.$$

For selected-controls, $Y(0)$ is observed and monotonicity implies membership in $\mathcal{A}$. Therefore, prediction intervals for the missing treated potential outcome $Y(1)$ among always-selected units can be translated directly into prediction intervals for $\tau_i$.

Let $p_d = \Pr(S(d) = 1)$ for $d \in \{0,1\}$. Under Assumption (ref), these are identified by $p_d = \Pr(S = 1 \mid D=d)$ for $d \in \{0,1\}$. Furthermore, under Assumption (ref), $p_1 \ge p_0$. Define $\pi = p_0 / p_1 = \Pr( \mathcal{A} \mid S(1) = 1 )$. Then, $\pi$ is the fraction of treated-selected units who are always-selected. Equivalently, $1 - \pi$ is the fraction of treated-selected units who are marginal-in.

This assumption ensures that the selected populations under treatment and control are nondegenerate and that $\pi = p_0 / p_1$ is well-defined.

Lee-Type Distributional Identification

To state the identification problem, Let $Z = (X,Y(1)) \in \mathcal{Z} = \mathcal{X} \times \mathcal{Y}$, where $(\mathcal{Z}, \mathcal{B})$ is a standard Borel space. Let $\mathcal{P}(\mathcal{Z}, \mathcal{B})$ denote the set of probability measure on $(\mathcal{Z}, \mathcal{B})$. The observed treated-selected distribution is $$P = \mathcal{L}( Z \mid S(1)=1).$$ By random assignment, this law is identified from the treated-selected observations: $$ P = \mathcal{L}( X,Y \mid D=1, S=1). $$ The target law is the true distribution of treated potential outcomes for always-selected units: $$ Q_0 = \mathcal{L}( Z \mid S(0)=1) = \mathcal{L}( Z \mid \mathcal{A}). $$ The key problem is that $Q_0$ is not observed. Among treated-selected units, only a fraction $\pi$ are always-selected; the remaining fraction $1 - \pi$ are marginal-in. Hence the observable treated-selected law can be written as $$ P= \pi Q_0 + (1 - \pi) R, $$ where $$ R = \mathcal{L} (Z \mid S(0)=0, S(1)=1) $$ is the distribution of $(X,Y(1))$ among marginal-in units. Because $R$ is unrestricted, the data do not point identify $Q_0$. Instead, they identify a class of possible target laws. The mixture representation implies the contaminated sampling model for the treated-selected sample (see, e.g., huber1981robust, horowitz1995identification).

This mixture representation is the distributional version of Lee’s trimming argument. Lee bounds trim the treated-selected outcome distribution to account for the unknown identity of always-selected units. Here, the same logic is applied to the joint law of $(X,Y(1))$, because the later conformal procedure calibrates nonconformity scores using treated-selected observations drawn from $P$, while the desired prediction guarantee concerns a future always-selected unit drawn from $Q_0$. Furthermore, the nesting $\{ S(0)=1 \} \subseteq \{ S(1)=1 \}$ implies that $Q_0$ must be dominated by $P$. More precisely, the always-selected law cannot put mass on events that have zero probability under the treated-selected law, and it cannot upweight any event by more than $1 / \pi$.

For $P \in \mathcal{P}(\mathcal{Z}, \mathcal{B})$ and $\pi \in (0,1]$, we define the Lee ambiguity set.

Proposition (ref) formalizes the necessity of this class under Assumptions (ref)--(ref). It shows that the true always-selected treated distribution $Q_0$ must lie in the resulting identification class $\mathcal{Q}(P, \pi)$. Lee ambiguity set can therefore be interpreted as the identified set of possible target distributions under selection-induced distribution shift. The treated-selected law $P$ is the source or calibration distribution, while each $Q\in\mathcal Q(P,\pi)$ is a possible always-selected target distribution. This is the first step toward conformalized Lee inference. Later sections will construct prediction sets that are valid uniformly over all target laws satisfying this likelihood-ratio bound.

Sharpness of the Lee Ambiguity Set Relative to Reduced Observables

Proposition (ref) shows that monotone selection imposes a likelihood ratio restriction on the always-selected treated distribution. Equivalently, the target counterfactual law $Q_0$ must belong to $\mathcal{Q}(P, \pi)$. This result gives an outer bound on what the always-selected distribution can be, given the observable treated-selected law $P$ and the selection-rate ratio $\pi = p_0 / p_1$. However, $\mathcal{Q}(P,\pi)$ may still contain distributions that satisfy the likelihood ratio inequality but cannot be generated by any data-generating process consistent with monotone selection and the observed selection rates $(p_0,p_1)$.

The purpose of this subsection is to rule out that possibility. We show that the Lee ambiguity set is sharp relative to the reduced information $(P, p_0, p_1)$: every probability law $Q \in \mathcal{Q}(P, \pi)$ can be rationalized as the treated potential outcome distribution of always-selected units under some data-generating process satisfying Assumptions (ref)--(ref) and matching the same observables $(P, p_0, p_1)$. Thus, the likelihood ratio restriction from Proposition (ref) is not merely necessary but also sufficient.

The argument is constructive. Given any candidate $\widetilde{Q} \in \mathcal{Q}(P,\pi)$ and Borel measurable $B \in \mathcal{B}$, define the residual law $$ R(B) = \frac{P(B) - \pi \widetilde{Q}(B)}{1-\pi}, $$ when $\pi < 1$. The bound $0 \le d\widetilde{Q} / dP \le 1 / \pi$ guarantees that $R$ is a valid probability measure. Hence, the observed treated-selected distribution $P$ can be decomposed as $$ P = \pi \widetilde{Q} + (1 - \pi) R. $$ This mixture representation has a direct principal-strata interpretation: $\widetilde{Q}$ is assigned to the always-selected stratum, while $R$ is assigned to the marginal-in stratum. Because the marginal-in distribution is unrestricted under Assumptions (ref)--(ref), the observable law cannot distinguish among different choices of $\widetilde{Q}$ inside $\mathcal{Q}(P, \pi)$.

Proposition (ref) formalizes this construction and establishes that $\mathcal{Q}(P, \pi)$ is exactly the identified region for the always-selected treated distribution. Consequently, any inference procedure that is uniformly valid under Assumptions (ref)--(ref) must be valid uniformly over all $\widetilde Q \in \mathcal{Q}(P, \pi)$. Conversely, no uniformly tighter distributional restriction is available without imposing additional assumptions.

Distribution-free Counterfactual Intervals for Always-selected

The goal of this section is to construct prediction intervals for the unobserved treated potential outcome $Y(1)$ and marginal ITE among always-selected units. The central prediction problem is cross-population nonexchangeability because the available calibration sample is not drawn from the target law. Conditional on the fitted score, the calibration observations are exchangeable among themselves because they are drawn from $P$. The target observation, however, is drawn from an unknown $Q$. Thus, it is not generally exchangeable with the calibration observations. Applying ordinary split-conformal prediction without modification would therefore control the score tail under \(P\), but not necessarily under the target law \(Q\).

The identification results above provide the structure needed to restore a distribution-free guarantee. Under monotone selection, the target law satisfies $Q \in \mathcal{Q}(P, \pi)$, where $\pi = p_0 / p_1$ is the share of always-selected units among treated-selected units. Consequently, for any measurable event $A$, $Q(A) \le P(A) / \pi$. This likelihood ratio domination converts the counterfactual coverage problem into a robust calibration problem.

The method does not restore exchangeability between $P$ and $Q$. Instead, it combines a distribution-shift transfer inequality with an ordinary exchangeable-rank argument under $P$. For the score-tail event $$ A_t=\{(x,y):V(x,y)>t\}, $$ the likelihood ratio restriction implies $$ Q(A_t)\le\frac1\pi P(A_t). $$ Thus, target miscoverage is at most $\alpha$ whenever the corresponding treated-selected tail probability is at most $\alpha\pi$. Therefore, to guarantee $Q(A_t) \le \alpha$ uniformly over all $Q \in \mathcal{Q}(P,\pi)$, it is sufficient to choose $t$ so that $P(A_t) \le \alpha \pi$. The conformalized Lee procedure estimates this $(1 - \alpha \pi)-$quantile of the score distribution under the observable treated-selected law $P$.

Split-Conformal Algorithm

We now describe the split-conformal implementation. Let $I_1 = \{i: D_i=1, S_i=1\}$ and $I_0 = \{i: D_i=0, S_i=1\}$. Split $I_1$ into a training fold $I_{\mathrm train}$ and a calibration fold $I_{\mathrm cal}$. The training fold is used to fit any predictive model for $Y$ given $X$, using only treated-selected observations. Given the trained model, let $V : \mathcal{X} \times \mathcal{Y} \rightarrow \mathbb{R}$ be the resulting nonconformity score. Let $\mathcal{H}$ be the sigma-field generated by the training fold $I_{\mathrm train}$, selection sample $\{ (D_i,S_i) \}_{i=1}^n$, and random split process. Conditional on $\mathcal H$, the calibration scores $V_i=V(X_i, Y_i)$, $i \in I_{\mathrm cal}$, are i.i.d. draws from the score distribution induced by $P$.

Let $m = |I_{\text{cal}}|$ and write order statistics $V_{(1)} \le \cdots \le V_{(m)}$ with $V_{(m+1)} = +\infty$. Define the split-conformal index $k = \lceil (m+1)(1-\alpha\pi) \rceil$ and the split-conformal threshold $\hat{t} = V_{(k)}$. The counterfactual prediction set for the treated potential outcome of an always-selected unit with covariates $x$ is $\widehat{C}_1(x) = \{y: V (x,y) \leq \hat{t} \}$. When $\pi=1$, the target law $Q$ coincides with the observed treated-selected law $P$, and the procedure reduces to ordinary split conformal prediction. When $\pi < 1$, the threshold is more conservative: the method calibrates to the $(1-\alpha \pi)$-quantile rather than the usual $(1-\alpha)$-quantile, reflecting the fact that the always-selected distribution may concentrate more heavily in the upper tail of the score distribution than the observed treated-selected mixture.

Conditional on the sigma-field $\mathcal{H}$, let $Z_i = \{(X_i, Y_i)\}_{i=1}^m$ be i.i.d. calibration observations from $P = \mathcal{L}((X,Y(1)) \mid S(1)=1)$, and $Z_{m+1} = (X_{m+1}, Y_{m+1})$ be an independent test point drawn from $Q$. Theorem (ref) shows that this modification delivers finite-sample, distribution-free marginal coverage uniformly over all counterfactual laws $Q \in \mathcal{Q}(P,\pi)$.

Theorem (ref) states that if $\widehat\pi\le \pi$, the error term is zero and the usual $1-\alpha$ coverage guarantee remains valid. If $\widehat\pi > \pi$, coverage may fall below \(1-\alpha\) by at most $$ \alpha\left(\frac{\widehat\pi}{\pi}-1\right). $$ Thus, the error term $\widehat \Delta$ is one-sided. If $\widehat\pi \le \pi $, then the algorithm behaves as if the always-selected share were smaller than it truly is. Equivalently, it uses the looser likelihood-ratio bound $1/\widehat\pi \ge 1 / \pi$, so the resulting prediction set is conservative and $\widehat\Delta =0$. Coverage loss can occur only when $\widehat\pi > \pi$, because then the algorithm understates the worst-case amount by which the target law can upweight high-score events. \hyperref[sec:appendix]{Appendix} provides two practical strategies for estimating $\widehat\pi$: Clopper-Pearson construction and H\"{o}effding construction.

The coverage guarantee is monotone in the value of \(\widehat\pi\) used for calibration. If \(\widehat\pi\le \pi\), then the procedure calibrates against the larger ambiguity set \(\mathcal Q(P,\widehat\pi)\), since $\mathcal Q(P,\pi)\subseteq \mathcal Q(P,\widehat\pi)$. Thus, the true target law remains covered by the robustness class and the usual \(1-\alpha\) guarantee is preserved. However, this validity comes at the cost of wider prediction sets. As \(\widehat\pi\) decreases, the cutoff $V_{(k)} = V_{(\lceil(m+1)(1-\alpha\widehat\pi)\rceil)}$ moves to a higher treated-selected score quantile. In the limit \(\widehat\pi=0\), the cutoff is \(V_{(m+1)}=+\infty\), yielding the trivial prediction set with coverage one. Thus, an arbitrarily small \(\widehat\pi\) is valid but not sharp. For informative inference, \(\widehat\pi\) should be chosen as tightly as possible while remaining a valid lower bound for the true share \(\pi\), for example by using a lower confidence bound.

Theorem (ref) proves coverage for a target law that is not the same as the calibration law. In ordinary split conformal prediction, the test point and calibration points are exchangeable because they are all drawn from the same law. In our case, however, the calibration sample comes from $P$, while the target counterfactual test draw comes from $Q \in \mathcal{Q}(P, \pi)$. Thus, the proof of Theorem (ref) introduces an auxiliary independent draw $Z_{m+1}^{\circ} \sim P$ as a proof device. This draw converts the random tail mass $P( V(Z) > t )$ into the exchangeable event $\{ V(Z_{m+1}^{\circ}) > t \}$. Furthermore, the proof augments nonconformity scores $(V_1, \ldots, V_m, V_{m+1})$ with i.i.d. $U_i \sim \text{Unif}(0,1)$ to randomize ties while preserving exchangeability (see, e.g., kuchibhotla2020exchangeability). Ties can occur with positive probability because the score distribution may have atoms. To solve the problem, the augmented scores $W_i = (V_i,U_i)$ are assumed to be ordered lexicographically to implement random tie-breaking. They preserve exchangeability and make the augmented scores almost surely distinct. Thus, the lexicographic rank is uniform.

Let $\Gamma = \mathcal{L}( X, Y(0), Y(1) \mid \mathcal{A})$ denote a joint law of potential outcomes and covariates in the always-selected population, and let $Q_{\Gamma} = \mathcal{L}(X,Y(1) \mid \mathcal{A})$ be its $(X,Y(1))$-marginal distribution. Under Assumptions (ref)--(ref), $Q_\Gamma \in \mathcal Q (P, \pi)$. For the prediction set $\widehat{C}_1$ from Theorem (ref), we define the shifted ITE prediction set.

Corollary (ref) converts marginal counterfactual coverage for $Y(1)$ into valid marginal predictive interval for the realized individual treatment effect of a randomly drawn selected control unit.

Optimality of the Split-Conformal Threshold

Theorem (ref) shows that the split-conformal threshold $\hat{t} = V_{(k)}$ delivers finite-sample coverage for every counterfactual law $Q \in \mathcal{Q}(P, \pi)$. This subsection explains why the same adjustment from the confidence level $1 - \alpha$ to $1 - \alpha \pi$ is not merely a convenient sufficient correction, but the sharp population target implied by the identification region.

To isolate the identification issue from sampling variability, fix a trained nonconformity score function $V$ and consider prediction sets of the form $C_t=\{z: V(z) \le t \}$. For such sets, the only remaining question is how large $t$ must be to guarantee $Q(V \le t) \ge 1 - \alpha$ uniformly over all admissible counterfactual laws $Q \in \mathcal{Q}(P, \pi)$. Since $Q$ is not identified, this is a minimax calibration problem: $$ \inf_{Q \in \mathcal{Q}(P,\pi)} Q(V \le t) \ge 1 - \alpha,$$or equivalently, $$ \sup_{Q \in \mathcal{Q}(P,\pi)} Q(V > t) \le \alpha.$$ The likelihood-ratio bound defining $\mathcal{Q}(P,\pi)$ implies that a counterfactual law can concentrate up to $1 / \pi$ times as much mass as $P$ on any score-tail event. Hence, for every threshold $t$, $$Q(V>t) \le \frac{1}{\pi} P(V>t).$$

Proposition (ref) shows that this upper bound is sharp over the reduced-information class $\mathcal{Q}(P, \pi)$: for every score-tail event $\{V>t \}$, there exists an admissible law $Q \in \mathcal{Q}(P,\pi)$ that attains the worst-case tail probability. Therefore, the minimax coverage constraint is equivalent to $$P(V>t) \le \alpha \pi.$$ This immediately identifies the smallest uniformly valid population threshold as the lower $(1 - \alpha \pi)-$quantile of the score distribution under the observable treated-selected law $P$. Relative to the full observed law, the procedure remains valid but may be conservative.

Let $F(t) = P(V \le t)$ and $F^{-1}(u) = \inf \{ t \in \overline{\mathbb{R}} : F(t) \ge u\}$, where $\overline{\mathbb{R}} = \mathbb{R} \cup \{+\infty\}$. We define the population minimax threshold over the Lee ambiguity set.

Proposition (ref) translates Proposition (ref) into an explicit minimax target threshold. Thus, among all threshold rules that form prediction sets by accepting scores below a cutoff, $t^*$ is the least conservative threshold that is uniformly valid over the sharp identification region. Any smaller threshold fails for some admissible counterfactual law $Q \in \mathcal{Q}(P,\pi)$. In this sense, the split-conformal threshold $\hat{t}$ used in Theorem (ref) is a finite-sample, distribution-free estimate of the sharp population minimax threshold.

This minimax interpretation connects the present construction to the robust conformal sensitivity-analysis framework of jin2023sensitivity. Their probably approximately correct (PAC) robust conformal procedure chooses a threshold through a conservative lower envelope for the worst-case distribution function of the nonconformity score, and their sharpness analysis studies when this envelope coincides with the true worst-case CDF over the relevant identification set. In the present monotone-selection problem, the ambiguity set is not generated by an analyst-chosen sensitivity parameter. Instead, monotonicity and the observed selection rates identify the likelihood-ratio restriction $0 \le dQ/dP \le 1/\pi,$ where $P$ is the treated-selected law, $Q$ is the always-selected target law, and $\pi$ is the selection rate. This yields the Lee-specific worst-case CDF: $$ G_{\mathrm Lee}(t) = \inf_{Q \in \mathcal{Q}(P, \pi)} Q(V \le t) = \max \Big\{ 0, 1 - \frac{P(V>t)}{\pi} \Big\}. $$ Thus, the conformalized Lee cutoff can be viewed as a closed-form robust conformal threshold obtained from the identified monotone-selection ambiguity set. Algorithm 2 of jin2023sensitivity constructs a PAC lower envelope $\hat{G}_n$ and threshold at $\inf\{t: \hat{G}_n(t) \ge 1 - \alpha \}$. However, the conformalized Lee procedure instead exploits the special scalar upper-tail structure to obtain the explicit split-conformal order statistic.

Simulation Studies

We conduct a Monte Carlo study of conformalized Lee inference under the conditional score-tail selection design. The goal is to test the mechanism of Algorithm 1 in the setting where ordinary conformal prediction is expected to fail.

Table (ref) summarizes the Monte Carlo design. The target sample in each replication is drawn independently from the always-selected population and is used only to evaluate coverage and length. The outcome designs for $Y(d), d \in \{0,1\}$ is heteroskedastic and linear. We draw $$ X\sim \mathrm{Unif}([0,1]^4), $$ and generate $$ U_1\mid X\sim N(0,\sigma^2(X)),\qquad \sigma^2(X)=1+\frac12(2.5X_1)^2. $$ The treated potential outcome is $$ Y(1)=\beta^\top X+U_1,\qquad \beta=(-0.531,0.126,-0.312,0.018)^\top. $$ For selected-control treatment-effect evaluation, we also generate $$ Y(0) = \beta_0^\top X+0.5U_1+U_0,\qquad U_0\sim N(0,1), \qquad \beta_0=(0.2,-0.1,0.1,0.05)^\top. $$ The shared component $U_1$ induces dependence between $Y(1)$ and $Y(0)$.

Table (ref) lists the principal-stratum assignment rules. In all designs, we first draw $$ D \sim \mathrm{Bernoulli}(1/2),\qquad S(1) \sim \mathrm{Bernoulli}(0.8), $$ then assign the always-selected indicator $A$ only among units with $S(1)=1$. We set $S(0) = \operatorname*{\mathbbm{1}} \{A=1\}$ Thus, monotonicity $S(1)\ge S(0)$ holds by construction. The four DGPs differ only in how $A$ is assigned. The benign DGP draws $A$ independently of $(X,U_1,Y(1))$, so the always-selected target law coincides with the treated-selected calibration law. The conditional-tail DGP assigns always-selected units to the conditional upper tail of $|U_1|$. The unconditional-tail DGP assigns always-selected units to the unconditional upper tail of $|U_1|$, which also induces covariate shift because $\sigma(X)$ depends on $X_1$. The smooth DGP uses a logistic selection probability depending on both $X_1$ and the standardized absolute residual $|U_1|/\sigma(X)$, with $(b_X,b_Y)=(1,1.5)$.

We consider the five prediction methods for the selection share $\pi$. Naive conformal uses $\pi_{\mathrm{used}}=1$. Oracle Lee uses the true value $\pi_{\mathrm{used}}=\pi$. Plug-in Lee uses $\widehat \pi= \min\{1, \widehat p_0/ \widehat p_1\}$, where $\widehat p_d$, $d \in \{0,1\}$ is the empirical selection rate. CP-Lee uses a Clopper--Pearson lower confidence bound $\hat \pi_{\mathrm CP}$, and H\"{o}effding-Lee uses a H\"{o}effding lower confidence bound $\widehat \pi_{H}$. For the lower-bound methods, the simulations use the same conformal miscoverage $\alpha$ as the other methods and set the confidence-budget parameter to $\delta_\pi=0.05$. Thus, CP-Lee and H\"{o}effding-Lee are interpreted under the formal target $1-\alpha-\delta_\pi.$ This is the same-$\alpha$ convention used in the reported tables and figures.

Finally, we consider three nonconformity scores. The oracle residual score is $$ V(x,y)=|y-\beta^\top x|, $$ where $\beta$ is the true parameter. This score isolates the Lee calibration logic from model-estimation error. The fitted residual score is $$ V(x,y)=|y-\widehat\mu_1(x)|, $$ where $\widehat\mu_1$ is fit on the treated-selected training fold. The conformalized quantile regression (CQR) score (romano2019conformalized) is $$ V(x,y)= \max\{\widehat q_{\alpha/2}(x)-y,\; y-\widehat q_{1-\alpha/2}(x)\}, $$ where the lower and upper conditional quantile functions are fit on the treated-selected training fold (see, meinshausen2006quantile). These three scores allow us to separate the ideal oracle calibration experiment from more practical implementations using estimated prediction functions.

Table (ref) gives the overall coverage summary across the full focused grid. Since this table averages over the five target coverage levels, the coverage column should be interpreted together with the reported gap column. Under benign selection, naive conformal is valid and its average coverage gap is $0.003$. This is expected because benign selection does not create a difference between the treated-selected calibration law and the always-selected target law. In the same benign design, Lee-adjusted methods over-cover. This overcoverage is the cost of guarding against target laws that are possible under the monotone-selection model but are not realized in the benign DGP.

The pattern is different under selection-induced distribution shift. In the conditional tail DGP, naive conformal has an average gap of $-0.273$. In the unconditional tail DGP, the average gap is $-0.284$. In the smooth DGP, the average gap is $-0.135$. Thus, ordinary conformal prediction fails whenever always-selected units are more likely to have large nonconformity scores than the treated-selected calibration sample suggests. The failure is strongest in the two tail designs and more moderate in the smooth design, which is less adversarial.

Oracle Lee and plug-in Lee largely correct this failure. In the conditional-tail DGP, oracle Lee and plug-in Lee both have average gaps around $0.014$. In the unconditional-tail DGP, their average gaps are around $0.011$ and $0.012$. In the smooth DGP, the average gaps are around $0.066$ and $0.067$. The systematic pattern is that the Lee adjustment restores coverage close to the target, whereas naive conformal remains substantially below target in the shifted designs.

As expected, the lower-bound methods are conservative. CP-Lee has positive average gaps in every DGP, and H\"{o}effding-Lee is even more conservative. This ordering reflects the construction of the lower confidence bounds. Smaller values of $\pi_{\mathrm{used}}$ move the cutoff farther into the upper tail of the treated-selected score distribution, increasing coverage and interval length. The H\"{o}effding bound is looser in these sample sizes, so H\"{o}effding-Lee tends to produce the largest coverage.

Table (ref) reports the same comparison at nominal target coverage $1-\alpha=0.90$. In the benign DGP, naive conformal has coverage $0.903$, while oracle and plug-in Lee cover around $0.953$ and $0.954$. In the shifted designs, naive conformal under-covers: its coverage is $0.771$ in the conditional tail DGP, $0.769$ in the unconditional tail DGP, and $0.821$ in the smooth DGP. Oracle Lee and plug-in Lee recover coverage close to $0.90$, with empirical coverage between $0.909$ and $0.919$ across the shifted designs. CP-Lee and H\"{o}effding-Lee are evaluated against the formal target $0.85$ under the same-$\alpha$ convention and cover well above this target.

Table (ref) summarizes selection-rate diagnostics at nominal coverage $0.90$. The plug-in estimator is nearly unbiased in these designs: for example, when $\pi=0.25$, the average $\widehat\pi$ is about $0.251$ for $m=100$ and $0.251$ for $m=200$. The CP and H\"{o}effding lower bounds are deliberately smaller. When $m=100$ and $\pi=0.25$, the average CP lower bound is $0.181$, while the average H\"{o}effding lower bound is $0.120$. This explains why H\"{o}effding-Lee is the most conservative method. It also explains the only notable infinite-interval issue: for $m=100$ and $\pi=0.25$, H\"{o}effding-Lee has $\Pr(k=m+1)=0.242$ on average. These infinite intervals are mathematically valid but should be interpreted as uninformative. For this reason, the tables report finite average length separately from the probability of $k=m+1$.

Table (ref) summarizes score robustness at nominal coverage $0.90$, averaging over the three shifted DGPs and excluding the benign no-shift design. Naive conformal under-covers for all three scores, with coverage around $0.782$, $0.783$, and $0.794$ for oracle residual, fitted residual, and CQR scores, respectively. Oracle Lee and plug-in Lee cover around $0.91$ across all three score classes. CP-Lee and H\"{o}effding-Lee are conservative relative to their formal target. These results indicate that the Lee correction is not an artifact of oracle residual scoring; it remains effective with fitted residual and CQR scores.

The four figures for coverage-curve consolidate the graphical evidence by DGP. Figure (ref) shows the no-shift benchmark. In the naive column, all rows lie essentially on the diagonal for all values of $\pi$ and for both calibration sizes. This is the expected result because benign selection assigns always-selected status independently of $(X,U_1,Y(1))$ among treated-selected units, so the treated-selected calibration law and always-selected target law coincide. The Lee-adjusted columns lie above the diagonal, with larger overcoverage as $\pi$ decreases. This overcoverage is not a failure of the method; it is the price of robustness when the Lee ambiguity set is larger than the realized target shift.

Figure (ref) gives the sharpest visual test of the Lee correction. In the naive column, coverage is far below the diagonal, and the shortfall grows as $\pi$ decreases. The deterioration is especially severe for $\pi=0.25$, where always-selected units are concentrated in the highest-score part of the treated-selected distribution. In contrast, plug-in Lee is close to the diagonal across oracle residual, fitted residual, and CQR scores. CP-Lee and H\"{o}effding-Lee are conservative, with H\"{o}effding-Lee generally lying highest because the H\"{o}effding lower bound for $\pi$ is the most conservative. This figure provides the clearest evidence that the usual conformal cutoff is calibrated to the wrong law, while the Lee-adjusted cutoff corrects the tail inflation.

Figure (ref) shows the same conclusion under joint covariate and outcome shift. The naive column again exhibits substantial undercoverage, particularly for smaller $\pi$. This is consistent with the construction of the unconditional-tail DGP: selection on the unconditional tail of $|U_1|$ shifts the residual distribution and, because the residual variance depends on $X_1$, also shifts the covariate distribution. Plug-in Lee restores coverage close to the target, while CP-Lee and H\"{o}effding-Lee remain conservative. The similarity between the conditional-tail and unconditional-tail figures indicates that the Lee correction handles shifts in the joint law of $(X,Y(1))$, not only shifts in marginal covariates.

Figure (ref) reports the smooth heterogeneous-selection design. Relative to the two tail DGPs, naive conformal under-covers less severely, but the undercoverage remains systematic and becomes larger as $\pi$ decreases. This pattern is expected because smooth selection is probabilistic rather than a hard tail rule. Plug-in Lee remains close to the target line, and the lower-bound methods over-cover. The results are similar across the oracle residual, fitted residual, and CQR rows, suggesting that the Lee adjustment is not tied to a particular score construction.

Across the four figures, the simulation results are internally coherent. Naive conformal is valid in the benign DGP, where $P=Q$, but fails under conditional-tail, unconditional-tail, and smooth selection, where the always-selected target law differs from the treated-selected calibration law. Plug-in Lee performs close to the oracle-Lee benchmark reported in the tables, while CP-Lee and H\"{o}effding-Lee provide conservative lower-bound implementations. The solid and dashed curves are close in most panels, indicating that increasing the target calibration size from $m=100$ to $m=200$ mainly improves finite-sample stability rather than changing the qualitative conclusions. Overall, the figures and tables support the same message: conformalized Lee inference restores coverage under monotone selection by calibrating to the higher treated-selected quantile implied by the identified always-selected share.

Conclusion

This paper develops conformalized Lee inference for distribution-free predictive inference under monotone sample selection. The starting point is the observation that, when treatment affects selection, the treated-selected sample is generally not drawn from the same population as the selected controls. Under Lee monotonicity, selected controls are always-selected units, while treated-selected observations are a mixture of always-selected and marginal-in units. This mixture structure identifies the share of always-selected units among treated-selected units, $\pi=p_0/p_1$, but not the identity of those units. As a result, the counterfactual treated-outcome law for the always-selected population is only partially identified.

The paper shows that this partial identification problem has a simple distributional form. The always-selected treated law $Q_0$ must be absolutely continuous with respect to the observed treated-selected law $P$, with likelihood ratio bounded above by $1/\pi$. This Lee ambiguity set is shown to be sharp relative to the reduced observables: every law satisfying the likelihood-ratio bound can arise from a data-generating process satisfying random assignment, monotone selection, and the observed selection rates. Thus, the ambiguity set is not an artifact of proof technique, but the exact distributional uncertainty generated by the monotone-selection model.

Building on this identification result, the paper proposes a split-conformal procedure that replaces the usual $(1-\alpha)$ calibration quantile with the higher $(1-\alpha\pi)$ treated-selected score quantile. This adjustment accounts for the fact that the always-selected target law may concentrate up to $1/\pi$ times more probability on high-score events than the treated-selected calibration law. The resulting prediction sets have finite-sample marginal coverage uniformly over all target laws in the Lee ambiguity set. For selected-control units, the observed untreated outcome can be subtracted from the counterfactual treated-outcome interval, yielding distribution-free prediction intervals for realized individual treatment effects in the always-selected population.

The paper also establishes a minimax interpretation of the conformalized Lee cutoff. The worst-case law in the Lee ambiguity set places as much mass as allowed on the upper tail of the nonconformity score. Consequently, the $(1-\alpha\pi)$ population quantile is the smallest score threshold that guarantees uniform coverage over the sharp identification region. Any smaller threshold fails for some target law that is observationally indistinguishable under the maintained monotone-selection assumptions. This result clarifies that the Lee adjustment is not merely conservative; it is the sharp robust calibration rule implied by the identified selection structure.

The simulation evidence supports the theoretical results. Ordinary conformal prediction performs well when selection is benign and the treated-selected and always-selected laws coincide, but it under-covers substantially when always-selected units are more likely to have large nonconformity scores. The Lee-adjusted procedures restore coverage in conditional-tail, unconditional-tail, and smooth selection designs, including designs that induce both outcome-tail shift and covariate shift. The lower-bound implementations based on Clopper--Pearson and Hoeffding confidence bounds are conservative, as expected, and illustrate the practical tradeoff between guaranteed coverage and interval informativeness.

Several directions for future research follow naturally. First, the main procedure uses the reduced information $(P,p_0,p_1)$, but the full observed law contains additional structure through the selected-control covariate distribution and the conditional selection rates $s_0(x)$ and $s_1(x)$. A promising next step is to develop fully covariate-adaptive conformalized Lee methods based on the conditional ambiguity set \[ Q_x \ll P_x, \qquad 0 \leq \frac{dQ_x}{dP_x}(y) \leq \frac{1}{\pi(x)}, \qquad \pi(x)=\frac{s_0(x)}{s_1(x)}. \] Such methods could exploit the identified covariate marginal of the always-selected population and may deliver shorter intervals than the reduced-information procedure while preserving distribution-free validity.

Second, future work should study the statistical problem of estimating local selection shares $\pi(x)$ in high-dimensional settings. Flexible machine-learning estimators, cross-fitting, and one-sided confidence bands for selection probabilities could make the full-law version practically useful. The main challenge is to retain finite-sample or high-probability coverage while avoiding excessive conservativeness from weak lower bounds on $\pi(x)$.

Third, the framework can be extended beyond binary treatment and monotone sample selection. Continuous or multi-valued treatments, dynamic treatment regimes, attrition over time, and competing selection mechanisms all generate related principal-stratum prediction problems. Extending the Lee ambiguity set logic to these settings would broaden the scope of distribution-free causal prediction under post-treatment selection.

Fourth, there is room to connect conformalized Lee inference with sensitivity analysis for violations of monotonicity. The present paper treats Lee monotonicity as the maintained identifying restriction. In applications, however, monotonicity may be credible only approximately. A useful extension would allow a controlled amount of defier mass or bidirectional selection and derive the corresponding robust conformal cutoff. This would place the method between sharp Lee-style identification and analyst-chosen sensitivity models.

Finally, empirical implementation raises important questions about diagnostics and reporting. Because the Lee adjustment becomes more conservative as $\pi$ decreases, applied work should report selection rates, estimated always-selected shares, probabilities of infinite or uninformative intervals, and sensitivity of interval length to the choice of lower confidence bound for $\pi$. Future empirical applications can clarify when conformalized Lee intervals are informative in realistic sample-selection settings and how their conclusions differ from average-effect Lee bounds.

Overall, the paper shows that Lee's monotone-selection logic can be used not only to bound average treatment effects, but also to construct finite-sample, distribution-free prediction intervals for counterfactual outcomes and individual treatment effects. The broader research agenda is to make this form of robust causal prediction more adaptive, more informative, and applicable to richer forms of selection while preserving the central advantage of the present approach: coverage guarantees driven by identified features of the selection problem rather than by unverifiable model specification.