Effect Identification and Unit Categorization in the Multi-Score Regression Discontinuity Design with Application to LED Manufacturing

TL;DR

This paper extends Regression Discontinuity Design to multi-score settings (MRD) by formalizing unit behavior under general Boolean cutoff rules and introducing an expanded set of compliance types, including indecisive units. It develops an identification framework for the local complier effect at the multi-dimensional cutoff, along with results on when subset removal of non-changing units preserves identification. The authors analyze how decomposing complex rules into subrules affects unit classification (inheritance) and provide practical guidance for estimating subset-complier effects in MRD. The empirical application to LED manufacturing demonstrates how multi-score rules and operator behavior influence treatment effects and suggests that removing identifiable noncompliers can improve estimation variance, especially when combined with ML-based adjustments. Overall, the work broadens MRD theory and offers actionable methodology for optimizing complex, multi-criteria decision policies in manufacturing and other settings.

Abstract

RDD (Regression discontinuity design) is a widely used framework for identifying and estimating causal effects at the cutoff of a single running variable. In practice, however, decision-making often involves multiple thresholds and criteria, especially in production systems. Standard MRD (multi-score RDD) methods address this complexity by reducing the problem to a one-dimensional design. This simplification allows existing approaches to be used to identify and estimate causal effects, but it can introduce non-compliance by misclassifying units relative to the original cutoff rules. We develop theoretical tools to detect and reduce "fuzziness" when estimating the cutoff effect for units that comply with individual subrules of a multi-rule system. In particular, we propose a formal definition and categorization of unit behavior types under multi-dimensional cutoff rules, extending standard classifications of compliers, alwaystakers, and nevertakers, and incorporating defiers and indecisive units. We further identify conditions under which cutoff effects for compliers can be estimated in multiple dimensions, and establish when identification remains valid after excluding nevertakers and alwaystakers. In addition, we examine how decomposing complex Boolean cutoff rules (such as AND- and OR-type rules) into simpler components affects the classification of units into behavioral types and improves estimation by making it possible to identify and remove non-compliant units more accurately. We validate our framework using both semi-synthetic simulations calibrated to production data and real-world data from opto-electronic semiconductor manufacturing. The empirical results demonstrate that our approach has practical value in refining production policies and reduces estimation variance. This underscores the usefulness of the MRD framework in manufacturing contexts.

Figures (10)

• Figure 1: Cutoff rules $D = I_1 \land I_2$ and $T = I_1$. The decision boundaries coincide with the coordinate axes. When $X_2 > 0$, $T$ complies with $D$; otherwise, $D = 0$ regardless of the value of $X_1$, which governs the behavior of $T$.
• Figure 2: General case with $D$ not being a cutoff rule. The decision boundaries of $T$ and $D$ are indicated with orange and blue dashed lines.
• Figure 3: All possible category changes when transitioning from the complex rule $T = G \land H$ to the simpler subrule $G$ given that $\mathop{\mathrm{supp}}\nolimits(G) \oplus \mathop{\mathrm{supp}}\nolimits(H) = \mathop{\mathrm{supp}}\nolimits(T)$ and assuming that there are no indecisive units. The figure summarizes Propositions \ref{['prop:bound:atnt']}, \ref{['prop:operation_simple']}, \ref{['prop:bound:comp']} and \ref{['prop:bound:def']}. The OR-case can be described analogously.
• Figure 4: Left: $X_D$ is defined as the distance between the current mean color point $C_1$ and the target point $C_P$, which is the closest the lot can technically get to the initial target $C_T$. The slope of the dashed curve is given by a physical process and thus it is not possible to reach $C_T$ once the curve deviates. Right: $X_Y$ evaluates the expected improvement by calculating the share of in-specification chips in the lot. This is done by moving the current distribution of color points to the target.
• Figure 5: Real data plot with respect to the score components $X_D$ and $X_Y$. The decision boundary $T$ is dashed. The actual treatment assignment (red and blue) follows an unobserved rule $D$, rendering the MRD fuzzy.

Theorems & Definitions (37)

• Definition 1
• Definition 2
• Lemma 1
• Proposition 1
• Definition 3
• Proposition 2
• Proposition 3
• Proposition 4
• Theorem 1
• Theorem 2