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Multivariate Tie-breaker Designs

Tim P. Morrison, Art B. Owen

TL;DR

The paper studies tie-breaker designs (TBDs) that interpolate between regression discontinuity designs (RDDs) and randomized controlled trials (RCTs) when interventions are costly, extending the TBD framework to multivariate covariates and a multivariate regression model. It introduces prospective D-optimality (EW D-optimality) to optimize treatment-probability schemes under fixed or random covariates, proving that, under random covariates, the RCT is prospectively optimal within a broad class of designs; it also analyzes symmetric designs and derives explicit efficiency expressions. A convex optimization framework is developed to compute p(X) under practical constraints (budget, monotonicity, gain, covariate balance), with a theoretical result showing monotonicity constraints induce sparsity in the number of distinct treatment levels. The methodology is demonstrated on a MIMIC-IV-ED triage dataset, illustrating how to balance long-term statistical efficiency against short-term gains and ethical constraints, and highlighting the potential of TBDs for efficient causal learning in constrained settings.

Abstract

In a tie-breaker design (TBD), subjects with high values of a running variable are given some (usually desirable) treatment, subjects with low values are not, and subjects in the middle are randomized. TBDs are intermediate between regression discontinuity designs (RDDs) and randomized controlled trials (RCTs). TBDs allow a tradeoff between the resource allocation efficiency of an RDD and the statistical efficiency of an RCT. We study a model where the expected response is one multivariate regression for treated subjects and another for control subjects. We propose a prospective D-optimality, analogous to Bayesian optimal design, to understand design tradeoffs without reference to a specific data set. For given covariates, we show how to use convex optimization to choose treatment probabilities that optimize this criterion. We can incorporate a variety of constraints motivated by economic and ethical considerations. In our model, D-optimality for the treatment effect coincides with D-optimality for the whole regression, and, without constraints, an RCT is globally optimal. We show that a monotonicity constraint favoring more deserving subjects induces sparsity in the number of distinct treatment probabilities. We apply the convex optimization solution to a semi-synthetic example involving triage data from the MIMIC-IV-ED database.

Multivariate Tie-breaker Designs

TL;DR

The paper studies tie-breaker designs (TBDs) that interpolate between regression discontinuity designs (RDDs) and randomized controlled trials (RCTs) when interventions are costly, extending the TBD framework to multivariate covariates and a multivariate regression model. It introduces prospective D-optimality (EW D-optimality) to optimize treatment-probability schemes under fixed or random covariates, proving that, under random covariates, the RCT is prospectively optimal within a broad class of designs; it also analyzes symmetric designs and derives explicit efficiency expressions. A convex optimization framework is developed to compute p(X) under practical constraints (budget, monotonicity, gain, covariate balance), with a theoretical result showing monotonicity constraints induce sparsity in the number of distinct treatment levels. The methodology is demonstrated on a MIMIC-IV-ED triage dataset, illustrating how to balance long-term statistical efficiency against short-term gains and ethical constraints, and highlighting the potential of TBDs for efficient causal learning in constrained settings.

Abstract

In a tie-breaker design (TBD), subjects with high values of a running variable are given some (usually desirable) treatment, subjects with low values are not, and subjects in the middle are randomized. TBDs are intermediate between regression discontinuity designs (RDDs) and randomized controlled trials (RCTs). TBDs allow a tradeoff between the resource allocation efficiency of an RDD and the statistical efficiency of an RCT. We study a model where the expected response is one multivariate regression for treated subjects and another for control subjects. We propose a prospective D-optimality, analogous to Bayesian optimal design, to understand design tradeoffs without reference to a specific data set. For given covariates, we show how to use convex optimization to choose treatment probabilities that optimize this criterion. We can incorporate a variety of constraints motivated by economic and ethical considerations. In our model, D-optimality for the treatment effect coincides with D-optimality for the whole regression, and, without constraints, an RCT is globally optimal. We show that a monotonicity constraint favoring more deserving subjects induces sparsity in the number of distinct treatment probabilities. We apply the convex optimization solution to a semi-synthetic example involving triage data from the MIMIC-IV-ED database.
Paper Structure (17 sections, 7 theorems, 70 equations, 2 figures, 1 table)

This paper contains 17 sections, 7 theorems, 70 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Tie-breaker designs
  3. Marginally constrained experimental design
  4. Sequential experimentation
  5. Setup
  6. Efficiency and D-optimality
  7. D-optimality for nonrandom X and Z
  8. Prospective D-optimality
  9. Symmetric Distributions with Symmetric Designs
  10. Short-term Gain
  11. Convex Formulation
  12. Some useful convex constraints
  13. Piecewise constant optimal designs
  14. MIMIC-IV-ED Example
  15. Discussion
  16. ...and 2 more sections

Key Result

Theorem 1

For data following model eq:linmod, assume that $U$ is full rank. Then the D-optimality criteria for $\tilde{\gamma}$, $\tilde{\beta}$ and $\delta$ are equivalent.

Figures (2)

  • Figure 1: Efficiency and gain of the MIMIC-IV-Ed simulation as a function of the width of the randomization window $\Delta$.
  • Figure 2: Optimal solutions for MIMIC-IV-ED treatment probabilities under various constraints. The treatment constraint imposed $\overline{p} = 0.2$ for the average treatment rate, and the gain constraint imposed $\rho = 0.7$, i.e., at least $70\%$ of the maximum possible gain.

Theorems & Definitions (16)

  • Theorem 1
  • proof
  • Definition 1
  • Theorem 2
  • Definition 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • proof : Proof of \ref{['thm:rctisoptimal']}
  • proof : Proof of \ref{['thm:keeppathalf']}
  • ...and 6 more