🔍 What Was Studied

Many experimenters gather a sample and baseline covariates before assigning treatments. The central question addressed here is how to allocate treatments across that fixed sample to produce the most accurate estimate of an average treatment effect (ATE).

🧠 A Decision-Theory Approach to Design

Framing experimental design as a statistical decision problem changes the usual prescription. If the goal is to estimate the ATE and estimates are judged by squared error, random assignment need not be optimal. Instead, treatment assignment should be chosen to minimize the expected mean squared error (MSE) of the estimator.

📐 What Is Minimized and How

• Objective: minimize the expected MSE of the ATE estimator under the available baseline information.
• Result: explicit expressions for the expected MSE are derived, showing how covariates and the chosen estimator enter the objective.
• Practical implication: these formulas identify deterministic or constrained assignment rules that outperform pure randomization on the MSE criterion.

🛠️ Matlab Implementation and Practical Steps

• The analytic expressions lead directly to concrete, implementable procedures for experimental design.
• A Matlab implementation accompanies the derivations to translate the MSE formulas into assignment algorithms that can be applied to a collected sample and its baseline covariates.

💡 Why It Matters

This approach shows that, under clear decision-theoretic criteria (ATE estimation, squared-error loss), experimenters can and should consider optimized assignment schemes instead of defaulting to randomization. The provided expressions and Matlab tools make such optimized designs practical for applied researchers seeking more precise treatment-effect estimates.