Retrospective Orthogonal Design: Response-Surface Reconstruction from Observational Data

2026-07-28

AI Summary

This study addresses the instability of regression estimates and inconsistency in sequential sums of squares arising from multicollinearity and term-order dependence in observational data. The authors propose a conditional mean surface reconstruction method based on probabilistic balanced lattices, which preserves observed cell means, imputes unsupported cells, and employs Freudenthal simplicial subdivision for piecewise affine interpolation. The approach innovatively introduces norm-invariant contrast effects and an order-independent decomposition of sums of squares, complemented by a response-agnostic projection calibration to mitigate resolution loss, along with a dependency-aware sample size planning criterion. Evaluated across 6,480 simulation scenarios, the method achieves superior performance in threshold detection, sign interaction, and localized surface recovery tasks. When applied to the Mincer wage equation, it yields the highest out-of-sample R² and produces sum-of-squares allocations entirely invariant to term ordering.

Abstract

Regression estimates from observational data can depend on specification under multicollinearity, while sequential sums of squares (SS) depend on term order. We introduce Retrospective Orthogonal Design (ROD), which reconstructs conditional mean surfaces on a probability-balanced lattice. ROD preserves observed cell means, completes unsupported cells, applies weighted tensor-product contrasts, and evaluates the reconstructed surface through piecewise-affine interpolation over Freudenthal polyhedra. Resolution and completion are selected jointly by validation among rank-admissible candidates, followed by refitting and evaluation on an untouched test set. For an admissible lattice, $\mathbf{X}^{\top}\mathbf{W}\mathbf{X}=c\mathbf{I}$, yielding specification-invariant contrast effects and unique, order-independent SS within the retained contrast space. Response-free projection calibration maps the fixed reconstruction onto a declared scientific basis and corrects finite-resolution recovery loss. Across 6,480 simulation conditions spanning nine data-generating processes, ROD matched or exceeded polynomial regression in five processes and performed strongest on threshold, sign-interaction, and localized surfaces. For the quadratic-interaction process, mean out-of-sample $R^2$ differed by only $0.0001$, while calibrated coefficient bias remained small across prespecified targets. A Rao-based information adjustment provides dependence-aware sample-size guidance for ROD planning. In a weighted Mincer application, ROD produced the highest out-of-sample $R^2$ point estimate, with substantial interval overlap with polynomial regression, and provided exhaustive SS allocations invariant to term-entry order.

Problem

• multicollinearity
• observational data
• sequential sums of squares
• specification dependence
• response-surface reconstruction

Innovation

• Retrospective Orthogonal Design
• response-surface reconstruction
• specification invariance
• order-independent sums of squares
• Freudenthal interpolation