Abstract

The Box–Cox transformation is widely used to improve normality, stabilize variance, and enable Gaussian-based modelling in a transformed scale. After model fitting, conditional summaries are often mapped back to the original scale by applying the inverse transformation. This paper shows that this transform–fit–inverse procedure has a structural limitation: nonlinear inverse transformations do not, in general, preserve conditional expectations. Equivalently, conditional expectation and nonlinear inversion do not commute. Within the Box–Cox Gaussian framework, the admissible domain of the inverse transformation leads naturally to a truncated normal formulation in the transformed scale. Under this formulation, we derive a second-order decomposition showing that the original-scale conditional mean differs from the inverse-transformed truncated conditional mean by a curvature-driven correction term depending on the truncated conditional variance. The usual untruncated Gaussian expression is recovered as a local approximation when the inadmissible probability is negligible. A numerical sensitivity analysis, focused on 0≤λY≤1, illustrates how the distortion depends on the transformation parameter, correlation, and conditional dispersion. A real-data illustration using medical insurance charges further shows that the discrepancy can be visible in an applied regression setting and is not removed by changing the transformation of the explanatory variable. The results distinguish this structural invariance problem from classical retransformation bias and show that inverse-transformed fitted curves should be interpreted as transformation-induced structural curves, not automatically as conditional mean functions on the original scale.

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