Summary: What is the interpretation of a confidence interval following estimation of a Box-Cox transformation parameter \(\lambda\)? Several authors have argued that confidence intervals for linear model parameters \(\psi\) can be constructed as if \(\lambda\) were known in advance, rather than estimated, provided the estimand is interpreted conditionally given \(\widehat\lambda\). If the estimand is defined as \(\psi(\widehat\lambda)\), a function of the estimated transformation, can the nominal confidence level be regarded as a conditional coverage probability given \(\widehat\lambda\), where the interval is random and the estimand is fixed? Or should it be regarded as an unconditional probability, where both the interval and the estimand are random?
This article investigates these questions via large-\(n\) approximations, small-\(\sigma\) approximations, and simulations. It is shown that, when model assumptions are satisfied and \(n\) is large, the nominal confidence level closely approximates the conditional coverage probability. When \(n\) is small, this conditional approximation is still good for regression models with small error variance. The conditional approximation can be poor for regression models with moderate error variance and single-factor ANOVA models with small to moderate error variance. In these situations the nominal confidence level still provides a good approximation for the unconditional coverage probability. This suggests that, while the estimand may be interpreted conditionally, the confidence level should sometimes be interpreted unconditionally.