Improving Transportability of Regression Calibration Under the Main/External Validation Study Design

In epidemiology, obtaining accurate individual exposure measurements can be costly and challenging. Thus, these measurements are often subject to error. Regression calibration with a validation study is widely employed as a study design and analysis method to correct for measurement error in the main study due to its broad applicability and simple implementation. However, relying on an external validation study to assess the measurement error process carries the risk of introducing bias into the analysis. Specifically, if the parameters of regression calibration model estimated from the external validation study are not transportable to the main study, the subsequent estimated parameter describing the exposure-disease association will be biased. In this work, we improve the regression calibration method for linear regression models using an external validation study. Unlike the original approach, our proposed method ensures that the regression calibration model is transportable by estimating the parameters in the measurement error generating process using the external validation study and obtaining the remaining parameter values in the regression calibration model directly from the main study. This guarantees that parameter values in the regression calibration model will be applicable to the main study. We derived the theoretical properties of our proposed method. The simulation results show that the proposed method effectively reduces bias and maintains nominal confidence interval coverage. We applied this method to data from the Health Professionals Follow-Up Study (main study) and the Men’s Lifestyle Validation Study (external validation study) to assess the effects of dietary intake on body weight.

💡 Research Summary

In epidemiologic studies, measurement error in exposure variables can lead to biased estimates of exposure–outcome relationships. Regression calibration (RC) is a popular method that uses a validation study to model the relationship between the true exposure (X) and its error‑prone surrogate (Z), then applies this model to correct the main study where only Z is observed. Traditional RC assumes that the calibration model estimated in an external validation study is fully transportable to the main study—that is, the parameters of the conditional expectation E(X|Z,W) are identical in both datasets. This “double transportability” often fails because the distribution of true exposures given covariates (X|W) may differ between the external validation cohort and the main cohort, leading to biased corrected estimates.

The authors propose a new RC approach that requires only the weaker “single transportability” assumption: the measurement‑error generating mechanism Z = c₀ + C₁X + C₂W + εₑ (a classical‑like error model) is the same in both studies. Under this assumption, they estimate the transportable parameters (c₀, C₁, C₂, Σₑ) from the external validation data. In the main study, they estimate the marginal distribution of Z given the error‑free covariates W, i.e., Z = b₀ + B₂ᵀW + ε_z, using only the observed Z and W. By applying Bayes’ theorem and introducing arbitrary linear operators L₁ and L₂, they combine the two sets of parameters to derive the main‑study calibration model X = γ₀ + Γ₁ᵀZ + Γ₂ᵀW + ε_x, where γ₀, Γ₁, and Γ₂ are expressed as functions of (c₀, C₁, C₂) and (b₀, B₂). This construction avoids the need for the full joint distribution of (X,Z,W) to be identical across studies.

The methodology is semiparametric; normality of the error terms is not required because the linear‑operator framework accommodates non‑Gaussian distributions (important for dietary intake variables). The authors prove that the resulting estimators are consistent and asymptotically normal, and they provide explicit formulas for the asymptotic variance.

Simulation studies compare the proposed method with the conventional RC that uses only the external calibration model. When the double transportability assumption is violated, the conventional RC exhibits substantial bias in the exposure coefficient and under‑coverage of confidence intervals. In contrast, the new method dramatically reduces bias and restores nominal 95 % coverage across a range of sample sizes and error structures.

The approach is illustrated with real data from the Health Professionals Follow‑Up Study (main cohort) and the Men’s Lifestyle Validation Study (external validation). The authors assess the effect of dietary intake on body weight. The conventional RC yields attenuated effect estimates, whereas the transportable RC provides larger, more plausible coefficients with appropriate confidence interval widths, demonstrating the practical advantage of the method.

In summary, this paper introduces a robust regression‑calibration technique for the main/external validation design that only requires the measurement‑error model to be transportable, not the full calibration model. By leveraging information from both studies and using a flexible linear‑operator formulation, the method achieves unbiased estimation and valid inference even when the external validation cohort differs in exposure distribution from the main cohort. This contribution fills a methodological gap and broadens the applicability of RC in epidemiology and related fields.