Sample Size and Bias Approximations For Continuous Exposures Measured with Error

Measurement error is a pervasive challenge across many disciplines, yet its impact on sample size determination and the accuracy and precision of estimators remains understudied in real-world complex scenarios. These include heteroskedastic continuous exposures, error-prone measurements, multiple exposure time points, and the use of calibrated exposure variables. This article develops approximation equations for sample size calculations, estimator accuracy, and standard errors. The framework accommodates non-linear effect estimation using polynomials and addresses non-differential, autocorrelated, and differential additive or multiplicative measurement errors in distributed lag models for heteroskedastic exposures in the absence or presence of exposure validation data. The proposed theory and methods provide practical tools for efficient research design and a deeper understanding of measurement error impacts on research, while seamlessly integrating uncertainty analyses.