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.

翻译：测量误差是跨多个学科普遍存在的挑战，然而在现实世界的复杂场景中，其对样本量确定以及估计量准确性与精确度的影响仍未得到充分研究。这些场景包括异方差连续暴露、易错测量、多个暴露时间点以及校准暴露变量的使用。本文推导了用于样本量计算、估计量准确度及标准误的近似方程。该框架适用于使用多项式进行的非线性效应估计，并针对异方差暴露的分布滞后模型，在存在或不存在暴露验证数据的情况下，处理了非差分、自相关以及差分加性或乘性测量误差。所提出的理论和方法为高效的研究设计提供了实用工具，并深化了对测量误差影响研究的理解，同时无缝整合了不确定性分析。