Identifying Sample Size and Accuracy and Precision of the Estimators in Case-Crossover Analysis with Distributed Lags of Heteroskedastic Time-Varying Continuous Exposures Measured with Simple or Complex Error

Power analyses help investigators design robust and reproducible research. Understanding of determinants of statistical power is helpful in interpreting results in publications and in analysis for causal inference. Case-crossover analysis, a matched case-control analysis, is widely used to estimate health effects of short-term exposures. Despite its widespread use, understanding of sample size, statistical power, and the accuracy and precision of the estimator in real-world data settings is very limited. First, the variance of exposures that exhibit spatiotemporal patterns may be heteroskedastic (e.g., air pollution and temperature exposures, impacted by climate change). Second, distributed lags of the exposure variable may be used to identify critical exposure time-windows. Third, exposure measurement error is not uncommon, impacting the accuracy and/or precision of the estimator, depending on the measurement error mechanism. Exposure measurement errors result in covariate measurement errors of distributed lags. All these issues complicate the understanding. Therefore, I developed approximation equations for sample size, estimates of the estimators and standard errors, and identified conditions for applications. I discussed polynomials for non-linear effect estimation. I analyzed air pollution estimates in the United States (U.S.), developed by U.S. Environmental Protection Agency to examine errors, and conducted statistical simulations. Overall, sample size can be calculated based on external information about exposure variable validation, without validation data in hand. For estimators of distributed lags, calculations may perform well if residual confounding due to covariate measurement error is not severe. This condition may sometimes be difficult to identify without validation data, suggesting investigators should consider validation research.