Weighted Trigonometric Regression for Suboptimal Designs in Circadian Transcriptome Studies

Circadian transcriptome studies often use trigonometric regression to model gene expression over time. Ideally, protocols in these studies would collect tissue samples at evenly distributed and equally spaced time points over a 24-hour period. This sample collection protocol is known as an equispaced design, which is considered the optimal experimental design for trigonometric regression under multiple statistical criteria. However, implementing equispaced designs in studies involving individuals is logistically challenging, and failure to employ an equispaced design could introduce variability in the statistical power of a hypothesis test relative to a model's phase-shift parameter estimates. This article is motivated by the variability in power for hypothesis testing when tissue samples are not collected under an equispaced design, and considers a weighted trigonometric regression as a remedy. Specifically, the weights for this regression are the normalized reciprocals of estimates derived from a kernel density estimator for sample collection time, which inflates the weight of samples collected at underrepresented time points. A search procedure is also introduced to identify the hyperparameter for kernel density estimation that relates to maximizing the smallest eigenvalue of the Hessian of weighted squared loss, which is motivated by the $E$ -optimality criterion from experimental design literature. Simulation studies consistently demonstrate that this weighted regression mitigates variability in power for hypothesis tests performed with an estimated model. Illustrations with six circadian transcriptome datasets further indicate that this weighted regression consistently yields larger test statistics than its unweighted counterpart for first-order trigonometric regression, or cosinor regression, which is prevalent in circadian transcriptome studies.