On Weighted Trigonometric Regression for Suboptimal Designs in Circadian Biology Studies

Studies in circadian biology often use trigonometric regression to model phenomena over time. Ideally, protocols in these studies would collect 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 cause a loss of statistical power when performing hypothesis tests with an estimated model. This paper is motivated by the potential loss of statistical power during hypothesis testing, 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 concentration hyperparameter for kernel density estimation that maximizes the Hessian of weighted squared loss, which relates to both maximizing the $D$-optimality criterion from experimental design literature and minimizing the generalized variance. Simulation studies consistently demonstrate that this weighted regression mitigates variability in inferences produced by an estimated model. Illustrations with three real circadian biology data sets 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 biology studies.