Response to the Commentary on "Experimental Design and Power Calculation in Omics Circadian Rhythmicity Detection...

We agree and highly appreciate the correction of the non-central parameter (ncp) in the power calculation by Dr. Westermark. We acknowledge the mathematical mistake in our derivation. As indicated in the commentary, under the "evenly spaced sampling design" (e.g., sampling every 2 h for 24 h with four biological replicates at each time point; $12\times 4=48$ samples), discrepancy of the non-central parameter and the resulting statistical power between the corrected calculation and ours is 0. Below, we conduct further simulations to investigate the degree of statistical power miscalculation in "unevenly-spaced sampling design" in case potential users may have applied our formula in their experimental planning. We consider two types of sampling designs throughout the 24 h of a day: (i) Uniformly distributed (uniform): $t_i\stackrel{\mathrm{iid}}{\sim }\text{UNIF}\left(0,24\right)$ ; (ii) Unimodal Gaussian distributed (unimodal): $t_i\stackrel{\mathrm{iid}}{\sim }N\left(0,,,{\delta }^2\right)$ with $\delta =2,4,6,8$ . The following figure shows results for amplitude-to-noise ratio $r=A/\sigma =0.4,0.6,0.8$ and phase $\varphi =0,3,6,9$ . Note that results of $\varphi =12,15,18,21$ are identical to $\varphi =0,3,6,9$ due to symmetry. As expected, for uniformly or almost uniformly distributed designs (i.e., Uniform, Unimodal-sd6, and Unimodal-sd8), the difference between the corrected and previous power calculations is 0 or close to 0. For moderate to extremely uneven distributed design (Unimodal-sd2 and Unimodal-sd4), the discrepancies become more and more substantial. The previous incorrect npc tends to inflate the estimated statistical power. We have updated the CircaPower R package on the Github ( https://github.com/circaPower/circaPower), highlighted the correction, and cited the commentary paper.