  Many biological processes display oscillatory behavior based on an
approximately 24 hour internal timing system specific to each individual. One
process of particular interest is gene expression, for which several circadian
transcriptomic studies have identified associations between gene expression
during a 24 hour period and an individual's health. A challenge with analyzing
data from these studies is that each individual's internal timing system is
offset relative to the 24 hour day-night cycle, where day-night cycle time is
recorded for each collected sample. Laboratory procedures can accurately
determine each individual's offset and determine the internal time of sample
collection. However, these laboratory procedures are labor-intensive and
expensive. In this paper, we propose a corrected score function framework to
obtain a regression model of gene expression given internal time when the
offset of each individual is too burdensome to determine. A feature of this
framework is that it does not require the probability distribution generating
offsets to be symmetric with a mean of zero. Simulation studies validate the
use of this corrected score function framework for cosinor regression, which is
prevalent in circadian transcriptomic studies. Illustrations with three real
circadian transcriptomic data sets further demonstrate that the proposed
framework consistently mitigates bias relative to using a score function that
does not account for this offset.
