Circadian rhythm analysis when output is collected at intervals

It has long been recognized that many physiological and biochemical parameters show a repeating pattern of variation over 24 hours, i.e., a circadian rhythm. Halberg, Tong and Johnson (1965) used the sinusoidal model with 24-hour cycle to describe this variation. Tong (1976) described the polar coordinate transformation by which the sinusoidal regression problem can be treated as a linear regression problem. When the output of a system following diurnal variation, e.g. the human kidney, is collected at regular intervals and assayed, the expected quantity of substance present corresponds to the integral of the underlying output function. This paper shows that the polar coordinate transformation also linearizes this regression problem. More importantly, the covariance structure of Halberg et al. does not include interindividual variation. An alternative and more general model is proposed here based upon Rao's (1959) growth curve analyses. The latter method allows testing for adequacy of the sinusoidal model and leads to inferences about population parameters. An example is given.