Nonlinear estimation and statistical testing of periods in nonsinusoidal longitudinal time series with unequidistant observations

The analysis of multiple components is often used to model biological variables that show nonsinusoidal predictable changes of known periods. In general, to anticipate the periods is not easy, and even in cases when we have some a priori information, it is advisable to have a statistical tool to test the chosen periods. In this work, we introduce a statistical procedure to estimate periods of longitudinal series by applying nonlinear regression techniques to the multiple sinusoidal model, as well as to the general linear model. Approximate inferences about the parameters of the model are carried out under the usual hypothesis of normality, independence, and constant variance of the errors. Confidence intervals (CIs) for each individual parameter, as well, as for the amplitude-acrophase pair or for any other subgroup of parameters of interest, can be computed. As in the linear analysis of multiple components, it is possible to check the existence of rhythm by means of a zero-amplitude test. The method also allows statistical testing of several hypotheses related to the periods. For example, it is possible to test if the periods are equal to certain values of chronobiologic interest and to check if some components included in the model are harmonically related. On the other hand, when the fitted components have proximal periods, the method allows one to verify if they are modeling the same or different spectral peaks. The method, which was validated by a simulation study for a model of two components and is illustrated by an example of modeling the diastolic blood pressure of two subjects, represents a new step in the development of statistical procedures in chronobiology.