  Rhythmic data are ubiquitous in the life sciences. Biologists need reliable
statistical tests to identify whether a particular experimental treatment has
caused a significant change in a rhythmic signal. When these signals display
nonstationary behaviour, as is common in many biological systems, the
established methodologies may be misleading. Therefore, there is a real need
for new methodology that enables the formal comparison of nonstationary
processes. As circadian behaviour is best understood in the spectral domain,
here we develop novel hypothesis testing procedures in the (wavelet) spectral
domain, embedding replicate information when available. The data are modelled
as realisations of locally stationary wavelet processes, allowing us to define
and rigorously estimate their evolutionary wavelet spectra. Motivated by three
complementary applications in circadian biology, our new methodology allows the
identification of three specific types of spectral difference. We demonstrate
the advantages of our methodology over alternative approaches, by means of a
comprehensive simulation study and real data applications, using both published
and newly generated circadian datasets. In contrast to the current standard
methodologies, our method successfully identifies differences within the
motivating circadian datasets, and facilitates wider ranging analyses of
rhythmic biological data in general.
