I have a dataset of microbial counts measured in 8 mice over 6 timepoints each. 4 of the mice were exposed to control conditions and the other 4 were exposed to sleep apnea conditions (IHC). I am interested in determining the effect of IHC on counts of a given microbe while accounting for the circadian nature of the data.

I have model that I’ve fit but I’m a bit confused about how the resultant parameters should be interpreted. IHC is encoded as a fixed effect and subject ID is encoded as a random effect.

where x_i is binary exposure status, t = time, P = period, \psi = phase shift

My understanding is that \beta_{\textrm{IHC}} can be interpreted as the log-fold change of the microbial abundance between IHC and control. However, I am less certain of how to interpret A. Would it just be the log amplitude associated with the sinusoidal signal?

Furthermore, if I wanted to extend this model to determine whether amplitude varies by exposure, would this be as simple as specifying the amplitude as \left( A_{\textrm{Intercept}} + x_i A_{\textrm{IHC}} \right) rather than just A? Where A_{\textrm{IHC}} is the log-fold change in amplitude associated with IHC status?

Please feel free to ask for clarification.

Thanks!