On 2015 Jul 06, Adrian Barnett commented:
I have used the random effects models on skewed exercise data with just 2 to 3 observations per person and the model converged and gave a meaningful answer. That may not always happen, but given the potential change in interpretation it is worth trying.
On 2015 May 29, Chris Wallace commented:
Thank you for your interest in our paper. We used the cosinor model because we wanted a parsimonious model for testing for seasonality. I completely agree, that to model and understand the specific seasonal patterns more precisely we will need to consider models with more parameters which do not enforce symmetry or a sinusoidal shape. Your point regarding individual random effects for the cosine and sine terms is interesting, but I wonder, does this require many repeated measures per individual to fit well? Typically the datasets we accessed, although longitudinal, had a limited number of observations per individual.
On 2015 May 16, Adrian Barnett commented:
This is an interesting and highly novel paper. The cosinor model uses only two parameters to create a seasonal pattern and is therefore parsimonious, however more complex seasonal models are likely to provide further understanding of the seasonal patterns shown.
Firstly the cosinor model used gives the seasonal pattern at a population level, but the seasonal pattern in individuals is likely to be stronger. As an extreme example, imagine half a study sample had a seasonal peak in early January and the other half in early July. At the population level these patterns would cancel out despite their being potentially strong individual seasonality. The strength of the individual seasonal pattern can be estimated by including random effects (at the individual level) for the cosine and sine terms. The greater the heterogeneity in individual seasonal patterns, the greater the difference between the individual and population seasonal pattern.
Secondly the cosinor model is symmetric in terms of both the peak and trough, and the rate of seasonal increase and decrease. More general models, such as splines or models that use a categorical variable for month, use more parameters but are able to show a wider variety of patterns, such as a steep increase during the autumn to winter transition and slower decrease during the winter to spring transition.
On 2015 May 16, Adrian Barnett commented:
This is an interesting and highly novel paper. The cosinor model uses only two parameters to create a seasonal pattern and is therefore parsimonious, however more complex seasonal models are likely to provide further understanding of the seasonal patterns shown.
Firstly the cosinor model used gives the seasonal pattern at a population level, but the seasonal pattern in individuals is likely to be stronger. As an extreme example, imagine half a study sample had a seasonal peak in early January and the other half in early July. At the population level these patterns would cancel out despite their being potentially strong individual seasonality. The strength of the individual seasonal pattern can be estimated by including random effects (at the individual level) for the cosine and sine terms. The greater the heterogeneity in individual seasonal patterns, the greater the difference between the individual and population seasonal pattern.
Secondly the cosinor model is symmetric in terms of both the peak and trough, and the rate of seasonal increase and decrease. More general models, such as splines or models that use a categorical variable for month, use more parameters but are able to show a wider variety of patterns, such as a steep increase during the autumn to winter transition and slower decrease during the winter to spring transition.