A Note on Harmonic Underspecification in Log-Normal Trigonometric Regression
Michael T. Gorczyca
TL;DR
The paper addresses how harmonic underspecification affects parameter estimation in log-normal trig regression versus GLMs for non-normally distributed biological rhythm data. It proves that, when responses follow a generalized gamma distribution, log-normal trig regression provides unbiased estimates of oscillation harmonics under expectation for $K\le K^*$, while GLMs with a log link can be biased unless the model is correctly specified. A key theoretical result (Proposition) derives the bias-canceling constant $c_0$ in $\mathbb{E}(\hat{\beta})$, clarifying the advantage of transforming the response. An empirical cortisol dataset illustrates that log-normal trig regression yields invariant harmonic estimates with respect to the number of specified harmonics, whereas GLM-based estimates vary with $K$, with convergence occurring when sufficiently many harmonics are included.
Abstract
Analysis of biological rhythm data often involves performing least squares trigonometric regression, which models the oscillations of a response over time as a sum of sinusoidal components. When the response is not normally distributed, an investigator will either transform the response before applying least squares trigonometric regression or extend trigonometric regression to a generalized linear model (GLM) framework. In this note, we compare these two approaches when the number of oscillation harmonics is underspecified. We assume data are sampled under an equispaced experimental design and that a log link function would be appropriate for a GLM. We show that when the response follows a generalized gamma distribution, least squares trigonometric regression with a log-transformed response, or log-normal trigonometric regression, produces unbiased parameter estimates for the oscillation harmonics, even when the number of oscillation harmonics is underspecified. In contrast, GLMs require correct specification to produce unbiased parameter estimates. We apply both methods to cortisol level data and find that only log-normal trigonometric regression produces parameter estimates that are invariant to the number of specified oscillation harmonics. Additionally, when a sufficiently large number of oscillation harmonics is specified, both methods produce identical parameter estimates for the oscillation harmonics.