A Note on Harmonic Underspecification in Log-Normal Trigonometric Regression

TL;DR

This paper compares least squares trigonometric regression and generalized linear models for biological rhythm data, showing that log-normal trigonometric regression yields unbiased and invariant harmonic estimates even when the model is underspecified.

Contribution

It demonstrates that log-normal trigonometric regression provides unbiased harmonic estimates under underspecification, unlike GLMs which require correct model specification.

Findings

Log-normal trigonometric regression produces unbiased harmonic estimates.

Estimates are invariant to the number of oscillation harmonics specified.

Both methods converge when 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.