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.

翻译：生物节律数据分析通常涉及最小二乘三角回归，该方法将响应随时间的变化建模为正弦分量的叠加。当响应不服从正态分布时，研究者通常会在应用最小二乘三角回归前对响应进行变换，或将三角回归扩展至广义线性模型框架。本文比较了在振荡谐波数量欠指定情况下这两种方法。我们假设数据在等间距实验设计下采样，且对数连接函数适用于广义线性模型。我们证明，当响应服从广义伽马分布时，即使振荡谐波数量欠指定，采用对数变换响应的最小二乘三角回归（即对数正态三角回归）仍能产生无偏的振荡谐波参数估计。相比之下，广义线性模型需要正确指定模型才能获得无偏参数估计。我们将两种方法应用于皮质醇水平数据，发现仅对数正态三角回归产生的参数估计对指定的振荡谐波数量具有不变性。此外，当指定足够多的振荡谐波数量时，两种方法对振荡谐波的参数估计结果完全一致。