Daily dynamics and weekly rhythms: A tutorial on seasonal ARMA models combined with day-of-week effects

Daily diary data of emotional experiences are typically modeled with a first-order autoregressive model to account for possible day-to-day dynamics. However, our emotional experiences are likely influenced by the weekly rhythm of our activities, which may be reflected by: (a) day-of-week effects (DOWEs), where different weekdays are characterized by different means; and (b) week-to-week dynamics, where weekday-specific activities and experiences have a delayed effect on the emotions that we experience on the same weekday a week later. While DOWEs have been studied occasionally, week-to-week dynamics have been largely ignored in psychological research. We present a set of complementary visualization techniques for detecting weekly rhythms and day-to-day dynamics in time series data. Subsequently, we introduce the family of seasonal autoregressive--moving average (SARMA) models from the econometrics literature, extend them with DOWEs models, and show how their components appear in visualizations. We then provide a tutorial on fitting these models in R, discuss model fit and model selection, and apply them to a daily diary dataset of 56-101 daily measures from 98 individuals. The results suggest that most individuals in the sample may be characterized by patterns and dynamics that the current practices in psychological research cannot capture adequately, and we discuss their implications for current psychological research practices. Reflecting critically on the limitations of our approach, we regard our findings as an initial step to encourage researchers to move beyond the ubiquitous paradigm of lag-1 autoregressive modeling and consider other types of dynamics at different timescales, and put forth ways forward.