Describing variability of intensively collected longitudinal ordinal data with latent spline models.

Population health studies increasingly collect longitudinal, patient-reported symptom data via mobile devices, offering unique insights into experiences outside clinical settings, such as pain, fatigue or mood. However, such data present challenges due to ordinal measurement scales, irregular sampling and temporal autocorrelation. This paper introduces two novel summary measures for analysing ordinal outcomes: (1) the mean absolute deviation from the median (MADM) for cross-sectional analyses and (2) the mean absolute deviation from expectation (MADE) for longitudinal data. The latter is based on a latent cumulative model with penalized splines, enabling smooth transitions between irregular time points while accounting for the ordinal nature of the data. Unlike black-box machine learning approaches, this method is interpretable, computationally efficient and easy to implement in standard statistical software. Through simulations, we demonstrate that the proposed measures outperform standard methods when the assumptions of normality or stationarity are violated. Application to real-world data from a national smartphone study, Cloudy with a Chance of Pain, highlights the utility of these measures in characterising symptom variability and trends over time. The methods developed here provide intuitive tools for analysing patient-reported outcomes in longitudinal studies, with potential applications in prediction modelling, causal discovery and evaluation of interventions.