Population-average trajectories and slopes in smooth random changepoint mixed models
Abstract
Two-slope mixed-effects models with subject-specific changepoints are widely used to describe biphasic longitudinal trajectories. Treatment-arm summaries are often reported using a single "typical subject" curve obtained by plugging mean parameter values into the subject-specific model. This can be misleading when changepoint timing varies across individuals, because averaging subject-specific curves is not the same as evaluating one curve at average parameter values. A related issue is that individual trajectories may bend gradually rather than change slope through a literal kink, with transition sharpness varying between subjects. We propose population-average estimands for two-slope mixed models with random changepoints and smooth subject-level transitions. These estimands target the treatment-arm mean trajectory and instantaneous slope by averaging over fitted heterogeneity in changepoint timing and transition sharpness. We use a baseline-anchored normal-convolution soft hinge, which preserves the broken stick interpretation, includes the abrupt changepoint model as a limiting case, and allows analytic evaluation of the changepoint-marginalized kernels under a truncated normal changepoint distribution for fixed transition sharpness. Averaging over the fitted transition sharpness distribution is then performed by deterministic one-dimensional quadrature. In an application to HIV-1 RNA data from ACTG Protocol 315, with censoring and heavy-tailed residuals, the marginal summaries reveal a more gradual population-level transition than conventional plug-in summaries. The marginal slope remains more negative for longer, and the marginal mean continues to decline after the plug-in curve has begun to flatten. These results illustrate the main reporting implication: in random changepoint models, representative subject curves should not be treated as population-average trajectories or slopes.
Related
- Optimal Covariate Adjustment beyond the Average Treatment Effect: Treated-Population and Overlap-Weighted Estimands
- Generalized Linear Mixed Models for Changepoint Analysis of Biomedical Time Series Data
- Changepoint detection methods as a practical tool for assessment of ecological restoration
- Adaptive online kernel changepoint detection
- Changepoint localization via nested exchangeability tests: from point estimates to confidence intervals
BibTeX
@misc{burger2026populationaverage,
title = {{Population-average trajectories and slopes in smooth random changepoint mixed models}},
author = {Divan A. Burger and Sean van der Merwe and Emmanuel Lesaffre},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34165},
eprint = {2610.34165},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34165},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Population-average trajectories and slopes in smooth random changepoint mixed models
AU - Divan A. Burger
AU - Sean van der Merwe
AU - Emmanuel Lesaffre
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34165
UR - https://arxiv.org/abs/2610.34165
AB - Two-slope mixed-effects models with subject-specific changepoints are widely used to describe biphasic longitudinal trajectories. Treatment-arm summaries are often reported using a single "typical subject" curve obtained by plugging mean parameter values into the subject-specific model. This can be misleading when changepoint timing varies across individuals, because averaging subject-specific curves is not the same as evaluating one curve at average parameter values. A related issue is that individual trajectories may bend gradually rather than change slope through a literal kink, with transition sharpness varying between subjects. We propose population-average estimands for two-slope mixed models with random changepoints and smooth subject-level transitions. These estimands target the treatment-arm mean trajectory and instantaneous slope by averaging over fitted heterogeneity in changepoint timing and transition sharpness. We use a baseline-anchored normal-convolution soft hinge, which preserves the broken stick interpretation, includes the abrupt changepoint model as a limiting case, and allows analytic evaluation of the changepoint-marginalized kernels under a truncated normal changepoint distribution for fixed transition sharpness. Averaging over the fitted transition sharpness distribution is then performed by deterministic one-dimensional quadrature. In an application to HIV-1 RNA data from ACTG Protocol 315, with censoring and heavy-tailed residuals, the marginal summaries reveal a more gradual population-level transition than conventional plug-in summaries. The marginal slope remains more negative for longer, and the marginal mean continues to decline after the plug-in curve has begun to flatten. These results illustrate the main reporting implication: in random changepoint models, representative subject curves should not be treated as population-average trajectories or slopes.
ER -