Joint Modeling of Conditional Mean and Variability in Multilevel Functional Data: Application to Postprandial Glucose Trajectories in the AEGIS Study

Yang Xiao; Francisco Gude; Marcos Matabuena

stat.MLNov 5, 2026

Joint Modeling of Conditional Mean and Variability in Multilevel Functional Data: Application to Postprandial Glucose Trajectories in the AEGIS Study

Authors: Yang Xiao, Francisco Gude, Marcos Matabuena

Organizations: Division of Biological and Life Sciences, Mohamed bin Zayed University of Artificial Intelligence, Abu Dhabi, United Arab Emirates · Instituto de Investigación Sanitaria de Santiago de Compostela (IDIS), Santiago de Compostela, Spain · Department of Medicine, University of Santiago de Compostela, Santiago de Compostela, Spain

Abstract

Functional data analysis provides a framework for studying time-varying associations between covariates and high-frequency biological signals. In digital health, participants often contribute repeated functional trajectories, yet existing multilevel regression methods largely focus on the conditional mean, overlooking variability and its role in biological heterogeneity. We propose a heteroskedastic functional mixed-effects regression framework that jointly models the conditional mean and variance of repeated trajectories, with participant-specific random effects in both components. We also develop a participant-level bootstrap to quantify uncertainty in estimated functional parameters. We apply the method to postprandial glucose trajectories from the AEGIS study. Premeal glucose, carbohydrate intake, and meal type show distinct time-dependent associations with glucose variability, whereas HbA1c is more strongly associated with the conditional mean than with variability. Substantial between-participant differences in residual variability remain even after accounting for observed covariates. These findings show that modeling conditional variability alongside the mean provides insights into glucose regulation and individual heterogeneity that mean-focused analyses may miss. More broadly, the proposed framework enables joint characterization of average responses, variability, and participant-specific heterogeneity in repeated biological signals.

Figures & tables

Figure 1: Postprandial glucose trajectories of two AEGIS participants after breakfast, lunch and dinner on five study days (D1–D5), on a five-minute grid from the start of each meal. Solid lines are the observed glucose and dashed lines the participant’s fitted mean for that meal. Shading is an approximate pointwise 95% prediction band with asymptotic approximation, the fitted mean $\pm 1.96$ times the fitted residual standard deviation; crosses mark observations outside it.

(a) Participant characteristics
Characteristic	Summary
Age (years)	47.2 (14.2)
BMI (kg/m 2 )	28.1 (5.0)
HbA1c (%)	5.35 (0.33)
Female, n (%)	230 (63.4)

Table 1: Characteristics of the 363 participants and their 5,237 meals. Continuous variables are summarized as mean (SD); female participants are summarized as count (percentage). Participant characteristics are calculated once per participant, and meal covariates are calculated within each meal type.

Figure 2: Mean-glucose associations over postmeal time and meal type. Panels show 11 conditional-mean coefficient effects in mg/dL, with covariate increments in the labels. Dark shading shows pointwise nominal 95% intervals and light shading simultaneous bands from 500 participant-bootstrap refits. Meal contrasts compare lunch and dinner with breakfast at 50 g carbohydrate; female (sex coded 0) is the recorded-sex reference. Calories and carbohydrate adjust for each other. The three carbohydrate panels share a vertical scale, as do the two meal-contrast panels.

Figure 3: Residual-SD associations over postmeal time. Effects are displayed as $100\{\exp(d_{a}\gamma_{a}(t))-1\}$ percent change in residual SD, with increments $d_{a}$ in the panel labels; effects and band limits are calculated on the log-SD scale before transformation. Dark and light shading show pointwise nominal 95% intervals and simultaneous bands from 500 participant-bootstrap refits. Meal contrasts use breakfast at 50 g carbohydrate, and female (sex coded 0) is the recorded-sex reference. The three carbohydrate panels share a vertical scale, as do the two meal contrasts; each simultaneous band covers one coefficient curve.

	Conditional mean, $\beta_{a}(t)$		Log residual SD, $\gamma_{a}(t)$
Coefficient	$p_{\mathrm{BH}}$	Windows	$p_{\mathrm{BH}}$	Windows
Age	0.003	0 (+); 40–105 (+)	0.074	none
BMI	0.017	90–180 (+)	0.117	none
HbA1c	0.033	170–180 (+)	0.329	none
Recorded sex	0.124	none	0.707	none
Calories	0.003	0–55 ( $-$ )	0.004	0–20 ( $-$ )

Table 2: AEGIS whole-curve tests and exclusion windows. BH adjustment is applied within each model’s 11 coefficient curves. Values of

p_{\mathrm{BH}}

below 0.05 are in bold. Windows are in minutes and use each curve’s nominal 95% simultaneous band, with signs in parentheses; they are not jointly simultaneous across coefficients. Covariate increments follow Figures 2 – 3 , and meal contrasts are evaluated at 50 g carbohydrate.

Figure 4: Dinner has a broader late elevation in the reference mean and residual variance. Part (a) shows $m(x,t)$ and part (b) shows $\bar{q}(x,t)$ at the sample covariate means, female sex and 50 g carbohydrate. The participant-averaged residual variance excludes between-participant mean variation. The three meal panels within each part share a vertical scale. Dark and light shading show pointwise nominal 95% intervals and simultaneous bands from 500 participant-bootstrap refits; variance limits are constructed on the log scale and exponentiated.

Figure 5: Participant mean effects reduce model-based squared error by a median of 5.4% of the common population variance ( $D=361.5$ (mg/dL) 2 ). (a) Each point pairs $R_{i,\mathrm{fixed}}^{2}$ and $R_{i,\mathrm{participant}}^{2}$ of one participant; axes have the same range and the solid line is the identity line. (b) Distribution of the paired contribution $\Delta R_{i}^{2}$ , with its median marked. All 363 participants and negative values are retained. Both values retain the fitted participant scale effect and treat participant effects as known; the nonnegative contribution follows from the definition. No individual uncertainty intervals are shown.

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Abstract

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