Scalar-on-distribution regression via generalized odds with applications to accelerometry-assessed disability in multiple sclerosis

TL;DR

The paper addresses how to leverage distributional tail information from digital health data to predict clinical outcomes, proposing a generalized odds (GO) framework that represents subject-specific distributions through odds ratios over regions of the sample space. It develops scalar-on-GO regression with spline-based functional covariates for 1-index, 2-index, and 4-index GO objects, estimated via penalized likelihood in a GLM setting. The approach is validated on HEAL-MS wrist accelerometry data to predict EDSS in MS, showing substantial gains over scalar and hazard-based representations, with the four-index GO achieving about 0.21 cross-validated $R^2$. The results highlight the value of tail-focused, distributional covariates for modeling gait/activity in MS and suggest broad applicability to other domains where extreme versus typical behavior carries clinical meaning.

Abstract

Distributional representations of data collected using digital health technologies have been shown to outperform scalar summaries for clinical prediction, with carefully quantified tail-behavior often driving the gains. Motivated by these findings, we propose a unified generalized odds (GO) framework that represents subject-specific distributions through ratios of probabilities over arbitrary regions of the sample space, subsuming hazard, survival, and residual life representations as special cases. We develop a scale-on-odds regression model using spline-based functional representations with penalization for efficient estimation. Applied to wrist-worn accelerometry data from the HEAL-MS study, generalized odds models yield improved prediction of Expanded Disability Status Scale (EDSS) scores compared to classical scalar and survival-based approaches, demonstrating the value of odds-based distributional covariates for modeling DHT data.

Figures (3)

• Figure 1: Two randomly selected subjects are selected. Subject 1: 47 years old female with EDSS = 1 (light blue). Subject 2: 62 years old female with EDSS = 6 (light red). The average log-activity counts for the two subjects are 0.0979 and 0.1007, respectively. Panel A represents the density function of the log-activity counts for two subjects. Panel B represents the 1-index odds $h_{1, F_{i}}$ for subject $i = 1, 2$. For example, we fix $u = 4.5$. The density function below $u$ is similar, but it's much different than the tail of the distribution. $F_{1}(4.5) = 0.17$ and $F_{2}(4.5) = 0.09$, whereas the difference between odds is different ($h_{1, F_{1}}(4.5) = 4.72$ and $h_{1, F_{2}}(4.5) = 9.8$).
• Figure 2: In panels (a) and (b), the top and right figures depict the density functions of the log-activity for Subjects 1 and 2, where light blue and light red represent Subjects 1 and 2, respectively. In each panel, the image plot illustrates the proposed 2-index function $h_{2, F}$. The green solid vertical line marks the sedentary activity level, while the green solid horizontal line represents the moderate-to-vigorous activity level. In the image plot, colors with higher identity indicate a higher values of $h_{2, F}$.
• Figure 3: Residual life distribution for two subjects. Subject 1: 47 years old female with EDSS = 1 (light blue). Subject 2: 62 years old female with EDSS = 6 (light red).

Theorems & Definitions (1)

• Remark 1