Model of hidden heterogeneity in longitudinal data

Abstract

Variables measured in longitudinal studies of aging and longevity do not exhaust the list of all factors affecting health and mortality transitions. Unobserved factors generate hidden variability in susceptibility to diseases and death in populations and in age trajectories of longitudinally measured indices. Effects of such heterogeneity can be manifested not only in observed hazard rates but also in average trajectories of measured indices. Although effects of hidden heterogeneity on observed mortality rates are widely discussed, their role in forming age patterns of other aging-related characteristics (average trajectories of physiological state, stress resistance, etc.) is less clear. We propose a model of hidden heterogeneity to analyze its effects in longitudinal data. The approach takes the presence of hidden heterogeneity into account and incorporates several major concepts currently developing in aging research (allostatic load, aging-associated decline in adaptive capacity and stress-resistance, age-dependent physiological norms). Simulation experiments confirm identifiability of model's parameters.

Fig. 1

Estimated trajectories of logarithms of baseline hazard (ln μ₀(Z, t)), quadratic hazard terms (Q(Z, t)), optimal values of a physiological index (f (Z, t)) and age-related changes in homeostatic capacity (−a(Z, t)) in two hypothetical subpopulations (‘‘Z = 1, est.’’ and ‘‘Z = 0, est.’’) for 100 simulated data sets with hidden heterogeneity evaluated by QHM with heterogeneity. Respective true trajectories in two subpopulations are denoted as ‘‘Z = 1, true’’ and ‘‘Z = 0, true’’.

Fig. 2

Estimated population trajectories of logarithm of baseline hazard (ln μ₀(t)), quadratic hazard term (Q(t)), optimal values of a physiological index (f (t)) and age-related changes in homeostatic capacity (−a(t)) for QHM without heterogeneity (‘‘no Z, est.’’) evaluating 100 data sets simulated by QHM with heterogeneity. Respective true trajectories in two hypothetical subpopulations for QHM with heterogeneity are denoted as ‘‘Z = 1, true’’ and ‘‘Z = 0, true’’.

Fig. 3

Estimated trajectories of logarithms of baseline hazard (ln μ₀(Z, t)), quadratic hazard terms (Q)Z, t)), optimal values of a physiological index (f (Z, t)) and age-related changes in homeostatic capacity (−a)Z, t)) in two hypothetical subpopulations for QHM with heterogeneity (“Z = 1, QHM” and “Z = 0, QHM”) evaluating 100 data sets simulated by Cox model. Respective true trajectories for Cox model are denoted as “Z = 1, Cox” and “Z = 0, Cox”.

Fig. 4

Estimated population trajectories of logarithm of baseline hazard (ln μ₀(t)), quadratic hazard term (Q(t)), optimal values of a physiological index (f (t)) and age-related changes in homeostatic capacity (−a(t)) for QHM without heterogeneity (“no Z, QHM”) evaluating 100 data sets with hidden heterogeneity simulated by Cox model. Respective true trajectories for Cox model are denoted as “Z = 1, Cox” and “Z = 0, Cox”.

Fig. 5

Estimated trajectories of logarithms of baseline hazard (ln μ₀(Z, t)), quadratic hazard terms (Q(Z, t)), optimal values of a physiological index (f (Z, t)) and age-related changes in homeostatic capacity (−a(Z, t)) in two subpopulations (“Z = 1” and “Z = 0”) for QHM with heterogeneity applied to the FHS data on BMI for females. Population estimates of respective characteristics for QHM model without heterogeneity (“no Z”) are shown for comparison.

Fig. 6

Estimated mortality rates (μ( Z, t, Y_t)) and relative risks of death (RR (Z, t, Y_t), logarithmic scale) over age (t) and values of a physiological index at age t (Y_t) in two subpopulations (“Z = 1” and “Z = 0”) for QHM with heterogeneity applied to the FHS data on BMI for females. Thick black lines denote optimal age-trajectories of BMI (f (Z, t)) in respective subpopulations.