Multiplicative Decomposition of Heterogeneity in Mixtures of Continuous Distributions

Abstract

A system’s heterogeneity (diversity) is the effective size of its event space, and can be quantified using the Rényi family of indices (also known as Hill numbers in ecology or Hannah–Kay indices in economics), which are indexed by an elasticity parameter q≥0. Under these indices, the heterogeneity of a composite system (the γ-heterogeneity) is decomposable into heterogeneity arising from variation within and between component subsystems (the α- and β-heterogeneity, respectively). Since the average heterogeneity of a component subsystem should not be greater than that of the pooled system, we require that γ≥α. There exists a multiplicative decomposition for Rényi heterogeneity of composite systems with discrete event spaces, but less attention has been paid to decomposition in the continuous setting. We therefore describe multiplicative decomposition of the Rényi heterogeneity for continuous mixture distributions under parametric and non-parametric pooling assumptions. Under non-parametric pooling, the γ-heterogeneity must often be estimated numerically, but the multiplicative decomposition holds such that γ≥α for q>0. Conversely, under parametric pooling, γ-heterogeneity can be computed efficiently in closed-form, but the γ≥α condition holds reliably only at q=1. Our findings will further contribute to heterogeneity measurement in continuous systems.

A scoping review and comparison of approaches for measuring genetic heterogeneity in psychiatric disorders

“…For example, the effective number of underlying factors is an immediate candidate, insofar as a genetically homogenous set of K phenotypes might plausibly be modeled using few (or even one) latent factors. It may also be of interest to measure the effective hypervolume of the latent space, which can be done with the latent variable covariance matrix Q (Nunes et al, 2020a, 2020b).…”