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 . Under these indices, the heterogeneity of a composite system (the -heterogeneity) is decomposable into heterogeneity arising from variation within and between component subsystems (the - and -heterogeneity, respectively). Since the average heterogeneity of a component subsystem should not be greater than that of the pooled system, we require that . 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 -heterogeneity must often be estimated numerically, but the multiplicative decomposition holds such that for . Conversely, under parametric pooling, -heterogeneity can be computed efficiently in closed-form, but the condition holds reliably only at . Our findings will further contribute to heterogeneity measurement in continuous systems.
Highlights
Measurement of heterogeneity is important across many scientific disciplines
We require that γ ≥ α, since it is counterintuitive that the heterogeneity of the overall ensemble should be less than any of its constituents, let alone the
This paper provided approaches for multiplicative decomposition of heterogeneity in continuous mixture distributions, thereby extending the earlier work on discrete space heterogeneity decomposition presented by Jost [9]
Summary
Measurement of heterogeneity is important across many scientific disciplines. Ecologists are interested in the heterogeneity of ecosystems’ biological composition (biodiversity) [1], economists are interested in the heterogeneity of resource ownership (wealth equality) [2], and medical researchers and physicians are interested in the heterogeneity of diseases and their presentations [3]. We require that γ ≥ α, since it is counterintuitive that the heterogeneity of the overall ensemble should be less than any of its constituents, let alone the “average” subsystem [8,9] Such a decomposition was introduced by Jost [9] for systems represented on discrete event spaces (such as representations of organisms by species labels). In this case, which amounts to a Gaussian mixed-effects model (as commonly implemented in biomedical meta-analyses), we show that γ ≥ α will hold at q = 1, though not necessarily at q 6= 1.
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