Whittaker Review

Ecological partition

Local diversity, regional diversity, and turnover

The alpha-beta-gamma vocabulary was developed for ecological communities across space. It remains useful for microbiome cohorts because each sample can be treated as a local community and the study population as the broader assemblage.

Alpha diversity

Alpha diversity describes diversity within a local unit: a plot, site, host, body site, or sample. It is the within-community component of the partition.

Gamma diversity

Gamma diversity describes diversity across the larger region or cohort. It asks what diversity is represented by the pooled or collective assemblage.

Beta diversity

Beta diversity describes differentiation among local units. It increases when communities are compositionally distinct from each other.

Classical forms

Whittaker diversity has multiplicative and additive expressions

For species richness, Whittaker's classical beta diversity is often expressed as a ratio between regional and mean local diversity. Later diversity-partitioning work also uses additive decompositions.

Multiplicative partition

$$\beta_{\mathrm{W}} = \frac{\gamma}{\bar\alpha}.$$

This form asks how many effective local communities are represented by the regional assemblage. When all local communities are similar, $\beta_{\mathrm{W}}$ is close to 1.

Additive partition

$$\beta_{\mathrm{add}} = \gamma - \bar\alpha.$$

This form measures the amount of regional diversity not explained by average within-community diversity. It is especially natural when alpha and gamma are defined on the same bounded scale.

Connection to HRIC

HRIC carries Whittaker's structure into sparse compositional geometry

The HRIC framework keeps the ecological logic of within-cohort partitioning, but it changes the measurement scale. Diversity is not computed from raw richness counts; it is computed from Hellinger-Riemann intrinsic coordinates of compositions.

Whittaker concept	HRIC analogue	Interpretation
Alpha diversity	$\alpha_i = 1 - \\|Z_i\\|_2^2/A_p^2$	Within-sample evenness measured by displacement from the uniform composition.
Gamma diversity	$\gamma = 1 - \\|\bar Z\\|_2^2/A_p^2$	Cohort-level evenness measured at the HRIC coordinate center.
Beta diversity	$\beta = \gamma - n^{-1}\sum_i\alpha_i$	Among-sample turnover remaining after average within-sample evenness.

Why this distinction matters

Microbiome diversity needs both ecological language and compositional geometry

Whittaker's terms provide the conceptual question: what is within-sample diversity, what is cohort-level diversity, and how much turnover separates samples or conditions? HRIC provides a coordinate system where that question can be asked for sparse relative-abundance data without first replacing exact zeros.

Practical consequence

The alpha-beta-gamma language transfers naturally to microbiome cohorts, but the measurement scale changes. HRIC's additive partition is a geometry-aware diversity partition, not a direct replacement for species-richness ratios.

Return to HRIC framework

References