Tsagris 2018 - Dirichlet Regression with Zeros - Overview
Summary
Tsagris and Stewart modify the Dirichlet regression log-likelihood so compositions containing exact zeros can be modelled without imputing them: each zero pattern is a “population” whose non-zero components follow a Dirichlet with the same coefficients and precision. The resulting ZADR model competes with Aitchison’s imputed log-ratio regression and the Kent/square-root model, and wins when zeros are numerous.
Overview
Question. Aitchison’s log-ratio regression and Dirichlet regression both fail when a component is exactly zero. Imputation (Palarea-Albaladejo and Martin-Fernandez 2008) assumes zeros are rounded values, alters every other component of an affected vector, and is sensitive to the assumed detection limit.
Contribution. ZADR (simple and full) built on Stewart and Field (2011) and Bear and Billheimer (2015), plus a four-step maximisation using the Aitchison fit as starting values.
Key result. Simple ZADR beats Aitchison+imputation on Arctic-lake data with injected zeros and on foraminifera (KL 3.126 vs 3.556), and in simulation reaches proportion 1 of runs with lower KL as grows.
Motivating PyMC examples
E70 (Aitchison’s Arctic lake sediments: sand/silt/clay vs depth; Dirichlet regression in mean-precision form) at https://redam94.github.io/pymc-challenges/examples.html; see Greenacre 2023 - Compositional Outcome Models in PyMC.
Main Content — Source Map
| Paper section | Pages | Covered in |
|---|---|---|
| 1 Introduction | 1 | this note |
| 2 Dirichlet regression (Eqs. 1-4) | 2-3 | Tsagris 2018 - Dirichlet Regression with Zeros - Dirichlet Regression |
| 3.1 Zeros and current approaches | 3-4 | Tsagris 2018 - Dirichlet Regression with Zeros - Discussion and Alternatives |
| 3.2 Zero values and Dirichlet regression (Eqs. 5-9) | 4-6 | Tsagris 2018 - Dirichlet Regression with Zeros - Zero Adjusted Dirichlet Regression |
| 3.3 Maximisation of the log-likelihood | 6-7 | Tsagris 2018 - Dirichlet Regression with Zeros - Zero Adjusted Dirichlet Regression |
| 4.1 Florida election; 4.2 glass; 4.3 foraminifera; 4.4 Arctic lake | 7-12 | Tsagris 2018 - Dirichlet Regression with Zeros - Examples and Simulation Results |
| 4.5 Simulation study | 11-14 | Tsagris 2018 - Dirichlet Regression with Zeros - Examples and Simulation Results |
| 5 Discussion; 6 Conclusions | 14-15 | Tsagris 2018 - Dirichlet Regression with Zeros - Discussion and Alternatives |
Relevance to marketing measurement
Share outcomes with structural zeros (a channel not run, a competitor absent in a market) are common. ZADR-style handling keeps observed shares unchanged rather than imputing an arbitrary epsilon.
Connections
Foundation in Greenacre 2023 - Wide Data, Zeros and Quasi-Coherence; Generalized Linear Models (multinomial-logit link).
See Also
Cluster notes under Greenacre 2023 - Aitchison CoDA 40 Years On - Overview.