Tsagris 2018 - Dirichlet Regression with Zeros - Discussion and Alternatives
Summary
The authors’ answer to “should zeros be treated as rounded and imputed, or naturally?” is naturally: imputing changes every other component of the vector. Limitations: the Dirichlet allows only negative correlations, cannot capture curvature without covariates, is not in the linear exponential family, and ZADR assumes common parameters across zero patterns. A Bayesian version is left as future work.
Overview
Sec. 5 compares ZADR with alternatives; Sec. 6 concludes.
Main Content
Assumption of common parameters. With components there are possible zero patterns. Separate coefficients per pattern would need many observations in each, and parameter count grows quickly; hence the shared-parameter assumption is “rather necessary”. Linking covariates to the zero-pattern probabilities is also impractical. Usually some patterns recur.
Dirichlet limitations.
- Covariance matrix allows only negative correlations (like multinomial logit).
- With no covariates the Dirichlet cannot capture the curvature of the simplex (Aitchison 2003, p. 59): a straight line in maps to a curve in ; the logistic-normal can, the Dirichlet only linear relationships. With covariates and the multivariate logit link (1) this is less of a problem, and Campbell and Mosimann (1987) found Dirichlet regression to work well; see also Maier (2014), Gueorguieva et al. (2008).
- Not a linear exponential family (Gourieroux et al. 1984): only correct specification gives consistency; the extra assumption that zero-containing vectors share the full Dirichlet’s parameters is not proved.
Compared with alternatives. Bear and Billheimer (2015) log-ratio mixture requires a component never zero; Scealy and Welsh (2011) is harder to fit and interpret; inflated beta (Ospina and Ferrari 2010) is unrelated to the conditional Dirichlet; a vector of all zeros or an all-ones vector is not allowed (point mass on 1 can be handled by a separate point mass).
Reading the conclusions. For the first two real examples Aitchison is slightly better; for the other two simple ZADR is slightly or much better; in simulation with injected zeros simple ZADR performs better.
Examples
Deciding between imputation and ZADR in practice
If zeros arise from a detection limit and are few (Tellus: 1.1%), imputation is harmless (Greenacre et al.). If zeros are structural (competitor absent) or frequent, use a zero-pattern model such as ZADR, or amalgamate; check sensitivity to the imputation constant.
Connections
Greenacre 2023 - Wide Data, Zeros and Quasi-Coherence, Greenacre 2023 - Compositional Outcome Models in PyMC. For counts with zero shares allowed, Dirichlet-multinomial (see the PyMC note) sidesteps the problem because zero counts have positive probability.