Compositional Covariate Importance Testing via Partial Conjunction of Bivariate Hypotheses

Abstract

Compositional data (i.e., data comprising random variables that sum up to a constant) arises in many applications including microbiome studies, chemical ecology, political science, and experimental designs. Yet when compositional data serve as covariates in a regression, the sum constraint renders every covariate automatically conditionally independent of the response given the other covariates, since each covariate is a deterministic function of the others. Since essentially all covariate importance tests and variable selection methods, including parametric ones, are at their core testing conditional independence, they are all completely powerless on regression problems with compositional covariates. In fact, compositionality causes ambiguity in the very notion of relevant covariates. To address this problem, we identify a natural way to translate the typical notion of relevant covariates to the setting with compositional covariates and establish that it is intuitive, well-defined, and unique. We then develop corresponding hypothesis tests and controlled variable selection procedures via a novel connection with \emph{bivariate} conditional independence testing and partial conjunction hypothesis testing. Finally, we provide theoretical guarantees of the validity of our methods, and through numerical experiments demonstrate that our methods are not only valid but also powerful across a range of data-generating scenarios.

Figures (14)

• Figure 1: Comparison of methods for (a) single testing and (b) variable selelction with Dirichlet covariates.
• Figure 2: Comparison of methods for (a) single testing and (b) multiple testing with multivariate normal covariates.
• Figure 3: Effect of conditioning out the sparse covariates on power for single testing (left) and average power for FWER control (middle) and FDR control (right), respectively. Error bars correspond to $\pm2$ Monte Carlo standard errors.
• Figure 4: Comparison of FWER and average power for variable selection with Dirichlet covariates. The target FWER level is 10% and error bars correspond to $\pm2$ Monte Carlo standard errors.
• Figure 5: Comparison of methods for (a) single testing, (b) FWER control and (c) FDR control with Logistic-normal covariates.

Theorems & Definitions (31)

• Example 1
• Definition 2.1: Markov boundary PEARLbook
• Example 2
• Remark 1
• Lemma 2.1
• proof
• Definition 2.2: Coordinatewise connectivity (through $A$ and $B$) Peters
• Definition 2.3: Equivalence (through $A$ and $B$) Peters
• Lemma 2.2
• Definition 2.4: Set action