[Paper Review] Regression Analysis for Microbiome Compositional Data

Pixu Shi, Anru R. Zhang+1 more|arXiv (Cornell University)|Mar 3, 2016

Geochemistry and Geologic Mapping23 references16 citations

TL;DR

This paper proposes a constrained regression framework for microbiome compositional data that ensures subcompositional coherence by imposing linear constraints on regression coefficients, enabling valid variable selection and inference. The method uses penalized estimation and de-biased inference to produce asymptotically normal, confidence-interval-ready estimates, identifying *Oscillibacter* as associated with BMI after adjusting for diet.

ABSTRACT

One important problem in microbiome analysis is to identify the bacterial taxa that are associated with a response, where the microbiome data are summarized as the composition of the bacterial taxa at different taxonomic levels. This paper considers regression analysis with such compositional data as covariates. In order to satisfy the subcompositional coherence of the results, linear models with a set of linear constraints on the regression coefficients are introduced. Such models allow regression analysis for subcompositions and include the log-contrast model for compositional covariates as a special case. A penalized estimation procedure for estimating the regression coefficients and for selecting variables under the linear constraints is developed. A method is also proposed to obtain de-biased estimates of the regression coefficients that are asymptotically unbiased and have a joint asymptotic multivariate normal distribution. This provides valid confidence intervals of the regression coefficients and can be used to obtain the $p$-values. Simulation results show the validity of the confidence intervals and smaller variances of the de-biased estimates when the linear constraints are imposed. The proposed methods are applied to a gut microbiome data set and identify four bacterial genera that are associated with the body mass index after adjusting for the total fat and caloric intakes.

Motivation & Objective

• To address the challenge of regression analysis with high-dimensional, compositional microbiome data where proportions sum to one.
• To ensure subcompositional coherence—meaning results remain consistent when analyzing subcompositions—by imposing linear constraints on regression coefficients.
• To develop a penalized estimation procedure for variable selection under these constraints in high-dimensional settings.
• To provide de-biased estimates of regression coefficients that are asymptotically normal, enabling valid confidence intervals and p-values.
• To apply the method to real gut microbiome data to identify taxa associated with body mass index (BMI), adjusting for total fat and caloric intake.

Proposed method

• Imposes linear constraints on regression coefficients to ensure subcompositional coherence, generalizing the log-contrast model.
• Uses a penalized likelihood approach with constraints for high-dimensional variable selection, combining Lasso-type regularization with linear equality constraints.
• Applies a coordinate descent method of multipliers to efficiently solve the constrained optimization problem.
• Develops a de-biasing procedure to correct estimation bias in high-dimensional settings, yielding asymptotically normal estimates.
• Employs convex optimization to compute de-biased estimates efficiently, with computational times of ~36s for p=100 and ~300s for p=200 on standard hardware.
• Derives asymptotic normality of de-biased estimates, enabling construction of confidence intervals and p-values for inference.

Experimental results

Research questions

• RQ1 Can a regression model for compositional microbiome data maintain subcompositional coherence when analyzing subcompositions of taxa?
• RQ2 How can variable selection and inference be performed under linear constraints on regression coefficients in high-dimensional compositional data?
• RQ3 What is the impact of correctly specified linear constraints on the precision and coverage of confidence intervals for regression coefficients?
• RQ4 Can de-biased estimates of regression coefficients be constructed under constraints to enable valid statistical inference?
• RQ5 Which bacterial taxa are significantly associated with BMI after adjusting for total fat and caloric intake in gut microbiome data?

Key findings

• The proposed method ensures subcompositional coherence by design through linear constraints on regression coefficients.
• Simulation results show that confidence intervals under correct linear constraints are shorter and have better coverage than under incorrect or no constraints, especially with small sample sizes.
• De-biased estimates are approximately normally distributed and yield valid confidence intervals and p-values.
• The method identifies *Oscillibacter* as significantly associated with BMI after adjusting for total fat and caloric intake in a real gut microbiome dataset.
• Penalized estimation under constraints improves prediction performance when the constraints are correctly specified.
• The de-biasing algorithm is computationally feasible, taking approximately 36 seconds for p=100 and 300 seconds for p=200 on a standard PC.