Generalized Linear Models With Linear Constraints for Microbiome Compositional Data

Abstract

Motivated by regression analysis for microbiome compositional data, this article considers generalized linear regression analysis with compositional covariates, where a group of linear constraints on regression coefficients are imposed to account for the compositional nature of the data and to achieve subcompositional coherence. A penalized likelihood estimation procedure using a generalized accelerated proximal gradient method is developed to efficiently estimate the regression coefficients. A de-biased procedure is developed to obtain asymptotically unbiased and normally distributed estimates, which leads to valid confidence intervals of the regression coefficients. Simulations results show the correctness of the coverage probability of the confidence intervals and smaller variances of the estimates when the appropriate linear constraints are imposed. The methods are illustrated by a microbiome study in order to identify bacterial species that are associated with inflammatory bowel disease (IBD) and to predict IBD using fecal microbiome.

Compositional knockoff filter for high-dimensional regression analysis of microbiome data

“…In order to account for this structure, the log-linear contrast model is often used for microbiome data (Lin et al, 2014;Shi et al, 2016). Without loss of generality, we assume that X ij > 0 by replacing the zero proportions by a tiny pseudo positive value as routinely performed in practice (Lin et al, 2014;Shi et al, 2016;Cao et al, 2017;Lu et al, 2019;Zhang et al, 2019). Let Z p ∈ R n×(p−1) be a log-ratio transformation of the matrix X, where Z p ij = log(X ij /X ip ) and p denotes the reference covariate.…”

“…We first generated the microbiome counts from the Dirichlet-multinomial distribution following previous designs (Zhao et al, 2015;Zhan et al, 2017a,b). Zero counts were first replaced by a pseudo count of 0.5, as commonly suggested in microbiome data analysis (Lin et al, 2014;Cao et al, 2017;Weiss et al, 2017;Lu et al, 2019;Zhang et al, 2019), and then microbiome counts were transformed to relative abundances. Next, we Under this scheme, it is easy to check that the coefficient vector always satisfies the sumto-zero constraint under each of the four sparsity levels.…”

“…The proposed compositional screening procedure has much better performance than the other two competing methods (Fan and Lv, 2008;Li et al, 2012), which have been widely used in statistic literature. This is another example that classic statistical methods may be inefficient for microbiome data without accounting for the compositional nature (Lin et al, 2014;Shi et al, 2016;Cao et al, 2017;Lu et al, 2019;Zhang et al, 2019). By incorporating the compositional constraint, the proposed CSP achieves the sure screening property for microbiome data as the proportion of true features retained in the screened set is always one based on Table 1.…”

“…However, the existing knockoff filter framework does not take into account the compositional structure of microbiome data. In the literature of many other statistical inference (e.g., regression-based modelling, two-sample testing and statistical casual mediation analysis), it has been observed that applying classic statistical methods to analyze microbiome composition data is usually underpowered and sometimes can render inappropriate results (Aitchison, 2003;Shi, Zhang, and Li, 2016;Cao, Lin, and Li, 2017;Sohn and Li, 2019;Lu, Shi and Li, 2019;Zhang et al, 2019). Thus, new methods are desired rather than directly applying knockoff filter to microbiome data.…”

MITRE: predicting host status from microbiota time-series data

“…A more subtle issue, particularly relevant to microbiome ratio-based data, is that linear models introduce mathematical/statistical difficulties when analyzing compositional or proportional data. While special constraints are needed for linear models to overcome these difficulties [17], nonlinear models do not suffer from the same limitations because they can inherently learn nonlinear transformations of the data. It is particularly straightforward to understand the transformations produced by the MITRE detectors described above: the learned thresholds effectively discretize ratio data into distinct levels, a transformation that renders the data mathematically noncompositional.…”