Variable selection for sparse Dirichlet-multinomial regression with an application to microbiome data analysis

Jun Chen; Hongzhe Li

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Variable selection for sparse Dirichlet-multinomial regression with an application to microbiome data analysis

Jun Chen, Hongzhe Li

arXiv:1305.5355v1May 23, 2013stat.AP

TL;DR

Microbiome covariate selection must handle overdispersed counts and high-dimensional nutrient–taxon associations without losing power through multiple testing. The paper uses sparse group ℓ1-penalized DM regression, finding stronger variable-selection performance than alternative count or proportion models and identifying nutrient–microbiome associations in human gut data.

Problem
High-dimensional microbiome studies need to associate environmental covariates with overdispersed taxa counts while avoiding power loss from multiple testing.
Method
The paper fits DM regression with a sparse group ℓ1 penalty that induces group-level and within-group sparsity, tuning penalties from training data and test-set likelihood.
Results
The sparse-group DM procedure outperforms multinomial, Dirichlet, and ℓ1-only procedures in reported variable-selection comparisons, and the gut-data analysis selects 11 nutrients and 13 associated genera.
Takeaways & Limitations
The method selects relevant covariates together with associated taxa and supports nutrient–gut microbiome association analysis, with selected nutrients showing generally stable bootstrap selection.
Takeaways & Limitations
The DM model does not accommodate zero-inflation or flexible covariance structures among taxa counts, motivating future extensions.

Abstract

from arXiv · show

With the development of next generation sequencing technology, researchers have now been able to study the microbiome composition using direct sequencing, whose output are bacterial taxa counts for each microbiome sample. One goal of microbiome study is to associate the microbiome composition with environmental covariates. We propose to model the taxa counts using a Dirichlet-multinomial (DM) regression model in order to account for overdispersion of observed counts. The DM regression model can be used for testing the association between taxa composition and covariates using the likelihood ratio test. However, when the number of covariates is large, multiple testing can lead to loss of power. To address the high dimensionality of the problem, we develop a penalized likelihood approach to estimate the regression parameters and to select the variables by imposing a sparse group $\ell_1$ penalty to encourage both group-level and within-group sparsity. Such a variable selection procedure can lead to selection of the relevant covariates and their associated bacterial taxa. An efficient block-coordinate descent algorithm is developed to solve the optimization problem. We present extensive simulations to demonstrate that the sparse DM regression can result in better identification of the microbiome-associated covariates than models that ignore overdispersion or only consider the proportions. We demonstrate the power of our method in an analysis of a data set evaluating the effects of nutrient intake on human gut microbiome composition. Our results have clearly shown that the nutrient intake is strongly associated with the human gut microbiome.

4.2. Tuning parameter selection.

The procedure tunes the composite sparse-group penalty with data by separating overall sparsity from the group-versus-within-group penalty balance. It searches λ from a sparsest starting model using warm starts along a log-scale grid.

Tuning parameter selection.: λ1 and λ2 are tuned using v-fold cross-validation or a BIC criterion, then reparameterized through λ and c.λ controls overall sparsity, while c controls the proportion of the group ℓ1 penalty.
Tuning parameter selection.: c = 0 gives the lasso, whereas c = 1 gives the group lasso.The simulations evaluate c values from {0, 0.05, 0.1, 0.2, 0.4}.
Tuning parameter selection.: The λ search starts at λmax, producing an intercept-only sparsest model, and then decreases λ using the previous estimate as a warm start.λmax is chosen so the initial estimate is a stationary point after the first iteration.
Tuning parameter selection.: A log-scale grid can use λj = 0.96^jλmax until λmin = 0.2λmax, or stop when the allowed number of nonzero coefficients is exceeded.The grid length m is chosen to reach the specified minimum penalty or the coefficient-count limit.

5.1. Simulation strategies.

The simulations vary covariate, taxon, growth, dispersion, and sequencing settings to evaluate sparse DM variable selection against alternative models. They assess group and within-group selection using replicated precision, recall, and F1 measures, with tuning selected by test-set likelihood.

5.1. Simulation strategies.: Relevant nutrient effects involve qr taxa per nutrient, with coefficients equally spaced over [0.6f,0.9f] and alternating signs.The nutrient intake vector follows a multivariate normal distribution with covariance Σi,j = ρ^|i−j|.
5.1. Simulation strategies.: The exponential-growth simulations allow base taxon abundances to differ up to 100 folds, while the linear-growth setting uses a separate intercept range with the same abundance spread.The exponential model uses intercepts from (−2.3,2.3), whereas the linear model uses (0.02,2).
5.1. Simulation strategies.: Counts are generated from a DM model with common dispersion θ and sample-specific sequencing depths uniformly distributed over (m,2m), despite using data-generation conditions that differ from the fitted model assumptions.The misspecified generation is intended to assess robustness of the proposed model.
5.1. Simulation strategies.: The main simulation uses n = 100 samples, p = 100 nutrients, q = 40 taxa, four relevant nutrients, four associated taxa per nutrient, m = 1000, θ = 0.025, and ρ = 0.4.This creates 4000 nutrient-taxon variables for comparing sparse DM, lasso-DM, multinomial, and Dirichlet procedures.
5.1. Simulation strategies.: Selection performance is evaluated with precision, recall, and F1, whose averages are reported over 100 replications.F1 weights precision and recall equally; TP, FN, and FP denote true positives, false negatives, and false positives.
5.1. Simulation strategies.: A small c efficiently identifies groups, while within-group selection follows a unimodal pattern in c, so both c and λ are tuned by test-set likelihood.Figure 1 evaluates group and within-group recall, precision, and F1 across c values from 0 to 0.4.
5.1. Simulation strategies.: The sparse-group DM procedure has higher group-selection precision than the ℓ1-only DM procedure, with similar recall and better within-group recall and F1.Compared with sparse Dirichlet and multinomial regression, DM performs better in variable selection, especially within groups; LRT selects more variables but has worse precision at 5% FDR.

5.3. Effects of overdispersion and model misspecification.

Greater overdispersion makes variable selection harder, but the DM model performs better when overdispersion is substantial. Dirichlet and DM models remain robust to misspecification, while multinomial performance declines markedly.

Effects of overdispersion: At θ = 0.1, DM performs much better than Dirichlet and multinomial models for both group and within-group selection.At θ = 0.005, the three models have similar selection performance; larger overdispersion lowers F1 values for all models.
Effects of overdispersion: Larger overdispersion makes selection more difficult for all three models, as shown by smaller F1 values.The simulations compare different degrees of overdispersion.
Effects of model misspecification: Under model misspecification, Dirichlet and DM selection performances do not decrease significantly.Counts were simulated using a linear growth model and compared with an exponential growth model.
Effects of model misspecification: The multinomial model’s group-selection F1 decreases from 0.79 to 0.56 under model misspecification.This comparison uses the F1 measure for group selection.
Effects of model misspecification: Even after increasing total counts tenfold, DM remains better than the proportion-based Dirichlet model.The result indicates improved performance for DM over models that consider only proportions in deeper sequencing settings.

5.4. Effects of the number of the covariates and the relevant taxa.

The sparse group-penalized DM model performs well as the number of covariates and relevant taxa increases, while modeling overdispersion improves fit and interpretability in the nutrient–microbiome application.

Effects of relevant taxa: When each relevant group contains one relevant taxon, sparse group and ℓ1-penalized DM models perform similarly for group selection.Grouping becomes more useful when relevant groups contain multiple relevant taxa.
Effects of relevant taxa: When each relevant group contains 8 relevant taxa, the sparse group regularized DM model performs much better than the ℓ1-penalized DM model.The comparison concerns selection of relevant groups.
Effects of covariates: Increasing the covariate dimension to 400, or 16,000 variables in total, does not deteriorate variable-selection performance.The authors describe this as demonstrating efficiency in handling high-dimensional data.
Nutrient–microbiome application: The fitted sparse DM model fits the nutrient data well, with r2 = 0.79, after selecting 11 nutrients and 13 associated genera.The comparison uses fitted counts against observed counts.
Nutrient–microbiome application: The analysis identified nutrient–genus associations, including a positive association between Naringenin and Faecalibacterium.Faecalibacterium is described as an anti-inflammatory commensal bacterium identified in gut microbiota analysis of Crohn’s disease patients.
Nutrient–microbiome application: Simulated counts from the fitted sparse DM model resemble observed counts better than those from the sparse multinomial model, whose simulations are apparently over-smoothed.This supports considering overdispersion when modeling gut microbiome counts.

APPENDIX

The appendix establishes structural properties of the sparse-group penalized quadratic minimization used in the optimization procedure, including conditions forcing components to zero and sign constraints on nonzero components.

Theorem 1: Theorem 1 characterizes the minimizer of a quadratic objective with group and overall ℓ1 penalties through zero-pattern and sign properties.Coordinates with sufficiently small |b_i| are forced to zero, while nonzero coordinates have signs opposite to the corresponding b_i.
Theorem 1: For coordinates with |b_i| < λ2, any minimizer must set x_i = 0.The proof constructs a lower-objective point by removing a nonzero coordinate in this set, contradicting optimality.
Theorem 1: For coordinates with |b_i| ≥ λ2, nonzero minimizer components have signs opposite to b_i.The unconstrained minimizer on the complementary coordinates satisfies the required sign constraint after accounting for the ℓ1 penalty.
Corollary 1: Corollary 1 transfers these zero and sign arguments to a shifted objective involving β + d.The relevant coordinate partition is defined using |b_i − β_i| < λ2, with the sign function determining the transformed linear term.