A two-stage hidden Markov model design for biomarker detection, with application to microbiome research

Abstract

It has been recognized that for appropriately ordered data, hidden Markov models (HMM) with local false discovery rate (FDR) control can increase the power to detect significant associations. For many high-throughput technologies, the cost still limits their application. Two-stage designs are attractive, in which a set of interesting features or biomarkers is identified in a first stage, and then followed up in a second stage. However, to our knowledge no two-stage FDR control with HMMs has been developed. In this paper, we study an efficient HMM-FDR based two-stage design, using a simple integrated analysis procedure across the stages. Numeric studies show its excellent performance when compared to available methods. A power analysis method is also proposed. We use examples from microbiome data to illustrate the methods.

Keywords: Biomarker; False discovery rates; Hidden Markov model; Metagenomics; Metatranscriptomics; PCR.

Figure 1 (A)

Heatmap of the HMP data, consisting of tag counts for 748 Operational Taxonomic Units (OTUs), transformed as log_e(count+0.5), with 103 males and 88 females.

Figure 1 (B)

Heat map of sample correlations of log(count+0.5) between OTUs. Data were from a metagenomic analysis of NIH human microbiome project with 103 males and 88 females, with OTUs ordered by phylogenetic relationships. The correlations are primarily block structured, with less extreme negative correlations than positive correlations. Solid lines indicate family-level boundaries in the ordered taxa.

Figure 2

Average empirical FDR, FNR, and Average Total Positives (ATP) for various a₁₁ at fixed δ = 1.5 (Row 1) and as a function of effect size δ (Row 2) for m = 500 at nominal FDR of 0.05. Column 1 compares the empirical FDR. Column 2 compares the empirical FNR. Column 3 compares the ATP. Methods include mHMM (○), Z approach (Δ), Fisher’s combination (×), full data HMM (◇), and one-stage Benjamini-Hochberg procedure (+).

Figure 3

Average empirical FDR, FNR, and Average Total Positives for various a₁₁ at fixed δ = 1.5 (Row 1) and as a function of effect size δ (Row 2) for m = 1000 at nominal FDR of 0.05. Column 1 compares the empirical FDR. Column 2 compares the empirical FNR. Column 3 compares the average total positives. Methods include mHMM (○), Z approach (Δ), Fisher’s combination (×), full data HMM (◇), and one-stage Benjamini-Hochberg procedure (+).

Figure 4

Average empirical FDR, FNR, and Average Total Positives when the number of components for nonnull is misspecified, for various a₁₁ at fixed δ = 1.5 (Row 1) and as a function of effect size δ (Row 2), with m = 500 and the nominal FDR of 0.05. Column 1 compares the empirical FDR. Column 2 compares the empirical FNR. Column 3 compares the ATP. Methods include mHMM (○), Z approach (Δ), Fisher’s combination (×), full data HMM (◇), and one-stage Benjamini-Hochberg procedure (+).

Figure 5

Average empirical FDR, FNR, and Average Total Positives when the number of components for nonnull is misspecified, for various a₁₁ at fixed δ = 1.5 (Row 1) and as a function of effect size δ (Row 2) , with m = 1000 and the nominal FDR of 0.05. Column 1 compares the empirical FDR. Column 2 compares the empirical FNR. Column 3 compares the Average Total Positives. Methods include mHMM (○), Z approach (Δ), Fisher’s combination (×), full data HMM (◇), and one-stage Benjamini-Hochberg procedure (+).

Figure 6. Phylum, class, order, and family for the 15 significant genera identified using the two-stage sampling (genera averaged within each family)

Using the full dataset, ratios of (female mean count)/(male mean count) are shown, corrected for a slight (1.1) male:female bias among the utilized 748 genera/taxa. Bold-face indicates microbiome order/families that appeared among significant sex-based genera in Markle et al.[ 23].