On Stein's Identity and Near-Optimal Estimation in High-dimensional Index Models

We consider estimating the parametric components of semi-parametric multiple index models in a high-dimensional and non-Gaussian setting. Such models form a rich class of non-linear models with applications to signal processing, machine learning and statistics. Our estimators leverage the score function based first and second-order Stein's identities and do not require the covariates to satisfy Gaussian or elliptical symmetry assumptions common in the literature. Moreover, to handle score functions and responses that are heavy-tailed, our estimators are constructed via carefully thresholding their empirical counterparts. We show that our estimator achieves near-optimal statistical rate of convergence in several settings. We supplement our theoretical results via simulation experiments that confirm the theory

Additional file 1: of Gut microbiota and physiologic bowel 18F-FDG uptake

Figure S1. Comparisons of community alpha diversities among groups for the intestinal 18F-FDG. Table S1. Association between alpha diversity and intestinal 18F-FDG uptake. Figure S2. Principal coordinate analysis (PCoA) plot with Bray-Curtis distance. Table S2-1. Associations of intestinal 18F-FDG uptake with gut microbiota at the phylum level: Group 1 vs. 2. Table S2-2. Associations of intestinal 18F-FDG uptake with gut microbiota at the phylum level: Group 1 vs. 3. Table S2-3. Associations of intestinal 18F-FDG uptake with gut microbiota at the phylum level: Group 2 vs. 3. Table S2-4. Associations of intestinal 18F-FDG uptake with gut microbiota at the phylum level: Group 1 vs. 2 + 3. Table S2-5. Associations of intestinal 18F-FDG uptake with gut microbiota at the phylum level: Group 1 + 2 vs. 3. Table S3. Differing abundance of specific populations of gut microbiota between group 2 and group 3 for intestinal 18F-FDG uptake at the genus level. (PDF 1250 kb

(a) A 57-year-old women with right breast cancer underwent ¹⁸F–FDG PET/CT. Mild ¹⁸F–FDG uptake (inferior to the liver) resulted in her being classified into the low uptake group. Her BMI was 20.0, and triglyceride level was 45 mg/dL. (b) A 64-year-old women with left breast cancer underwent ¹⁸F–FDG PET/CT. Intense ¹⁸F–FDG uptake along the intestine was classified into the high uptake group. Her BMI was 27.3, and triglyceride level was 393 mg/dL.

<p>(a) A 57-year-old women with right breast cancer underwent ¹⁸F–FDG PET/CT. Mild ¹⁸F–FDG uptake (inferior to the liver) resulted in her being classified into the low uptake group. Her BMI was 20.0, and triglyceride level was 45 mg/dL. (b) A 64-year-old women with left breast cancer underwent ¹⁸F–FDG PET/CT. Intense ¹⁸F–FDG uptake along the intestine was classified into the high uptake group. Her BMI was 27.3, and triglyceride level was 393 mg/dL.</p

Results of univariate and multivariate regression analyses.

<p>HDL, high-density lipoprotein; LDL, low-density lipoprotein</p><p>*<i>p</i><0.05</p><p>Results of univariate and multivariate regression analyses.</p

Additional file 5: Table S4. of Comparative analysis of gut microbiota associated with body mass index in a large Korean cohort

Comparison of regression analysis with or without adjustment of T2DM or T2DM under medication as covariates. (DOCX 19 kb

Scatter plots of age (a), body mass index (b), triglyceride (c), cholesterol (d), low-density lipoprotein (e), and high-density lipoprotein (f) according TB SUV_max.

<p>Scatter plots of age (a), body mass index (b), triglyceride (c), cholesterol (d), low-density lipoprotein (e), and high-density lipoprotein (f) according TB SUV_max.</p

The number of mice used in each experiment.

<p>*Not determined.</p><p>The number of mice used in each experiment.</p

Lung microbiota influences alveolar morphology and mucus production. A. Histology of murine lungs from GF, SPF, non-SPF C57BL6 or wild-derived mice.

<p>Sections were stained with periodic acid-Schiff (PAS) and analyzed by light microscopy. Black bars indicate the magnification used (500 µm = 40X, 200 µm = 100X, 50 µm = 400X). Pictures show representative sections of 3–6 mice analyzed per group. B–C) Alveolar size as calculated from area measurements and the number of alveolae per microscopical field (400X magnification). Each dot represents the mean of either the area (µm²) from 10 alveolae (B) or the number of alveolae from 10 fields per section (C). Results from three to six mice per group are shown. Results are expressed as mean ± SEM, two tailed Student's t-test followed by Mann Whitney test.</p

Bacterial compositions with respect to mouse origin.

<p>Evaluation of the distribution of bacterial phyla or classes from 454 sequencing data derived from each of the four groups of mice - 3 SPF, 4 non-SPF C57BL/6, 8 wild-derived, and 15 wild-caught mice.</p

Bacterial diversity with respect to mouse origin.

<p>Rarefaction analysis of 16S rRNA gene sequences from 454 pyrosequencing data using Shannon's index (A) and Chao1 (B). The results of statistical analyses are provided in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0113466#pone.0113466.s002" target="_blank">Table S1</a>. Principle coordinate analysis of the unweighted UniFrac distance (C). Significant separation is present with respect to the origin of the mouse samples (<i>adonis, r²</i> = 0.15, <i>p</i> = 0.002).</p