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Multiple Hypothesis Testing To Estimate The Number Of Communities in Stochastic Block Models

Chetkar Jha, Mingyao Li, Ian Barnett

TL;DR

A simple likelihood-based approach for extracting stochastic block models (SBMs) out of scRNA-seq datasets and a new sequential multiple testing (SMT) method for estimating the number of communities in SBMs are presented.

Abstract

Clustering of single-cell RNA sequencing (scRNA-seq) datasets can give key insights into the biological functions of cells. Therefore, it is not surprising that network-based community detection methods (one of the better clustering methods) are increasingly being used for the clustering of scRNA-seq datasets. The main challenge in implementing network-based community detection methods for scRNA-seq datasets is that these methods \emph{apriori} require the true number of communities or blocks for estimating the community memberships. Although there are existing methods for estimating the number of communities, they are not suitable for noisy scRNA-seq datasets. Moreover, we require an appropriate method for extracting suitable networks from scRNA-seq datasets. For addressing these issues, we present a two-fold solution: i) a simple likelihood-based approach for extracting stochastic block models (SBMs) out of scRNA-seq datasets, ii) a new sequential multiple testing (SMT) method for estimating the number of communities in SBMs. We study the theoretical properties of SMT and establish its consistency under moderate sparsity conditions. In addition, we compare the numerical performance of the SMT with several existing methods. We also show that our approach performs competitively well against existing methods for estimating the number of communities on benchmark scRNA-seq datasets. Finally, we use our approach for estimating subgroups of a human retina bipolar single cell dataset.

Multiple Hypothesis Testing To Estimate The Number Of Communities in Stochastic Block Models

TL;DR

A simple likelihood-based approach for extracting stochastic block models (SBMs) out of scRNA-seq datasets and a new sequential multiple testing (SMT) method for estimating the number of communities in SBMs are presented.

Abstract

Clustering of single-cell RNA sequencing (scRNA-seq) datasets can give key insights into the biological functions of cells. Therefore, it is not surprising that network-based community detection methods (one of the better clustering methods) are increasingly being used for the clustering of scRNA-seq datasets. The main challenge in implementing network-based community detection methods for scRNA-seq datasets is that these methods \emph{apriori} require the true number of communities or blocks for estimating the community memberships. Although there are existing methods for estimating the number of communities, they are not suitable for noisy scRNA-seq datasets. Moreover, we require an appropriate method for extracting suitable networks from scRNA-seq datasets. For addressing these issues, we present a two-fold solution: i) a simple likelihood-based approach for extracting stochastic block models (SBMs) out of scRNA-seq datasets, ii) a new sequential multiple testing (SMT) method for estimating the number of communities in SBMs. We study the theoretical properties of SMT and establish its consistency under moderate sparsity conditions. In addition, we compare the numerical performance of the SMT with several existing methods. We also show that our approach performs competitively well against existing methods for estimating the number of communities on benchmark scRNA-seq datasets. Finally, we use our approach for estimating subgroups of a human retina bipolar single cell dataset.

Paper Structure

This paper contains 21 sections, 5 theorems, 14 equations, 1 figure, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation
  3. Stochastic Block Models
  4. Estimating the Number of Communities in SBM
  5. Preliminaries
  6. Stochastic Block Model
  7. Main Idea
  8. Second Eigenvalues of Erdös Rényi graphs
  9. Sequential Multiple Tests
  10. Main Results
  11. Asymptotic Null
  12. Asymptotic Power
  13. Investigations into Type I error and Power
  14. Comparisons
  15. Methodological Comparison
  16. ...and 6 more sections

Key Result

Theorem 2.1

Let $\epsilon > 0$, $N_i p_{ii} \ge N_i^{1/3}, \mu_i = N_i p_{ii}$ and $\tilde{\mu} = N_i (1 - p_{ii})$. Let $A^{(i)}_{\star}$ be the adjacency matrix generated from Erdös Rényi graph with $N_i$ nodes. Define $M^{(i)}$ as a scaled adjacency matrix defined in (scale.M). Then the second eigenvalue of where $TW_1(\cdot)$ is the Tracy-Widom distribution with Dyson parameter one.

Figures (1)

  • Figure 1: The figure draws the tsne plot for the $17$ estimated clusters of the human retina bipolar cells. Here, the scRNA-seq network was extracted using the likelihood method while optimizing for the hyper-parameters and the estimated number of cluster was estimated using the SMT method.

Theorems & Definitions (6)

  • Theorem 2.1
  • Definition 2.2: (Consistency of Community Detection Methods)
  • Theorem 3.1
  • Corollary 3.2
  • Theorem 3.3
  • Corollary 3.4