Bioinformatic and Statistical Analysis of Microbiome Data. From Raw Sequences to Advanced Modeling with QIIME 2 and R

Preface
Acknowledgments
Contents
About the Authors
Chapter 1: Introduction to QIIME 2
	1.1 Overview of QIIME 2
	1.2 Core Concepts in QIIME 2
		1.2.1 Artifacts
		1.2.2 Visualizations
		1.2.3 Semantic Type
		1.2.4 Plugins
	1.3 Install QIIME 2
	1.4 Store and Track Data
	1.5 Extract Data from QIIME 2 Archives
	1.6 Summary
	References
Chapter 2: Introduction to R for Microbiome Data
	2.1 Some Useful R Functions
		2.1.1 Save Data into R Data Format
			2.1.1.1 Use SaveRDS() to Save Single R Object
			2.1.1.2 Use Save() to Save Multiple R Objects
			2.1.1.3 Use Save.Image() to Save Workspace Image
		2.1.2 Read Back Data from R Data Format Files
			2.1.2.1 Use ReadRDS() to Read Back Single R Object
			2.1.2.2 Use Load() to Read Back Multiple Objects from R File
		2.1.3 Use Download.File() to Download File from Website
	2.2 Some Useful R Packages for Microbiome Data
		2.2.1 Readr
		2.2.2 Ggpubr
			2.2.2.1 Box Plots
			2.2.2.2 Violin Plots
			2.2.2.3 Density Plots
			2.2.2.4 Histogram Plots
		2.2.3 Tidyverse
	2.3 Specifically Designed R Packages for Microbiome Data
		2.3.1 Phyloseq
			2.3.1.1 Create a phyloseq Object
			2.3.1.2 Merge Samples in a phyloseq Object
			2.3.1.3 Merge Two phyloseq Objects
		2.3.2 Microbiome
			2.3.2.1 Create a phyloseq Object
			2.3.2.2 Summarize the Contents of phyloseq Object
			2.3.2.3 Retrieve the Data Elements of phyloseq Object
			2.3.2.4 Operate on Taxa
			2.3.2.5 Aggregate Taxa to Higher Taxonomic Levels
			2.3.2.6 Operate on Sample
			2.3.2.7 Operate on Metadata
			2.3.2.8 Data Transformations
			2.3.2.9 Export phyloseq Data into CSV Files
			2.3.2.10 Import CSV, Mothur, and BIOM Format Files into a phyloseq Object
		2.3.3 ampvis2
		2.3.4 curatedMetagenomicData
	2.4 Some R Packages for Analysis of Phylogenetics
		2.4.1 Ape
		2.4.2 Phytools
		2.4.3 Castor
	2.5 BIOM Format and Biomformat Package
	2.6 Creating Analysis Microbiome Dataset
	2.7 Summary
	References
Chapter 3: Basic Data Processing in QIIME 2
	3.1 Importing Data into QIIME 2
		3.1.1 Import FASTA Format Data
		3.1.2 Import FASTQ Format Data
		3.1.3 Import Feature Table
		3.1.4 Import Phylogenetic Trees
	3.2 Exporting Data from QIIME 2
		3.2.1 Export Feature Table
		3.2.2 Export Phylogenetic Trees
	3.3 Extracting Data from QIIME 2 Archives
		3.3.1 Extract Data Using the Qiime Tools Export Command
		3.3.2 Extract Data Using Unzip Program on macOS
	3.4 Filtering Data in QIIME 2
		3.4.1 Filter Feature Table
			3.4.1.1 Total-Frequency-Based Filtering
			3.4.1.2 Contingency-Based Filtering
			3.4.1.3 Identifier-Based Filtering
			3.4.1.4 Metadata-Based Filtering
		3.4.2 Taxonomy-Based Tables and Sequences Filtering
			3.4.2.1 Filter Tables Based on Taxonomy
			3.4.2.2 Filter Sequences Based on Taxonomy
		3.4.3 Filter Distance Matrices
			3.4.3.1 Filtering Distance Matrix Based on Identifiers
			3.4.3.2 Filter Distance Matrix Based on Sample Metadata
	3.5 Introducing QIIME 2 View
	3.6 Communicating Between QIIME 2 and R
		3.6.1 Export QIIME 2 Artifacts into R Using qiime2R Package
			3.6.1.1 Read a .qza File
			3.6.1.2 Create a phyloseq Object
		3.6.2 Prepare Feature Table and Metadata in R and Import into QIIME 2
	3.7 Summary
	References
Chapter 4: Building Feature Table and Feature Representative Sequences from Raw Reads
	4.1 Analyzing Demultiplexed Paired-End FASTQ Data
		4.1.1 Prepare Sample Metadata
		4.1.2 Prepare Raw Sequence Data
		4.1.3 Import Data Files as Qiime Zipped Artifacts(.qza)
		4.1.4 Examine and Visualize the Qualities of the Sequence Reads
	4.2 Analyzing Demultiplexed Paired-End FASTQ Data Using DADA2 and q2-dada2 Plugin
		4.2.1 Introduction to DADA2 and q2-dada2 Plugin
		4.2.2 Denoise Sequences to Construct Feature Table and Feature Data with q2-dada2 Plugin
		4.2.3 Summarize the Feature Table and Feature Data from q2-dada2 Plugin
	4.3 Analyzing Multiplexed Paired-End FASTQ Data Using q2-dada2 Plugin
		4.3.1 Prepare Sample Metadata
		4.3.2 Prepare Raw Sequence Data
		4.3.3 Import Data Files as Qiime Zipped Artifacts(.qza)
		4.3.4 Demultiplexing Sequences
		4.3.5 Summarize the Demultiplexing Results and Examine Quality of the Reads
	4.4 Analyzing Demultiplexed Paired-End FASTQ Data Using Deblur and q2-deblur Plugin
		4.4.1 Introduction to Deblur and q2-deblur Plugin
		4.4.2 Process Initial Quality Filtering
		4.4.3 Preliminary Works for Denoising with Deblur
		4.4.4 Denoise Sequences with Deblur to Construct Feature Table and Feature Data
		4.4.5 Summarize the Feature Table and Feature Data from Deblur
		4.4.6 Remarks on DADA2 and Deblur
	4.5 Summary
	References
Chapter 5: Assigning Taxonomy, Building Phylogenetic Tree
	5.1 Assigning Taxonomy
		5.1.1 Bioinformatics Tools and Reference Databases
		5.1.2 QIIME 2-Formatted and Maintained Taxonomic Reference Databases
		5.1.3 DADA2-Formatted and Maintained Taxonomic Reference Databases
		5.1.4 Introduction to q2-Feature-Classifier
		5.1.5 Assign Taxonomy Using the q2-Feature-Classifier
		5.1.6 Remarks on Taxonomic Classification
	5.2 Building Phylogenetic Tree
		5.2.1 Introduction to Phylogenetic Tree
		5.2.2 Build a Phylogenetic Tree Using the Alignment and Phylogeny Commands
		5.2.3 Remarks on the Taxonomic and Phylogenetic Trees
	5.3 Summary
	References
Chapter 6: Clustering Sequences into OTUs
	6.1 Introduction to Clustering Sequences into OTUs
		6.1.1 Merge Reads
		6.1.2 Remove Non-biological Sequences
		6.1.3 Trim Reads Length
		6.1.4 Discard Low-Quality Reads
		6.1.5 Dereplicate Sequences
	6.2 Introduction to VSEARCH and q2-vsearch
	6.3 Closed-Reference Clustering
		6.3.1 Introduction
		6.3.2 Implement Cluster-Features-Closed-Reference
		6.3.3 Remarks on Closed-Reference Clustering
	6.4 De Novo Clustering
		6.4.1 Introduction
		6.4.2 Implement Cluster-Features-De-Novo
		6.4.3 Remarks on De Novo Clustering
	6.5 Open-Reference Clustering
		6.5.1 Introduction
		6.5.2 Implement Cluster-Features-Open-Reference
		6.5.3 Remarks on Open-Reference Clustering
	6.6 Summary
	References
Chapter 7: OTU Methods in Numerical Taxonomy
	7.1 Brief History of Numerical Taxonomy
	7.2 Principles of Numerical Taxonomy
		7.2.1 Definitions of Characters and Taxa
		7.2.2 Sample Size Calculation: How Many Characters?
		7.2.3 Equal Weighting Characters
		7.2.4 Taxonomic Rank
	7.3 Phenetic Taxonomy: Philosophy of Numerical Taxonomy
		7.3.1 Phenetics: Numerical-Based Empiricism Versus Aristotle´s Essentialism
			7.3.1.1 Aristotle´s Metaphysics of Essentialism
			7.3.1.2 Natural Classification: Numerical-Based Empiricism
		7.3.2 Classification: Inductive Theory Versus Darwin´ Theory of Evolution
		7.3.3 Biological Classifications: Phenetic Approach Versus Cladistic Approach
	7.4 Construction of Taxonomic Structure
		7.4.1 Defining the Operational Taxonomic Units
		7.4.2 Estimation of Taxonomic Resemblance
			7.4.2.1 Data Matrix/OTU Table
			7.4.2.2 R-Technique and Q-Technique
			7.4.2.3 Similarity Coefficients
		7.4.3 Commonly Clustering-Based OTU Methods
			7.4.3.1 Similarity/Resemblance Matrix
			7.4.3.2 Definitions of Cluster in Numerical Taxonomy
			7.4.3.3 Commonly Clustering Methods in Defining Groups of OTUs
				7.4.3.3.1 Single Linkage Clustering
				7.4.3.3.2 Complete Linkage Clustering
				7.4.3.3.3 Average Linkage Clustering
				7.4.3.3.4 Central or Nodal Clustering
				7.4.3.3.5 Weighted Versus Unweighted Clustering
				7.4.3.3.6 Arithmetic Average Versus Centroid Clustering
			7.4.3.4 Factor Analysis
			7.4.3.5 Ordination Methods
			7.4.3.6 Discriminant Analysis in Identifying OTUs
				7.4.3.6.1 Discriminant Function Analysis (DFA)
				7.4.3.6.2 Multiple Discriminant Analysis (MDA)
				7.4.3.6.3 Canonical Variate Analysis (CVA)
	7.5 Statistical Hypothesis Testing of OTUs
		7.5.1 Hypothesis Testing on Similarity Coefficients of OTUs
			7.5.1.1 Heterogeneity of the Column Vectors of the Data Matrix
			7.5.1.2 Hypothesis Testing of Individual Similarity Coefficients
		7.5.2 Hypothesis Testing on Clustering OTUs
	7.6 Some Characteristics of Clustering-Based OTU Methods
		7.6.1 Some Basic Questions in Numerical Taxonomy
		7.6.2 Characteristics of Numerical Taxonomy
			7.6.2.1 Summary of Characteristics of Numerical Taxonomy
			7.6.2.2 Remarks on Characteristics of Numerical Taxonomy
			7.6.2.3 Remarks on Impact of Numerical Taxonomy on Microbiome Research
	7.7 Summary
	References
Chapter 8: Moving Beyond OTU Methods
	8.1 Clustering-Based OTU Methods in Microbiome Study
		8.1.1 Common Clustering-Based OTU Methods
		8.1.2 Hierarchical Clustering OTU Methods
		8.1.3 Heuristic Clustering OTU Methods
		8.1.4 Limitations of Clustering-Based OTU Methods
			8.1.4.1 Assumptions of OTU Methods in Numerical Taxonomy and Microbiome
			8.1.4.2 Challenges of Defining Species Using OTU Methods
			8.1.4.3 Criteria of Defining the Levels of Taxonomy
			8.1.4.4 Clustering Algorithms and the Accuracy of OTU Clustering Results
		8.1.5 Purposes of Using OTUs in Microbiome Study
			8.1.5.1 Defining Taxonomy
			8.1.5.2 Reducing the Impact of Sequencing Error
		8.1.6 Defining Species and Species-Level Analysis
			8.1.6.1 Eukaryote Species Concepts
			8.1.6.2 Prokaryote or Bacterial Species Concepts
			8.1.6.3 16S rRNA Method and Definition of Species
			8.1.6.4 16S rRNA Method and Physiological Characteristics
	8.2 Moving Toward Single-Nucleotide Resolution-Based OTU Methods
		8.2.1 Concept Shifting in Bioinformatic Analysis
		8.2.2 Single-Nucleotide Resolution Clustering-Based OTU Methods
			8.2.2.1 Distribution-Based Clustering (DBC)
			8.2.2.2 Swarm2
	8.3 Moving Beyond the OTU Methods
		8.3.1 Entropy-Based Methods: Oligotyping
		8.3.2 Denoising-Based Methods
			8.3.2.1 Denoising-Based Methods Versus Clustering-Based Methods
			8.3.2.2 Pyrosequencing Flowgrams
			8.3.2.3 Cluster-Free Filtering (CFF)
			8.3.2.4 DADA2
			8.3.2.5 UNOISE2 and UNOISE3
			8.3.2.6 Deblur
			8.3.2.7 SeekDeep
			8.3.2.8 Remarks on Denoising-Based Methods
	8.4 Discussion on Moving Beyond OTU Methods
		8.4.1 Necessity of Targeting Single-Base Resolution
		8.4.2 Possibility of Moving Beyond Traditional OTU Methods
		8.4.3 Issues of Sub-OTU Methods
		8.4.4 Prediction of Sequence Similarity to Ecological Similarity
		8.4.5 Functional Analysis and Multi-omics Integration
	8.5 Summary
	References
Chapter 9: Alpha Diversity
	9.1 Abundance-Based Alpha Diversity Metrics
		9.1.1 Chao 1 Richness and Abundance-Based Coverage Estimator (ACE)
			9.1.1.1 The Measures of Chao 1 Richness
			9.1.1.2 The Measures of ACE
			9.1.1.3 Calculating Chao 1 Richness and ACE Using Ampvis2 Package
			9.1.1.4 Calculating Chao 1 Richness and ACE Using Microbiome Package
		9.1.2 Shannon Diversity
			9.1.2.1 The Measures of Shannon Diversity
			9.1.2.2 Calculating Shannon Diversity Using ampvis2 Package
			9.1.2.3 Calculating Shannon Diversity Using Microbiome Package
		9.1.3 Simpson Diversity
			9.1.3.1 The Measures of Simpson Diversity
			9.1.3.2 Calculating Simpson Diversity and Inverse Simpson Diversity Using Ampvis2 Package
			9.1.3.3 Calculating Simpson Diversity, Inverse Simpson Diversity, and Simpson Evenness Using Microbiome Package
		9.1.4 Pielou´s Evenness
			9.1.4.1 The Measures of Pielou´s Evenness
			9.1.4.2 Calculating Pielou Evenness Using Microbiome Package
	9.2 Phylogenetic Alpha Diversity Metrics
		9.2.1 Phylogenetic Diversity
		9.2.2 Phylogenetic Entropy
		9.2.3 Phylogenetic Quadratic Entropy (PQE)
	9.3 Exploring Alpha Diversity and Abundance
		9.3.1 Heatmap
		9.3.2 Boxplot
			9.3.2.1 Generating Boxplot Using Ampvis2 Package
			9.3.2.2 Generating Boxplot Using Microbiome Package
		9.3.3 Violin Plot
	9.4 Statistical Hypothesis Testing of Alpha Diversity
		9.4.1 Summarize the Diversity Measures
		9.4.2 Plot Histogram of the Diversity Distributions
		9.4.3 Kruskal-Wallis Test
		9.4.4 Perform Multiple Comparisons
	9.5 Alpha Diversity Analysis in QIIME 2
		9.5.1 Calculate Alpha Diversity Using Core-Metrics-Phylogenetic Method
		9.5.2 Calculate Alpha Diversity Using Alpha Method
			9.5.2.1 Shannon Index
			9.5.2.2 Chao1 Index and Chao1 Confidence Interval
			9.5.2.3 Observed Features
			9.5.2.4 Simpson Index and Simpson´s Evenness
			9.5.2.5 Pielou´s Evenness
		9.5.3 Calculate Alpha Diversity Using Alpha-Phylogenetic Method
		9.5.4 Test for Differences of Alpha Diversity Between Groups
		9.5.5 Alpha Rarefaction in QIIME 2
	9.6 Summary
	References
Chapter 10: Beta Diversity Metrics and Ordination
	10.1 Abundance-Based Beta Diversity Metrics
		10.1.1 Bray-Curtis Dissimilarity
			10.1.1.1 The Measures of Bray-Curtis Index
			10.1.1.2 Calculating Bray-Curtis Index Using the vegan Package
		10.1.2 Jaccard Dissimilarity
			10.1.2.1 The Measures of Jaccard Index
			10.1.2.2 Calculate Jaccard Index Using the vegan Package
		10.1.3 Sørensen Dissimilarity
			10.1.3.1 The Measures of Sørensen Index
			10.1.3.2 Calculate Sørensen Index Using the vegan Package
			10.1.3.3 Calculate Matrices of Bray-Curtis, Jaccard, and Sørensen Indices Using the vegan Package
	10.2 Phylogenetic Beta Diversity Metrics
		10.2.1 Unweighted UniFrac
		10.2.2 Weighted UniFrac
		10.2.3 GUniFrac
		10.2.4 pldist
		10.2.5 Calculate (Un)Weighted UniFrac and GUniFrac Distances Using the GUniFrac Package
		10.2.6 Remarks on Rarefaction for Alpha and Beta Diversity Analysis
	10.3 Ordination Methods
		10.3.1 Introduction to Ordination
			10.3.1.1 Brief History of Ordination
			10.3.1.2 Ordination Plots
		10.3.2 Ordination Plots in the ampvis2 Package
		10.3.3 Principal Component Analysis (PCA)
			10.3.3.1 Introduction to PCA
			10.3.3.2 Implement PCA
		10.3.4 Principal Coordinate Analysis (PCoA)
			10.3.4.1 Introduction to PCoA
			10.3.4.2 Implement PCoA
		10.3.5 Nonmetric Multidimensional Scaling (NMDS)
			10.3.5.1 Introduction to NMDS
			10.3.5.2 Implement NMDS
		10.3.6 Correspondence Analysis (CA)
			10.3.6.1 Introduction to CA
			10.3.6.2 Implement CA
		10.3.7 Detrended Correspondence Analysis (DCA)
			10.3.7.1 Introduction to DCA
			10.3.7.2 Implement DCA
		10.3.8 Redundancy Analysis (RDA)
			10.3.8.1 Introduction to RDA
			10.3.8.2 Implement RDA
		10.3.9 Canonical Correspondence Analysis (CCA)
			10.3.9.1 Introduction to CCA
			10.3.9.2 Implement CCA
	10.4 Beta Diversity Metrics and Ordination in QIIME 2
		10.4.1 Calculate Beta Diversity Measures
		10.4.2 Explore Principal Coordinates (PCoA) Using Emperor Plots
	10.5 Remarks on Ordination and Clustering
	10.6 Summary
	References
Chapter 11: Statistical Testing of Beta Diversity
	11.1 Introduction to Nonparametric MANOVA Using Permutation Tests
	11.2 Analysis of Similarity (ANOSIM)
		11.2.1 Introduction of ANOSIM
		11.2.2 Perform ANOSIM Using the Vegan Package
		11.2.3 Remarks on ANOSIM
	11.3 Permutational MANOVA (PERMANOVA)
		11.3.1 Introduction to PERMANOVA
		11.3.2 Perform PERMANOVA Using the Vegan Package
		11.3.3 Remarks on PERMANOVA
	11.4 Analysis of Multivariate Homogeneity of Group Dispersions
		11.4.1 Introduction to the Function betadisper()
		11.4.2 Implement the Function betadisper()
	11.5 Pairwise PERMANOVA
		11.5.1 Introduction to Pairwise PERMMANOVA
		11.5.2 Implement Pairwise PERMMANOVA Using the RVAideMemoire Package
	11.6 Identify Core Microbial Taxa Using the Microbiome Package
	11.7 Statistical Testing of Beta Diversity in QIIME 2
		11.7.1 Significant Testing of Bray-Curtis Distance
		11.7.2 Significant Testing of Jaccard Distance
		11.7.3 Significant Testing of Unweighted UniFrac Distance
		11.7.4 Significant Testing of Weighted UniFrac Distance
	11.8 Summary
	References
Chapter 12: Differential Abundance Analysis of Microbiome Data
	12.1 Zero-Inflated Gaussian (ZIG) and Zero-Inflated Log-Normal (ZILN) Mixture Models
		12.1.1 Total Sum Scaling (TSS)
		12.1.2 Cumulative Sum Scaling (CSS)
		12.1.3 ZIG and ZILN Models
	12.2 Implement ZILN via metagenomeSeq
	12.3 Some Additional Statistical Tests in metagenomeSeq
		12.3.1 Log-Normal Permutation Test of Abundance
		12.3.2 Presence-Absence Testing of the Proportion/Odds
		12.3.3 Discovery Odds Ratio Testing of the Proportion of Observed Counts
		12.3.4 Perform Feature Correlations
	12.4 Illustrate Some Useful Functions in metagenomeSeq
		12.4.1 Access the MRexperiment Object
		12.4.2 Subset the MRexperiment Object
		12.4.3 Filter the MRexperiment Object or Count Matrix
		12.4.4 Merge the MRexperiment Object
		12.4.5 Call the Normalized Counts Using the cumNormMat() and MRcounts()
		12.4.6 Calculate the Normalization Factors Using the calcNormFactors()
		12.4.7 Access the Library Sizes Using the libSize()
		12.4.8 Save the Normalized Counts Using the exportMat()
		12.4.9 Save the Sample Statistics Using the exportStats()
		12.4.10 Find Unique OTUs or Features
		12.4.11 Aggregate Taxa
		12.4.12 Aggregate Samples
	12.5 Remarks on CSS Normalization, ZIG, and ZILN
	12.6 Summary
	References
Chapter 13: Zero-Inflated Beta Models for Microbiome Data
	13.1 Zero-Inflated Beta Modeling Microbiome Data
	13.2 Zero-Inflated Beta Regression (ZIBSeq)
		13.2.1 Introduction to ZIBseq
			13.2.1.1 ZIBseq Model
			13.2.1.2 Statistical Hypothesis Testing of Microbiome Composition and Outcome
		13.2.2 Implement ZIBseq
		13.2.3 Remarks on ZIBSeq
	13.3 Zero-Inflated Beta-Binomial Model (ZIBB)
		13.3.1 Introduction to ZIBB
			13.3.1.1 Zero and Count Models in ZIBB
			13.3.1.2 Statistical Hypothesis Testing in ZIBB
		13.3.2 Implement ZIBB
		13.3.3 Remarks on ZIBB
	13.4 Summary
	References
Chapter 14: Compositional Analysis of Microbiome Data
	14.1 Introduction to Compositional Data
		14.1.1 What Are Compositional Data?
		14.1.2 Microbiome Data Are Treated as Compositional
		14.1.3 Aitchison Simplex
		14.1.4 Challenges of Analyzing Compositional Data
		14.1.5 Fundamental Principles of Compositional Data Analysis
		14.1.6 The Family of Log-Ratio Transformations
			14.1.6.1 Additive Log-Ratio (alr) Transformation
			14.1.6.2 Centered Log-Ratio (clr) Transformation
			14.1.6.3 Isometric Log-Ratio (ilr) Transformation
			14.1.6.4 Inter-quartile Log-Ratio (iqlr) Transformation
		14.1.7 Remarks on Log-Ratio Transformations
	14.2 ANOVA-Like Compositional Differential Abundance Analysis
		14.2.1 Introduction to ALDEx2
		14.2.2 Implement ALDEx2 Using R
		14.2.3 Remarks on ALDEx2
	14.3 Analysis of Composition of Microbiomes (ANCOM)
		14.3.1 Introduction to ANCOM
		14.3.2 Implement ANCOM Using QIIME 2
		14.3.3 Remarks on ANCOM
	14.4 Analysis of Composition of Microbiomes-Bias Correction (ANCOM-BC)
		14.4.1 Introduction to ANCOM-BC
		14.4.2 Implement ANCOM-BC Using the ANCOMBC Package
		14.4.3 Remarks on ANCOM-BC
	14.5 Remarks on Compositional Data Analysis Approach
	14.6 Summary
	References
Chapter 15: Linear Mixed-Effects Models for Longitudinal Microbiome Data
	15.1 Introduction to Linear Mixed-Effects Models (LMMs)
		15.1.1 Advantages and Disadvantages of LMMs
		15.1.2 Fixed and Random Effects
		15.1.3 Definition of LMMs
		15.1.4 Statistical Hypothesis Tests
		15.1.5 How to Fit LMMs
	15.2 Identifying the Significant Taxa Using the nlme Package
		15.2.1 Introduction to LMMs in Microbiome Research
		15.2.2 Longitudinal Microbiome Data Structure
		15.2.3 Fit LMMs Using the Read Counts as the Outcome
	15.3 Modeling the Diversity Indices Using the lme4 and LmerTest Packages
		15.3.1 Introduction to the lme4 and lmerTest Packages
		15.3.2 Fit LMMs Using the Diversity Index as the Outcome
	15.4 Implement LMMs Using QIIME 2
		15.4.1 Introduction to the QIIME Longitudinal Linear-Mixed-Effects Command
		15.4.2 Fit LMMs in QIIME 2
		15.4.3 Perform Volatility Analysis
	15.5 Remarks on LMMs
	15.6 Summary
	References
Chapter 16: Introduction to Generalized Linear Mixed Models
	16.1 Generalized Linear Models (GLMs) and Generalized Nonlinear Models (GNLMs)
	16.2 Generalized Linear Mixed Models (GLMMs)
	16.3 Model Estimation in GLMMs
	16.4 Algorithms for Parameter Estimation in GLMMs
		16.4.1 Penalized Quasi-Likelihood-Based Methods Using Taylor-Series Linearization
		16.4.2 Likelihood-Based Methods Using Numerical Integration
			16.4.2.1 Laplace Approximation
			16.4.2.2 Gauss-Hermite Quadrature (GHQ)
			16.4.2.3 Adaptive Gauss-Hermite Quadrature (AGQ)
		16.4.3 Markov Chain Monte Carlo-Based Integration
		16.4.4 IWLS and EM-IWLS Algorithms
			16.4.4.1 Extension of IWLS Algorithm for Fitting NBMMs
			16.4.4.2 Extension of IWLS Algorithm for Fitting Longitudinal NBMMs
			16.4.4.3 EM-IWLS Algorithm for Fitting Longitudinal ZINBMMs
	16.5 Statistical Hypothesis Testing and Modeling in GLMMs
		16.5.1 Model Selection in Statistics
		16.5.2 Model Selection in Machine Learning
		16.5.3 Information Criteria for Model Selection
			16.5.3.1 AIC (Akaike´s Information Criterion)
			16.5.3.2 AICc (Finite-Sample Corrected AIC)
			16.5.3.3 QAIC and QAICc (Quasi Akaike Information Criterion and Corrected Quasi-AIC)
			16.5.3.4 BIC (Bayesian Information Criterion)
			16.5.3.5 BC (Bridge Criterion)
			16.5.3.6 DIC (Deviance Information Criterion)
			16.5.3.7 GICλ (Generalized Information Criterion)
		16.5.4 Likelihood-Ratio Test
		16.5.5 Vuong Test
	16.6 Summary
	References
Chapter 17: Generalized Linear Mixed Models for Longitudinal Microbiome Data
	17.1 Generalized Linear Mixed Models (GLMMs) in Microbiome Research
		17.1.1 Data Transformation Versus Using GLMMs
		17.1.2 Model Selection in Microbiome Data
		17.1.3 Statistical Hypothesis Testing in Microbiome Data
	17.2 Generalized Linear Mixed Modeling Using the glmmTMB Package
		17.2.1 Introduction to glmmTMB
			17.2.1.1 Conditional Model Formula
			17.2.1.2 Distribution for the Conditional Model
			17.2.1.3 Dispersion Model Formula
			17.2.1.4 Zero-Inflation Model Formula
		17.2.2 Implement GLMMs via glmmTMB
		17.2.3 Remarks on glmmTMB
	17.3 Generalized Linear Mixed Modeling Using the GLMMadaptive Package
		17.3.1 Introduction to GLMMadaptive
		17.3.2 Implement GLMMs via GLMMadaptive
		17.3.3 Remarks on GLMMadaptive
	17.4 Fast Zero-Inflated Negative Binomial Mixed Modeling (FZINBMM)
		17.4.1 Introduction to FZINBMM
			17.4.1.1 Zero-Inflated Negative Binomial Mixed Models (ZINBMMs)
			17.4.1.2 EM-IWLS Algorithm for Fitting the ZINBMMs
		17.4.2 Implement FZINBMM Using the NBZIMM Package
		17.4.3 Remarks on FZINBMM
	17.5 Remarks on Fitting GLMMs
	17.6 Summary
	References
Chapter 18: Multivariate Longitudinal Microbiome Models
	18.1 Overview of Multivariate Longitudinal Microbiome Analysis
		18.1.1 Multivariate Distance/Kernel-Based Longitudinal Models
		18.1.2 Multivariate Integration of Multi-omics Methods
		18.1.3 Univariate Analysis Versus Multivariate Analysis
	18.2 Nonparametric Microbial Interdependence Test (NMIT)
		18.2.1 Introduction to NMIT
			18.2.1.1 Calculate Interdependence Correlation Matrix
			18.2.1.2 Calculate Distance Matrix
			18.2.1.3 Perform Statistical Hypothesis Testing of Distance Matrix
		18.2.2 Implement NMIT Using R
		18.2.3 Implement NMIT Using QIIME 2
		18.2.4 Remarks on NMIT
	18.3 The Large P Small N Problem
	18.4 Summary
	References
Index