Age K. Smilde is Emeritus-Professor of Biosystems Data Analysis at the Swammerdam Institute for Life Sciences at the University of Amsterdam. He also holds a part-time position at the Department of Plant and Environmental Sciences at the University of Copenhagen. Federico Marini is Professor of Analytical Chemistry at the Department of Chemistry of the University of Rome “La Sapienza”. Johan A. Westerhuis is Assistant Professor at the Swammerdam Institute for Life Sciences, University of Amsterdam, The Netherlands. Kristian H. Liland is Professor of Statistics at the Faculty of Science and Technology, Norwegian University of Life Sciences, Norway.

• About the Authors xiForeword xiiPreface xiii1 Introduction 11.1 Types of Data 11.2 Statistical Design of Experiments 41.3 High-Dimensional Data 61.4 Examples 81.4.1 Metabolomics 81.4.2 Genomics 101.4.3 Microbiome 111.4.4 Proteomics 151.4.5 Food Science 161.4.6 Sensory Science 171.4.7 Chemistry 181.5 Complexities 201.5.1 Normalization 201.5.2 Different Measurement Scales 211.5.3 Different Distributions 211.5.4 Heteroscedastic Error 221.5.5 Comparability 231.5.6 Sparseness, Non-detects, and Missing Values 231.5.7 Unbalancedness 241.6 Direct Versus Indirect Methods 241.7 Some History 251.A Appendix 251.A.1 Types of Measurements 251.A.2 Notation and Terminology 251.A.3 Some Definitions 271.A.4 Abbreviations 282 Basic Theory and Concepts 312.1 Mathematical Background 312.1.1 Vector Spaces and Subspaces 312.1.2 Matrix Decompositions 332.1.3 Inverses and Generalized Inverses 342.1.4 Distances and Projections 352.1.4.1 Formal Description of Distances 352.1.4.2 Projections 382.1.5 Principal Component Analysis 422.2 Statistical Background 442.2.1 Estimation Methods 442.2.1.1 Least Squares 452.2.1.2 Maximum Likelihood 462.2.2 Regression Methods 472.2.2.1 Multiple Linear Regression: Full Rank Case 482.2.2.2 Multiple Linear Regression Using Dummy Variables 482.2.2.3 Multiple Linear Regression: Rank Deficient Case 482.2.2.4 Penalized Regression 492.2.2.5 Principal Component Regression 492.2.2.6 Partial Least Squares 502.2.2.7 Redundancy Analysis 512.2.3 Significance Tests 532.2.3.1 Classical Tests 532.2.3.2 Permutation Tests 542.2.3.3 Likelihood Ratio Tests 552.3 Association Measures 552.3.1 Pearson and Spearman Correlation Coefficients 552.3.2 Problems with Correlations 562.A Appendix 583 Linear Models 613.1 Introduction 613.2 Simple ANOVA Models 623.2.1 One-Way ANOVA 623.2.2 Two-Way ANOVA 653.2.2.1 Crossed Designs 653.2.2.2 Nested Designs 683.2.3 Unbalanced Designs 703.2.3.1 One-Way ANOVA 703.2.3.2 Two-Way ANOVA for Crossed Designs 713.2.3.3 Nested ANOVA 733.3 Regression Formulation, Estimability, and Contrasts 733.4 Coding Schemes 743.4.1 Codings for Balanced Designs 743.4.1.1 One-Way Layout 743.4.1.2 Two-Way Crossed Designs 793.4.1.3 Two-Way Nested Designs 853.4.2 Codings for Unbalanced Designs 853.5 Advanced Models 933.5.1 Variance Component Models 933.5.2 Linear Mixed Models 953.5.2.1 General Idea 953.5.2.2 Estimation of Model Parameters 953.5.2.3 Repeated Measures ANOVA 973.5.2.4 Cross-over Designs and Models 983.5.2.5 Longitudinal LMMs 983.6 Hasse Diagrams 1053.6.1 Building a Hasse Diagram 1053.7 Validation 1073.7.1 Classical Tests Revisited 1073.7.2 Expected Mean Squares from Hasse Diagrams 1083.7.3 Permutation Tests 1103.7.3.1 Exact Tests 1113.7.3.2 Approximate Tests 1133.8 Miscellaneous Models 1143.8.1 Multivariate Analysis of Variance 1143.8.1.1 Traditional Multivariate Analysis of Variance 1143.8.1.2 Significance Testing in MANOVA 1153.8.1.3 Regression Formulation of MANOVA 1163.8.2 Multivariate LMMs 1173.A Appendix 1173.A.1 Proof 1173.A.2 Relationships Between Codings 1183.A.3 Practical Aspects of Codings 1204 ASCA and Related Methods 1254.1 ASCA 1254.1.1 Basic Idea of ASCA 1254.1.2 Properties of ASCA 1294.1.3 Permutation Tests for ASCA 1304.1.4 Back-Projection 1314.1.5 Scaling in ASCA 1324.1.6 Group-wise ASCA 1394.1.7 Variable-Selection ASCA 1404.1.8 REP-ASCA 1404.1.9 ASCA as a Multivariate Multiple Regression Model 1424.1.10 Geometry of ASCA 1424.1.10.1 Geometry of ASCA in Row-Space 1434.1.10.2 Geometry of ASCA in Column-Space 1454.2 APCA 1484.2.1 Basic Idea of APCA 1484.2.2 Comparing APCA with ASCA 1504.3 ASCA+ 1514.3.1 Confidence Ellipsoids for ASCA 1524.3.2 ASCA and ASCA+ as RDA Models 1554.4 Principal Response Curves 1564.5 SMART 1604.6 ASCA, PRC, and SMART Compared 1624.7 MSCA 1674.A Appendix 1714.A.1 Proof of Equation 4.20 1714.A.2 Proof of Equation 4.6 1725 Alternative Methods 1735.1 General Introduction 1735.2 PLSR-Based Methods 1735.2.1 ANOVA-TP 1735.2.2 ANOVA Multiblock Orthogonal Partial Least Squares (AMOPLS) 1805.3 LMM-Based Methods 1855.3.1 RM-ASCA+ with Qualitative Time Models 1855.3.2 Validation of the RM-ASCA+ Model 1865.3.2.1 Validation of RM-ASCA+ Models with Nonparametric Bootstrap 1865.3.2.2 Validation of RM-ASCA+ Models with Permutation Testing 1875.3.2.3 Visualization 1885.3.2.4 RM-ASCA+ with Quantitative Time Models 1895.3.3 LiMM-PCA 1925.3.3.1 Validation 1935.3.3.2 Visualization of Effects in LiMM-PCA 1945.4 Miscellaneous Methods 1965.4.1 PC-ANOVA 1965.4.1.1 Basic Idea of PC-ANOVA 1965.4.1.2 Comparing PC-ANOVA with ASCA 2005.4.2 PARAFASCA 2005.4.3 PE-ASCA 2035.4.4 rMANOVA 2095.4.5 Fifty–Fifty MANOVA 2115.4.6 AComDim 2125.4.7 General Effect Modeling (GEM) 2186 Distance-based Methods 2196.1 Introduction 2196.1.1 Double Zeros 2206.1.2 Horseshoe Effect 2216.1.3 Compositionality 2216.2 Methods 2246.2.1 Principal Coordinate Analysis 2246.2.2 PERMANOVA 2286.2.2.1 PERMANOVA Calculated from the Gower Matrix 2326.2.2.2 PERMANOVA of Non-Euclidean Dissimilarity Matrices 2356.2.3 Effect Sizes in PERMANOVA 2366.2.4 Permutations in PERMANOVA 2376.2.5 Assumptions for PERMANOVA 2376.3 ANOSIM 2387 Reviews and Reflections 2397.1 Reviews 2397.1.1 Metabolomics 2397.1.1.1 Plant Science 2397.1.1.2 Microbiology and Biotechnology 2417.1.1.3 Animal Science 2427.1.1.4 Human Science 2437.1.2 Microbiome 2457.1.3 Genomics 2467.1.4 Proteomics 2467.1.5 Food Science 2467.1.6 Sensory Science 2487.1.7 Chemistry 2487.2 Reflections 2497.2.1 Summary of Reviews 2497.2.2 Overview of Methods 2507.2.3 Remaining Challenges 2517.2.3.1 ASCA+ and Partial RDA 2517.2.3.2 Permutations: Correlations and Unbalancedness 2527.2.3.3 PERMANOVA and Effect Sizes 2527.2.3.4 Back-Projection Approach 2527.2.3.5 Inferential Statistics 2527.2.3.6 Advanced HD-ANOVA Methods 2528 Software 2558.1 HD-ANOVA Software 2558.2 R Package HDANOVA 2558.3 Installing and Starting the Package 2568.4 Data Handling 2568.4.1 Read from File 2568.4.2 Data Pre-processing 2578.4.2.1 Re-coding Categorical data 2588.4.3 Data Structures for Analysis Including Blocks 2598.4.3.1 Create List of Blocks 2598.4.3.2 Create data.frame of Blocks 2608.5 Analysis of Variance (ANOVA) 2628.5.1 Simulated Data 2628.5.2 Fixed Effect Models 2638.5.2.1 One-Way ANOVA 2638.5.2.2 Two-Way Crossed Effects ANOVA 2658.5.2.3 Types of Sums of Squares 2658.5.2.4 Coding Schemes 2668.5.2.5 Fixed Effect Nested ANOVA 2678.5.3 Linear Mixed Models 2688.5.3.1 Least Squares − mixlm 2688.5.3.2 Restrictions 2698.5.3.3 Repeated Measures 2698.5.3.4 REML 2718.5.4 Multivariate ANOVA (MANOVA) 2728.6 Basic ASCA Family 2748.6.1 Fit ASCA Model 2758.6.1.1 Permutation Testing 2758.6.1.2 Random Effects 2768.6.1.3 Scores and Loadings 2768.6.1.4 Data Ellipsoids and Confidence Ellipsoids 2798.6.1.5 Combined Effects 2808.6.1.6 Quantitative Effects 2818.6.2 ANOVA-PCA (APCA) 2828.6.3 PC-ANOVA 2838.6.4 MSCA 2858.6.5 LiMM-PCA 2868.6.6 Repeated Measures ASCA 2888.7 Alternative Methods 2898.7.1 Principal Response Curves (PRC) 2908.7.2 Permutation-Based MANOVA (PERMANOVA) 2918.8 Software Packages 2918.8.1 R Packages 2918.8.2 MATLAB Toolboxes 2938.8.3 Python 293References 295Index 319

Age K. Smilde is Emeritus-Professor of Biosystems Data Analysis at the Swammerdam Institute for Life Sciences at the University of Amsterdam. He also holds a part-time position at the Department of Plant and Environmental Sciences at the University of Copenhagen. Federico Marini is Professor of Analytical Chemistry at the Department of Chemistry of the University of Rome “La Sapienza”. Johan A. Westerhuis is Assistant Professor at the Swammerdam Institute for Life Sciences, University of Amsterdam, The Netherlands. Kristian H. Liland is Professor of Statistics at the Faculty of Science and Technology, Norwegian University of Life Sciences, Norway.

• About the Authors xiForeword xiiPreface xiii1 Introduction 11.1 Types of Data 11.2 Statistical Design of Experiments 41.3 High-Dimensional Data 61.4 Examples 81.4.1 Metabolomics 81.4.2 Genomics 101.4.3 Microbiome 111.4.4 Proteomics 151.4.5 Food Science 161.4.6 Sensory Science 171.4.7 Chemistry 181.5 Complexities 201.5.1 Normalization 201.5.2 Different Measurement Scales 211.5.3 Different Distributions 211.5.4 Heteroscedastic Error 221.5.5 Comparability 231.5.6 Sparseness, Non-detects, and Missing Values 231.5.7 Unbalancedness 241.6 Direct Versus Indirect Methods 241.7 Some History 251.A Appendix 251.A.1 Types of Measurements 251.A.2 Notation and Terminology 251.A.3 Some Definitions 271.A.4 Abbreviations 282 Basic Theory and Concepts 312.1 Mathematical Background 312.1.1 Vector Spaces and Subspaces 312.1.2 Matrix Decompositions 332.1.3 Inverses and Generalized Inverses 342.1.4 Distances and Projections 352.1.4.1 Formal Description of Distances 352.1.4.2 Projections 382.1.5 Principal Component Analysis 422.2 Statistical Background 442.2.1 Estimation Methods 442.2.1.1 Least Squares 452.2.1.2 Maximum Likelihood 462.2.2 Regression Methods 472.2.2.1 Multiple Linear Regression: Full Rank Case 482.2.2.2 Multiple Linear Regression Using Dummy Variables 482.2.2.3 Multiple Linear Regression: Rank Deficient Case 482.2.2.4 Penalized Regression 492.2.2.5 Principal Component Regression 492.2.2.6 Partial Least Squares 502.2.2.7 Redundancy Analysis 512.2.3 Significance Tests 532.2.3.1 Classical Tests 532.2.3.2 Permutation Tests 542.2.3.3 Likelihood Ratio Tests 552.3 Association Measures 552.3.1 Pearson and Spearman Correlation Coefficients 552.3.2 Problems with Correlations 562.A Appendix 583 Linear Models 613.1 Introduction 613.2 Simple ANOVA Models 623.2.1 One-Way ANOVA 623.2.2 Two-Way ANOVA 653.2.2.1 Crossed Designs 653.2.2.2 Nested Designs 683.2.3 Unbalanced Designs 703.2.3.1 One-Way ANOVA 703.2.3.2 Two-Way ANOVA for Crossed Designs 713.2.3.3 Nested ANOVA 733.3 Regression Formulation, Estimability, and Contrasts 733.4 Coding Schemes 743.4.1 Codings for Balanced Designs 743.4.1.1 One-Way Layout 743.4.1.2 Two-Way Crossed Designs 793.4.1.3 Two-Way Nested Designs 853.4.2 Codings for Unbalanced Designs 853.5 Advanced Models 933.5.1 Variance Component Models 933.5.2 Linear Mixed Models 953.5.2.1 General Idea 953.5.2.2 Estimation of Model Parameters 953.5.2.3 Repeated Measures ANOVA 973.5.2.4 Cross-over Designs and Models 983.5.2.5 Longitudinal LMMs 983.6 Hasse Diagrams 1053.6.1 Building a Hasse Diagram 1053.7 Validation 1073.7.1 Classical Tests Revisited 1073.7.2 Expected Mean Squares from Hasse Diagrams 1083.7.3 Permutation Tests 1103.7.3.1 Exact Tests 1113.7.3.2 Approximate Tests 1133.8 Miscellaneous Models 1143.8.1 Multivariate Analysis of Variance 1143.8.1.1 Traditional Multivariate Analysis of Variance 1143.8.1.2 Significance Testing in MANOVA 1153.8.1.3 Regression Formulation of MANOVA 1163.8.2 Multivariate LMMs 1173.A Appendix 1173.A.1 Proof 1173.A.2 Relationships Between Codings 1183.A.3 Practical Aspects of Codings 1204 ASCA and Related Methods 1254.1 ASCA 1254.1.1 Basic Idea of ASCA 1254.1.2 Properties of ASCA 1294.1.3 Permutation Tests for ASCA 1304.1.4 Back-Projection 1314.1.5 Scaling in ASCA 1324.1.6 Group-wise ASCA 1394.1.7 Variable-Selection ASCA 1404.1.8 REP-ASCA 1404.1.9 ASCA as a Multivariate Multiple Regression Model 1424.1.10 Geometry of ASCA 1424.1.10.1 Geometry of ASCA in Row-Space 1434.1.10.2 Geometry of ASCA in Column-Space 1454.2 APCA 1484.2.1 Basic Idea of APCA 1484.2.2 Comparing APCA with ASCA 1504.3 ASCA+ 1514.3.1 Confidence Ellipsoids for ASCA 1524.3.2 ASCA and ASCA+ as RDA Models 1554.4 Principal Response Curves 1564.5 SMART 1604.6 ASCA, PRC, and SMART Compared 1624.7 MSCA 1674.A Appendix 1714.A.1 Proof of Equation 4.20 1714.A.2 Proof of Equation 4.6 1725 Alternative Methods 1735.1 General Introduction 1735.2 PLSR-Based Methods 1735.2.1 ANOVA-TP 1735.2.2 ANOVA Multiblock Orthogonal Partial Least Squares (AMOPLS) 1805.3 LMM-Based Methods 1855.3.1 RM-ASCA+ with Qualitative Time Models 1855.3.2 Validation of the RM-ASCA+ Model 1865.3.2.1 Validation of RM-ASCA+ Models with Nonparametric Bootstrap 1865.3.2.2 Validation of RM-ASCA+ Models with Permutation Testing 1875.3.2.3 Visualization 1885.3.2.4 RM-ASCA+ with Quantitative Time Models 1895.3.3 LiMM-PCA 1925.3.3.1 Validation 1935.3.3.2 Visualization of Effects in LiMM-PCA 1945.4 Miscellaneous Methods 1965.4.1 PC-ANOVA 1965.4.1.1 Basic Idea of PC-ANOVA 1965.4.1.2 Comparing PC-ANOVA with ASCA 2005.4.2 PARAFASCA 2005.4.3 PE-ASCA 2035.4.4 rMANOVA 2095.4.5 Fifty–Fifty MANOVA 2115.4.6 AComDim 2125.4.7 General Effect Modeling (GEM) 2186 Distance-based Methods 2196.1 Introduction 2196.1.1 Double Zeros 2206.1.2 Horseshoe Effect 2216.1.3 Compositionality 2216.2 Methods 2246.2.1 Principal Coordinate Analysis 2246.2.2 PERMANOVA 2286.2.2.1 PERMANOVA Calculated from the Gower Matrix 2326.2.2.2 PERMANOVA of Non-Euclidean Dissimilarity Matrices 2356.2.3 Effect Sizes in PERMANOVA 2366.2.4 Permutations in PERMANOVA 2376.2.5 Assumptions for PERMANOVA 2376.3 ANOSIM 2387 Reviews and Reflections 2397.1 Reviews 2397.1.1 Metabolomics 2397.1.1.1 Plant Science 2397.1.1.2 Microbiology and Biotechnology 2417.1.1.3 Animal Science 2427.1.1.4 Human Science 2437.1.2 Microbiome 2457.1.3 Genomics 2467.1.4 Proteomics 2467.1.5 Food Science 2467.1.6 Sensory Science 2487.1.7 Chemistry 2487.2 Reflections 2497.2.1 Summary of Reviews 2497.2.2 Overview of Methods 2507.2.3 Remaining Challenges 2517.2.3.1 ASCA+ and Partial RDA 2517.2.3.2 Permutations: Correlations and Unbalancedness 2527.2.3.3 PERMANOVA and Effect Sizes 2527.2.3.4 Back-Projection Approach 2527.2.3.5 Inferential Statistics 2527.2.3.6 Advanced HD-ANOVA Methods 2528 Software 2558.1 HD-ANOVA Software 2558.2 R Package HDANOVA 2558.3 Installing and Starting the Package 2568.4 Data Handling 2568.4.1 Read from File 2568.4.2 Data Pre-processing 2578.4.2.1 Re-coding Categorical data 2588.4.3 Data Structures for Analysis Including Blocks 2598.4.3.1 Create List of Blocks 2598.4.3.2 Create data.frame of Blocks 2608.5 Analysis of Variance (ANOVA) 2628.5.1 Simulated Data 2628.5.2 Fixed Effect Models 2638.5.2.1 One-Way ANOVA 2638.5.2.2 Two-Way Crossed Effects ANOVA 2658.5.2.3 Types of Sums of Squares 2658.5.2.4 Coding Schemes 2668.5.2.5 Fixed Effect Nested ANOVA 2678.5.3 Linear Mixed Models 2688.5.3.1 Least Squares − mixlm 2688.5.3.2 Restrictions 2698.5.3.3 Repeated Measures 2698.5.3.4 REML 2718.5.4 Multivariate ANOVA (MANOVA) 2728.6 Basic ASCA Family 2748.6.1 Fit ASCA Model 2758.6.1.1 Permutation Testing 2758.6.1.2 Random Effects 2768.6.1.3 Scores and Loadings 2768.6.1.4 Data Ellipsoids and Confidence Ellipsoids 2798.6.1.5 Combined Effects 2808.6.1.6 Quantitative Effects 2818.6.2 ANOVA-PCA (APCA) 2828.6.3 PC-ANOVA 2838.6.4 MSCA 2858.6.5 LiMM-PCA 2868.6.6 Repeated Measures ASCA 2888.7 Alternative Methods 2898.7.1 Principal Response Curves (PRC) 2908.7.2 Permutation-Based MANOVA (PERMANOVA) 2918.8 Software Packages 2918.8.1 R Packages 2918.8.2 MATLAB Toolboxes 2938.8.3 Python 293References 295Index 319