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Multivariate Data
Integration Using R
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Introduction to Bioinformatics with R: A Practical Guide for Biologists

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Multivariate Data Integration Using R: Methods and Applications with the mixOmics Package
Kim-Anh LeCao, Zoe Marie Welham
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Multivariate Data
Integration Using R
Methods and Applications with the
mixOmics Package
Kim-Anh Lê Cao
Zoe Marie Welham
First edition published 2022
by CRC Press
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ISBN: 9780367460945 (hbk)

ISBN: 9781032128078 (pbk)
ISBN: 9781003026860 (ebk)
DOI: 10.1201/9781003026860
This book has been prepared from camera-ready copy provided by the authors.
From Kim-Anh Lê Cao:
To my parents, Betty and Huy Lê Cao
To the mixOmics team and our mixOmics users,
And to my co-author Zoe Welham without whom this book would not have existed.
For Zoe Welham:

To Joy and Tamara Welham, for their continual support
Contents
Preface xv
Authors xxi
I Modern biology and multivariate analysis 1

1 Multi-omics and biological systems 3
1.1 Statistical approaches for reductionist or holistic analyses . . . . . . . . . . 3
1.2 Multi-omics and multivariate analyses . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 More than a ‘scale up’ of univariate analyses . . . . . . . . . . . . . 5
1.2.2 More than a fishing expedition . . . . . . . . . . . . . . . . . . . . 5
1.3 Shifting the analysis paradigm . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Challenges with high-throughput data . . . . . . . . . . . . . . . . . . . . . 6
1.4.1 Overfitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.2 Multi-collinearity and ill-posed problems . . . . . . . . . . . . . . . 7
1.4.3 Zero values and missing values . . . . . . . . . . . . . . . . . . . . 7
1.5 Challenges with multi-omics integration . . . . . . . . . . . . . . . . . . . . 8
1.5.1 Data heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5.2 Data size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5.3 Platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5.4 Expectations for analysis . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5.5 Variety of analytical frameworks . . . . . . . . . . . . . . . . . . . 9
1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 The cycle of analysis 11

2.1 The Problem guides the analysis . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Plan in advance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 What affects statistical power? . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Sample size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.3 Identify covariates and confounders . . . . . . . . . . . . . . . . . . 13
2.2.4 Identify batch effects . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Data cleaning and pre-processing . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 Normalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.2 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.3 Missing values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Analysis: Choose the right approach . . . . . . . . . . . . . . . . . . . . . . 15
2.4.1 Descriptive statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.2 Exploratory statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.3 Inferential statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.4 Univariate or multivariate modelling? . . . . . . . . . . . . . . . . . 16
2.4.5 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
vii
viii Contents
2.5 Conclusion and start the cycle again . . . . . . . . . . . . . . . . . . . . . . 18

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Key multivariate concepts and dimension reduction in mixOmics 19

3.1 Measures of dispersion and association . . . . . . . . . . . . . . . . . . . . 19
3.1.1 Random variables and biological variation . . . . . . . . . . . . . . 19
3.1.2 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.3 Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.4 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.5 Covariance and correlation in mixOmics context . . . . . . . . . . . 22
3.1.6 R examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Dimension reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1 Matrix factorisation . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.2 Factorisation with components and loading vectors . . . . . . . . . 24
3.2.3 Data visualisation using components . . . . . . . . . . . . . . . . . 24
3.3 Variable selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.1 Ridge penalty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.2 Lasso penalty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.3 Elastic net . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.4 Visualisation of the selected variables . . . . . . . . . . . . . . . . . 26
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Choose the right method for the right question in mixOmics 29

4.1 Types of analyses and methods . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1.1 Single or multiple omics analysis? . . . . . . . . . . . . . . . . . . . 29
4.1.2 N − or P −integration? . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.1.3 Unsupervised or supervised analyses? . . . . . . . . . . . . . . . . . 31
4.1.4 Repeated measures analyses . . . . . . . . . . . . . . . . . . . . . . 32
4.1.5 Compositional data . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Types of data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.1 Classical omics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.2 Microbiome data: A special case . . . . . . . . . . . . . . . . . . . . 33
4.2.3 Genotype data: A special case . . . . . . . . . . . . . . . . . . . . . 33
4.2.4 Clinical variables that are categorical: A special case . . . . . . . . 33
4.3 Types of biological questions . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3.1 A PCA type of question (one data set, unsupervised) . . . . . . . . 34
4.3.2 A PLS type of question (two data sets, regression or unsupervised) 34
4.3.3 A CCA type of question (two data sets, unsupervised) . . . . . . . 35
4.3.4 A PLS-DA type of question (one data set, classification) . . . . . . 35
4.3.5 A multiblock PLS type of question (more than two data sets,
supervised or unsupervised) . . . . . . . . . . . . . . . . . . . . . . 36
4.3.6 An N −integration type of question (several data sets, supervised) . 36
4.3.7 A P −integration type of question (several studies of the same omic
type, supervised or unsupervised) . . . . . . . . . . . . . . . . . . . 37
4.4 Examplar data sets in mixOmics . . . . . . . . . . . . . . . . . . . . . . . . 37
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.A Appendix: Data transformations in mixOmics . . . . . . . . . . . . . . . . . 38
4.A.1 Multilevel decomposition . . . . . . . . . . . . . . . . . . . . . . . . 38
4.A.2 Mixed-effect model context . . . . . . . . . . . . . . . . . . . . . . 39
4.A.3 Split-up variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.A.4 Example of multilevel decomposition in mixOmics . . . . . . . . . . 40
Contents ix
4.B Centered log ratio transformation . . . . . . . . . . . . . . . . . . . . . . . 41

4.C Creating dummy variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
II mixOmics under the hood 45

5 Projection to latent structures 47
5.1 PCA as a projection algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.1.2 Calculating the components . . . . . . . . . . . . . . . . . . . . . . 48
5.1.3 Meaning of the loading vectors . . . . . . . . . . . . . . . . . . . . 49
5.1.4 Example using the linnerud data in mixOmics . . . . . . . . . . . 49
5.2 Singular Value Decomposition (SVD) . . . . . . . . . . . . . . . . . . . . . 50
5.2.1 SVD algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2.2 Example in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2.3 Matrix approximation . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.3 Non-linear Iterative Partial Least Squares (NIPALS) . . . . . . . . . . . . . 54
5.3.1 NIPALS pseudo algorithm . . . . . . . . . . . . . . . . . . . . . . . 55
5.3.2 Local regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.3.3 Deflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.3.4 Missing values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.4 Other matrix factorisation methods in mixOmics . . . . . . . . . . . . . . . 57
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6 Visualisation for data integration 59

6.1 Sample plots using components . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.1.1 Example with PCA and plotIndiv . . . . . . . . . . . . . . . . . . 59
6.1.2 Sample plot for the integration of two or more data sets . . . . . . 60
6.1.3 Representing paired coordinates using plotArrow . . . . . . . . . . 63
6.2 Variable plots using components and loading vectors . . . . . . . . . . . . . 65
6.2.1 Loading plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.2.2 Correlation circle plots . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.2.3 Biplots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2.4 Relevance networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.2.5 Clustered Image Maps (CIM) . . . . . . . . . . . . . . . . . . . . . 73
6.2.6 Circos plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.A Appendix: Similarity matrix in relevance networks and CIM . . . . . . . . 76
6.A.1 Pairwise variable associations for CCA . . . . . . . . . . . . . . . . 76
6.A.2 Pairwise variable associations for PLS . . . . . . . . . . . . . . . . 76
6.A.3 Constructing relevance networks and displaying CIM . . . . . . . . 77
7 Performance assessment in multivariate analyses 79

7.1 Main parameters to choose . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.2 Performance assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.2.1 Training and testing: If we were rich . . . . . . . . . . . . . . . . . 80
7.2.2 Cross-validation: When we are poor . . . . . . . . . . . . . . . . . . 81
7.3 Performance measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.3.1 Evaluation measures for regression . . . . . . . . . . . . . . . . . . 82
7.3.2 Evaluation measures for classification . . . . . . . . . . . . . . . . . 83
7.3.3 Details of the tuning process . . . . . . . . . . . . . . . . . . . . . . 83
7.4 Final model assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
x Contents
7.4.1 Assessment of the performance . . . . . . . . . . . . . . . . . . . . 86

7.4.2 Assessment of the signature . . . . . . . . . . . . . . . . . . . . . . 86
7.5 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.5.1 Prediction of a continuous response . . . . . . . . . . . . . . . . . . 87
7.5.2 Prediction of a categorical response . . . . . . . . . . . . . . . . . . 88
7.5.3 Prediction is related to the number of components . . . . . . . . . 90
7.6 Summary and roadmap of analysis . . . . . . . . . . . . . . . . . . . . . . . 90
III mixOmics in action 93

8 mixOmics: Get started 95
8.1 Prepare the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.1.1 Normalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.1.2 Filtering variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
8.1.3 Centering and scaling the data . . . . . . . . . . . . . . . . . . . . 96
8.1.4 Managing missing values . . . . . . . . . . . . . . . . . . . . . . . . 100
8.1.5 Managing batch effects . . . . . . . . . . . . . . . . . . . . . . . . . 101
8.1.6 Data format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8.2 Get ready with the software . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.2.1 R installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.2.2 Pre-requisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.2.3 mixOmics download . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.2.4 Load the package . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.3 Coding practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.3.1 Set the working directory . . . . . . . . . . . . . . . . . . . . . . . 103
8.3.2 Good coding practices . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.4 Upload data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.4.1 Data sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.4.2 Dependent variables . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.4.3 Set up the outcome for supervised classification analyses . . . . . . 105
8.4.4 Check data upload . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8.5 Structure of the following chapters . . . . . . . . . . . . . . . . . . . . . . . 106
9 Principal Component Analysis (PCA) 109

9.1 Why use PCA? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9.1.1 Biological questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9.1.2 Statistical point of view . . . . . . . . . . . . . . . . . . . . . . . . 109
9.2 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
9.2.1 PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
9.2.2 Sparse PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
9.3 Input arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
9.3.1 Center or scale the data? . . . . . . . . . . . . . . . . . . . . . . . . 112
9.3.2 Number of components (choice of dimensions) . . . . . . . . . . . . 112
9.3.3 Number of variables to select in sPCA . . . . . . . . . . . . . . . . 113
9.4 Key outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
9.5 Case study: Multidrug . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
9.5.1 Load the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
9.5.2 Quick start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
9.5.3 Example: PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
9.5.4 Example: Sparse PCA . . . . . . . . . . . . . . . . . . . . . . . . . 121
9.5.5 Example: Missing values imputation . . . . . . . . . . . . . . . . . 125
Contents xi
9.6 To go further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

9.6.1 Additional processing steps . . . . . . . . . . . . . . . . . . . . . . 129
9.6.2 Independent component analysis . . . . . . . . . . . . . . . . . . . 129
9.6.3 Incorporating biological information . . . . . . . . . . . . . . . . . 130
9.7 FAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
9.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9.A Appendix: Non-linear Iterative Partial Least Squares . . . . . . . . . . . . 132
9.A.1 Solving PCA with NIPALS . . . . . . . . . . . . . . . . . . . . . . 132
9.A.2 Estimating missing values with NIPALS . . . . . . . . . . . . . . . 132
9.B Appendix: sparse PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.B.1 sparse PCA-SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.B.2 sPCA pseudo algorithm . . . . . . . . . . . . . . . . . . . . . . . . 134
9.B.3 Other sPCA methods . . . . . . . . . . . . . . . . . . . . . . . . . . 134
10 Projection to Latent Structure (PLS) 137

10.1 Why use PLS? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
10.1.1 Biological questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
10.2 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
10.2.1 Univariate PLS1 and multivariate PLS2 . . . . . . . . . . . . . . . 139
10.2.2 PLS deflation modes . . . . . . . . . . . . . . . . . . . . . . . . . . 140
10.2.3 sparse PLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
10.3 Input arguments and tuning . . . . . . . . . . . . . . . . . . . . . . . . . . 142
10.3.1 The deflation mode . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
10.3.2 The number of dimensions . . . . . . . . . . . . . . . . . . . . . . . 143
10.3.3 Number of variables to select . . . . . . . . . . . . . . . . . . . . . 143
10.4 Key outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
10.4.1 Graphical outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
10.4.2 Numerical outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
10.5 Case study: Liver toxicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
10.5.1 Load the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
10.5.2 Quick start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
10.5.3 Example: PLS1 regression . . . . . . . . . . . . . . . . . . . . . . . 147
10.5.4 Example: PLS2 regression . . . . . . . . . . . . . . . . . . . . . . . 152
10.6 Take a detour: PLS2 regression for prediction . . . . . . . . . . . . . . . . . 163
10.7 To go further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
10.7.1 Orthogonal projections to latent structures . . . . . . . . . . . . . . 165
10.7.2 Redundancy analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 166
10.7.3 Group PLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
10.7.4 PLS path modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 166
10.7.5 Other sPLS variants . . . . . . . . . . . . . . . . . . . . . . . . . . 167
10.8 FAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
10.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
10.A Appendix: PLS algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
10.A.1 PLS Pseudo algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 169
10.A.2 Convergence of the PLS iterative algorithm . . . . . . . . . . . . . 170
10.A.3 PLS-SVD method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
10.B Appendix: sparse PLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
10.B.1 sparse PLS-SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
10.B.2 sparse PLS pseudo algorithm . . . . . . . . . . . . . . . . . . . . . 171
10.C Appendix: Tuning the number of components . . . . . . . . . . . . . . . . . 172
xii Contents
10.C.1 In PLS1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

10.C.2 In PLS2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
11 Canonical Correlation Analysis (CCA) 177

11.1 Why use CCA? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
11.1.1 Biological question . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
11.2 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
11.2.1 CCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
11.2.2 rCCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
11.3.1 CCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
11.3.2 rCCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
11.4 Key outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
11.5 Case study: Nutrimouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
11.5.1 Load the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
11.5.2 Quick start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
11.5.3 Example: CCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
11.5.4 Example: rCCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
11.6 To go further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
11.7 FAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
11.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
11.A Appendix: CCA and variants . . . . . . . . . . . . . . . . . . . . . . . . . . 196
11.A.1 Solving classical CCA . . . . . . . . . . . . . . . . . . . . . . . . . 196
11.A.2 Regularised CCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
12 PLS-Discriminant Analysis (PLS-DA) 201

12.1 Why use PLS-DA? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
12.2 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
12.2.1 PLS-DA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
12.2.2 sparse PLS-DA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
12.3.1 PLS-DA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
12.3.2 sPLS-DA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
12.3.3 Framework to manage overfitting . . . . . . . . . . . . . . . . . . . 205
12.4 Key outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
12.5 Case study: SRBCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
12.5.1 Load the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
12.5.2 Quick start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
12.5.3 Example: PLS-DA . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
12.5.4 Example: sPLS-DA . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
12.5.5 Take a detour: Prediction . . . . . . . . . . . . . . . . . . . . . . . 223
12.5.6 AUROC outputs complement performance evaluation . . . . . . . . 225
12.6 To go further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
12.6.1 Microbiome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
Contents xiii
12.6.2 Multilevel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

12.6.3 Other related methods and packages . . . . . . . . . . . . . . . . . 228
12.7 FAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
12.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
12.A Appendix: Prediction in PLS-DA . . . . . . . . . . . . . . . . . . . . . . . 229
12.A.1 Prediction distances . . . . . . . . . . . . . . . . . . . . . . . . . . 229
12.A.2 Background area . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
13 N −data integration 233

13.1 Why use N −integration methods? . . . . . . . . . . . . . . . . . . . . . . . 233
13.1.2 Statistical point of view and analytical challenges . . . . . . . . . . 234
13.2 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
13.2.1 Multiblock sPLS-DA . . . . . . . . . . . . . . . . . . . . . . . . . . 234
13.2.2 Prediction in multiblock sPLS-DA . . . . . . . . . . . . . . . . . . 236
13.4 Key outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
13.5 Case Study: breast.TCGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
13.5.1 Load the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
13.5.2 Quick start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
13.5.3 Parameter choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
13.5.4 Final model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
13.5.5 Sample plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
13.5.6 Variable plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
13.5.7 Model performance and prediction . . . . . . . . . . . . . . . . . . 251
13.6 To go further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
13.6.1 Additional data transformation for special cases . . . . . . . . . . . 255
13.6.2 Other N −integration frameworks in mixOmics . . . . . . . . . . . . 255
13.6.3 Supervised classification analyses: concatenation and ensemble
methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
13.6.4 Unsupervised analyses: JIVE and MOFA . . . . . . . . . . . . . . . 256
13.7 FAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
13.8 Additional resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
13.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
13.A Appendix: Generalised CCA and variants . . . . . . . . . . . . . . . . . . . 258
13.A.1 regularised GCCA . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
13.A.2 sparse GCCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
13.A.3 sparse multiblock sPLS-DA . . . . . . . . . . . . . . . . . . . . . . 260
14 P −data integration 261

14.1 Why use P −integration methods? . . . . . . . . . . . . . . . . . . . . . . . 261
14.2 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
14.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
14.2.2 Multi-group sPLS-DA . . . . . . . . . . . . . . . . . . . . . . . . . 263
14.3.1 Data input checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
14.3.2 Number of components . . . . . . . . . . . . . . . . . . . . . . . . . 265
xiv Contents
14.3.3 Number of variables to select per component . . . . . . . . . . . . 265

14.4 Key outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
14.5 Case Study: stemcells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
14.5.1 Load the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
14.5.2 Quick start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
14.5.3 Example: MINT PLS-DA . . . . . . . . . . . . . . . . . . . . . . . 268
14.5.4 Example: MINT sPLS-DA . . . . . . . . . . . . . . . . . . . . . . . 271
14.5.5 Take a detour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
14.6 Examples of application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
14.6.1 16S rRNA gene data . . . . . . . . . . . . . . . . . . . . . . . . . . 280
14.6.2 Single cell transcriptomics . . . . . . . . . . . . . . . . . . . . . . . 280
14.7 To go further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
14.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
Glossary of terms 283
Key publications 285
Bibliography 287
Index 299
Preface
Context and scope
Modern high-throughput technologies generate information about thousands of biological

molecules at different cellular levels in a biological system, leading to several types of
omics studies (e.g. transcriptomics as the study of messenger RNA molecules expressed
from the genes of an organism, or proteomics as the study of proteins expressed by a
cell, tissue, or organism). However, a reductionist approach that considers each of these
molecules individually does not fully describe an organism in its environment. Rather, we
use multivariate analysis to investigate the simultaneous and complex relationships that
occur in molecular pathways. In addition, to obtain a holistic picture of a complete biological
system, we propose to integrate multiple layers of information using recent computational
tools we have developed through the mixOmics project.
mixOmics is an international endeavour that encompasses methodological developments,
software implementation, and applications to biological and biomedical problems to address
some of the challenges of omics data integration. We have trained students and researchers in
essential statistical and data analysis skills via our numerous multi-day workshops to build
capacity in best practice statistical analysis and advance the field of computational statistics
for biology. The goal of this book is to provide guidance in applying multivariate dimension
reduction techniques for the integration of high-throughput biological data, allowing readers
to obtain new and deeper insights into biological mechanisms and biomedical problems.
Who is this book for?
This book is suitable for biologists, computational biologists, and bioinformaticians who
generate and work with high-throughput omics data. Such data include – but are not
restricted to – transcriptomics, epigenomics, proteomics, metabolomics, the microbiome,
and clinical data. Our book is dedicated to research postgraduate students and scientists at
any career stage, and can be used for teaching specialised multi-disciplinary undergraduate
and Masters’s courses. Data analysts with a basic level of R programming will benefit most
from this resource. The book is organised into three distinct parts, where each part can be
skimmed according to the level and interest of the reader. Each chapter contains different
levels of information, and the most technical chapters can be skipped during a first read.
Overview of methods in mixOmics
The mixOmics package focuses on multivariate analysis which examines more than two
variables simultaneously to integrate different types of variables (e.g. genes, proteins, metabo-
lites). We use dimension reduction techniques applicable to a wide range of data analysis
types. Our analyses can be descriptive, exploratory, or focus on modeling or prediction. Our
xv
xvi Preface
Single ‘omics N-integration P-integration
INPUT
PCA*
MULTIVARIATE
PLS*
METHODS
IPCA* DIABLO* MINT*

rCCA
PLS-DA*
10
SAMPLE PLOTS
-5
-10
GRAPHICS
-15
-5
0 5
0
5 -5
-10
A
VARIABLE PLOTS
RN
i
m
mRNA
pr
te
in
o
supervised method *variable selection
FIGURE 1: Overview of the methods implemented in the mixOmics package for the explo-
ration and integration of multiple data sets. This book aims to guide the data analyst in
constructing the research question, applying the appropriate multivariate techniques, and
interpreting the resulting graphics.
aim is to summarise these large biological data sets to elucidate similarities between sam-
ples, between variables, and the relationship between samples and variables. The mixOmics
package provides a range of methods to answer different kinds of biological questions, for
example to:
• Highlight patterns pertaining to the major sources of variation in the data (e.g. Principal
Component Analysis),
• Segregate samples according to their known group and predict group membership of new
samples (e.g. Partial Least Squares Discriminant Analysis),
• Identify agreement between multiple data sets (e.g. Canonical Correlation Analysis,
Partial Least Squares regression, and other variants),
• Identify molecular signatures across multiple data sets with sparse methods that achieve
variable selection.
Key methodological concepts in mixOmics
Methods in mixOmics are based on matrix factorisation techniques, which offer great
flexibility in analysing and integrating multiple data sets in a holistic manner. We use
dimension reduction combined with feature selection to summarise the main characteristics
of the data and posit novel biological hypotheses.
Preface xvii
Dimension reduction is achieved by combining all original variables into a smaller number of
artificial components that summarise patterns in the original data.
The mixOmics package is unique in providing novel multivariate techniques that enable
feature selection to identify molecular signatures. Feature selection refers to identifying
variables that best explain, or predict, the outcome variable (e.g. group membership, or
disease status) of interest. Variables deemed irrelevant according to the specific statistical
criterion we use in the methods are not taken into account when calculating the components.
Data integration methods use data projection techniques to maximise the covariance, or the
correlation between, omics data sets. We propose two types of data integration, whether on
the same N samples, or on the same P variables (Figure 1).
Finally, our methods can provide either unsupervised or supervised analyses. Unsupervised
analyses are exploratory: any information about sample group membership, or outcome,
is disregarded, and data are explored based on their variance or correlation structure.
Supervised analyses aim to segregate sample groups known a priori (e.g. disease status,
treatments) and identify variables (i.e. biomarker candidates, or molecular signatures) that
either explain or separate sample groups.
These concepts will be explained further in Part I.
To aid in interpreting analysis results, mixOmics provides insightful graphical plots designed
to highlight patterns in both the sample and variable dimensions uncovered by each method
(Figure 1).
Concepts not covered
Each mixOmics method corresponds to an underlying statistical model. However, the methods
we present radically differ from univariate formulations as they do not test one variable at a
time, or produce p-values. In that sense, multivariate methods can be considered exploratory
as they do not enable statistical inference. Our wide range of methods come in many different
flavours and can be applied also for predictive purposes, as we detail in this book. ‘Classical’
univariate statistical inference methods can still be used in our analysis framework after
the identification of molecular signatures, as our methods aim to generate novel biological
hypotheses.
Who is ‘mixOmics’?
The mixOmics project has been developed between France, Australia and Canada since 2009,
when the first version of the package was submitted to the CRAN1 . Our team is composed of
core members from the University of Melbourne, Australia, and the Université de Toulouse,
France. The team also includes several key contributors and collaborators.
The package implements more than nineteen multivariate and sparse methodologies for
omics data exploration, integration, and biomarker discovery for different biological settings,
amongst which thirteen were developed by our team (see our list of publications in Section
14.8). Originally, all methods were designed for omics data, however, their application is
not limited to biological data only. Other applications where integration is required can be
considered, but mostly for cases where the predictor variables are continuous.
1 The Comprehensive R Architecture Network https://www.cran.r-project.org
xviii Preface
The package is currently available from Bioconductor2 , with a development version available
on GitHub3 . We continue to maintain and improve the package via new methods, code
optimisation and efficient memory storage of R objects.
About this book
Part I: Modern biology and multivariate analysis introduces fundamental concepts

in multivariate analysis. Multi-omics and biological systems (Chapter 1) compares and
contrasts multivariate and univariate analysis, and outlines the advantages and challenges
of multivariate analyses. The Cycle of Analysis (Chapter 2) details the necessary steps
in planning, designing and conducting multivariate analyses. Key multivariate concepts
and dimension reduction in mixOmics (Chapter 3) describes measures of dispersion and
association, and introduces key methods in mixOmics to manage large data, such as dimension
reduction using matrix factorisation and feature selection. Choose the right method for the
right question in mixOmics (Chapter 4) provides an overview of the methods available in
mixOmics and the types of biological questions these methods can answer.
Part II: mixOmics under the hood provides a deeper understanding of the statistical
concepts underlying the methods presented in Part III. Projection to Latent Structures
(PLS) (Chapter 5) illustrates the different types of algorithms used to solve Principal
Component Analysis. We detail in particular the iterative PLS algorithm that projects data
onto latent structures (components) for matrix decomposition and dimension reduction, as
this algorithm forms the basis of most of our methods. Visualisation for data integration
(Chapter 6) showcases the variety of graphical outputs offered in mixOmics to complement
each method. Performance assessment in supervised analyses (Chapter 7) describes the
techniques employed to evaluate the results of the analyses.
Part III: mixOmics in action provides detailed case studies that apply each method
in mixOmics to answer pertinent biological questions, complete with example R code and
insightful plots. We begin with mixOmics: get started (Chapter 8) to guide the novice analyst
in using the R platform for data analysis. Each subsequent chapter is dedicated to one method
implemented in mixOmics. In Principal Component Analysis (PCA) (Chapter 9) and PLS -
Discriminant Analysis (PLS-DA) (Chapter 12), we introduce different multivariate methods
for single omics analysis. The N -integration framework is introduced in PLS (Chapter 10)
and Canonical Correlation Analysis (Chapter 11) for two omics, and N −data integration
(DIABLO, Chapter 13) for multi-omics integration. P −data integration (MINT, Chapter 14)
introduces our latest developments for P -integration to combine independent omics studies.
Each of these chapters is organised as follows:
• Aim of the method,
• Research question framed biologically and statistically,
• Principles of the method,
• Input arguments and key outputs,
• Introduction of the case study,
• Quick start R command lines,
• Further options to go deeper into the analysis,
• Frequently Asked Questions,
• Technical methodological details in each Appendix.
2 https://www.bioconductor.org/packages/release/bioc/html/mixOmics.html
3 https://github.com/mixOmicsTeam/
Preface xix
Additional resources related to this book
In addition to the R package, the mixOmics project includes a website with extensive
tutorials in http://www.mixOmics.org. The R code of each chapter is also available on
the website. Our readers can also register for our newsletter mailing list, and be part of
the mixOmics community on GitHub and via our discussion forum https://mixomics-
users.discourse.group/.
Authors
Dr Kim-Anh Lê Cao develops novel methods, software and tools to interpret big biological
data and answer research questions efficiently. She is committed to statistical education to
instill best analytical practice and has taught numerous statistical workshops for biologists
and leads collaborative projects in medicine, fundamental biology or microbiology disciplines.
Dr Kim-Anh Lê Cao has a mathematical engineering background and graduated with a PhD
in Statistics from the Université de Toulouse, France. She is currently an Associate Professor
in Statistical Genomics at the University of Melbourne. In 2019, Kim-Anh received the
Australian Academy of Science’s Moran Medal for her contributions to Applied Statistics in
multidisciplinary collaborations. She has contributed to a leadership program for women
in STEMM, including the international Homeward Bound which culminated in a trip to
Antarctica, and Superstars of STEM from Science Technology Australia.
Zoe Welham completed a BSc in molecular biology and during this time developed an
interest in the analysis of big data. She completed a Master of Bioinformatics with a focus
on the statistical integration of different omics data in bowel cancer. She is currently a PhD
candidate at the Kolling Institute in Sydney where she is furthering her research into bowel
cancer with a focus on integrating microbiome data with other omics to characterise early
bowel polyps. Her research interests include bioinformatics and biostatistics for many areas
of biology and making that information accessible to the general public.
xxi
Part I
Modern biology and

multivariate analysis
1
Multi-omics and biological systems
Technological advances such as next-generation sequencing and mass spectrometry generate a

wealth of diverse biological information, allowing for the monitoring of thousands of variables,
or dimensions, that describe a given sample or individual, hence the term ‘high dimensional
data’. Multi-omics variables represent molecules from different functional levels: for example,
transcriptomics for the study of transcripts, proteomics for proteins, and metabolomics
for metabolites. However, their complex nature requires an integrative, multidisciplinary
approach to analysis that is not yet fully established.
Historically, the scientific community has adopted a reductionist approach to data analysis by
characterising a very small number of genes or proteins in one experiment to assess specific
hypotheses. A holistic approach allows for a deeper understanding of biological systems by
adding two new facets to analysis (Figure 1.1): Firstly, by integrating data from different omic
functional levels, we move from clarifying a linear process (e.g. the dysregulation of one or
two genes) towards understanding the development, health, and disease of an ever-changing,
dynamic, hierarchical system. Secondly, by adopting a hypotheses-free, data-driven approach,
we can build integrated and coherent models to address novel, systems-level hypotheses that
can be further validated through more traditional hypotheses.
1.1 Statistical approaches for reductionist or holistic analyses
Compared to a traditional reductionist analysis, multivariate multi-omics analysis drastically

differs in its viewpoint and aims. We briefly introduce three types of analysis to illustrate
this point:
A univariate analysis is a fundamentally reductionist, hypothesis-driven approach that is
related to inferential statistics (introduced in Section 2.4). A hypothesis test is conducted
on one variable (e.g. gene expression or protein abundance) independently from the other
variables. Univariate methods make inferences about the population and measure the certainty
of this inference through test statistics and p-values. Linear models, t-tests, F -tests, or
non-parametric tests fit into a univariate analysis framework. Although interactions between
variables are not considered in univariate analyses, when one variable is manipulated in a
controlled experiment, we can often attribute the result to that particular variable. In omics
studies, where multiple variables are monitored simultaneously, it is difficult to determine
which variables influence the biology of interest.
A bivariate analysis considers two variables simultaneously, for example, to assess the
association between the expression levels of two genes via correlation or linear regression.
Such an analysis is often supported by visualisation through scatterplots but can quickly
DOI: 10.1201/9781003026860-1 3
4 Multi-omics and biological systems
RNA CS
MI
IP TO
SCR
AN
TR
PROTEIN CS
MI
EO
OT
PR
METABOLITE ICS
OM
B OL
TA
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MICROBE ICS
OM
EN
AG
MET
Conventional molecular
biology Single-omics Multi-omics
(Reductionist) (Hypothesis free) (Holistic)
H yp ot g
h e s is g e n e r a t i n
FIGURE 1.1: From reductionism to holism. Until recently, only a few molecules of a
given omics type were analysed and related to other omics. The advent of high-throughput
biology has ushered in an era of hypothesis-free approaches within a single type of omics
data, and across multiple omics from the same set of samples. A holistic approach is now
required to understand the different omic functional layers in a biological system and posit
novel hypotheses that can be further validated with a traditional reductionist approach.
We have omitted DNA as this data type needs to be handled differently in mixOmics, see
Section 4.2.3.
become cumbersome when dealing with thousands of variables that are considered in a
pairwise manner.
A multivariate analysis examines more than two variables simultaneously and potentially
thousands at a time. In omics studies, this approach can lead to computational issues and
inaccurate results, especially when the number of samples is much smaller than the number of
variables. Several computational and statistical techniques have been revisited or developed
for high-dimensional data. This book focuses on multivariate analyses and extends this to
include the integration of multi-omics data sets.
1.2 Multi-omics and multivariate analyses

The aim of omics data integration is to identify associations and patterns amongst different
variables and across different omics collected from a sample. Provided appropriate data
analysis is conducted, the integration of multiple data sources may also consolidate our
confidence in the results when consensus is observed from different experiments.
Shifting the analysis paradigm 5
1.2.1 More than a ‘scale up’ of univariate analyses
The fundamental difference between multivariate and univariate analysis lies in the scope of
the results obtained. Multivariate analysis can unravel groups of variables that share similar
patterns in expression across different phenotypes, thus complementing each other to describe
an outcome. A univariate analysis may declare the same variables as non-significant, as a
variable’s ability to explain the phenotype may be subtle and can be masked by individual
variation, or confounders (Saccenti et al., 2014). However, with sufficiently powered data,
univariate and multivariate methods are complementary and can help make sense of the
data. For example, several multivariate and exploratory methods presented in this book can
suggest promising candidate variables that can be further validated through experiments,
reductionist approaches, and inferential statistics.
1.2.2 More than a fishing expedition
Multivariate analyses, which examine up to thousands of variables simultaneously, are

often considered to be ‘fishing expeditions’. This somewhat pejorative term refers to either
conducting analyses without first specifying a testable hypothesis based on prior research,
or, conducting several different analyses on the same data to ‘fish’ for a significant result
regardless of its domain relevance. Indeed, examining a large number of variables can lead
to statistically significant results purely by chance.
However, the integration of multi-omics data, with an appropriate experimental design set
in an exploratory, rather than predictive approach, offers a tremendous opportunity for
discovering associations between omics molecules (whether genes, transcripts, proteins, or
metabolites), in normal, temporal or spatial changes, or in disease states. For example,
one of our studies identified pathways that were never previously identified as relevant to
ontogeny during the first week of human life (Lee et al., 2019). Multi-omics data integration
has deepened our understanding of gene regulatory networks by including information
from related molecules prior to validation of gene associations with a functional approach
(Gligorijević and Pržulj, 2015) and has also efficiently improved functional annotations to
proteins instead of using expensive and time-consuming experimental techniques (Ma et al.,
2013). Multi-omics can also more easily characterise the relatively small number of genes
associated with a particular disease by integrating multiple sources of information (Žitnik
et al., 2013). Finally, it has further developed precision medicine by integrating patient-
and disease-specific information with the aim to improve prognosis and clinical outcomes
(Ritchie et al., 2015).
1.3 Shifting the analysis paradigm
Despite the potential advantages of high-dimensional data, we should keep in mind that
quantity does not equal quality. Multivariate data integration is not straightforward: the
analyses cannot be reduced to a mere concatenation of variables of different types, or by
overlapping information between single data sets, as we illustrate in Figure 1.2. As such, we
must shift our traditional view of analysing data.
Biological experimentation often employs univariate statistics to answer clear hypotheses
Factorisation
≈ ×
Matrix
prior
Bayesian
prior Inferred
ICS posterior
M
TO
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IP prior
AN
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MIC
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Network
based
ICS
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Multiple
steps
Single-omics Correlation/
analysis overlap
FIGURE 1.2: Types of methods for data integration. Methods for multi-omics data
integration are still in active development, and can be broadly categorised into matrix
factorisation techniques (the focus of this book), Bayesian, network-based, and multiple-
step approaches. The latter deviates from data integration as it considers each data set
individually before combining the results.
about the potential causal effect of a given molecule of interest. In high-dimensional data
sets, this reductionist approach may not hold due to the sheer amount of molecules that are
monitored, and their interactions that might be of biological interest. Therefore, exploratory,
data-driven approaches are needed to extract information from noise and generate new
hypotheses and knowledge. However, the lack of a clear, causal-driven hypothesis presents a
challenging new paradigm in statistical analyses.
In univariate hypothesis testing, we report p-values to determine the significance of a
statistical test conducted on a single variable. In a multivariate setting, however, a p-
value assesses the statistical significance of a result while taking into account all variables
simultaneously. In such analyses, permutation-based tests are common to assess how far
from random a result is when the data are reshuffled, but other inference-based methods
are currently being developed in the field of multivariate analysis (Wang and Xu, 2021). In
mixOmics we do not offer such tests, but related methods propose permutation approaches
to choose the parameters in the method (see Section 10.7.5).
1.4 Challenges with high-throughput data

There are multiple challenges associated with managing large amounts of biological data,
pertaining to specific types of data as well as statistical analysis. To make reliable, valid,
and meaningful interpretations, these challenges must be considered, ideally before data
collection.
Challenges with high-throughput data 7
1.4.1 Overfitting
Multivariate omics analysis assesses many molecules that individually, or in combination, can
explain the biological outcome of interest. However, these associations may be spurious, as the
large number of features can often be combined in different ways to explain the outcome well,
despite having no biological relevance. Overfitting occurs when a statistical model captures
the noise along with the underlying pattern in the data: if we apply the same statistical
model fitted on a high-dimensional data set to a similar but external study, we might obtain
different results.1 The problem of overfitting is a well-known issue in high-throughput biology
(Hawkins, 2004). We can assess the amount of overfit using cross-validation or subsampling
of the data, as described in Chapter 7.
1.4.2 Multi-collinearity and ill-posed problems
As the number of variables increase, the number of pairwise correlations also increases. Multi-
collinearity poses a problem in most statistical analyses as these variables bring redundant
and noisy information that decreases the precision of the statistical model. Correlations
in high-throughput data sets are often spurious, especially when the number of biological
samples, or individuals N , is small compared to the number of variables P 2 . The ‘small N
large P ’ problem is defined as ill-posed, as standard statistical inference methods assume N
is much greater than P to generalise the results to the population the sample was drawn
from. Ill-posed problems also lead to inaccurate computations.
1.4.3 Zero values and missing values
Data sets may contain a large number of zeros, depending on the type of omics studied
and the platform that is used. This is particularly the case for microbiome, proteomics, and
metabolomics data: a large number of zeros results in zero-inflated (skewed) data, which
can impair methods that assume a normal distribution of the data. Structural zeros, or true
zeros, reflect a true absence of the variable in the biological environment while sampling
zeros, or false zeros, may not reflect reality due to experimental error, technological reasons,
or an insufficient sample size (Blasco-Moreno et al., 2019). The challenge is whether to
consider these zeros as a true zero or missing (coded as NA in R).
Methods that can handle missing values often assume they are ‘missing at random’, i.e. miss-
ingness is not related to a specific sample, individual, molecule, or type of omics platform.
Some methods can estimate missing values, as we present in Appendix 9.A.
1 Statistical models that overfit have low bias and high variance, meaning that they tend to be complex to
fit the training data well, but do not predict well on test data (more details about the bias-variance tradeoff
can be found in Friedman et al. (2001) Chapter 2).
2 In our context, N can also refer to the number of cells in single cell assays, as we briefly mention in
Section 14.6.
1.5 Challenges with multi-omics integration
Examining data holistically may lead to better biological understanding, but integrating
multiple omics data sets is not a trivial task and raises another series of challenges.
1.5.1 Data heterogeneity
Different omics rely on different laboratory techniques and data extraction platforms, resulting
in data sets of different formats, complexity, dimensionalities, information content, and scale,
and may be processed using different bioinformatics tools. Therefore, data heterogeneity
arises from biological and technical reasons and is the main analytical challenge to overcome.
1.5.2 Data size
Integrating multiple omics results in a drastic increase in the number of variables. A filtering
step is often applied to remove irrelevant and noisy variables (see Section 8.1). However, the
number of variables P still remains extremely large compared to the number of samples N ,
which raises computational as well as analytical issues.
1.5.3 Platforms
The data integration field is constantly evolving due to ever-advancing technologies with new
platforms and protocols, each containing inherent technical biases and analytical challenges.
It is crucial that data analysts swiftly adapt their analysis framework to keep apace with
these omics-era demands. For example, single cell techniques are rapidly advancing, as are
new protocols for their multi-omics analysis.
1.5.4 Expectations for analysis
The field of data integration has no set definition. Data integration can be managed
biologically, bioinformatically, statistically, or at the interpretation steps (i.e. by overlapping
biological interpretation once the statistical results are obtained). Therefore, the expectations
for data integration are diverse; from exploration, and from a low to high-level understanding
of the different omics data types. Despite recent advances in single cell sequencing, current
technologies are still limited in their ability to parse omics interactions at precise functional
levels. Thus, our expectations for data integration are limited, not only by the statistical
methods but also by the technologies available to us.
Summary 9
1.5.5 Variety of analytical frameworks
Integrative techniques fully suited to multi-omics biological data are still in development and
continue to expand3 . Different types of techniques can be considered and broadly categorised
into (Huang et al. (2017), Figure 1.2):
• Matrix factorisation techniques, where large data sets are decomposed into smaller
sub-matrices to summarise information. These techniques use algebra and analysis to
optimise specific statistical criteria and integrate different levels of information. Methods
in mixOmics fit into this category and will be detailed in Chapter 3 and subsequent
chapters,
• Bayesian methods, which use assumptions of prior distributions for each omics type to
find correlations between data layers and infer posterior distributions,
• Network-based approaches, which use visual and symbolic representations of biological
systems, with nodes representing molecules and edges as correlations between molecules,
if they exist. Network-based methods are mostly applied for detecting significant genes
within pathways, discovering sub-clusters, or finding co-expression network modules,
• Multiple-step approaches that first analyse each single omics data set individually
before combining the results based on their overlap (e.g. at the gene level of a molecular
signature) or correlation. This type of approach technically deviates from data integration
but is commonly used.
1.6 Summary
Modern biological data are high dimensional; they include up to thousands of molecular
entities (e.g. genes, proteins, or epigenetic markers) per sample. Integrating these rich
data sets can potentially uncover the hierarchical and holistic mechanisms that govern
biological pathways. While classical, reductionist, univariate methods ignore these molecular
interactions, multivariate, integrative methods offer a promising alternative to obtain a
more complete picture of a biological system. Thus, univariate and multivariate methods
are different approaches with very little overlap in results but have the advantage of
complementarity.
The advent of high-throughput technology has revealed a complex world of multi-omics
molecular systems that can be unraveled with appropriate integration methods. However,
multivariate methods able to manage high-dimensional and multi-omics data are yet to
be fully developed. The methods presented in this book mitigate some of these challenges
and will help to reveal patterns in omics data, thus forging new insights and directions for
understanding biological systems as a whole.
3 A comprehensive list of multi-omics methods and software is available at https://github.com/mikelove/
awesome-multi-omics.
2
The cycle of analysis
The Problem, Plan, Data, Analysis, Conclusion (PPDAC) cycle is a useful framework for
answering an experimental question effectively (Figure 2.1). The mixOmics project emphasises
crafting a well-defined biological question (Chapter 4), as this guides data acquisition
and preparation (Chapter 8), as well as choosing appropriate multivariate techniques for
analysis (Chapter 4). Although this book is focused on analysis and interpretation, careful
consideration of each step will maximise a successful analytical outcome.
PROBLEM
CONCLUSIONS
PLAN
ANALYSIS
DATA
FIGURE 2.1: PPDAC. The Problem, Plan, Data, Analysis, Conclusion cycle proposed
by MacKay and Oldford (2000) will guide our multivariate analysis process.
2.1 The Problem guides the analysis
Multivariate analysis is appropriate for large data sets where the biological question en-
compasses a broad domain, rather than parsing the action of a single or small number of
variables. Thus, we often require a hypothesis-free investigation based on a data-driven
approach. However, this does not imply that multivariate analysis is a fishing expedition
with no underlying biological question. The experimental design, driven by a well-formulated
biological question and the choice of statistical method, will ensure a successful analysis
(Shmueli, 2010). Chapter 4 lists several types of biological questions that can be answered
with multivariate and integrative methods.
DOI: 10.1201/9781003026860-2 11
12 The cycle of analysis
2.2 Plan in advance
To consult the statistician after an experiment is finished is often merely to

ask him to conduct a post mortem examination. He can perhaps say what the
experiment died of.
– Sir Ronald Fisher, Statistician, Presidential Address to the First Indian Statistical
Congress, 1938
Thoughtful experimental design is essential to ensure reproducible and replicable results as

much as possible (Nichols et al., 2021). Once the biological question is formulated, care must
be taken in choosing appropriate omics technologies, statistical methods, and a sufficient
sample size to parse biological variation from individual differences.
2.2.1 What affects statistical power?
The underlying biological question will narrow the choice of appropriate omics technology
(see Chapter 4), keeping in mind that each technology has its own artefacts and generates
noise that may mask the effect of interest. The type of organism under study will also affect
the amount of biological variation we expect to uncover. Similarly, the nature of the effect of
interest, whether subtle, or strong, will also impact the amount of variation we can extract
from the data. Taken together, these issues will affect the sample size needed to detect the
effect of interest (as discussed in Saccenti and Timmerman (2016) and Lenth (2001)).
2.2.2 Sample size
Statistical power analyses are mostly valid for inferential and univariate tests but difficult to
estimate when the method considered for statistical analysis does not assume any specific
data distribution. As such, methods in mixOmics that are exploratory in nature do not fit
into a classical power analysis framework. This limitation is a double-edged sword. On the
one hand, any exploration on any sample size can be conducted to understand the amount
of variation present in the data, and whether this variation coincides with the biological
variation of interest. Such an approach can be useful for a pilot study to choose an omics
technology, for example. On the other hand, a small sample size will limit any follow-up
analysis: as the number of variables becomes very large, a small sample size is likely to lead
to overfitting if the analysis goes beyond exploration (discussed further in Section 1.4).
Appropriate sample sizes can be calculated if pilot data are available. However, if the aim
Plan in advance 13
TABLE 2.1: Example of a confounding factor in a poorly designed experiment.

The number of samples with respect to the variable of interest (treatment A or B) and the
covariate (sex) highlights a confounding factor (zeros outside the diagonal).
A B
F 10 0
M 0 10
TABLE 2.2: Example where the number of samples with respect to the variable
of interest (treatment) and the covariate (sex) are balanced across treatments.
A B
F 5 5
M 5 5
is to use multivariate methods, the calculation will rely on an empirical approach using
permutation tests1 .
2.2.3 Identify covariates and confounders
Covariates are observed variables that affect the outcome of interest but may not be of
primary interest. For example, consider an experiment examining the effect of weight (primary
variable of interest) on gene expression. Subject sex, age, ethnicity, or medication intake are
all examples of covariates, that if assessed as having an effect on the data, must be taken
into account in the analysis.
Confounding occurs when a covariate is intimately linked with the primary variable of
interest. A typical example is when all females receive treatment A and all males treatment
B (Table 2.1). In this case, we are not able to differentiate whether gene expression is affected
by treatment, or sex, or both. Confounders can be avoided during the experimental planning
stage by ensuring that both treatments are evenly assigned across sex (Table 2.2).
2.2.4 Identify batch effects
Batch effects refer to variation introduced by factors that are unrelated to the biological
variable of interest. These factors can be technical (e.g. day of experiment, technician,
sequencing run), computational (bioinformatics methods) or biological (birth dam, animal
facility), as described in Figure 2.2 and Table 2.3. The nature of the variation is defined as
systematic across batches, however, depending on the omics technology, this may not be the
case. For example, specific genes or micro-organisms might be more affected by one type of
batch compared to others (Wang and Lê Cao, 2019).
Batch effects can be mitigated with an appropriate experimental design so that their variation
does not overwhelm the biological variation. When strong batch effects cannot be avoided,
methods exist to correct or to account for batch effects (see Section 8.1).
1 Permutation tests build a sampling distribution based on the existing data by resampling them randomly.
TABLE 2.3: Example of a batch factor. Number of samples with respect to the
variable of interest (treatment) and the batch (sequencing run). Such a table gives a better
understanding of the experimental design when the batch information is recorded.
A B
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FIGURE 2.2: Potential sources of batch effects. Examples of factors that may affect
the quality of microbiome data, from Wang and Lê Cao (2019).
2.3 Data cleaning and pre-processing
Prior to analysis, the data sets should be normalised and pre-filtered using appropriate
techniques specific to the type of omics technology platform used. We briefly describe these
steps and further discuss relevant techniques in Chapter 8.
2.3.1 Normalisation
High-throughput biological data often results in samples with different sequencing depths
(RNA-sequencing) or overall expression levels (microarrays) due to technological platform
artefacts rather than true biological differences. Thus, data must be normalised for samples
to be comparable to one another. Moreover, highly-expressed genes may take up a large part
of sequenced reads in an experiment, with fewer reads remaining for other genes, leading to
a possibly incorrect conclusion that the latter genes have log expression levels (Robinson and
Oshlack, 2010). Many normalisation techniques have been proposed that are platform-specific
and constantly evolve with the latest technological, computational, and statistical advances.
Analysis: Choose the right approach 15
2.3.2 Filtering
Whilst most multivariate methods can manage a large number of molecules (several tens of
thousands), it is often beneficial to remove those that are not expressed, or vary very little,
across samples. These variables are usually not relevant for the biological problem, add noise
to the multivariate analyses, and may increase computational time during parameter tuning.
2.3.3 Missing values
In biological terms, missing values refer to data that are not measured (see Section 1.4).
However, in practice, the definition of missing is often unclear. For example, a value might
be missing because it is not relevant biologically or because it did not pass the detection
threshold in a mass spectrometry experiment. Therefore, the data analyst must make
the choice of setting the missing value to a numerical 0 (i.e. the ‘absence’ is biologically
relevant), or ‘NA’ (i.e. technically missing), keeping in mind that these decisions will affect
the statistical analysis.
2.4 Analysis: Choose the right approach
Data analysis can be descriptive, exploratory, inferential, or include modelling and prediction.
A well-framed biological question will clarify which type of analysis is better suited to address
a biological problem.
2.4.1 Descriptive statistics
Descriptive statistics precede any other type of analysis. They help to anticipate future
analytical challenges and obtain a basic understanding of the data. Descriptive statistics
solely describe the properties or characteristics of the data without making any assumptions
that the data were sampled from a population. In descriptive analysis, we use summary
statistics or visualisations to either obtain an initial description of the data, or investigate
a particular aspect of the data. Univariate summary statistics can include calculating the
sample mean and standard deviation, or graphing boxplots of a single variable. Bivariate
summary statistics can include the calculation of a correlation coefficient, or a scatterplot
representing the expression values of two genes. When dealing with a large number of
variables, such summaries can become cumbersome and difficult to interpret, which may be
overcome by using exploratory methods.
2.4.2 Exploratory statistics
Exploratory statistics aim to summarise and provide insights into the data before proceeding
with the analysis. This step does not lay any emphasis on an underlying biological hypothesis
and may not require a specific data distribution. Typically, such analyses can be conducted
to better understand the major sources of variation in the data by using dimension reduction
methods such as Principal Component Analysis (Jolliffe, 2005). More sophisticated integrative
methods can also be applied to highlight the correlation structure between variables from
different data sets, for example with Canonical Correlation Analysis (Vinod, 1976), or
Projection to Latent Structures, also called Partial Least Squares (Wold, 1966).
2.4.3 Inferential statistics
Inferential statistics allow us to infer from a sample taken from a population. They go
together with univariate statistics. If we assume that a sample is representative of the
population it was drawn from, then the conclusions reached from the statistical test should
generalise to the whole population. In univariate methods, we conduct hypothesis testing,
calculate test statistics and compare them to a theoretical statistical distribution in order to
infer that our conclusions indeed could be true for the whole population.
In the omics era, statistical inference is difficult to achieve and is often unreliable due to
an insufficient sample size. Inferential statistics can be achieved with multivariate methods,
but only when the number of samples, or individuals, is much larger than the number of
variables, and when the multivariate methods fit into an inferential statistical framework.
2.4.4 Univariate or multivariate modelling?
Another approach in data analysis is to fit a statistical model, i.e. a mathematical and
often simplified formula, to the data. The ultimate aim of modelling is to make predictions
based on the then-fitted model to a new set of data, when applicable. Here is an example of
an equation for a simple linear regression that models a dependent variable (or outcome
variable, e.g. a phenotype) with an explanatory variable (also referred to as an independent
variable, or predictor, e.g. the expression levels of a gene):
phenotype = intercept + a ∗ gene expression (2.1)
In model (2.1), a is a coefficient representing the slope of the line fitted if we were to plot
the gene expression levels along the x-axis and the phenotype along the y-axis (Figure 2.3).
Therefore, we assume the relationship between the dependent variable phenotype and the
independent variable gene expression is linear. Using an ordinary least squares approach
(OLS2 ), we can estimate the intercept or offset value (here estimated to −58.95), and the
slope a of this linear relationship (here estimated to 6.71). We often refer to this model as a
univariate linear regression.
A multivariate linear regression model includes several explanatory variables or predictors.
For example in a transcriptomics experiment, if we had measured the expression levels of P
genes:
phenotype = intercept + a1 ∗ gene1 + a2 ∗ gene2 + · · · + aP ∗ geneP (2.2)
where the intercept and regression coefficients (a1 , a2 , . . . , aP ) can be estimated using the
2 OLS estimates the unknown parameters in a linear regression model (e.g. the slope (a) and the intercept)
by estimating values for these parameters that minimise the sum of the squares of the differences between
the observed values of the variable and those predicted by the linear equation.
Analysis: Choose the right approach 17
fitted line: –58.95 + 6.71 x gene expr value
240
prediction of phenotype
220
(unit of response)
200
phenotype
180
prediction of phenotype
160
140
new gene expr value new gene expr value

35 40 45
gene expression levels

(unit of expression)
FIGURE 2.3: Example of a linear regression fit between the expression levels of a
given gene (along the x-axis) and the phenotype value (along the y-axis) for 20 individuals.
The model from the linear regression equation is fitted to the data points and is represented
as a blue line using the Ordinary Least Squares method. Red full circles represent new gene
expression values from which we can predict the phenotype of two new individuals, based
on this fitted model.
OLS method. However, when the number of predictors P becomes much larger than the
number of samples, the algebra to estimate these parameters becomes complex and sometimes
numerically impossible to solve due to multi-collinearity (see Section 1.4). A scatterplot
representation of all possible pairs of variables is often impractical and not easy to interpret.
The methods available in mixOmics use similar types of models as Equation (2.2), but
instead of a separate OLS approach, we use Partial Least Square approaches to manage high
dimensional data, as described in detail in Chapter 5. We also use other techniques such as
ridge and lasso regularisation to manage the large number of variables (see Section 3.3).
2.4.5 Prediction
Predictive statistics refers to the process of applying a statistical model or data mining
algorithm to training data to make predictions about new test data, often in terms of a
phenotype or disease status of those new samples3 . In the univariate regression model in
Equation (2.1) that is fitted on training data, let us assume we also measured the expression
levels of that same gene but in another cohort of patients. We can then predict the phenotype
of those new patients based on that same equation. The new gene expression values for
the new samples are represented in full red circles in Figure 2.3, with their y-axis values
corresponding to the predicted phenotype.
3 In our methods we focus on point estimate, i.e. a single value estimate of a parameter rather than
interval estimate, i.e. a range of values where the parameter is expected to lie.
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and by CRC Press

© 2022 Taylor & Francis Group, LLC

CRC Press is an imprint of Taylor & Francis Group, LLC

ISBN: 9780367460945 (hbk)

For Zoe Welham:

I Modern biology and multivariate analysis 1

2 The cycle of analysis 11

2.5 Conclusion and start the cycle again . . . . . . . . . . . . . . . . . . . . . . 18

3 Key multivariate concepts and dimension reduction in mixOmics 19

4 Choose the right method for the right question in mixOmics 29

4.B Centered log ratio transformation . . . . . . . . . . . . . . . . . . . . . . . 41

II mixOmics under the hood 45

6 Visualisation for data integration 59

7 Performance assessment in multivariate analyses 79

7.4.1 Assessment of the performance . . . . . . . . . . . . . . . . . . . . 86

III mixOmics in action 93

9 Principal Component Analysis (PCA) 109

9.6 To go further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

10 Projection to Latent Structure (PLS) 137

10.C.1 In PLS1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

11 Canonical Correlation Analysis (CCA) 177

12 PLS-Discriminant Analysis (PLS-DA) 201

12.6.2 Multilevel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

13 N −data integration 233

14 P −data integration 261

14.3.3 Number of variables to select per component . . . . . . . . . . . . 265

Glossary of terms 283

Key publications 285

Context and scope

Modern high-throughput technologies generate information about thousands of biological

Who is this book for?

Overview of methods in mixOmics

IPCA* DIABLO* MINT*

Key methodological concepts in mixOmics

Concepts not covered

About this book

Part I: Modern biology and multivariate analysis introduces fundamental concepts

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