Multivariate Analysis for Ecologists: Step-by-Step

Multivariate Analysis for Ecologists: Step-by-Step is a book by Dr. JeriLynn E Peck that answers your questions: What are multivariate data? How should I prepare my data? Which analysis tools should I use? What do they do? Do they have weaknesses? How do I interpret the output?

This richly illustrated second edition, published by MjM Software Design 2016, will help you answer those questions by providing a step-by-step process for approaching your multivariate community data analysis project. Simple explanations and diagrams explain the tools, recipes walk you though the process using the PC-ORD software, and guidance is given on everything from ensuring sampling independence to when to rotate an ordination. Each technique has sections on:

  • what it's good for

  • what it actually does

  • what it means

  • what you need in order to run it

  • what you get in the output

  • what you should know about its strengths and weaknesses

  • how to run it in PC-ORD v.7

  • when you should use it




PeckMefford MAEcover_400px.png

Cover graphic: The analysis of community data is a process involving many steps, an example of only a few of which are portrayed here from top left to bottom right.

  • Samples: The multivariate community data analysis pathway begins with field observations from different samples.

  • Matrix: Based on these observations, a square distance matrix is computed to assess the dissimilarity of sample units to one another.

  • Flowchart: A decision is made to select the most appropriate analysis tool to address the stated analysis objective: in this case, the group-testing tool MRPP.

  • Analysis: MRPP calculates within-group distances for comparison to random assignments to groups.


Preface i
Why Multivariate? 1
The 10-Step Process 5
Step 1. Getting yourself ready 8
1. Embracing statistics 8
2. Articulating goals 16
Analysis objectives 17
Step 2. Getting your data ready 19
3. Meeting matrices 19
4. Starting in PC-ORD 22
5. Inputting and screening your data 27
Entering your data 27
Importing your data 27
From a spreadsheet 27
From a database 31
Screening your data 32
Screening with Summary 32
Screening with Species Lists 33
Step 3. Structuring your data 34
6. Sampling sufficiently 34
Is N enough? 34
Species Area Curves (SPA) 34
7. Structuring your matrices 36
Restructuring with Modify Data 36
Step 4. Exploring and preparing your data 39
8. What have you actually measured? 39
Sparsity (zeros) 40
Checking sparsity with Summary 41
Checking sparsity with Profile 41
Reducing sparsity with Delete Columns 41
Non-zero values 41
Response comparability 43
Standardizing with General Relativization 44
Standardizing with Relativization by Maximum 45
9. How variable are your data? 46
Heterogeneity 46
Checking sparsity with SUMMARY 46
Checking sparsity with PROFILE 48
Reducing sparsity with Functional Diversity 48
Outliers 51
Checking for outliers with Summary 51
Checking for outliers with Outlier Analysis 51
Checking for outliers with Profile 51
Checking for outliers with Boxplots 52
Response distributions 52
Checking for normality with Distributions 52
Checking for normality with Boxplots 53
Step 5. Selecting the tools 54
10. Distance measures 54
Euclidean Distance 55
Chi-square Distance 57
City-block Distance (e.g., Sørensen) 58
11. Model form 60
Hypothesizing relationships 60
Exploring relationships with Scatterplot 60
12. Analysis tools 62
13. What is ordination? 64
Step 6. Modifying your data 68
14. Meeting parametric assumptions 68
Transforming to a Power 68
Transforming to the Logarithmic 69
Transforming to Arcsinesquareroot 69
To Multiply or add a constant 69
15. Reweighting responses 70
Checking influence with Dominance Curves 70
Adjusting influence with Relativizations 70
Step 7a: Guiding pattern (guided ordination) 73
16. WA: Weighted averaging 73
How to run it 75
17. Polar (Bray-Curtis) ordination 77
How to run it 79
18. CCA: Canonical correspondence analysis 81
How to run it 88
19. RDA: Redundancy analysis 91
How to run it 95
20. FSO: Fuzzy set ordination 97
How to run it 100
21. Interpreting guided ordinations 102
Step 7b: Seeking pattern (free ordination) 104
22. PCA: Principal components analysis 104
How to run it 109
23. NMS: Nonmetric multidimensional scaling 112
How to run it 117
Choosing dimensionality using the stress test 118
Verifying the final solution 119
NMS Scores 121
24. Interpreting free ordinations 124
Step 7c: Looking for groups (classification) 126
25. Cluster analysis 126
How to run it 130
Two-way cluster 131
26. TWINSPAN 135
How to run it 136
Step 7d: Testing among groups 137
27. MRPP: Multi-response permutation procedure 137
28. PerMANOVA: Distance-based MANOVA 141
How to run it 144
29. SumF 147
How to run it 148
30. ISA: Indicator species analysis 150
How to run it 153
Blocked indicator species analysis 153
Step 7e: Assessing associations 154
31. Mantel test 154
How to run it 156
Partial Mantel test 156
32. FCA: Fourth corner analysis 157
How to run it 160
Step 8: Confirming your results 161
Step 9: Using interpretive tools 163
33. Finding your story 163
Interpreting ordination diagrams 163
Simple scatterplot 164
Overlay Main Matrix 165
Overlay Second Matrix 167
Joint plots 168
Convex hulls and centroids 169
% of variance 170
Successional vectors 170
Ordered Main Matrix 173
Step 10: Presenting your story 175
34. Communicating your message 175
Presenting Summaries 175
Presenting Graphics 175
Rotating and reflecting ordinations 177
Drawing outlines on ordered tables 179
Appendix A: Other PC-ORD Tools 180
Appendix B: Matrix Algebra Unplugged 181
Appendix C: How to Report 183
Appendix D: Further Reading 186
Appendix E: Available Supplements 187
Appendix F: Acronyms & Symbols 188
Index 189



Is this book for you?

    This book is a guide for individuals who are making their first attempts at applying multivariate analysis tools to community datasets. This means that although the analysis tools are described in what I hope is a fairly comprehensive manner, the organization of the book, instructions for how to run the analyses, and the examples are all geared toward the beginner. You will find this book helpful if any of the following apply:

  • you are unfamiliar with multivariate statistics in general
  • you are unfamiliar with community datasets in particular
  • you are unfamiliar with the PC-ORD software
  • you are not satisfied with the explanations given in the multivariate references you already have


What’s in this book

    This book is the required companion book for the workshops/webinars I teach on this subject, used in lieu of a handout.  The structure thus parallels the ten step analysis process I propose.  Step 1 is intended to get you ready to conduct the analysis—refreshing your statistical background and clarifying your analysis objectives.  Step 2 is getting your dataset ready for analysis using the PC-ORD software.  Step 3 is to structure your matrices to address your analysis objectives.  Step 4 focuses on exploring your dataset to make sure you understand its properties, while Step 5 is about selecting the analysis tools that will best achieve your objectives.  Step 6 involves manipulating the dataset, when needed, to meet those objectives and fit those tools.

    Each chapter in Step 7 focuses on a different analysis tool, allowing you to direct your attention to only those you wish to apply.  For each procedure, I provide a conceptual description of what you can do with it and when you might use it, what it actually does, what the results mean, how to run it, what you get in the output, what you should know about its strengths and weaknesses, how to run it in PC-ORD version 7, and then a little on how to interpret it.   There are certainly more ways to use the tools discussed in the book than the few presented here, but I hope the descriptions provided will at least get you thinking about the application of these tools.

    Finally, Step 8 briefly reinforces the need to seek confirmation of your results before turning to interpretative tools in Step 9 and communication in Step 10.  Supplemental information, including instructional materials, is provided in the Appendices and online.