Contents

Preface 

i 
Why Multivariate? 

1 
The 10Step 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 PCORD 

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 
Nonzero 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 
Chisquare Distance 

57 
Cityblock 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 (BrayCurtis) 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 
Twoway cluster 

131 
26. TWINSPAN 

135 
How to run it 

136 
Step 7d: Testing among groups 

137 
27. MRPP: Multiresponse permutation procedure 

137 
28. PerMANOVA: Distancebased 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 PCORD 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 