1. An Introduction to Multivariate Analysis
Lectures 14-15
Drs. Alan S.L. Leung and Kenneth M.Y. Leung

2. Multivariate analysis
An extension to univariate (with a single variable) and bivariate (with two variables) analysis
Dealing with a number of samples and species/environmental variables simultaneously

3. Multivariate Data Set Data usually in a form of data matrix…..
Morphological measurement of organisms (e.g. length)
Physiological measurement of organisms (e.g. blood pressure)
Physiochemical measurement of the environment (e.g. air temperature)
Species abundance
Species richness etc……
Data usually in a form of data matrix…..

7. Similarity (S) between samples
Ranged from 0 to 100 % or 0 to 1
S = 100% if two samples are totally similar (i.e. the entries in two samples are identical)
S = 0 if two samples are totally dissimilar (i.e. the two samples has no species in common)

8. Bray-Curtis coefficient (Bray & Curtis, 1957)
First developed in terrestrial ecology
Where,
yij represented the abundance of species i in sample j,
yik represented the abundance of species i in sample k, and
n represented the total number of samples.

9. Please calculate the Bray-Curtis Similarity between samples:
where, yij represented the abundance of species i in sample j, yik represented the
abundance of species i in sample k, and n represented the total number of samples.
Please calculate the Bray-Curtis Similarity between samples:
X2 and X3
X3 and Y1

10. } = 84 } = 38 SX2 X3 = 100 { 1 - SX3 Y1 = 100 { 1 - 3+0+0+2+8
} = 84
SX3 Y1 = 100 { 1 -
} = 38

11. Species similarity matrix

12. Transformation Two distinct roles:
To validate statistical assumptions for parametric analysis (e.g. variance heterogeneity in ANOVA)
To weight the contributions of common and rare species in non-parametric multivariate analysis

13. Why Transforming the data?
To weight the contributions of common and rare species
Transformed and untransformed data can give different results on the computation of dissimilarities between samples
Affect the final outcome (solution) of nMDS

14. Choice of transformation in multivariate analysis
Intermediate abundance species
Square-root
Fourth-root / Log (1+y)
Presence/Absence
Degree of severity
Rare species
Not commonly used

15. Species similarity matrix – Fourth-root transformed
Some patterns can be seen, but…

16. Multivariate Techniques
The most widely used multivariate techniques included:
Cluster Analysis
Ordination
E.g. Multiple discriminant analysis

17. Cluster Analysis
Put samples (sites, species, or environmental variables) into groups based on their similarity.
Samples within the same group are more similar to each other than samples in different groups

18. Dendrogram
Samples
Statistical Software: PRIMER 5 for Windows

19. Ordination Graphical presentation technique
Ordination map (usually two or three-dimensional)
The relatively distances among points in the ordination map represent the similarity among samples (say species composition)

20. Two Types of Ordination Techniques
Indirect gradient analysis
Only includes biological data
- Species abundance by samples matrix
Environmental data can be correlated with the ordination axes subsequently
Direct gradient analysis
Includes both environmental and biological data

21. Non-metric Multi-dimensional Scaling (nMDS)
Indirect gradient analysis
Including:
Principle Component Analysis (PCA)
Correspondence Analysis (CA)
Detrended Correspondence Analysis (DCA)
Non-metric Multi-dimensional Scaling (nMDS)
Principle Component Analysis (PCA)
Direct gradient analysis
Including:
Redundancy Analysis (RD)
Canonical Correspondence Analysis (CCA)
Detrended Canonical Correspondence Analysis (DCCA)
Non-metric Multi-dimensional Scaling (nMDS)

22. PCA Use original data matrix Best-fit curve
First Principle Component Axis (PC1)
Source: Clarke, K. R. & Warwick, R. M. (1994) Change in Marine Communities: an Approach to Statistical Analysis and Interpretation.
Plymouth Marine Laboratory, Plymouth: 144pp.

23. Rotation
Second principal component axis (PC2) – perpendicular to PC1 (i.e. uncorrelated / orthogonal)

24. Third principal component axis (PC3)
Theoretically, many more species can be added

26. The variances extracted by the PCs
Eigenvalues
PC Eigenvalues %Variation Cum.%Variation
Eigenvectors
(Coefficients in the linear combinations of variables
making up PC's)
Variable PC1 PC2 PC3 PC4 PC5 A B C D E
Species

28. Ecological data which can fulfill these assumptions are rare…..
PCA Assumptions
Linear relationships between variables
Normality of the variables
Ecological data which can fulfill these assumptions are rare…..

29. Multidimensional Scaling
A technique for analyzing multivariate data
Visualization of the relationships between samples to facilitate interpretation in a low dimensional space
There are two types of MDS:
Metric
Non-metric

30. Metric MDS: Non-metric MDS (nMDS)
Assume the input data is either interval or ratio during measurement
Quantitative
Non-metric MDS (nMDS)
The data should be in the form of rank
Quantitative and/or Qualitative

31. Major Advantages of nMDS
Ordination is based on the ranked similarities/dissimilarities between pairs of samples
Ordinal data could be used
The actual values of data are not being used in the ordination, few (no?) assumptions on the nature and quality of the data
e.g. 1 = very low; 2 = low; 3 = mid; 4 = high; 5 = very high

32. Bray-Curtis similarity
Modified from Clarke & Warwick, 1994

33. An Ecological Example Spatial and temporal variability in benthic
macroinvertebrate communities in Hong
Kong Streams

34. Macroinvertebrate communities

35. Macroinvertebrate communities

36. Macroinvertebrate communities

37. Study Sites (HK map)

38. Spatial

39. Temporal

40. Macroinvertebrate Sampling & Identification

41. Nested analysis of variance (ANOVA)
Statistical Analysis
Nested analysis of variance (ANOVA)
Regions (Random, orthogonal)
Sites (Random, nested within Regions)
Sections (Random, nested within Sites)
Spatial
Years (Random, orthogonal)
Seasons (Fixed, orthogonal)
Days (Random, nested within Years and Seasons)
Temporal
Interactions between them

42. Non-parametric multivariate analysis
Statistical Analysis
Non-parametric multivariate analysis
Non-metric multidimensional scaling (NMDS)
Analysis of similarities (ANOSIM)
Display the stream community data in ordination diagrams intended to reveal underlying patterns in the community structure
Compare the community structure among spaces and times

43. Species Abundance vs. Samples

44. Fourth – root transformed

45. A measure of goodness-of-fit

50. Multivariate analysis - Temporal
Years [All samples in all sites; Each Region; Each Site; Each Section in each Site]
Seasons (all years & each year) [All samples in all sites; Each Region; Each Site; Each Section in each Site]
Dates within Seasons in each year

54. Day 1 A 1 –
Day 1 B 2
n.s.
Day 1 C 3
Day 2 A 4
0.281
Day 2 B 5
Day 2 C 6
Day 3 A 7
0.531
Day 3 B 8
0.500
0.698
Day 3 C 9
Day 4 A 10
0.771
0.729
0.813
0.844
0.833
0.792
Day 4 B 11
0.688
0.667
0.521
0.333
0.469
0.615
Day 4 C 12
0.406
0.448
0.417
ANOSIM
R statistics:
R = 1 only if all replicates within sites are more similar to each other than any replicates from different sites
R is approximately zero if the similarities between and within sites are the same on average
Results of one-way ANOSIM between the Lam Tsuen site sampling sections within the dry season in The pairs that are significantly different (at 5% significant level) are shown with the R statistics values.

55. Day 1 A 1 –
Day 1 B 2
n.s.
Day 1 C 3
Day 2 A 4
0.281
Day 2 B 5
Day 2 C 6
Day 3 A 7
0.531
Day 3 B 8
0.500
0.698
Day 3 C 9
Day 4 A 10
0.771
0.729
0.813
0.844
0.833
0.792
Day 4 B 11
0.688
0.667
0.521
0.333
0.469
0.615
Day 4 C 12
0.406
0.448
0.417
Day 2
Day 3
ANOSIM
R statistics:
R = 1 only if all replicates within sites are more similar to each other than any replicates from different sites
R is approximately zero if the similarities between and within sites are the same on average

56. ANOSIM
LT 1997 Dry Season
The number of pairs of sections significantly different (percentage)
Average R statistics of significantly different pairs
The same section between different days
9/18 (50%)
0.578
Among all sections within the same day
0/12 (0%) ––
Among all sections between different days
20/36 (56%)
0.602

57. ANOSIM
LT 1997 Wet Season
The number of pairs of sections significantly different (percentage)
Average R statistics of significantly different pairs
The same section between different days
15/18 (83%)
0.674
Among all sections within the same day
2/12 (17%)
0.662
Among all sections between different days
29/36 (81%)
0.647

58. Implications The macroinvertebrate community structures are,
on average:
more similar within the same region 
more similar within the same site
    …. and the patterns are more obvious in the dry seasons
Sites of the same region are more similar to each others
Samples of the same site are more similar to each others

59. Implications
There is no obvious pattern on the community structure between sections within a site
The community structures of the study sites are, in general, similar between years
Seasonality
The spatial scale “Sections” is not an important factor
However, in some sites, variation between years could be high
There are STRONG seasonality patterns. However, within
season variation (days) is also noticeable

60. Implications
Patterns in the community structure are uncovered. Regions, Sites and Seasons are important factors to our understanding of the stream communities in Hong Kong
Although there is small scale variability (within site), large scale variability (among sites and between regions) is playing a more important role in the macroinvertebrate communities