An Introduction to Multivariate Analysis

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

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 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. 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 SX2 X3= 100{1 - 3+0+0+2+8 11+0+0+14+58 }= 38 SX3 Y1= 100{1 - 0+6+8+2+30 14+6+8+10+36

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

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. 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 Best-fit curve • Use original data matrix 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. Second principal component axis (PC2) – perpendicular to PC1 (i.e. uncorrelated / orthogonal) Rotation

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 1 3.39 67.8 67.8 2 0.92 18.4 86.1 3 0.56 11.2 97.4 4 0.11 2.1 99.5 5 0.02 0.5 100.0 Eigenvectors (Coefficients in the linear combinations of variables making up PC's) Variable PC1 PC2 PC3 PC4 PC5 A 0.269 0.823 0.485 -0.088 -0.092 B 0.521 -0.264 -0.018 -0.143 -0.799 C 0.515 -0.226 0.082 -0.635 0.523 D -0.499 0.227 -0.292 -0.739 -0.261 E -0.377 -0.388 0.820 -0.150 -0.109 Species

28. 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: • 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. 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. 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