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

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Presentation on theme: "An Introduction to Multivariate Analysis Drs. Alan S.L. Leung and Kenneth M.Y. Leung Lectures 14-15."— Presentation transcript:

1 An Introduction to Multivariate 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…..

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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, y ij represented the abundance of species i in sample j, y ik represented the abundance of species i in sample k, and n represented the total number of samples.

9 where, y ij represented the abundance of species i in sample j, y ik 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 S X2 X3 = 100 { } = 84 S X3 Y1 = 100 { } = 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 Square-root Fourth-root / Log (1+y) Presence/Absence Degree of severity Intermediate abundance species 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 Statistical Software: PRIMER 5 for Windows Samples

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

22 PCA Use original data matrix First Principle Component Axis (PC1) Best-fit curve 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

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26 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 The variances extracted by the PCs Species

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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

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37 Study Sites (HK map)

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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) Years (Random, orthogonal) Seasons (Fixed, orthogonal) Days (Random, nested within Years and Seasons) Spatial 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

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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

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54 Day 1 A 1– Day 1 B 2n.s.– Day 1 C 3n.s. – Day 2 A 4n.s.0.281n.s.– Day 2 B 5n.s. – Day 2 C 6n.s. – Day 3 A 7n.s.0.531n.s. – Day 3 B 8n.s n.s. – Day 3 C 9n.s.0.500n.s. – Day 4 A – Day 4 B 11n.s n.s n.s.– Day 4 C n.s n.s. – 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. 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

55 Day 1 A 1– Day 1 B 2n.s.– Day 1 C 3n.s. – Day 2 A 4n.s.0.281n.s.– Day 2 B 5n.s. – Day 2 C 6n.s. – Day 3 A 7n.s.0.531n.s. – Day 3 B 8n.s n.s. – Day 3 C 9n.s.0.500n.s. – Day 4 A – Day 4 B 11n.s n.s n.s.– Day 4 C n.s n.s. – 123Day 2Day 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 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 LT 1997 Dry Season

57 LT 1997 Wet Season ANOSIM 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 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 Implications 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


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