1. Multivariate Statistical Methods
Measuring and Testing
Multivariate Distances
by Jen-pei Liu, PhD
Division of Biometry, Department of Agronomy,
National Taiwan University
and
Division of Biostatistics and Bioinformatics
National Health Research Institutes
2019/2/4
Copyright by Jen-pei Liu, PhD

2. Measuring and Testing Multivariate Distances
Introduction
Distances between Individual Observations
Distances between Populations and Samples
Distances Based on Proportions
Presence-absence Data
The Mantel Randomization Test
Summary
2019/2/4
Copyright by Jen-pei Liu, PhD

3. Copyright by Jen-pei Liu, PhD
Introduction
Multivariate Problems in terms of distances
Between single observations
Between samples of observations
Between populations of observations
2019/2/4
Copyright by Jen-pei Liu, PhD

4. Copyright by Jen-pei Liu, PhD
Introduction
Mandible measurements of canine groups
Dogs, wolves, jackals, cuons, and dingos
How far one of these groups from the other groups
Two groups are close if two animals have similar mandible measurements
Distance measures for representation of similar measurements
2019/2/4
Copyright by Jen-pei Liu, PhD

5. Copyright by Jen-pei Liu, PhD
Introduction
Different types of measurements
Example of 16 colonies of a butterfly species
Two sets of distances
environmental
genetic
Relationship between these two sets of distances
2019/2/4
Copyright by Jen-pei Liu, PhD

6. Distances between Individual Observations
N objects with p variables X1,…,Xp
Object i: Xi1, Xi2, …, Xip
Object j: Xj1, Xj2, …, Xjp
Distance measures between object i
and object j
Graphical presentation of two or three variables
2019/2/4
Copyright by Jen-pei Liu, PhD

7. Distances between Individual Observations
2019/2/4
Copyright by Jen-pei Liu, PhD

8. Copyright by Jen-pei Liu, PhD
2019/2/4
Copyright by Jen-pei Liu, PhD

9. Copyright by Jen-pei Liu, PhD
2019/2/4
Copyright by Jen-pei Liu, PhD

10. Distances between Individual Observations
Example: Dogs and Related Species
Standardized Variables
Group X X X X X5 X6
Modern Dogs
Golden Jackals
Chinese Wolf
Indian Wolf
Cuon
Dingo
Prehistoric Dog
2019/2/4
Copyright by Jen-pei Liu, PhD

11. Distances between Individual Observations
Euclidean Distances between Seven Canine Group
Modern Golden Chinese Indian Prehistoric
dog jackal wolf wolf Cuon Dingo dog
Modern dog
Golden jackal
Chinese wolf
Indian wolf
Cuon
Dingo
Prehistoric dog
2019/2/4
Copyright by Jen-pei Liu, PhD

12. Distances between Populations and Samples
Information about populations
Means
Variances
Covariances
Measures
Between populations
Penrose distance
Mahalanobis distance
Between populations and samples
2019/2/4
Copyright by Jen-pei Liu, PhD

13. Distances between Populations and Samples
Penrose Distance
Variables: X1,…,Xp
The ith population means:1i,…, pi
The ith population variances: v1i,…,vpi
2019/2/4
Copyright by Jen-pei Liu, PhD

14. Distances between Populations and Samples
The Mahalanobis distance:
takes correlation into consideration
2019/2/4
Copyright by Jen-pei Liu, PhD

15. Distances between Populations and Samples
The Mahalanobis distance between the samples and population
2019/2/4
Copyright by Jen-pei Liu, PhD

16. Distances between Populations and Samples
Example: Distances between Egyptian skulls
Penrose’s distance between sample 1 and sample 2
P12 = ( )2/(4x21.112)
+ ( )2/(4x23.486)
+ ( )2/(4x24.180)
+ ( )2/(4x10.154)
=0.023
2019/2/4
Copyright by Jen-pei Liu, PhD

17. Distances between Populations and Samples
2019/2/4
Copyright by Jen-pei Liu, PhD

18. Distances between Populations and Samples
The inverse of sample covariance matrix
2019/2/4
Copyright by Jen-pei Liu, PhD

19. Distances between Populations and Samples
The Mahalanobis distance between sample 1 and sample 2
D122 = ( )0.0483( )
+( )0.0011( )
+…+( )( )( )
+( )0.1041( )
=0.091
2019/2/4
Copyright by Jen-pei Liu, PhD

20. Distances between Populations and Samples
Penrose Distances
(1) (2) (3) (4) (5)
Early predynastic (1)
Late predynastic (2)
12-13th dynastic (3)
Ptolemaic (4)
Roman (5)
2019/2/4
Copyright by Jen-pei Liu, PhD

21. Distances between Populations and Samples
Mahalanobis Distances
(1) (2) (3) (4) (5)
Early predynastic (1)
Late predynastic (2)
12-13th dynastic (3)
Ptolemaic (4)
Roman (5)
2019/2/4
Copyright by Jen-pei Liu, PhD

22. Distances Based on Proportions
Animals of a certain species might be classified into K genetic classes
Class Colony Colony Difference
1 p1 q p1-q1
p q p2-q2
k pk qk pk-qk
2019/2/4
Copyright by Jen-pei Liu, PhD

23. Distances Based on Proportions
Distance Measures
2019/2/4
Copyright by Jen-pei Liu, PhD

24. Distances Based on Proportions
Distance Measures
2019/2/4
Copyright by Jen-pei Liu, PhD

25. Presence-absence Data
Presence and absence of two species at 10 locations
Site
Species
Species
2019/2/4
Copyright by Jen-pei Liu, PhD

26. Presence-absence Data
Species 2
Species 1 Present Absent Total
Present a b a+b
Absent c d c+d
Total a+c b+d n
2019/2/4
Copyright by Jen-pei Liu, PhD

27. Presence-absence Data
Distance Measures
Simple matching index: (a+d)/n
Ochiai index: a/[(a+b)(a+c)]1/2
Dice-Sorensen index: 2a/(2a+b+c)
Jaccard index: a/(a+b+c)
2019/2/4
Copyright by Jen-pei Liu, PhD

28. The Mantel Randomization Test
Detection of space and time clustering of disease – whether cases of a disease that occur close in space also tend to be close in time
Two 4x4 distance matrices of 4 objects
Symmetric matrices
2019/2/4
Copyright by Jen-pei Liu, PhD

29. The Mantel Randomization Test
2019/2/4
Copyright by Jen-pei Liu, PhD

30. The Mantel Randomization Test
2019/2/4
Copyright by Jen-pei Liu, PhD

31. The Mantel Randomization Test
Mantel Test
Whether the elements in M and E show some significant correlation
Matching m12 with e12, m13 with e13, etc.
2019/2/4
Copyright by Jen-pei Liu, PhD

32. The Mantel Randomization Test
M stay as it is
Random order chosen for E
Order of 3,2,4,1
2019/2/4
Copyright by Jen-pei Liu, PhD

33. The Mantel Randomization Test
Time Distances
(1) (2) (3) (4) (5)
B.C. (1)
2999 – 2000 B.C. (2)
B.C. (3)
(4)
A.D (5)
2019/2/4
Copyright by Jen-pei Liu, PhD

34. The Mantel Randomization Test
A total of 5!=120 ways to re-order the five samples
A total elements in the randomization distribution in the correlation
The observed correlation is 0.954
There are only two correlations greater than or equal to 0.954
The p-value (1-sided) = 2/120 = 0.017
2019/2/4
Copyright by Jen-pei Liu, PhD

35. Copyright by Jen-pei Liu, PhD
Summary
Euclidean distances
Penrose distance
Mahalanobis distance
Distance measures for proportions
Mantel test for similarity between distance matrices
2019/2/4
Copyright by Jen-pei Liu, PhD