Email: [email protected] URL: michele.scardi.name
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Transcript of Email: [email protected] URL: michele.scardi.name
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Permutation tests in multivariate analysis
Michele Scardi
Department of Biology‘Tor Vergata’ University
Rome, Italy
Email: [email protected]: http://www.michele.scardi.name
The Fourth BIOSAFENET Seminar – January 12-15, 2009
International Centre for Genetic Engineering and Biosafety (ICGEB)Ca’ Tron di Roncade, Italy
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Ecology and agriculture
A few reasons why studyingGMO effects on NTO is relevant
to ecologists
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Agriculture and biodiveristy
Intermediatedisturbancehypothesis!
Modified from EEA Report 2/2006
Intensive farmland
High Nature Value farmland
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CAP and community ecology
• In the past, Common Agricultural Policy mainly promoted the expansion of agricultural production.
• A complete re-arrangement of the funding scheme now puts the environment at the center of farming policy.
• Preserving biodiversity (and therefore monitoring agroecosystems) is one of the CAP main goals.
• Biodiversity can only be monitored through community ecology studies.
• Similar targets have been set in other fields (e.g. Water Framework Directive).
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EU and GMOs
• In December 2008, the Council of the EU adopted the following conclusions on GMOs:
“GMOs, … , give rise to discussion and questions, within the scientific community and society at large regarding their possible impact on health, environment and ecosystems.”
• Therefore, there is a growing need for ecological research on agroecosystems.
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Permutation tests
Because bugs don’t know they’re supposed to be normally
distributed…
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A basic (univariate) permutation test
• Sample A: 28, 32, 45 (mA=35.00)
• Sample B: 22, 25, 29 (mB=25.33)
• H0: no difference between means
• Six data can be arranged in two groups of three in 20 different combinations:
2066
720
)!36(!3
!6
)!(!
!
rnr
n
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Sample A Sample BPermutatio
n a1 a2 a3 mA b1 b2 b3 mB mA-mB
1 28 32 45 35.00 25 22 29 25.33 9.672 28 32 25 28.33 45 22 29 32.00 -3.673 28 32 22 27.33 25 45 29 33.00 -5.674 28 32 29 29.67 25 22 29 25.33 4.335 28 25 45 32.67 32 22 29 27.67 5.006 28 22 45 31.67 25 32 29 28.67 3.007 28 29 45 34.00 25 22 32 26.33 7.678 25 32 45 34.00 28 22 29 26.33 7.679 22 32 45 33.00 25 28 29 27.33 5.6710 29 32 45 35.33 25 22 28 25.00 10.3311 25 22 45 30.67 28 32 29 29.67 1.0012 25 32 22 26.33 28 45 29 34.00 -7.6713 28 25 22 25.00 25 28 32 28.33 -3.3314 25 29 45 33.00 28 32 29 29.67 3.3315 25 32 29 28.67 28 22 45 31.67 -3.0016 28 25 29 27.33 25 32 45 34.00 -6.6717 22 29 45 32.00 28 32 29 29.67 2.3318 22 32 29 27.67 28 22 45 31.67 -4.0019 28 22 29 26.33 25 32 45 34.00 -7.6720 25 22 29 25.33 28 32 45 35.00 -9.67
A basic (univariate) permutation test
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• Difference between Sample A and Sample B means is mA-mB=9.67
• By permuting our data we obtained an empirical distribution of the values of this difference acting as if H0: equal means were true
• How large is this original difference relative to the empirical distribution?
• Is it so large that H0 is probably false?
A basic (univariate) permutation test
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mA-mB
9.67 -3.67 -5.67 4.33 5.00 3.00 7.67 7.67 5.6710.33 1.00 -7.67 -3.33 3.33 -3.00 -6.67 2.33 -4.00 -7.67 -9.67
mA-mB
In 2+1 casesout of 19+1:
|mA-mB| 9.67
P(H0: mA=mB) =
= (2+1)/(19+1) =
= 0.15 > 0.05
We cannot rejectH0: mA=mB
A basic (univariate) permutation testP-value cannot be
less than:
nspermutatio of n.
1
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A basic (univariate) permutation test
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What permutation tests do ecologist use?
• ANOSIM, MRPP, NPMANOVA, Mantel Test and Indicator Species Analysis are quite popular
• Other permutation have been developed to suit specific needs (e.g. testing significance of relationships between envrionmental variables and species in CCA)
• If you have to solve a particular ecological problem, chances are you can develop your own permutation test
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0
2
4
6
8
10
12
14
16
0 2 4 6 8 10 12 14 16 18 20 22 24
species 1
spec
ies
2
Treatment A Treatment B Treatment C Treatment D
A toy multivariate problem (4 treatments and a community with only 2 species)…
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…and a toy permutation test
• Differences between samples are measured as Euclidean distances
• The test statistics is the the average distance between group centroids (i.e. average responses to treatments)
• If the test statistics is large enough with respect to those obtained by data permutation, then differences between groups are significant.
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Species 1
Spe
cies
2
Distances between average responses to treatments (between centroids)
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Mantel test
A B C D EA 0.0 1.2 2.6 1.8 3.2B 1.2 0.0 3.1 0.5 2.7C 2.6 3.1 0.0 1.1 4.2D 1.8 0.5 1.1 0.0 3.4E 3.2 2.7 4.2 3.4 0.0
A B C D EA 0.00 0.29 0.56 0.45 0.49B 0.01 0.00 0.48 0.06 0.12C 0.21 0.17 0.00 0.27 0.59D 0.07 0.04 0.16 0.00 0.02E 0.45 0.34 0.78 0.21 0.00
X matrixe.g. geographical
distances
Y matrixe.g. species dissimilarity
Problem: are X and Y (in)dependent of each other?
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Mantel statistics
1
1 12
)()(
12
)(
1 n
i
n
ij y
ij
x
ij
s
yy
s
xx
nnR
1
1 1
n
i
n
ijijij yxZ
Range: [0,)
Range: [-1,1]i.e. correlation between xij and yij
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Mantel statistics
A B C D E
A 0.0 1.2 2.6 1.8 3.2
B 1.2 0.0 3.1 0.5 2.7
C 2.6 3.1 0.0 1.1 4.2
D 1.8 0.5 1.1 0.0 3.4
E 3.2 2.7 4.2 3.4 0.0
A B C D E
A 0.00 0.29 0.56 0.45 0.49
B 0.29 0.00 0.48 0.06 0.12
C 0.56 0.48 0.00 0.27 0.59
D 0.45 0.06 0.27 0.00 0.02
E 0.49 0.12 0.59 0.02 0.00
1
1 1
n
i
n
ijijij yxZ
Z=8.867
X
Y
xijyijyijxij
0.0680.023.4
2.4780.594.2
0.2970.271.1
0.3240.122.7
0.0300.060.5
1.4880.483.1
1.5680.493.2
0.8100.451.8
1.4560.562.6
0.3480.291.2
xijyijyijxij
0.0680.023.4
2.4780.594.2
0.2970.271.1
0.3240.122.7
0.0300.060.5
1.4880.483.1
1.5680.493.2
0.8100.451.8
1.4560.562.6
0.3480.291.2
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Mantel statistics
A B C D E
A 0.0 1.2 2.6 1.8 3.2
B 1.2 0.0 3.1 0.5 2.7
C 2.6 3.1 0.0 1.1 4.2
D 1.8 0.5 1.1 0.0 3.4
E 3.2 2.7 4.2 3.4 0.0
A B C D E
A 0.00 0.29 0.56 0.45 0.49
B 0.29 0.00 0.48 0.06 0.12
C 0.56 0.48 0.00 0.27 0.59
D 0.45 0.06 0.27 0.00 0.02
E 0.49 0.12 0.59 0.02 0.00
X
Y
1
1 1
n
i
n
ijijij yxZ
Z=8.867
xijyijyijxij
0.0680.023.4
2.4780.594.2
0.2970.271.1
0.3240.122.7
0.0300.060.5
1.4880.483.1
1.5680.493.2
0.8100.451.8
1.4560.562.6
0.3480.291.2
xijyijyijxij
0.0680.023.4
2.4780.594.2
0.2970.271.1
0.3240.122.7
0.0300.060.5
1.4880.483.1
1.5680.493.2
0.8100.451.8
1.4560.562.6
0.3480.291.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 1 2 3 4 5
X
Y
NB: Mantel statistics is maximum when large xij are multiplied by large yij, i.e. when the two matrices have the same structure
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Mantel testA B C D E
A 0.0 1.2 2.6 1.8 3.2
B 1.2 0.0 3.1 0.5 2.7
C 2.6 3.1 0.0 1.1 4.2
D 1.8 0.5 1.1 0.0 3.4
E 3.2 2.7 4.2 3.4 0.0
A B C D E
A 0.00 0.29 0.56 0.45 0.49
B 0.29 0.00 0.48 0.06 0.12
C 0.56 0.48 0.00 0.27 0.59
D 0.45 0.06 0.27 0.00 0.02
E 0.49 0.12 0.59 0.02 0.00
X
Y
Z=8.867
A B C D E
A 0.0 0.5 1.1 1.8 3.4
B 0.5 0.0 3.1 1.2 2.7
C 1.1 3.1 0.0 2.6 4.2
D 1.8 1.2 2.6 0.0 3.2
E 3.4 2.7 4.2 3.2 0.0
ZP1=8.365
Z = 8.867
ZP1 = 8.365
ZP2 = 7.834
ZP3 = 6.897
ZP4 = 8.531
ZP5= 8.885
…
ZPn = 7.852
actual value
permutations
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Z=8.867
3%
Z
Fre
quen
cy
Mantel test
Z = 8.867
ZP1 = 8.365
ZP2 = 7.834
ZP3 = 6.897
ZP4 = 8.531
ZP5= 8.885
…
ZPn = 7.852
actual value
permutations
H0: X and Y are independent
permutations
actual value
p(Z)=0.03
Therefore, we reject H0
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Mantel test
• The test can be performed on any couple of distance or similarity matrices, for instance:– two different groups of organisms– environmental data and species composition– etc.
• In case the matrices are large enough (D>30) the test statistics is approximately distributed like Student’s t
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W
E
1 0.5 0.9 0.1 0.6 1.0 0.3 0.9 0.7 0.8 0.8 0.8 0.6 0.9 0.0 0.6 0.4 0.6 0.2 0.2 0.7
0.5 1 0.3 0.7 0.3 0.2 0.4 0.6 0.9 0.1 0.0 0.8 0.5 0.9 0.1 0.8 0.1 0.8 0.5 0.9 0.7
0.9 0.3 1 0.0 0.4 0.7 0.1 0.9 0.5 0.0 0.4 0.4 0.9 0.5 0.6 0.5 0.0 0.1 0.4 0.3 0.5
0.1 0.7 0.0 1 0.7 1.0 0.4 0.3 0.1 0.8 0.1 0.5 0.2 0.5 0.7 0.9 0.3 0.6 0.9 0.7 0.5
0.6 0.3 0.4 0.6 1 0.7 0.6 0.2 0.8 0.3 0.9 0.2 0.9 0.1 0.7 0.6 0.9 0.6 0.6 0.2 0.6
1.0 0.2 0.7 1.0 0.7 1 0.0 0.6 0.3 0.0 0.2 0.8 0.5 0.2 0.1 0.5 0.7 0.6 0.8 0.7 0.6
0.3 0.4 0.1 0.4 0.6 0.0 1 0.6 0.3 0.2 0.4 0.6 0.9 0.5 0.0 0.1 0.7 0.3 0.5 0.6 0.5
0.9 0.6 0.9 0.3 0.2 0.6 0.6 1 0.4 0.6 0.6 0.2 0.4 0.6 0.4 0.5 0.4 0.3 0.6 0.2 0.1
0.7 0.9 0.5 0.1 0.8 0.3 0.3 0.4 1 0.7 0.6 0.7 0.5 0.0 0.7 0.9 0.3 0.8 0.1 0.5 0.8
0.8 0.1 0.0 0.8 0.3 0.0 0.2 0.6 0.7 1 0.9 0.8 0.7 0.2 0.5 0.2 0.6 0.8 0.6 0.6 0.6
0.8 0.0 0.4 0.1 0.9 0.2 0.4 0.6 0.6 0.9 1 0.3 0.3 0.3 0.0 0.8 0.2 0.4 0.8 0.1 0.8
0.8 0.8 0.4 0.5 0.2 0.8 0.6 0.2 0.7 0.8 0.3 1 0.9 0.1 0.2 1.0 0.6 0.6 0.4 0.6 1.0
0.6 0.5 0.9 0.2 0.9 0.5 0.9 0.4 0.5 0.7 0.3 0.9 1 0.6 0.9 0.7 0.0 0.9 0.9 0.8 0.9
0.9 0.9 0.5 0.5 0.1 0.2 0.5 0.6 0.0 0.2 0.3 0.1 0.6 1 0.8 0.6 0.5 0.1 0.1 0.5 0.6
0.0 0.1 0.6 0.7 0.7 0.1 0.0 0.4 0.7 0.5 0.0 0.2 0.9 0.8 1 0.5 0.4 0.2 0.7 0.2 0.3
0.6 0.8 0.5 0.9 0.6 0.5 0.1 0.5 0.9 0.2 0.8 1.0 0.7 0.6 0.5 1 0.3 0.9 0.1 0.8 0.6
0.4 0.1 0.0 0.3 0.9 0.7 0.7 0.4 0.3 0.6 0.2 0.6 0.0 0.5 0.4 0.3 1 0.5 0.1 0.9 0.3
0.6 0.8 0.1 0.6 0.6 0.6 0.3 0.3 0.8 0.8 0.4 0.6 0.9 0.1 0.2 0.9 0.5 1 0.6 0.7 0.2
0.2 0.5 0.4 0.9 0.6 0.8 0.5 0.6 0.1 0.6 0.8 0.4 0.9 0.1 0.7 0.1 0.1 0.6 1 0.1 0.2
0.2 0.9 0.3 0.7 0.2 0.7 0.6 0.2 0.5 0.6 0.1 0.6 0.8 0.5 0.2 0.8 0.9 0.7 0.1 1 0.5
0.7 0.7 0.5 0.5 0.6 0.6 0.5 0.1 0.8 0.6 0.8 1.0 0.9 0.6 0.3 0.6 0.3 0.2 0.2 0.5 1
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
W E
W
E
W E
W
E
Jaccard similarity Same basin? (1=yes, 0=no)
R=0.37
p<0.001
Years: 1969-1970Samples: 9 (W), 12 (E)Depth: 570-2550 m
H0: structure ofdeep copepod assemblages is independent of basin
W and E basins are separated by two
shallow sills
Mantel test
H0 was rejected, concluding that (according to common opinion)
Eastern and Western basins actually had different
deep copepod assemblages
But…More about this at the end
of the presentation!
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A few more facts about Mantel test
• Mantel test assumes that relationships between data in the two matrices are linear
• Therefore, it is sensitive to non-linearity (as well as to outliers)
• A rank-based Mantel test can be also performed, although it is not very popular
• Partial Mantel correlation (e.g. using 3 matrices) can be computed
• Mantel test can be performed on non-symmetrical matrices
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Measuring ecological resemblance• Similarity coefficients
– Range: [0,1]– S=1 when similarity between species lists is maximum
• Dissimilarity coefficients– D=1-S– Range: [0,1]– D=0 when similarity between species lists is maximum– Some of them are metric
• Distance coefficients– Range: [0,] or [0,Dmax]– D=0 when two species lists are identical (In some cases
proportional)– Most of them are metric
i.e. they can be used to arrange objects,
samples, etc. in an Euclidean space
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Measuring ecological resemblance
• Legendre & Legendre (1983,1998) list 25 similarity coefficients and 14 metric distances, but many others have been used.
• Different measures of ecological resemblance may lead to different results.
• Selection of an optimal way of measuring ecological resemblance is inherently subjective.
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A few examples…
cba
aS jk
Jaccard
cba
aS jk
2
2Sørensen
p
iikij
p
iikij
jk
xx
xx
S
1
1
),min(2Steinhaus(Bray-Curtis)
p
iikijjk xxD
1
2)(
p
iikijjk xxD
1
p
i ikij
ikij
ij xx
xxD
1
Euclidean
Manhattan
Canberra
A smart pick: asymmetric similarity coefficients ignore absence data (they rely upon the lowest level of information, i.e. presence)
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nst 6
x
1 2 3 4 5 6
1
2
3
4
5
6
7
8
9
10
11
12
0 1 4 0 0 0
0 0 4 10 0 0
0 1 5 1 16 29
0 0 0 0 0 1
0 0 4 14 0 0
0 1 0 7 0 0
0 0 4 6 15 18
0 0 0 0 8 7
0 0 0 12 8 9
15 8 0 0 12 20
17 19 0 7 0 0
0 1 0 9 0 0
nsp 12
D4j k
1
nsp
i
xi j x
i k
D4
0
13
53
84
67
86
13
0
48
77
72
97
53
48
0
61
62
87
84
77
61
0
95
118
67
72
62
95
0
27
86
97
87
118
27
0
S3j k
aj k
aj k b
j k cj k
S3
1
0.333
0
0.111
0.167
0.143
0.333
1
0.222
0.4
0.222
0.2
0
0.222
1
0.444
0.25
0.222
0.111
0.4
0.444
1
0.3
0.273
0.167
0.222
0.25
0.3
1
0.833
0.143
0.2
0.222
0.273
0.833
1
D5j k
1
nsp
i
xi j x
i k
xi j x
i k
D5
0
4.36
7
8.42
5.11
6.14
4.36
0
8.27
8.01
8.08
9.36
7
8.27
0
6.85
7.1
8.34
8.42
8.01
6.85
0
8.51
9.58
5.11
8.08
7.1
8.51
0
1.76
6.14
9.36
8.34
9.58
1.76
0
quantit
ativ
e
qualita
tive
Jaccard
Canberra
Manhattan
treatmentor impact
control
control
treatmentor impact
similaritieswithin groups
similaritiesbetween groups
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ANOSIM(ANalysis Of SIMilarities)
( 1) / 4b wr r
RN N
Problem: are differences between groups large enough as to say that groups are different from each other?
H0: mean rank of similarites between groups is equal to mean rank of similarities within groups, i.e. rb = rw
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sort
( 1) / 4b wr r
RN N
wr br
0.5
ANOSIM(ANalysis Of SIMilarities)
n=6 n=9
N=6
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0%
5%
10%
15%
20%
25%
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
n=6 n=9
( 1) / 4b wr r
RN N
n=6 n=9 n=6 n=9 n=6 n=9
R= 0.50
rw= 5.75rb=9.50
R= 0.20
rw= 7.08rb=8.61
R= 0.19
rw= 7.17rb=8.56
R= -0.26
rw= 9.17rb=7.22
...
0%
5%
10%
15%
20%
25%
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0%
20%
40%
60%
80%
100%
p(R)=0.10
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Final remarks about ANOSIM
• In case of a posteriori pairwise comparisons, don’t forget the Bonferroni correction:p(R) = p(R) * n. of pairwise comparisons
• Two-way analyses can also be performed
• The most particular feature of ANOSIM is that it takes into account ranks of similarities (not sensitive to outliers, but maybe too sensitive to small differences)
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OUTSIDE
INSIDE
Problem: is the fish assemblage compositionin a Marine Protected Area different from the
neighbouring sites?
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Other approaches
• ANOSIM is loosely related to ANOVA-like tests, but it can be easily integrated into explorative analyses based on ordination or clustering
• Other approaches are more similar to ANOVA and to ANOVA users’ way of thinking: MRPP and NPMANOVA
• Exactly like ANOVA, they support more complex experimental designs
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MRPP(MultiResponse Permutation Procedure)
• It is based on actual distances (usually Euclidean), not ranks
• The test statistic is Δ, a weighted average within group distance
• Significance of Δ is assessed by the empirical distribution of permuted Δs, but an approximation to Student’s t is also available (quite handy for very large data sets)
• There are different methods for weighting Δ according to group size
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MRPP• PRO: not using ranks, it can detect
differences in mean values as well as differences in data dispersion
• CON: biases due to distance metric are more pronounced than in ANOSIM
• Output– observed Δ– significance of observed Δ– expected Δ– a ratio between observed and expected Δ
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MRPP
T1 T2 T3 C1 C2 C3T1 0.0 8.2 5.4 17.4 16.1 15.1T2 8.2 0.0 5.7 15.9 14.2 12.4T3 5.4 5.7 0.0 15.3 13.9 11.9C1 17.4 16.1 15.1 0.0 5.6 6.9C2 15.9 14.2 12.4 5.6 0.0 5.8C3 15.3 13.9 11.9 6.9 5.8 0.0
T1 T2 T3 C1 C2 C3T1 8.2 5.4 17.4 16.1 15.1T2 5.7 15.9 14.2 12.4T3 15.3 13.9 11.9C1 5.6 6.9C2 5.8C3
T1 T2 T3 C1 C2 C3species 1 0 0 0 1 1 0species 2 0 0 0 0 1 1species 3 0 1 1 9 5 4species 4 1 0 1 0 0 0species 5 0 0 0 1 1 0species 6 8 12 9 0 1 2species 7 5 8 4 9 11 8species 8 0 0 1 4 6 2species 9 2 4 5 3 5 6
species 10 12 6 8 1 2 0
treatment control
Euclidean distances
distances within groups
distances between groups
=6.43*3/6+6.10*3/6=6.26
T=(8.2+5.4+5.7)/3=6.43
C=(5.6+6.9+5.8)/3=6.10
Weight = ni/sum(ni)
Expected = overall average distance (i.e. between and within groups) = 11.32
447.01expected
observed
R
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MRPP output------------------------------------------------- MRPP -- Multi-Response Permutation Procedures [Michele Scardi, 1999] -------------------------------------------------
Input data: 6 objects, 10 variables, 2 groupsWeighting option: C(I) = n(I)/sum(n(I))Distance measure: Euclidean
Group # 1, n = 3, avg(d) = 6.43 Group # 2, n = 3, avg(d) = 6.10
Test statistic, T = -2.95 Observed delta = 6.26 Expected delta = 11.32 Variance of delta = 2.94 Skewness of delta = -2.52 Within group agreement, R = 0.447 P-value of a <= delta = 0.022
R1 when D0i.e. R is large when
within group distance is small
groups are significantly more homegenous than expected (therefore, differences between groups are significant)
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Arpaia et al. (2007). Composition of Arthropod Species Assemblages inBt-expressing and Near Isogenic Eggplants in Experimental Fields.Environ. Entomol. 36(1): 213-227
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NPMANOVA
• Non-Parametric Multivariate ANalysis Of VAriance
• Test statistics
– SStotal = sum of squared distances between all observations and the overall centroid
– SSwithin = sum of squared differences between group observations and group centroid
– SSamong = SStotal-SSwithin
– Pseudo-F = ratio of SSamong to SSwithin
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NPMANOVA
Species 1
Spe
cies
2
SSamong
SSwithin
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NPMANOVA
• Test is based on permutation of a distance matrix and computation of new Pseudo-F
• Original Pseudo-F is then compared to the empirical distribution of permuted Pseudo-F values
• Traditional ANOVA output (Pseudo-F)
• One-way, two-ways and higher level ANOVA designs
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NPMANOVA
• It can be applied any time ANOSIM or MRPP can be used, and it seems more robust
• It allows higher level ANOVA designs, including designs with interactions
• Any distance coefficient can be used• Like ANOVA in multiple regression, it can be used
to evaluate model functioning: e.g. community structure=f(disturbance,time)
• Software available from the Author’s web site for complex analysis designs
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NPMANOVA
T1 T2 T3 C1 C2 C3species 1 0 0 0 1 1 0species 2 0 0 0 0 1 1species 3 0 1 1 9 5 4species 4 1 0 1 0 0 0species 5 0 0 0 1 1 0species 6 8 12 9 0 1 2species 7 5 8 4 9 11 8species 8 0 0 1 4 6 2species 9 2 4 5 3 5 6
species 10 12 6 8 1 2 0
treatment controlT1 T2 T3 C1 C2 C3
species 1 0 0 0 1 1 0species 2 0 0 0 0 1 1species 3 0 1 1 9 5 4species 4 1 0 1 0 0 0species 5 0 0 0 1 1 0species 6 8 12 9 0 1 2species 7 5 8 4 9 11 8species 8 0 0 1 4 6 2species 9 2 4 5 3 5 6
species 10 12 6 8 1 2 0
treatment control
Non-parametric Multivariate Analysis of Variance
Source df SS MS F P---------------------------------------------------Treatment 1 288.0 288.0 14.4 0.0981Residuals 4 80.0 20.0Total 5 368.0---------------------------------------------------
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NPMANOVA vs. MRPP
Species 1
Spe
cies
2
SSamong
SSwithin
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NPMANOVA vs. MRPP
Species 1
Spe
cies
2
expected
observed
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Indicator Species Analysis
• Indicator Species Analysis allows identifiying species that are significantly more frequent and/or abundant in a group of samples
• Each species is associated to a vector of Indicator Values (IVs), i.e. to an IV for each group of samples
• Significance of IVs is tested by permutation of the raw data matrix
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g
kkj
kjkj
x
xRA
1
kn
i k
ijkkj n
bRF
1
100 kjkjkj RFRAIV
Indicator Species AnalysisRelative abundance
of species j in group k
Average frequency of occurence of
species j in group k
The Indicator value (IV) is obtained by combining
relative abundances (RA) and average frequencies
of occurrence (RF)
Counts, biomass, etc.
Presence or absence
(0,1)
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Group: CTL IMP N: 43 20
n ID INDVAL p taxon----------------------------------------------------
1 EUPHAS 4 44 0.017 Euphasiacea
2 THYSAN 0 25 0.004 Thysanopoda aequalis
3 RESPES 26 1 0.046 Fish scales
4 RESCRO 0 20 0.009 Unid. Crustaceans
5 MISIDA 5 0 0.595 Misidacea
6 DECAPO 2 3 0.999 Decapoda
7 CEFALO 5 0 0.554 Cephalopoda
8 CHLORO 0 5 0.299 Chlorotocus crassicornis
9 CRANGO 2 0 0.999 Crangon sp
10 SARDIN 0 5 0.299 Sardina pilchardus
11 ROCINE 0 5 0.307 Rocinela sp
12 POLICH 2 0 0.999 Polychaeta
--------------------------------------------------------
Indicator Species AnalysisGut contents of
Merluccius merluccius Group: CTL IMP
N: 43 20
n ID INDVAL p taxon----------------------------------------------------
1 EUPHAS 4 44 0.017 Euphasiacea
2 THYSAN 0 25 0.004 Thysanopoda aequalis
3 RESPES 26 1 0.046 Fish scales
4 RESCRO 0 20 0.009 Unid. Crustaceans
5 MISIDA 5 0 0.595 Misidacea
6 DECAPO 2 3 0.999 Decapoda
7 CEFALO 5 0 0.554 Cephalopoda
8 CHLORO 0 5 0.299 Chlorotocus crassicornis
9 CRANGO 2 0 0.999 Crangon sp
10 SARDIN 0 5 0.299 Sardina pilchardus
11 ROCINE 0 5 0.307 Rocinela sp
12 POLICH 2 0 0.999 Polychaeta
--------------------------------------------------------
Fish don’t feed on polluted sea
bed!
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Arpaia et al. (2007). Composition of Arthropod Species Assemblages inBt-expressing and Near Isogenic Eggplants in Experimental Fields.Environ. Entomol. 36(1): 213-227
![Page 53: Email: mscardi@mclink.it URL: michele.scardi.name](https://reader030.fdocuments.in/reader030/viewer/2022012908/568149de550346895db70401/html5/thumbnails/53.jpg)
As for the deep copepods…Back to the Mantel test
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W
E
1 0.5 0.9 0.1 0.6 1.0 0.3 0.9 0.7 0.8 0.8 0.8 0.6 0.9 0.0 0.6 0.4 0.6 0.2 0.2 0.7
0.5 1 0.3 0.7 0.3 0.2 0.4 0.6 0.9 0.1 0.0 0.8 0.5 0.9 0.1 0.8 0.1 0.8 0.5 0.9 0.7
0.9 0.3 1 0.0 0.4 0.7 0.1 0.9 0.5 0.0 0.4 0.4 0.9 0.5 0.6 0.5 0.0 0.1 0.4 0.3 0.5
0.1 0.7 0.0 1 0.7 1.0 0.4 0.3 0.1 0.8 0.1 0.5 0.2 0.5 0.7 0.9 0.3 0.6 0.9 0.7 0.5
0.6 0.3 0.4 0.6 1 0.7 0.6 0.2 0.8 0.3 0.9 0.2 0.9 0.1 0.7 0.6 0.9 0.6 0.6 0.2 0.6
1.0 0.2 0.7 1.0 0.7 1 0.0 0.6 0.3 0.0 0.2 0.8 0.5 0.2 0.1 0.5 0.7 0.6 0.8 0.7 0.6
0.3 0.4 0.1 0.4 0.6 0.0 1 0.6 0.3 0.2 0.4 0.6 0.9 0.5 0.0 0.1 0.7 0.3 0.5 0.6 0.5
0.9 0.6 0.9 0.3 0.2 0.6 0.6 1 0.4 0.6 0.6 0.2 0.4 0.6 0.4 0.5 0.4 0.3 0.6 0.2 0.1
0.7 0.9 0.5 0.1 0.8 0.3 0.3 0.4 1 0.7 0.6 0.7 0.5 0.0 0.7 0.9 0.3 0.8 0.1 0.5 0.8
0.8 0.1 0.0 0.8 0.3 0.0 0.2 0.6 0.7 1 0.9 0.8 0.7 0.2 0.5 0.2 0.6 0.8 0.6 0.6 0.6
0.8 0.0 0.4 0.1 0.9 0.2 0.4 0.6 0.6 0.9 1 0.3 0.3 0.3 0.0 0.8 0.2 0.4 0.8 0.1 0.8
0.8 0.8 0.4 0.5 0.2 0.8 0.6 0.2 0.7 0.8 0.3 1 0.9 0.1 0.2 1.0 0.6 0.6 0.4 0.6 1.0
0.6 0.5 0.9 0.2 0.9 0.5 0.9 0.4 0.5 0.7 0.3 0.9 1 0.6 0.9 0.7 0.0 0.9 0.9 0.8 0.9
0.9 0.9 0.5 0.5 0.1 0.2 0.5 0.6 0.0 0.2 0.3 0.1 0.6 1 0.8 0.6 0.5 0.1 0.1 0.5 0.6
0.0 0.1 0.6 0.7 0.7 0.1 0.0 0.4 0.7 0.5 0.0 0.2 0.9 0.8 1 0.5 0.4 0.2 0.7 0.2 0.3
0.6 0.8 0.5 0.9 0.6 0.5 0.1 0.5 0.9 0.2 0.8 1.0 0.7 0.6 0.5 1 0.3 0.9 0.1 0.8 0.6
0.4 0.1 0.0 0.3 0.9 0.7 0.7 0.4 0.3 0.6 0.2 0.6 0.0 0.5 0.4 0.3 1 0.5 0.1 0.9 0.3
0.6 0.8 0.1 0.6 0.6 0.6 0.3 0.3 0.8 0.8 0.4 0.6 0.9 0.1 0.2 0.9 0.5 1 0.6 0.7 0.2
0.2 0.5 0.4 0.9 0.6 0.8 0.5 0.6 0.1 0.6 0.8 0.4 0.9 0.1 0.7 0.1 0.1 0.6 1 0.1 0.2
0.2 0.9 0.3 0.7 0.2 0.7 0.6 0.2 0.5 0.6 0.1 0.6 0.8 0.5 0.2 0.8 0.9 0.7 0.1 1 0.5
0.7 0.7 0.5 0.5 0.6 0.6 0.5 0.1 0.8 0.6 0.8 1.0 0.9 0.6 0.3 0.6 0.3 0.2 0.2 0.5 1
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
W E
W
E
W E
W
E
Jaccard similarity Same basin? (1=yes, 0=no)
R=0.37
p<0.001
Years: 1969-1970Samples: 9 (W), 12 (E)Depth: 570-2550 m
H0: structure ofdeep copepod assemblages is independent of basin
W and E basins are separated by two
shallow sills
Mantel test
H0 was rejected, concluding that (according to common opinion)
Eastern and Western basins actually had different
deep copepod assemblages
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But…
• Deep copepod density is very, very low• In many cases only a single specimen of a given
species was found in the whole sample• 45 out of 65 species were found in both basins,
and only 19 among them accounted for 96% of the specimens
• The number of samples was very small• When a very rare species was not found in a
sample it was actually absent in the sampling area or just too sparse to be always collected?
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Simulation
• Re-sampling was simulated 500 times, assuming that species found in a real sample with only 1 or 2 specimens had a 50% chance of being found again
• Mantel tests were performed after each simulated re-sampling
• H0 was rejected in only 73 out of 500 cases (14.6%)
• So, are the deep copepod assemblages from Western and Eastern Basins really different?
Happy ending!Recent studies confirmed that
there are no differences betweenW and E deep copepod
assemblages.
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Conclusions
Only two more slides!
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Permutation tests in 6 easy steps
1. Define a problem and a null hypothesis
2. Define an appropriate statistics
3. Compute that statistics for your data set
4. Permute many times your data in a way that is consistent with the null hypothesis
5. Obtain an empirical distribution of the test statistics
6. Compare the original value of the test statistics to the empirical distribution: is it large (or small) enough as to reject the null hypothesis?
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The bottom line• Permutation tests are very useful in ecological
studies, as they don’t rely upon some assumptions that are seldom met in ecological applications: e.g. (multi)normal distribution of data.
• However, some assumption are still to be met:– variances (dispersion) must be homogeneous– effects (in case of complex designs) must be additive– data (observations) must be independent
• You con design/adapt your own test, provided that is consistent with H0 (and other general assumptions)
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Thank you!More info about ecological data analysis(+Machine Learning and other related topics)?
Email: [email protected]: http://www.michele.scardi.name(or just google Michele Scardi)
PAST software package:
http://folk.uio.no/ohammer/past(or just google PAST)
Marti J. Anderson’s web site (NPMANOVA):
http://www.stat.auckland.ac.nz/~mja