Norman MacLeod

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Partial Least-Squares Analysis
Norman MacLeod Department of Palaeontology, The
Natural History Museum
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Partial Least-Squares Analysis
Consider the case in which it is desirable to
understand relations between two groups of data
Y1 a data matrix composed of n1 observations
and p1 variables
Y2 a data matrix composed of n2
observations and p2 variables
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Partial Least-Squares Analysis
In such instances the total covariance or
correlation matrix can be subdivided into blocks
R
Note Block R12 represents the structure of
covariances or correlations among the two blocks
of variables
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Partial Least-Squares Analysis
Basic Equation
R12 USV
where U matrix of eigenvectors ( major axis
multiple regression slopes) of Y1 on Y2 S
singular values (similar to eigenvalues) of R12 V
matrix of eigenvectors ( major axis multiple
regression slopes) of Y2 on Y1
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Partial Least-Squares Analysis
Applications
Partial least-squares (PLS) is a generalized
technique for comparing all types of relations
between datasets. The types of comparisons
possible are essentially unlimited and
applications of the method are only beginning to
be explored.
To date PLS has proven to be especially helpful
in

• Studies seeking to test correlations between
  morphological variation and variation in
  particular environmental, ecological, or
  life-history factors
• Studies seeking to document patterns of
  morphological integration

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Partial Least-Squares Analysis
Example 1
Cranial shape variation as a function of
preferred prey in plethodontid salamanders
Rohlf and Corti (2000)
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Partial Least-Squares Analysis
Example 1
Singular Values
Rohlf and Corti (2000)
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Partial Least-Squares Analysis
Example 1
Singular Values
Rohlf and Corti (2000)
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Partial Least-Squares Analysis
Example 1
Singular Values
Skull Shape
Prey Item
Rohlf and Corti (2000)
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Partial Least-Squares Analysis
Example 1
Singular Values
Skull Shape
Prey Item
Rohlf and Corti (2000)
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Partial Least-Squares Analysis
Example 1
Association Prey vs Shape
r 0.725
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Partial Least-Squares Analysis
Example 1
Association Prey vs Shape
Hymenoptera, Diptera
Coleoptera, Isopoda
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Partial Least-Squares Analysis
Example 2
Morphological Integration within the Talpid
Dentary
Barrow and MacLeod (2008)
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Partial Least-Squares Analysis
Example 2
Singular Values Coronoid vs. Condylar Processes
Barrow and MacLeod (2008)
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Partial Least-Squares Analysis
Example 2
Singular Values Coronoid vs. Condylar Processes
Barrow and MacLeod (2008)
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Partial Least-Squares Analysis
Example 2
Singular Values Coronoid vs. Condylar Processes
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Scalopus
Uropsilus
13
12
Euroscaptor
Desmana
11
Parascaptor
Galemys
10
9
Coronoid Process
Mongera
Urotrichus
8
Talpa
Neurotrichus
7
6
Condylura
Scapanus
5
r -0.409
Parascalopus
4
3
3
4
5
6
7
8
9
10
11
12
13
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Condylar Process
Barrow and MacLeod (2008)
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Partial Least-Squares Analysis
Example 2
Singular Values Coronoid vs. Condylar Processes
4.00,11.00
6.25,10.50
8.50,10.00
10.75,9.50
13.00,9.00
Barrow and MacLeod (2008)
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Partial Least-Squares Analysis
Example 2
Singular Values Coronoid vs. Condylar Processes
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Scalopus
Uropsilus
13
12
Euroscaptor
Desmana
11
Parascaptor
Galemys
10
9
Coronoid Process
Mongera
Urotrichus
8
Talpa
Neurotrichus
7
6
Condylura
Scapanus
5
r -0.409
Parascalopus
4
3
3
4
5
6
7
8
9
10
11
12
13
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Condylar Process
Barrow and MacLeod (2008)
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Partial Least-Squares Analysis
Norman MacLeod Department of Palaeontology, The
Natural History Museum