Multivariate Transformation

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Multivariate Transformation
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Multivariate Transformations

• Started in statistics of psychology and
  sociology.
• Also called multivariate analyses and
  multivariate statistics.
• Have been used by biological scientists since
  Fisher 1921.
• Different from all other forms of statistics.
• Explained in form of matrix algebra.

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Bus Time-Table
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Bus Time-Table
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Bus Time-Table
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Properties of Matrices
Sum of leading diagonal is called the trace
There is the possibility that a symmetrical
matrix may be singular
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Singular Matrices
4 7 -1 7 8 11 -1 11 -38
5 -3 -1
5a -3b -c 0
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Singular Matrices
If A and B are symmetrical matrices, of size p x
p then there are p values of A that make A - ?B
singular
These values are called latent-roots or
eigen-values
The multipliers (transformants) are called
eigen-vectors
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12 5 13 5 13 4 13 4 21
1 1 1 1 2 1 1 1 2
A
B
There should be 3 values of ? that make A - ?B
singular One would be 21
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12 5 13 5 13 4 13 4 21
1 1 1 1 2 1 1 1 2
A
B
There should be 3 values of ? that make A - ?B
singular One would be 21
21 21 21 21 42 21 21 21 42
-9 -16 -8 -16 -29 -17 -8 -17 -21
21 B
A- 21B
Which has the eigen-vector 8 -5 1
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12 5 13 5 13 4 13 4 21
1 1 1 1 2 1 1 1 2
A
B
There should be 3 values of ? that make A - ?B
singular Another one would be 6

• 6 6 6
• 12 6
• 6 6 12

6 -1 7 -1 1 -2 7 -2 9
6 B
A- 6 B
Which has the eigen-vector 1 -1 -1
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12 5 13 5 13 4 13 4 21
1 1 1 1 2 1 1 1 2
A
B
There should be 3 values of ? that make A - ?B
singular Another one would be 7
7 7 7 7 14 7 7 7 14
5 -2 6 -2 -1 -3 6 -3 7
7 B
A- 6 B
Which has the eigen-vector 4 1 -3
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Eigen values and eigen vectors
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Use of Singular Matrices

• Used in several multivariate transformations
  where A and B represent variability of sets of
  characters.
• Making A - ?B singular may be regarded as
  subtracting B from A as often as possible, until
  the determinential value is zero.

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What do plant scientists do?

• They test hypothesis Does this treatment affect
  the crop?
• They estimate a quantity in a hypothesis What
  is the expected yield increase resulting from
  adding 100 lbs of nitrogen?
• Multivariate transformations serve neither
  purpose, but rather they set hypothesis!

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Why use multivariate Transformations
Principal Components
Canonical Analyses
Reduce the dimensions of complex situations
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Matrix of Interest
XX
A
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Principal Components Example 1

• Extracted from the work of Moore.
• Concerned with the effect of size of apple trees
  at planting on future tree development
• Tree weight (w) trunk circumference squared (x)
  length of laterals (y) and length of central
  leader (z)

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Principal Components Example 1
Character Weight
Weight 1.00 Trunk
Trunk 0.75 1.00 Lateral
Lateral 0.78 0.67 1.00 Leader
Leader 0.55 0.60 0.30 1.00
A
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Principal Components Example 1
Character Weight
Weight 1.00 Trunk
Trunk 0 1.00 Lateral
Lateral 0 0 1.00 Leader
Leader 0 0 0 1.00
B
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Principal Components Example 1

• The sum of the eigen values equals the trace of A
  (the original correlation matrix).
• The trace of A is the total variance of the four
  variables.
• The value of the eigen value indicates the
  proportion of the total variation that is
  accounted for by that transformation.

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Principal Components Example 1
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Principal Components Example 1
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Principal Components Example 1
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Principal Components Example 2

• Twenty different Brassica cultivars.
• Effect of insect damage and plant morphology.
• Record 10 variables, three treatments.

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Principal Components Example 2
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Principal Components Example 2
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Principal Components Example 2
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Principal Components Example 2
S. alba
S. alba x B. napus
B. napus
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Problems for Statisticians

• It should be noted that multivariate
  transformations are often speculative.
• Analyses are laborious and require unique and
  specific computer software.
• There are large dangers that we let the computer
  reduce the dimensions of a problem but in a
  non-biological manner.

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Multivariate Transformations
Applicable to multiple dimension problems
Reduce the dimensions of complex problems
Must be treated with knowledge of biological
systems. Can be considered as a try it and see
technique
Can point researchers in the correct direction
and indicates possible hypothesis that might be
tested in future studies
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Summary

• Association between characters.
• Simple linear regression model.
• Estimation of parameters.
• Analysis of variance of regression.
• Testing regression parameters (t tests).

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Summary

• Prediction using regression.
• Outliers.
• Scatter diagrams.
• Making a curved line strait.
• Transformation, probit analysis.
• Optimal assent, where strait lines meet.

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Summary

• Correlation.
• Bi-variate distribution.
• Testing correlation coefficients.
• Transforming to z.
• Use of correlation.

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Summary

• Multiple regression.
• Analysis of variance.
• Forward step-wise regression.
• Polynomial regression.
• Multivariate transformation.

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Multiple Experiments Genotype x Environment
Interactions