Multivariate Transformation

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

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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

Singular Matrices a -3b -c = 0

If A and B are symmetrical matrices, of size p x p then there are p values of A that make A - ΔB singular These values are called latent- roots or eigen-values The multipliers (transformants) are called eigen-vectors Singular Matrices

A= B= There should be 3 values of Δ that make A - ΔB singular One would be 21

A= B= There should be 3 values of Δ that make A - ΔB singular One would be B= A- 21B = Which has the eigen-vector [8 -5 1]

A= B= There should be 3 values of Δ that make A - ΔB singular Another one would be B= A- 6 B = Which has the eigen-vector [ ]

A= B= There should be 3 values of Δ that make A - ΔB singular Another one would be B= A- 6 B = Which has the eigen-vector [4 1 -3]

Eigen values and eigen vectors

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

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!

Reduce the dimensions of complex situations Why use multivariate Transformations Principal Components Canonical Analyses

Matrix of Interest XX’ = = A

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)

CharacterWeight Weight1.00Trunk Trunk Lateral Lateral Leader Leader Principal Components Example # 1 = A

CharacterWeight Weight1.00Trunk Trunk01.00Lateral Lateral001.00Leader Leader Principal Components Example # 1 = B

 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. Principal Components Example # 1

 Twenty different Brassica cultivars.  Effect of insect damage and plant morphology.  Record 10 variables, three treatments. Principal Components Example # 2

+54%+23%

Principal Components Example # 2

S. alba B. napus S. alba x B. napus

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.

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

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

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

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

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

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