1. Multivariate Transformation

2. 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.

3. Bus Time-Table

6. Properties of Matrices Sum of leading diagonal is called the trace There is the possibility that a symmetrical matrix may be singular

7. Singular Matrices 4 7 -1 7 8 11 7 8 11 -1 11 -38 5 -3 -1 5a -3b -c = 0

8. 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

9. 12 5 13 5 13 4 5 13 4 13 4 21 A= 1 1 1 1 2 1 1 1 2 B= There should be 3 values of Δ that make A - ΔB singular One would be 21

10. 12 5 13 5 13 4 5 13 4 13 4 21 A= 1 1 1 1 2 1 1 1 2 B= There should be 3 values of Δ that make A - ΔB singular One would be 21 21 21 21 21 42 21 21 21 42 21 B= -9 -16 -8 -16 -29 -17 -8 -17 -21 -8 -17 -21 A- 21B = Which has the eigen-vector [8 -5 1]

11. 12 5 13 5 13 4 5 13 4 13 4 21 A= 1 1 1 1 2 1 1 1 2 B= There should be 3 values of Δ that make A - ΔB singular Another one would be 6 6 6 6 6 12 6 6 6 12 6 B= 6 -1 7 -1 1 -2 7 -2 9 A- 6 B = Which has the eigen-vector [1 -1 -1]

12. 12 5 13 5 13 4 5 13 4 13 4 21 A= 1 1 1 1 2 1 1 1 2 B= There should be 3 values of Δ that make A - ΔB singular Another one would be 7 7 7 7 7 14 7 7 7 14 7 B= 5 -2 6 -2 -1 -3 6 -3 7 A- 6 B = Which has the eigen-vector [4 1 -3]

13. Eigen values and eigen vectors

14. 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.

15. 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!

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

17. Matrix of Interest XX’ = = A

18. 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)

19. CharacterWeight Weight1.00Trunk Trunk0.751.00Lateral Lateral0.780.671.00Leader Leader0.550.600.301.00 Principal Components Example # 1 = A

20. CharacterWeight Weight1.00Trunk Trunk01.00Lateral Lateral001.00Leader Leader0001.00 Principal Components Example # 1 = B

21.  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

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

27. +54%+23%

28. Principal Components Example # 2

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

30. 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.

31. 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

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

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

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

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

36. Multiple Experiments Genotype x Environment Interactions