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Quantitative Methods for Model Selection and Design Experiments

This guide delves into the principles of model choice and designed experiments, covering factors like varying coefficients, polynomial approximations, and hierarchy in model formulas to aid in effective model selection.

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Quantitative Methods for Model Selection and Design Experiments

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  1. Quantitative Methods Model Selection I: principles of model choice and designed experiments

  2. Model Selection I: principles of model choice The problem of model choice

  3. Model Selection I: principles of model choice The problem of model choice

  4. Model Selection I: principles of model choice The problem of model choice Varying b Varying a Y = a + bX

  5. Model Selection I: principles of model choice The problem of model choice Varying c Y = a + bX + cX2

  6. Model Selection I: principles of model choice The problem of model choice Varying d, Part II Varying d, Part I Varying c Y = a + bX + cX2 Y = a + bX + cX2 + dX3 Any continuous curve can be sufficiently well approximately by a polynomial of high enough order.

  7. Model Selection I: principles of model choice The problem of model choice Y1 = -7.62 + 3.189*X1 + 0.825*X12

  8. Model Selection I: principles of model choice The problem of model choice Y1 = -15.75 + 6.179*X1 + 0.6169*X12 + 0.00500*X13

  9. Model Selection I: principles of model choice The problem of model choice Linear Quadratic Cubic … Y1 = X1 Y1 = X1|X1 Y1 = X1|X1|X1 … Y1 = -128.08 + 29.473*X1 Y1 = -7.62 + 3.189*X1 + 0.825*X12 Y1 = -15.75 + 6.179*X1 + 0.6169*X12 + 0.00500*X13 …

  10. Model Selection I: principles of model choice Principles of model choice

  11. Model Selection I: principles of model choice Principles of model choice • Economy of variables • Multiplicity of p-values • Marginality

  12. Model Selection I: principles of model choice Principles of model choice • Economy of variables • Multiplicity of p-values • Marginality • Hierarchies must be respected in model formulae • Significance of interactions includes importance of main effects • Do not test main effects with a SS that has been adjusted for the interaction

  13. Model Selection I: principles of model choice Principles of model choice What does marginal mean? A is marginal to A*B, A*B*C, A*X*X A is not marginal to B, B*C, B*C*X X is marginal to X*X, A*X, A*B*X X is not marginal to A, Z, Z*Z, A*B, A*B*Z

  14. Model Selection I: principles of model choice Principles of model choice Why marginal?

  15. Model Selection I: principles of model choice Principles of model choice • Economy of variables • Multiplicity of p-values • Marginality • Hierarchies must be respected in model formulae • Significance of interactions includes importance of main effects • Do not test main effects with a SS that has been adjusted for the interaction

  16. Model Selection I: principles of model choice Principles of model choice Y=X Y=X+X*X Y=X+X*X+X*X*X Hierarchical Y=X*X Y=X*X + X Y=X*X*X + X Not hierarchical Lower order term missing Lower order term after higher order term Lower order term missing and wrong order

  17. Model Selection I: principles of model choice Principles of model choice • Economy of variables • Multiplicity of p-values • Marginality • Hierarchies must be respected in model formulae • Significance of interactions includes importance of main effects • Do not test main effects with a SS that has been adjusted for the interaction

  18. Y B=1 B=2 1 2 3 A Model Selection I: principles of model choice Principles of model choice No main effect of A because the average value of Y at each level of A is the same. No main effect of B because the average value of Y at each level of B is the same. Yet there is an interaction, and this means A and B both affect Y.

  19. Y B=1 B=2 1 2 3 A Model Selection I: principles of model choice Principles of model choice No main effect of A because the average value of Y at each level of A is the same. No main effect of B because the average value of Y at each level of B is the same. Yet there is an interaction, and this means A and B both affect Y. (i) a significant interaction A*B means that A affects the way B affects Y, (ii) but then certainly B must affect Y. So if A*B is significant, conclude that A and B affect Y as well as the direct inference that A affects the way B affects Y.

  20. Model Selection I: principles of model choice Principles of model choice • Economy of variables • Multiplicity of p-values • Marginality • Hierarchies must be respected in model formulae • Significance of interactions includes importance of main effects • Do not test main effects with a SS that has been adjusted for the interaction

  21. Model Selection I: principles of model choice Principles of model choice

  22. Model Selection I: principles of model choice Principles of model choice

  23. Model Selection I: principles of model choice Principles of model choice

  24. Model Selection I: principles of model choice Principles of model choice

  25. Model Selection I: principles of model choice Choosing a model

  26. Model Selection I: principles of model choice Choosing a model: polynomials

  27. Model Selection I: principles of model choice Choosing a model: polynomials

  28. Model Selection I: principles of model choice Choosing a model: polynomials Y1 = -7.62 + 3.189*X1 + 0.825*X12 s = square-root(6010) = 77.52

  29. Model Selection I: principles of model choice Choosing a model: orthogonal design

  30. Model Selection I: principles of model choice Choosing a model: orthogonal design bottom up! pooling?

  31. Model Selection I: principles of model choice Choosing a model: non-orthogonality

  32. Model Selection I: principles of model choice Choosing a model: non-orthogonality

  33. Model Selection I: principles of model choice Choosing a model: non-orthogonality

  34. Model Selection I: principles of model choice Choosing a model: trends in a factor - Shape - Sensitivity to consistent effects

  35. Model Selection I: principles of model choice Choosing a model: trends in a factor

  36. Model Selection I: principles of model choice Choosing a model: trends in a factor

  37. Model Selection I: principles of model choice Choosing a model: trends in a factor

  38. Model Selection I: principles of model choice Choosing a model: trends in a factor Sensitivity

  39. Model Selection I: principles of model choice Choosing a model: trends in a factor Shape

  40. Model Selection I: principles of model choice Last words… • Model choice represents a whole extra layer of sophistication to use of GLM • Very powerful extensions: polynomials • Very important principles: economy, multiplicity • Very important cautions: marginality Model Selection II: datasets with several explanatory variables Read Chapter 11

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