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

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

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

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

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Model Selection I: principles of model choice The problem of model choice Varying c Y = a + bX + cX 2

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Model Selection I: principles of model choice The problem of model choice Varying c Varying d, Part I Varying d, Part II Y = a + bX + cX 2 + dX 3 Any continuous curve can be sufficiently well approximately by a polynomial of high enough order. Y = a + bX + cX 2

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Model Selection I: principles of model choice The problem of model choice Y1 = -7.62 + 3.189*X1 + 0.825*X1 2

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Model Selection I: principles of model choice The problem of model choice Y1 = -15.75 + 6.179*X1 + 0.6169*X1 2 + 0.00500*X1 3

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Model Selection I: principles of model choice The problem of model choice Y1 = -128.08 + 29.473*X1 Y1 = -7.62 + 3.189*X1 + 0.825*X1 2 Y1 = -15.75 + 6.179*X1 + 0.6169*X1 2 + 0.00500*X1 3 … Y1 = X1 Y1 = X1|X1 Y1 = X1|X1|X1 … Linear Quadratic Cubic …

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

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

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

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Model Selection I: principles of model choice Principles of model choice 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 What does marginal mean?

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

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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

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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

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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

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Model Selection I: principles of model choice Principles of model choice 123 A Y B=1 B=2 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.

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(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. Model Selection I: principles of model choice Principles of model choice 123 A Y B=1 B=2 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.

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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

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

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

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

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

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Model Selection I: principles of model choice Choosing a model

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

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

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Model Selection I: principles of model choice Choosing a model: polynomials Y1 = -7.62 + 3.189*X1 + 0.825*X1 2 s = square-root(6010) = 77.52

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

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

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

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

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

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

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

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

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

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

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

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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 Model Selection I: principles of model choice

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MA.912.A.4.2: Add, subtract, and multiply polynomials. Which of the following expressions is equivalent to (5x − 3) 2 ? A. 25x 2 − 30x + 9 B. 25x 2 −

MA.912.A.4.2: Add, subtract, and multiply polynomials. Which of the following expressions is equivalent to (5x − 3) 2 ? A. 25x 2 − 30x + 9 B. 25x 2 −

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