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

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

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

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

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

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

Model Selection I: principles of model choice The problem of model choice Y1 = *X *X1 2

Model Selection I: principles of model choice The problem of model choice Y1 = *X *X *X1 3

Model Selection I: principles of model choice The problem of model choice Y1 = *X1 Y1 = *X *X1 2 Y1 = *X *X *X1 3 … Y1 = X1 Y1 = X1|X1 Y1 = X1|X1|X1 … Linear Quadratic Cubic …

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

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

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?

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

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

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

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

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.

(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.

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

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

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

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

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

Model Selection I: principles of model choice Choosing a model

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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