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

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

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

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

The problem of model choice 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 + cX2 Y = a + bX + cX2 + dX3 Any continuous curve can be sufficiently well approximately by a polynomial of high enough order.

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

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

The problem of model choice 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 …

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

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

Principles of model choice 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

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

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

Principles of model choice 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

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

Principles of model choice 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. 1 2 3 A Y B=1 B=2 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.

Principles of model choice 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. 1 2 3 A Y B=1 B=2 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.

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

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

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

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

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

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

Choosing a model: polynomials 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

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

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

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

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

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

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

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

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

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

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

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

Model Selection II: datasets with several explanatory variables 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