quantitative methods
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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
QuickTime™ and aAnimation decompressor
are needed to see this picture.
QuickTime™ and aAnimation decompressor
are needed to see this picture.
Y = a + bX
Model Selection I: principles of model choice
The problem of model choice
QuickTime™ and aAnimation decompressor
are needed to see this picture.
Varying c
Y = a + bX + cX2
Model Selection I: principles of model choice
The problem of model choice
QuickTime™ and aAnimation decompressor
are needed to see this picture.
Varying c Varying d, Part I
QuickTime™ and aAnimation decompressor
are needed to see this picture.
QuickTime™ and aAnimation decompressor
are needed to see this picture.
Varying d, Part II
Y = a + bX + cX2 + dX3
Any continuous curve can be sufficiently well approximately by a polynomial of high enough order.
Y = a + bX + cX2
Model Selection I: principles of model choice
The problem of model choice
Y1 = -7.62 + 3.189*X1 + 0.825*X12
Model Selection I: principles of model choice
The problem of model choice
Y1 = -15.75 + 6.179*X1 + 0.6169*X12 + 0.00500*X13
Model Selection I: principles of model choice
The problem of model choice
Y1 = -128.08 + 29.473*X1Y1 = -7.62 + 3.189*X1 + 0.825*X12
Y1 = -15.75 + 6.179*X1 + 0.6169*X12 + 0.00500*X13
…
Y1 = X1Y1 = X1|X1Y1 = X1|X1|X1…
LinearQuadraticCubic…
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*XA is not marginal to B, B*C, B*C*XX is marginal to X*X, A*X, A*B*XX 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?
B
1 2 3 4
1 A1B1 A1B2 A1B3 A1B4
A 2 A2B1 A2B2 A2B3 A2B4
3 A3B1 A3B2 A3B3 A3B4
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=XY=X+X*XY=X+X*X+X*X*X
Hierarchical
Y=X*XY=X*X + XY=X*X*X + X
Not hierarchical
Lower order term missingLower order term after higher order termLower 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
1 2 3A
Y
B=1B=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
1 2 3A
Y
B=1B=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 = -7.62 + 3.189*X1 + 0.825*X12
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