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

Quantitative Methods

Model Selection I:principles of model choice and

designed experiments

Model Selection I: principles of model choice

The problem of model choice



Varying a Varying b

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Y = a + bX





Varying c

Y = a + bX + cX2





Varying c Varying d, Part I





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



Y1 = -7.62 + 3.189*X1 + 0.825*X12



Y1 = -15.75 + 6.179*X1 + 0.6169*X12 + 0.00500*X13



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…


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






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?



Why marginal?

B

1 2 3 4

1 A1B1 A1B2 A1B3 A1B4

A 2 A2B1 A2B2 A2B3 A2B4

3 A3B1 A3B2 A3B3 A3B4



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



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.



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.


Choosing a model


Choosing a model: polynomials


Choosing a model: polynomials

Y1 = -7.62 + 3.189*X1 + 0.825*X12

s = square-root(6010) = 77.52


Choosing a model: orthogonal design


Choosing a model: orthogonal design

bottom up!pooling?


Choosing a model: non-orthogonality


Choosing a model: trends in a factor

- Shape- Sensitivity to consistent effects



Sensitivity



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


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

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