Experimental design and analysis

Experimental design and analysis
[email protected]
University of Castilla-La Mancha
Department of Mathematics
Institute of Applied Mathematics to Science and Engineering
Inst. Appl. Math. Sci. & Eng.
Experimental design and analysis, Jesús López Fidalgo
OUTLINE
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THIS COURSE.
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1. MOTIVATING INTRODUCTION TO STATISTICS.
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2. IMPORTANCE OF DESIGNING AN EXPERIMENT.
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3. ANOVA.
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4. REGRESSION AND CORRELATION.
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5. OPTIMAL DESIGN: MOTIVATION AND CRITICISMS.
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6. OPTIMAL DESIGN THEORY (LINEAR MODELS).
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7. OPTIMAL DESIGNS FOR NONLINEAR MODELS.
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8. REAL APPLICATIONS.
Inst. Appl. Math. Sci. & Eng.
Experimental design and analysis, Jesús López Fidalgo
5. OPTIMAL DESIGN:
MOTIVATION AND
CRITICISMS
Inst. Appl. Math. Sci. & Eng.
Experimental design and analysis, Jesús López Fidalgo
Motivating example (weighing two objects)
y = P + ε with experimental error ε ≡ N (0, σ 2 ).
Classic design:
MLE: P̂A = y1 and P̂B = y2 , with variance σ 2 and independent.
Inst. Appl. Math. Sci. & Eng.
Experimental design and analysis, Jesús López Fidalgo
Optimal design
MLE: P̂A =
y1 +y2
2
and P̂B =
y1 −y2
2 ,
with variance σ 2 /2
and independent (uncorrelated):
cov(P̂A , P̂B ) = {cov(y1 , y1 ) − cov(y1 , y2 ) + cov(y1 , y2 ) − cov(y2 , y2 )}/4 = 0
Inst. Appl. Math. Sci. & Eng.
Experimental design and analysis, Jesús López Fidalgo
Simpson paradox
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Traditional treatment: Curamina.
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New drug: Fraudol.
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Experimentation:
Curamina
Fraudol
Improving
20
24
Inst. Appl. Math. Sci. & Eng.
No improving
20
16
%
50%
60%
Experimental design and analysis, Jesús López Fidalgo
Disaggregated
Curamina
Fraudol
Men:
Improving No improving
12
18
3
7
%
40%
30%
Curamina
Fraudol
Women:
Improving No improving
8
2
21
9
%
80%
70%
Inst. Appl. Math. Sci. & Eng.
Experimental design and analysis, Jesús López Fidalgo
Summary
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Linear Model,
E (y | x) = f T (x)θ, Var (y | x) = σ 2 , x ∈ χ compact
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Select experimental conditions x1 , . . . , xn (exact design).
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Observe the responses y1 , . . . , yn .
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Approximate design: Probability measure ξ.
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Information matrix: M(ξ) = χ f (x)f T (x)ξ(dx).
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Criterion function, Φ[M(ξ)] or Φ(ξ).
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Equivalence theorem.
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D–optimality (most popular): Determinant.
−1/m Φ[M(ξ ∗ )]
det M(ξ ∗ )
Efficiency: eff Φ (ξ) = Φ[M(ξ)] = det M(ξ)
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Inst. Appl. Math. Sci. & Eng.
Experimental design and analysis, Jesús López Fidalgo
Criticisms
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A priori choice of the model without data.
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Box (1979): “Models, of course, are never true, but fortunately it is
only necessary that they be useful”.
Experience, retrospective data, intuitions of the practitioner, some
are analytically derived ...
Criteria to discriminate between rival models (JRSSB 2007).
Nonlinear models: IM depending on the parameters.
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locally Φ–optimal design (explicit or numerical): sensitivity analysis.
Minimax designs.
Sequential designs.
Bayesian designs.
Inst. Appl. Math. Sci. & Eng.
Experimental design and analysis, Jesús López Fidalgo
Criticisms
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Criterion selection.
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Controversy exact (Box)/approximate (Kiefer) designs.
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Compound criteria.
Restricted criteria.
Exact designs: small sample size.
Frequently an optimal design demands extreme conditions
(“lo mejor es enemigo de lo bueno”).
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Reference to measure the efficiency.
Restricted criteria.
Inst. Appl. Math. Sci. & Eng.
Experimental design and analysis, Jesús López Fidalgo
Criticisms
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Difficult computation.
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Ok, but this is what we are here for.
Problem with some criteria: Scales of the parameters may be quite
different.
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Solutions: Standardized optimality criteria by the efficiencies of each
parameter (similar final efficiencies) or
by the coefficient of variation (JSPI 2007, CSTM 2007).
Inst. Appl. Math. Sci. & Eng.
Experimental design and analysis, Jesús López Fidalgo
Continued
Inst. Appl. Math. Sci. & Eng.
Experimental design and analysis, Jesús López Fidalgo