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 I THIS COURSE. I 1. MOTIVATING INTRODUCTION TO STATISTICS. I 2. IMPORTANCE OF DESIGNING AN EXPERIMENT. I 3. ANOVA. I 4. REGRESSION AND CORRELATION. I 5. OPTIMAL DESIGN: MOTIVATION AND CRITICISMS. I 6. OPTIMAL DESIGN THEORY (LINEAR MODELS). I 7. OPTIMAL DESIGNS FOR NONLINEAR MODELS. I 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 I Traditional treatment: Curamina. I New drug: Fraudol. I 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 I Linear Model, E (y | x) = f T (x)θ, Var (y | x) = σ 2 , x ∈ χ compact I Select experimental conditions x1 , . . . , xn (exact design). I Observe the responses y1 , . . . , yn . I Approximate design: Probability measure ξ. R Information matrix: M(ξ) = χ f (x)f T (x)ξ(dx). I I Criterion function, Φ[M(ξ)] or Φ(ξ). I Equivalence theorem. I D–optimality (most popular): Determinant. −1/m Φ[M(ξ ∗ )] det M(ξ ∗ ) Efficiency: eff Φ (ξ) = Φ[M(ξ)] = det M(ξ) I Inst. Appl. Math. Sci. & Eng. Experimental design and analysis, Jesús López Fidalgo Criticisms I A priori choice of the model without data. I I I I 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. I I I I 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 I Criterion selection. I I I Controversy exact (Box)/approximate (Kiefer) designs. I I Compound criteria. Restricted criteria. Exact designs: small sample size. Frequently an optimal design demands extreme conditions (“lo mejor es enemigo de lo bueno”). I I Reference to measure the efficiency. Restricted criteria. Inst. Appl. Math. Sci. & Eng. Experimental design and analysis, Jesús López Fidalgo Criticisms I Difficult computation. I I Ok, but this is what we are here for. Problem with some criteria: Scales of the parameters may be quite different. I 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