Nature-Inspired Metaheuristic Algorithms for Finding Efficient Experimental Designs Weng Kee Wong Department of Biostatistics Fielding School of Public Health October 19, 2012 DAE 2012 Department of Statistics University of Georgia October 17-20th 2012 Weng Kee Wong (Department of Biostatistics [email protected] School of Public Health ) October19,2012 1/50 Two Upcoming Statistics Conferences at UCLA Western Northern American Region (WNAR 2013) June 16-19 2013 http://www.wnar.org Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 2/50 Two Upcoming Statistics Conferences at UCLA Western Northern American Region (WNAR 2013) June 16-19 2013 http://www.wnar.org The 2013 Spring Research Conference (SRC) on Statistics in Industry and Technology June 20-22 2013 http://www.stat.ucla.edu/ hqxu/src2013 Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 2/50 Two Upcoming Statistics Conferences at UCLA Western Northern American Region (WNAR 2013) June 16-19 2013 http://www.wnar.org The 2013 Spring Research Conference (SRC) on Statistics in Industry and Technology June 20-22 2013 http://www.stat.ucla.edu/ hqxu/src2013 Beverly Hills, Brentwood, Bel Air, Westwood, Wilshire Corridor Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 2/50 Two Upcoming Statistics Conferences at UCLA Western Northern American Region (WNAR 2013) June 16-19 2013 http://www.wnar.org The 2013 Spring Research Conference (SRC) on Statistics in Industry and Technology June 20-22 2013 http://www.stat.ucla.edu/ hqxu/src2013 Beverly Hills, Brentwood, Bel Air, Westwood, Wilshire Corridor Close to the Pacific Ocean Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 2/50 Acknowledgements of Collaborators Ray-Bing Chen, PhD Department of Statistics National Cheng Kung University Tainan, Taiwan Jia-Heng(Joe) Qiu Department of Biostatistics UCLA Los Angeles, California, USA Weichung Wang, PhD Chien-ChihHuang Chung-WeiChen Ming-HsienWu Department of Mathematics National Taiwan University Taipei, Taiwan Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 3/50 Outline 1 Motivation 2 Metaheuristic algorithms: Particle Swarm Optimization (PSO) 3 Demonstrations using PSO with MATLAB 4 Discussion Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 4/50 1. Motivation Derivation of optimal designs for nonlinear models is usually tedious, difficult and method for one model does not usually generalize to another Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 5/50 1. Motivation Derivation of optimal designs for nonlinear models is usually tedious, difficult and method for one model does not usually generalize to another Formulae for optimal designs rarely exist and if they do, they are complicated and frequently unhelpful to the practitioners Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 5/50 1. Motivation Derivation of optimal designs for nonlinear models is usually tedious, difficult and method for one model does not usually generalize to another Formulae for optimal designs rarely exist and if they do, they are complicated and frequently unhelpful to the practitioners Algorithms are very helpful - available only for some types of optimal designs Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 5/50 1. Motivation Derivation of optimal designs for nonlinear models is usually tedious, difficult and method for one model does not usually generalize to another Formulae for optimal designs rarely exist and if they do, they are complicated and frequently unhelpful to the practitioners Algorithms are very helpful - available only for some types of optimal designs Issues - proof, speed of convergence, ease of use and availability of software Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 5/50 1. Motivation Derivation of optimal designs for nonlinear models is usually tedious, difficult and method for one model does not usually generalize to another Formulae for optimal designs rarely exist and if they do, they are complicated and frequently unhelpful to the practitioners Algorithms are very helpful - available only for some types of optimal designs Issues - proof, speed of convergence, ease of use and availability of software Is there an easy-to-use and efficient method for finding optimal designs for different types of optimal designs for any given model? Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 5/50 1.1 Locally D-optimal Designs for the Logistic Model on X =[ 1, 1] (from Silvey’s text, 1980) − log π(x) = θ + θ x, θ Θ= (θ ,θ ) : θ > 0 & θ > 0 . 1−π(x) 1 2 ∈ { 1 2 1 2 } Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 6/50 1.1 Locally D-optimal Designs for the Logistic Model on X =[ 1, 1] (from Silvey’s text, 1980) − log π(x) = θ + θ x, θ Θ= (θ ,θ ) : θ > 0 & θ > 0 . 1−π(x) 1 2 ∈ { 1 2 1 2 } Let a∗ solve exp(a) = (a + 1)/(a 1) and let u∗ solve − 2 + (u + 1)θ exp(θ + θ u)= 2 . 1 2 2 + (u + 1)θ − 2 Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 6/50 1.1 Locally D-optimal Designs for the Logistic Model on X =[ 1, 1] (from Silvey’s text, 1980) − log π(x) = θ + θ x, θ Θ= (θ ,θ ) : θ > 0 & θ > 0 . 1−π(x) 1 2 ∈ { 1 2 1 2 } Let a∗ solve exp(a) = (a + 1)/(a 1) and let u∗ solve − 2 + (u + 1)θ exp(θ + θ u)= 2 . 1 2 2 + (u + 1)θ − 2 condition locally D-optimal design a−θ1 −a−θ1 1 1 θ : θ2 θ1 a , ; , { − ≥ } { θ2 θ2 2 2 } θ2+1 ∗ 1 1 θ : θ2 θ1 < a, exp(θ1 + θ2) − 1, u ; , { − ≤ θ2 1 } {− 2 2 } θ2+1 1 1 θ : exp(θ1 + θ2) > − 1, 1; , { θ2 1 } {− 2 2 } Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 6/50 1.2 Amended Ford’s results on X =[ c, c], c > 0 − ∗ a a+1 ∗ Let a solve the equation e = a−1 (a = 1.5434), ∗ θ0+bc cb+1 let b solve the equation e = cb−1 ∗ (x+c)θ +2 and let x solve the equation eθ0+θ1x = 1 . (x+c)θ1−2 condition locally D-optimal design ∗ ∗ 1 ∗ −a −θ0 a −θ0 1 1 θ : θ1 > (θ0 + a ) , ; , { c } { θ1 θ1 2 2 } θ : b∗ <θ 1 (θ + a∗) c, x ∗; 1 , 1 { 1 ≤ c 0 } {− 2 2 } θ : 0 <θ b∗ c, c, ; 1 , 1 . { 1 ≤ } {− 2 2 } Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 7/50 1.2 Amended Ford’s results on X =[ c, c], c > 0 − ∗ a a+1 ∗ Let a solve the equation e = a−1 (a = 1.5434), ∗ θ0+bc cb+1 let b solve the equation e = cb−1 ∗ (x+c)θ +2 and let x solve the equation eθ0+θ1x = 1 . (x+c)θ1−2 condition locally D-optimal design ∗ ∗ 1 ∗ −a −θ0 a −θ0 1 1 θ : θ1 > (θ0 + a ) , ; , { c } { θ1 θ1 2 2 } θ : b∗ <θ 1 (θ + a∗) c, x ∗; 1 , 1 { 1 ≤ c 0 } {− 2 2 } θ : 0 <θ b∗ c, c, ; 1 , 1 . { 1 ≤ } {− 2 2 } Corrected results when X = [a, b] in Sebastiani and Settimi (JSPI, 1997) Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 7/50 1.2 Amended Ford’s results on X =[ c, c], c > 0 − ∗ a a+1 ∗ Let a solve the equation e = a−1 (a = 1.5434), ∗ θ0+bc cb+1 let b solve the equation e = cb−1 ∗ (x+c)θ +2 and let x solve the equation eθ0+θ1x = 1 . (x+c)θ1−2 condition locally D-optimal design ∗ ∗ 1 ∗ −a −θ0 a −θ0 1 1 θ : θ1 > (θ0 + a ) , ; , { c } { θ1 θ1 2 2 } θ : b∗ <θ 1 (θ + a∗) c, x ∗; 1 , 1 { 1 ≤ c 0 } {− 2 2 } θ : 0 <θ b∗ c, c, ; 1 , 1 . { 1 ≤ } {− 2 2 } Corrected results when X = [a, b] in Sebastiani and Settimi (JSPI, 1997) What is the E-optimal design for X = [3, 6]? Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 7/50 1.3 A 4-parameter Heteroscedastic Hill Model Di m (Econ b)( ) IC50 2λ yi = − + b + εi = η(Di ,θ)+ εi , εi N(0,σ(Eyi ) ) 1 + ( Di )m ∼ IC50 Di = dose of a drug assigned to subject i yi = drug effect of subject i Econ = the control effect at zero drug concentration b = background effect at infinite drug concentration IC50 = inflection point on the curve (a measure of the drug potency) = drug concentration that induces a 50% decrease in the maximal effect (Econ b) − m = slope parameter of the curve. Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 8/50 Weng Kee Wong (Dept. of Biostatistics [email protected] of Public Health ) October19,2012 9/50 1.5 Information matrix for the Hill Model 0 0 0 0 T Let the nominal value for θ be θ0 = (Econ, b , IC50, m ) and let T ∂η(x,θ) ∂η(x,θ) ∂η(x,θ) ∂η(x,θ) f (x,θ0) =( , , , ) θ0 ∂Econ ∂b ∂IC50 ∂m | m ∂η(x,θ) (x/IC50) where = m ∂Econ (1 + x/IC50) ∂η(x,θ) 1 = m ∂b 1 + (x/IC50) m ∂η(x,θ) (b Econ)(x/IC50) log(x/IC50) = − m 2 ∂IC50 − (1 + (x/IC50) ) m ∂η(x,θ) (b Econ)m(x/IC50) = − m 2 .