Background on Rune H B Christensen PhD:Sensometrics: Thurstonian and - PowerPoint PPT Presentation

Background Background on Rune H B Christensen PhD:“Sensometrics: Thurstonian and Statistical Models” November 2008 — April 2012 Psychometric and Statistical Models in the R packages sensR and ordinal Education: Engineer from DTU in 2008 — Statistics and Data Analysis Rune H B Christensen Research interests: Sensometrics DTU Informatics, IMM Likelihood methods Section for Statistics Technical University of Denmark Mixed effects models rhbc@imm.dtu.dk Computational statistics February 9th 2012 Applied statistics (food science, biology, . . . ) R-packages: sensR with Per Bruun Brockhoff ordinal binomTools with Merete Kjær Hansen ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 1 / 64 ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 2 / 64 Outline The sensR package Outline Outline The sensR package The sensR package 1 1 The ordinal package — overview 2 The ordinal package — overview 2 Implementation in the ordinal package 3 Implementation in the ordinal package 3 Assessment of estimation accuracy 4 Assessment of estimation accuracy 4 Cumulative link mixed models (CLMMs) 5 Cumulative link mixed models (CLMMs) 5 ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 3 / 64 ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 4 / 64

The sensR package The sensR package The sensR package Psychometric protocols supported in sensR Regression analysis “Estimation and inference in Thurstonian models for sensory discrimination” Difference test Similarity test d ′ estimation Likelihood CI On CRAN since July 2008: Sample size n o Simulation Replicated i t www.cran.r-project.org/packages=sensR a n i m Power i r c s Development on R-Forge: i D https://r-forge.r-project.org/projects/sensr/ Duo-Trio, Triangle � � � � � � � � � 2-AFC, 3-AFC � � � � � � � � � Vignettes: A-not A � � � � � � Statistical methodology for sensory discrimination tests and its Same-Different ( � ) � � � � � implementation in sensR 2-AC � � � � � � � Examples for papers A-not A w. Sureness ( � ) � � � � � ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 5 / 64 ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 6 / 64 The sensR package The sensR package Basic functions in sensR The Thurstonian model, 3-alternatives Power & Sample size Transformation Miscellaneous d': sensory difference d ′ , CI, tests Illustration A B discrim discrimPwr rescale plot findcr AnotA d.primePwr psyfun ROC clm2twoAC a1 a2 b samediff discrimSS psyinv AUC SDT twoAC d.primeSS pc2pd samdiffSim betabin twoACpwr pd2pc discrimSim What is the probability of a correct answer? Beyond the basics: glm family objects for Thurstonian models: It depends on the question — 3-AFC or Triangle. twoAFC() , threeAFC() , duotrio() , triangle() ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 7 / 64 ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 8 / 64

The sensR package The sensR package Psychometric functions Psychometric functions: Inverse link functions 1.0 � ∞ φ ( z − d ′ )Φ( z ) 2 d z f 3-AFC ( d ′ ) = −∞ 0.9 2−AFC � ∞ 3−AFC √ f 2-AFC ( d ′ ) = φ ( z − d ′ )Φ( z ) d z = Φ( d ′ / A GLM: 2) 0.8 −∞ � ∞ Duo−trio � � √ � � √ � � 3 + d ′ � 3 − d ′ � y ∼ binom ( p c , n ) f triangle ( d ′ ) = 2 Φ − z 2 / 3 + Φ − z 2 / 3 φ ( z ) d z 0.7 0 p c √ √ √ √ f duo-trio ( d ′ ) = 1 − Φ( d ′ / 2) − Φ( d ′ / 6) + 2Φ( d ′ / 2)Φ( d ′ / 6) . p c = f psy ( d ′ ) 0.6 Triangle d ′ = X β 0.5 Family objects: twoAFC() , threeAFC() , duotrio() , triangle() 0.4 Problem: d ′ ≥ 0 0.3 psyphy : mafc(m=3) : F ( η ) = 1 m (1 − λ − 1 m )Φ( η ) → only depends on no. alternatives. 0 1 2 3 4 5 6 d' ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 9 / 64 ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 10 / 64 The ordinal package — overview The ordinal package — overview Introduction Outline The ordinal package Regression models for ordinal data via cumulative link models The sensR package 1 On CRAN since March 2010: www.cran.r-project.org/packages=ordinal The ordinal package — overview 2 Development on R-Forge: Implementation in the ordinal package 3 https://r-forge.r-project.org/projects/ordinal/ Vignettes: Assessment of estimation accuracy 4 Analysis if ordinal data with cumulative link models (32 pages) Cumulative link mixed models (CLMMs) clm tutorial (18 pages) 5 clmm tutorial (9 pages) ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 11 / 64 ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 12 / 64

The ordinal package — overview Introduction The ordinal package — overview Introduction What is a cumulative link model (CLM)? Ordinal data — the wine data Ordinal data: “large” ,“medium” ,“small” Objective: Human assessments — subjective judgements (preference, grades) How does perceived bitternes depend on temperature and contact? Grouped continuous, e.g., age (15-24, 25-34, 35-50) Table: The wine data (Randall, 1989) , N=72 γ ij = P ( Y i ≤ j ) = F ( θ j − x T CLM: i β ) Variables Type Values bitterness response 1, 2, 3 ,4, 5 A regression model for an ordered variable less — more (Agresti, 2002; Greene and Hensher, 2010) temperature predictor cold, warm contact predictor no, yes Intuitively: judges random 1, . . . , 9 A logistic regression model for J ≥ 2 (ordered) categories Temperature and contact between juice and skins can be controlled when A linear model that respects the ordered categorical nature of the cruching grapes during wine production. response ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 13 / 64 ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 14 / 64 The ordinal package — overview Introduction The ordinal package — overview Introduction Interpretation of the cumulative link model Interpretation of the cumulative link model Y: 1 2 3 4 5 Latent bitterness follows a linear Latent bitterness follows a linear β β model: model: ε i ∼ N (0 , σ 2 ) ε i ∼ N (0 , σ 2 ) S i = α + x T S i = α + x T i β + ε i , i β + ε i , = α + β ( temp i ) + ε i = α + β ( temp i ) + ε i warm warm We only observe a grouped We only observe a grouped version of S i : version of S i : θ j − 1 ≤ S i < θ j → Y = j P(Y = 2|cold) P ( Y i ≤ j ) = F ( θ j − x T i β ) cold cold θ 1 θ 2 θ 3 θ 4 α α ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 15 / 64 ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 15 / 64

The ordinal package — overview Introduction The ordinal package — overview Introduction A teaser — Fitting cumulative link models with clm Likelihood ratio tests of CLMs > data(wine) > fm1 <- clm(rating ~ contact + temp, data=wine, link="probit") > summary(fm1) formula: rating ~ contact + temp > fm2 <- update(fm1, ~.-temp) data: wine > anova(fm1, fm2) Likelihood ratio tests of cumulative link models: link threshold nobs logLik AIC niter max.grad cond.H probit flexible 72 -85.76 183.52 5(0) 1.53e-13 2.2e+01 formula: link: threshold: fm2 rating ~ contact probit flexible Coefficients: fm1 rating ~ contact + temp probit flexible Estimate Std. Error z value Pr(>|z|) contactyes 0.8677 0.2669 3.251 0.00115 ** no.par AIC logLik LR.stat df Pr(>Chisq) tempwarm 1.4994 0.2918 5.139 2.77e-07 *** fm2 5 210.05 -100.026 --- fm1 6 183.52 -85.761 28.529 1 9.231e-08 *** Signif. codes: 0 ✬ *** ✬ 0.001 ✬ ** ✬ 0.01 ✬ * ✬ 0.05 ✬ . ✬ 0.1 ✬ ✬ 1 --- Signif. codes: 0 ✬ *** ✬ 0.001 ✬ ** ✬ 0.01 ✬ * ✬ 0.05 ✬ . ✬ 0.1 ✬ ✬ 1 Threshold coefficients: Estimate Std. Error z value 1|2 -0.7733 0.2829 -2.734 2|3 0.7360 0.2499 2.945 3|4 2.0447 0.3218 6.353 4|5 2.9413 0.3873 7.595 ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 16 / 64 ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 17 / 64 The ordinal package — overview Overview The ordinal package — overview Overview What is unique about the implementation in ordinal? Functions (exported) in ordinal Miscellaneous Former impl. Distributions Extensive model framework Fitting Efficient computational methods [pdqrg]gumbel C clm convergence clm2 clmm C clmm2 C [pdg]lgamma C Convergence assessment slice gnorm C clm.fit drop.coef clm2.control Carefully designed printing glogis C clm.control clmm2.control gcauchy C clmm.control C : Implementations in C ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 18 / 64 ➞ Rune H B Christensen (DTU) The sensR and ordinal packages Psychoco 2012 19 / 64