Models for Replicated Discrimination Tests Models for Replicated Discrimination Tests: A Synthesis of Latent Class Mixture Models and Generalized Linear Mixed Models Rune Haubo Bojesen Christensen & Per Bruun Brockhoff DTU Informatics Section for Statistics Technical University of Denmark rhbc@imm.dtu.dk August 13 2008 Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 1 / 19
Models for Replicated Discrimination Tests Motivation This Talk is About A non-standard type of mixed models Applicable to a range of discrimination tests Focus: Insight into models—not computational methods Motivated by examples from sensometrics and psychometrics (but also applicable in signal detection, medical decision making etc.) The models extend existing models by ◮ Modelling the covariance structure ◮ Having a close connection to psychological theory of cognitive decision making ◮ Providing inference for individuals via random effect estimates Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 2 / 19
Models for Replicated Discrimination Tests Outline Outline Background 1 Models for Independent Data 2 Models for Replicated Data 3 Examples 4 Summary 5 Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 3 / 19
Models for Replicated Discrimination Tests Background A Replicated Discrimination Test Example: Coke Inc. wants to substitute a sweetener in a diet coke, A with a cheaper alternative B . Coke Inc. employs 30 consumers in a discrimination (triangle) test Each consumer performs the test 10 times (replications) Can consumers distinguish between the two recipes? Why Replications? Advantages: ◮ Cheap and Easy: Substitute some assessors with replications ◮ Information on difference between assessors Challenges: ◮ Observations are often correlated and not independent Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 4 / 19
Models for Replicated Discrimination Tests Background The Triangle Test δ Two regular products and one 0.4 new product are presented to the consumer 0.3 A B Two A - products ( a 1 , a 2 ) and one B -product ( b 1 ) are 0.2 presented to the consumer 0.1 Task: Identify the odd product δ = µ B − µ A : A measure of 0.0 discriminal ability and difference between products. a1 a2 b1 Sensory magnitude Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 5 / 19
Models for Replicated Discrimination Tests Background The Triangle Test Answers are binomial: A proportion of correct answers 1.0 Triangle Psychometric Y i ∼ Bin ( π i ; n i ) π 0 ≤ π i < 1 Function 0.8 Guessing probability: π 0 = 1 / 3 0.6 Relation between π i and δ i : Triangle π Model � ∞ √ � � � 0.4 � A B π i = f ( δ i ) = Φ − z 3 + δ 2 / 3 0 √ � � � 0.2 � +Φ 3 − δ 2 / 3 φ ( z ) d z − z 0.0 a1 a2 b1 Psychometric function: Relates the probability of a 0 1 2 3 4 5 6 correct answer to the ability to δ discriminate Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 6 / 19
Models for Replicated Discrimination Tests Models for Independent Data The Basic (Naive) Model (GLM) Y i ∼ Bin ( π i , n i ) π i = f triangle ( δ i ) δ i = δ A Generalized Linear Model (GLM) with ◮ Binomial distribution ◮ Psychometric function as inverse link function ◮ Simple linear predictor Assumes π and δ identical for all individuals Family-object; triangle in package sensR for use with glm ◮ Extends discrimination tests to allow for explanatory variables ◮ Prepares the way for mixed effect models for discrimination tests Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 7 / 19
Models for Replicated Discrimination Tests Models for Replicated Data Models for Replicated Discrimination Tests Ignore covariance structure in data ◮ Basic GLM Marginal Models (adjust se’s for overdispersion) ◮ quasi-binomial GLM Latent Class Mixture model Conditional Models (model covariance structure) ◮ Generalized Linear Mixed Model (GLMM) ◮ Synthesis of Mixture and GLMM Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 8 / 19
Models for Replicated Discrimination Tests Models for Replicated Data Models for Replicated Discrimination Tests Ignore covariance structure in data ◮ Basic GLM Marginal Models (adjust se’s for overdispersion) ◮ quasi-binomial GLM Latent Class Mixture model Conditional Models (model covariance structure) ◮ Generalized Linear Mixed Model (GLMM) ◮ Synthesis of Mixture and GLMM Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 8 / 19
Models for Replicated Discrimination Tests Models for Replicated Data Latent Class Mixture Models Two-class model for discriminal ability 0.8 p = P ( δ i > 0 ) P i ∼ Bernoulli ( p ) Y i | p i ∼ Bin ( π i ; n i ) 0.6 � 0 if p i = 0 π i = f triangle ( δ i ) δ i = 1 − p = P ( δ i = 0 ) δ if p i = 1 0.4 No dispersion among subjects 0.2 with positive δ Often an inappropriate 0.0 assumption! δ 0 δ i Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 9 / 19
Models for Replicated Discrimination Tests Models for Replicated Data Generalized Linear Mixed Model 0.4 b i ∼ N ( 0 , σ 2 δ ) Y i | b i ∼ Bin ( π i ; n i ) 0.3 π i = f triangle ( δ i ) δ i = δ + b i σ δ 0.2 Assumes a continuous distribution for subjects 0.1 Allows for dispersion among subjects δ 0.0 Subjects can have δ < 0 ! Impossible in the triangle test −2 0 2 4 δ i Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 10 / 19
Models for Replicated Discrimination Tests Models for Replicated Data Generalized Linear Mixed Model 0.4 b i ∼ N ( 0 , σ 2 δ ) Y i | b i ∼ Bin ( π i ; n i ) 0.3 π i = f triangle ( δ i ) δ i = δ + b i σ δ 0.2 Assumes a continuous distribution for subjects 0.1 Allows for dispersion among subjects δ 0.0 Subjects can have δ < 0 ! Impossible in the triangle test −2 0 2 4 δ i Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 10 / 19
Models for Replicated Discrimination Tests An Appropriate Alternative Latent Class Mixed Model δ i ∼ F ( δ, σ 2 Y i | b i ∼ Bin ( π i ; n i ) δ ) 0.4 π i = f triangle ( δ i ) δ i = δ + b i P ( δ i = 0 ) = P ( δ i > > 0 ) = 0.3 Φ ( −δ σ δ ) − Φ ( − −δ σ δ ) � σ δ 1 − 1 − p , δ i = 0 σ δ f ( δ i ) = � � δ i − δ 1 σ δ φ , δ i > 0 0.2 σ δ p = 1 − Φ( − δ/σ δ ) 0.1 One-dimensional random δ effect with two attributes: 0.0 ◮ Class probabilities ˜ p i ◮ The magnitude of −2 0 2 4 δ i discriminal ability ˜ δ i Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 11 / 19
Models for Replicated Discrimination Tests An Appropriate Alternative Estimation in Latent Class Mixed Model Likelihood function ∼ marginal density of y f ( y i ) = ( 1 − p ) f 1 ( y i ) + pf 2 ( y i ) π y i � n i 0 ( 1 − π 0 ) ( n i − y i ) � Likelihood at δ i = 0 : f 1 ( y i ) = y i � ∞ f 2 ( y i ) = 1 Likelihood at δ i > 0 : f π ( y i | δ i ) φ (( δ i − δ ) /σ δ ) σ δ d δ i p 0 � n i π y i � i ( 1 − π i ) ( n i − y i ) where f π ( y i | δ i ) = y i Define likelihood function as R-function via integrate optimize with optim Structure motives an EM algorithm Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 12 / 19
Models for Replicated Discrimination Tests An Appropriate Alternative Attenuation Effect 1.0 Marginal link function: Original 0.9 link function π m = E δ i [ E [ y i | δ i ]] 0.8 Marginal = π 0 ( 1 − p ) link function 0.7 � ∞ π + f triangle ( δ i ) φ (( δ i − δ ) /σ δ ) /σ δ d δ i δ 0 Stationary point 0.6 0 0.5 Marginal estimates are closer to “stationary points” rather 0.4 than closer to zero 0.3 Marginal link function depends 0 1 2 3 4 5 6 7 on σ 2 δ δ Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 13 / 19
Models for Replicated Discrimination Tests Examples Example Triangle Data δ se( δ ) σ δ Model p Basic (GLM) 1.67 0.186 Overdisp. GLM 1.67 0.257 Proposed Model 1.62 0.234 1.08 93.3% Difference in estimate of δ (attenuation effect) Basic model gives too small se’s Clear variation between subjects Large proportion of discriminators Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 14 / 19
Models for Replicated Discrimination Tests Examples Example Triangle Data 0.4 ^ = 1.62 δ Random effect estimates as ^ δ = 1.08 σ 0.3 Conditional expectations: � ∞ ˜ δ i = E [ δ i | y i ] = δ i f ( δ i | y i ) d δ i 0.2 −∞ � ∞ −∞ δ i f ( y i | δ i ) f ( δ i ) d δ i 0.1 = f ( y i ) One extra integral per 0.0 individual −2 0 2 4 δ i Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 15 / 19
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