Models for Replicated Discrimination Tests: A Synthesis of Latent Class Mixture Models and Generalized Linear Mixed Models

Transcription

1 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 Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 1 / 19

2 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

3 Outline Outline 1 Background 2 Models for Independent Data 3 Models for Replicated Data 4 Examples 5 Summary Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 3 / 19

4 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

5 Background The Triangle Test Two regular products and one new product are presented to the consumer Two A- products (a1, a2) and one B-product (b1) are presented to the consumer Task: Identify the odd product δ = µ B µ A : A measure of discriminal ability and difference between products A δ a1 a2 b1 Sensory magnitude B Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 5 / 19

6 Background The Triangle Test π Triangle Psychometric Function A B Triangle Model Answers are binomial: A proportion of correct answers Y i Bin(π i ; n i ) π 0 π i < 1 Guessing probability: π 0 = 1/3 Relation between π i and δ i : π i = f (δ i ) = 0 +Φ { Φ ( z 3 + δ 2/3 ( z 3 δ 2/3 ) ) } φ(z)dz 0.0 a1 a2 b δ Psychometric function: Relates the probability of a correct answer to the ability to discriminate Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 6 / 19

7 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

8 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

9 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

10 Models for Replicated Data Latent Class Mixture Models Two-class model for discriminal ability P i Bernoulli(p) Y i p i Bin(π i ; n i ) { 0 if pi = 0 π i = f triangle (δ i ) δ i = δ if p i = p = P(δ i = 0) p = P(δ i > 0) No dispersion among subjects with positive δ Often an inappropriate assumption! δ δ i Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 9 / 19

11 Models for Replicated Data Generalized Linear Mixed Model b i N(0, σ 2 δ ) Y i b i Bin(π i ; n i ) 0.4 π i = f triangle (δ i ) δ i = δ + b i 0.3 Assumes a continuous distribution for subjects Allows for dispersion among subjects Subjects can have δ < 0! Impossible in the triangle test δ δ i σ δ Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 10 / 19

12 Models for Replicated Data Generalized Linear Mixed Model b i N(0, σ 2 δ ) Y i b i Bin(π i ; n i ) 0.4 π i = f triangle (δ i ) δ i = δ + b i 0.3 Assumes a continuous distribution for subjects Allows for dispersion among subjects Subjects can have δ < 0! Impossible in the triangle test δ δ i σ δ Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 10 / 19

13 An Appropriate Alternative Latent Class Mixed Model Y i b i Bin(π i ; n i ) δ i F(δ, σ 2 δ ) 0.4 π i = f triangle (δ i ) δ i = δ + b i { 1 p, δi = 0 f (δ i ) = ( ) 1 σ δ φ δi δ σ δ, δ i > 0 p = 1 Φ( δ/σ δ ) 0.1 One-dimensional random effect with two attributes: 0.0 δ Class probabilities pi The magnitude of discriminal ability δ δ i i P(δ i = 0)= Φ( δ σ δ ) σ δ P(δ i > 0)= 1 Φ( δ σ δ ) Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 11 / 19

14 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 ) Likelihood at δ i = 0: Likelihood at δ i > 0: f 1 (y i ) = ( n i y i ) π y i 0 (1 π 0) (n i y i ) f 2 (y i ) = 1 p 0 f π (y i δ i )φ((δ i δ)/σ δ ) σ δ dδ i where f π (y i δ i ) = ( n i y i ) π y i i (1 π i) (n 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

15 An Appropriate Alternative Attenuation Effect 1.0 Marginal link function: 0.9 π m = E δi [E[y i δ i ]] 0.8 = π 0 (1 p) f triangle (δ i )φ((δ i δ)/σ δ )/σ δ dδ π i Marginal estimates are closer to stationary points rather than closer to zero Marginal link function depends on σ 2 δ Original link function δ 0 Marginal link function Stationary point δ Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 13 / 19

16 Examples Example Triangle Data Model δ se(δ) σ δ p Basic (GLM) Overdisp. GLM Proposed Model % 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

17 Examples Example Triangle Data Random effect estimates as Conditional expectations: δ i = E[δ i y i ] = = δ i f (δ i y i )dδ i δ if (y i δ i )f (δ i ) dδ i f (y i ) One extra integral per individual δ^ = 1.62 σ^δ = δ i Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 15 / 19

18 Examples Example Triangle Data Class probabilities as Conditional expectations: p i = E[p i y i ] = = = P i (0,1) P i df(p i y i ) P i f (y i P i )f (P i )/f (y i ) pf 2 (y i ) (1 p)f 1 (y i ) + pf 2 (y i ) Random effects are almost normal Expect similar results from GLMM Observed Quantiles (δ i ) Normal Quantiles Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 16 / 19 1

19 Examples Example: 2-AFC Test Random effect estimates: δ i, p i Clear tail on the left Clear lower bound on δ Discrete nature of data more clear Observed Quantiles (δ i ) Normal Quantiles Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 17 / 19

20 Summary Summary and Challenges Summary Close connection to psychological theory Model-type apply to a range of discrimination test protocols (eg. triangle, duo-trio, 2-AFC and 3-AFC) A synthesis of Latent Class Mixture Models and GLMMs One-dimensional random effects with two attributes Class probabilities p i The magnitude of discriminal ability δ i Challenges Extension to additional fixed effects (changes area of integration) random effects (multi-dimensional integrals) Variance of random effect estimates Implementation with better computational methods Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 18 / 19

21 Summary Acknowledgments Thanks to the Program Committee Thanks to Professor Per Bruun Brockhoff Thank you for listening! Rune H B Christensen (DTU) Models for Replicated Discrimination Tests UseR! 2008, Dortmund 19 / 19