Mixed-effects Maximum Likelihood Difference Scaling

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1 Mixed-effects Maximum Likelihood Difference Scaling Kenneth Knoblauch Inserm U 846 Stem Cell and Brain Research Institute Dept. Integrative Neurosciences Bron, France Laurence T. Maloney Department of Psychology Center for Neural Science New York University New York, NY USA

2 Psychophysics, qu est-ce que c est? Gustav Fechner ( ) A body of techniques and analytic methods to study the relation between physical stimuli and the organism s (classification) behavior to infer internal states of the organism or their organization.

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4 Difference scaling is a psychophysical procedure used to estimate a perceptual (interval) scale for stimuli distributed along a physical continuum. Example: VQ compressed images, Up to what compression rate can the observer detect no loss of image quality? 1:1 6:1 9:1 12:1 15:1 18:1 21:1 24:1 27:1 30:1 Charrier, Maloney, Cherifi & Knoblauch, (2007) J Opt Soc Am A, 24,

5 Difference Scaling: Experimental Procedure From a set of p stimuli, {I 1 <I 2 <... < I p }, a random quadruple, {I a,i b ; I c,i d }, is chosen (w/out replacement) and presented to the observer as in this example, on each trial: Between which pair (upper/lower) is the perceived difference greatest?

6 The aim of the Maximum Likelihood Difference Scaling (MLDS) procedure is to estimate scale values, (ψ 1, ψ 2,...,ψ p ), that best capture the observer s judgments of the perceptual difference between the stimuli in each pair. The MLDS package, available on CRAN, provides tools for performing this analysis in R. An example scale obtained from an observer for the apples sequence of VQ compressed images is shown on the right: Difference Scale Value Image: apples Color Space: L*a*b* Observer: PYB Compression Rate

7 The decision model Given a quadruple, q =(a, b ; c, d), from a single trial, we assume that the observer chooses the upper pair to be further apart when (a, b ; c, d) = ψ d ψ c ψ b ψ a + ɛ > 0, where and ψ i σ are estimated scale values, a scale factor. ɛ N (0, σ 2 )

8 Estimation of Scale Values Maloney and Yang (2003) used a direct method for estimating the maximum likelihood scale values, ( ( n ) ) 1 Rk ( ( ( ) δ q k δ q k L(Ψ, σ) = Φ 1 Φ σ σ k=1 where Ψ =( ψ 2, ψ 3,...,ψ p 1 ) δ(q k )= ψ d ψ c ψ b ψ a )) Rk ( Φ is the cumulative standard Gaussian (a probit analysis) R k ψ 1 =0, ψ p =1 is 0/1 if the judgment is lower/upper leaving for identifiability, p 1 parameters to estimate Maloney LT, Yang JN (2003). Maximum Likelihood Difference Scaling. Journal of Vision, 3(8), URL

9 Estimation of Scale Values The problem can also be conceptualized as a GLM. Each level of the stimulus is treated as a covariate in the model matrix, taking on values of 0 or ± 1 in the design matrix, depending on the presence of the stimulus in a trial and its weight in the decision variable, with absolute value signs removed. resp S1 S2 S3 S p p" p# p$ p% p& p' p( p) p* p For model identifiability, we drop the first column (fixing and σ =1). ψ 1 =0 Knoblauch & Maloney (2008) J. Stat. Software, 25, 1-25

10 Estimation of Scale Values > kk.ix <- make.ix.mat(kk) > head(kk.ix) resp stim.2 stim.3 stim.4 stim.5 stim.6 stim.7 stim.8 stim.9 stim.10 stim η (E [Y ]) = Xβ > glm(resp ~. - 1, family = binomial( "probit" ), data = kk.ix)

11 Can the MLDS analysis be extended within a mixed-effects modeling framework to account for random sensitivity variations across sessions and across observers? FD SF NM BA Difference Scale Test Control FD Test Control SF Test Control NM Test Control BA Elevation Difference Scaling of the Watercolor Illusion Devinck & Knoblauch (in preparation)

12 Mixed-effects models with MLDS Three Strategies 1. Re-parameterize in terms of parametric decision variable 2. Normalize to common scale 3. Regression on estimated coefficients

13 Knoblauch & Maloney, J. Stat. Soft., 25, Difference Scaling: Correlation in scatterplots r = 0 r = 0.1 r = 0.2 r = 0.3 r = 0.4 r = 0.5 r = 0.6 r = 0.7 r = 0.8 r = 0.9 r = r Difference Scale Value ψ(r) r 2

14 Mixed-effects models with MLDS: Re-parameterize in terms of decision variable 6 indiv runs combined average = ψ d ψ c ψ b + ψ a re-parameterized as empirical decision variable: Difference Scale 4 2 DV = ρ 2 d ρ 2 c ρ 2 b + ρ 2 a then, fit GLMM 0 Φ 1 (E[Y ]) = (β + b i )DV, b N (0, σ 2 ) r

15 resp S1 S2 S3 S4 Obs DV OL S S S S S S > library(lme4)... > gm1 <- glmer( resp ~ DV + (DV + 0 Obs) + 0, allraw.df, binomial(probit) ) > summary( gm1 ) Generalized linear mixed model fit by the Laplace approximation Formula: resp ~ DV + (DV + 0 Obs) + 0 Data: allraw.df AIC BIC loglik deviance Random effects: Groups Name Variance Std.Dev. Obs DV Number of obs: 4620, groups: Obs, 7 Fixed effects: Estimate Std. Error z value Pr(> z ) DV <2e-16 ***

16 Mixed-effects models with MLDS: Re-parameterize in terms of decision variable Difference Scale indiv runs combined average glmm pred > coef(gm1) $Obs DV S S S S S S S r

17 Mixed-effects models with MLDS: Re-parameterize in terms of decision variable S5 S6 S Difference Scale 6 S1 S2 S3 0 S r

18 Mixed-effects models with MLDS: Normalize to common scale Experiment of Fleming, Jäkel and Maloney (2011). Perception of transparency as a function of rendered index of refraction Difference Scale Index of Refraction No simple functional description of relation because of kink in curve Use each individual s scale value to compute decision variables and fit GLMM to these value; normalizes out individual shape differences. DV o = ˆψ d,o ˆψ c,o ˆψ b,o + ˆψ a,o Fleming, RW, Jäkel, F, Maloney, LT (2011). Visual perception of thick transparent materials. Psychol Sci, 22, 6:

19 Mixed-effects models with MLDS: Normalize to common scale O4 O5 O Difference Scale O1 O2 O Index of Refraction

20 Mixed-effects models with MLDS: Regression on estimated coefficients For this approach we use lmer and fit the coefficients as a function of the stimulus level using MLDS directly. ˆψ(S) (β 1 + b 1 )S +(β 2 + b 2 )S ɛ By taking the log of the coefficients, we transform the multiplicative effect to additive. We use polynomials to fit the fixed effect but also to model random differences in the shapes of the function across observers log( ˆψ(S)) (β 0 + b 0)+(β 1 + b 1)S +(β 2 + b 2)S ɛ Difference Scale Index of Refraction

21 Mixed-effects models with MLDS: Regression on estimated coefficients log( ˆψ(S)) (β 0 + b 0)+(β 1 + b 1)S +(β 2 + b 2)S ɛ First, test random effects:

22 Then, test fixed effects: Mixed-effects models with MLDS: Regression on estimated coefficients

23 Mixed-effects models with MLDS: Regression on estimated coefficients Linear Scale Log Scale O4 O5 O6 O4 O5 O Difference Scale 15 O1 O2 O3 5 Log Difference Scale 1.0 O1 O2 O Index of Refraction Index of Refraction

24 - Difference Scaling is a psychophysical technique that permits estimation of interval perceptual scales by maximum likelihood - The approach is implemented in the R package MLDS on CRAN. - We can introduce mixed-effects into MLDS models using the lme4 package (and perhaps others) via a number of strategies. Thank you.

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