The Ratcliff diffusion model

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The Ratcliff diffusion model

• Extensions and applications

Joachim Vandekerckhove and Francis
Tuerlinckx Research Group Quantitative
Psychology University of Leuven, Belgium
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Overview

• Application areas
• A diffusion model for two-choice accuracy data
• Extensions of the diffusion model
• Example applications

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The empirical situation

• Experiment let participants perform some sort of
  speeded accuracy task
• Measure their performance
• Task easy ? high accuracy, low RTs
• Task difficult ? 50 accuracy, high RTs
• Goal Learn about cognitive processes governing
  task execution

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The data

• At each subtask (trial)
• Accuracy score (0 or 1)
• Reaction time (in seconds)

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The data

•  Two-choice reaction time data 
• Accuracy and RT are not independent

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The problem

• Standard analyses (e.g., GLM) make certain
  assumptions
• Local stochastic independence
• Gaussian distributions
• These assumptions are decidedly not met

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The Ratcliff diffusion model

• Basic idea
• Represents the decision process

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The Ratcliff diffusion model

• Basic idea
• Represents the decision process
• Internal representation of evidence

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The Ratcliff diffusion model

• Basic idea
• Represents the decision process
• Internal representation of evidence
• Sequential sampling from stimulus until response
  criterion is reached

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The Ratcliff diffusion model

• Main parameters
• Boundary separation a (speed-accuracy trade-off)
• Starting point z0 (response bias)
• Drift rate ? (quality of the stimulus)

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The Ratcliff diffusion model

• Predictions
• Separate distributions of RT for correct and
  incorrect responses
• Marginal probability of a correct response
• RT and accuracy in a trial are random variables

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The Ratcliff diffusion model

• Observable data
• Ones and zeros
• Reaction times
• (Covariates)

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Some basic extensions

• Also take motor response time into account
• RT ter DT
• Allow parameters to vary across trials
• z0 U(z-sz/2,zsz/2)
• ? N(v,?)
• ter U(Ter-st/2,Terst/2)

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A further extension

• Allow for guesses and delayed start-ups
• Mixture model

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Meet the model
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Fitting the model

• Quite costly to compute (2,000/s)
• Fitting requires many evaluations of the CDF ?
  binning
• Chi-square
• Multinomial likelihood

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More extensions

• Multiple diffusion models
• Usually assume independent diffusion processes
  in different conditions
• Our approach assume dependency between
  experimental conditions ? some constancy in
  parameters
• E.g. If only quality of the stimulus (v) is
  manipulated, no reason why boundary separation
  (a) should change

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Still more extensions

• Nested diffusion modeling
• Test theoretical assumptions regarding diffusion
  parameters, across conditions
• E.g., mean drift rate in condition i
• v(i) r iq
• E.g., mean nondecision time in condition i
• Ter(i) r if i lt 3
• q otherwise

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Imposing restrictions

• Flexible method use design matrices
• Matrix representation of diffusion model
  parameters

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Imposing restrictions
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Imposing restrictions
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Estimating the parameters

• Find the minimum of the deviance function
• Still costly to compute
• Numerical approximations reduce accuracy and
  underflow leads to discontinuities
• Parameter space is bounded and wrought with local
  minima

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Estimating the parameters

• Robustness against discontinuities
• Nelder-Mead simplex algorithm
• Reduce local minima risk
• Restart algorithm several times
• Identify suspected local minima and jump away
• As fast as possible
• Get a good starting point based on linearity
  assumptions

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Appeal to a wider audience

• Diffusion model analysis is not easy
• Need to write custom software
• Need to extensively test software
• Need lots of computing power / time
• Need technical background common to only the
  nerdiest of the nerdy

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The DMA Toolbox

• Diffusion Model Analysis Toolbox
• MATLAB
• Aimed at wide range of practitioners
• No programming knowledge required
• Freely downloadable
• Efficient
• Fast (1 minute)
• Accurate (Monte Carlo simulations)

http//ppw.kuleuven.be/okp/dmatoolbox
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The DMA Toolbox

• Many features
• Outlier treatment
• Design matrices
• Fix parameters
• Compare model families (queue)
• Custom settings
• Extra tools
• Manual demos

http//ppw.kuleuven.be/okp/dmatoolbox
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The DMA Toolbox

• Visualizations

http//ppw.kuleuven.be/okp/dmatoolbox
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Example applications

• An exploratory study
• Implicit Association Test
• Relations with covariates
• A hypothesis-driven study
• Change detection
• Relative influence of experimental manipulations

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Implicit Association Test

• General idea
• Measure implicit association between a concept
  and an attribute
• Cross response mapping through blocks and measure
  change in mean RT

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Implicit Association Test

• Effect on diffusion parameters?
• Investigate effect on each separate parameter
• Diffusion model construction
• Model 1 all parameters free
• Model 2 one parameter equal across blocks
• Repeat for each parameter (a, Ter, z, v)

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Implicit Association Test

• Results
• Pooled over participants

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Implicit Association Test

• Results
• Pooled over participants

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Implicit Association Test

• Results
• Individual analyses
• H0 across individuals, delta c(1)2

2.29
3.05
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Implicit Association Test

• Results
• H0 rejected for a and z
• IAT manipulations affect a and z

38.93
33.59
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Implicit Association Test

• Correlation parameters with covariates?
• Difference in a or z

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Implicit Association Test

• Conclusions
• IAT effect shows in scale of decision space
  (boundary separation and starting point)
• No evidence of correlations with covariates
• More work needed!

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Change detection

• Simple change detection experiment
• One participant
• 3 variables
• Not fully crossed

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Change detection

• Questions
• Effect of type on drift rate
• Effect above and beyond effect of quality?
• Effect independent of quality (interaction)?
• Any other effects?

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Change detection

• Diffusion model construction
• Variable of interest is drift rate

Model 5 All parameters free
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Change detection

• Results
• Compare fit of each model
• Step 1 Effect of manipulations
• Step 2 Effect of type over quality
• Step 3 Interaction between type and quality
• Step 4 Effect on other parameters besides v

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Change detection

• Conclusions
• Type has an effect on drift rate, above and
  beyond quality
• The effects appear independent
• There are no further effects in the data
• No more research needed