Fitting the diffusion model to experimental data

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Fitting the diffusion model to experimental data

• Methods and tools

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

• The Ratcliff diffusion model
• Handling outliers
• Parameter reduction
• Fitting the model
• The DMA Toolbox
• Statistical inference
• Simulations

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The Ratcliff diffusion model

• Produces bivariate predictions
• Reaction times
• Binary response
• Separate, interpretable parameters
• Speed-accuracy trade-off (a)
• Response bias (z)
• Stimulus quality (v)
• Nondecision time (Ter)

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The Ratcliff diffusion model
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Handling outliers

• Classical methods
• Absolute cut-off (e.g. 300-3000ms)
• Relative cut-off (e.g. 2.5 SD)
• Newer methods
• Weighted mixture of process distribution and
  contaminant distribution(s) (Mixture model)
• Exponentially weighted moving average (EWMA)

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Outliers Mixture Model

• Explicit modeling of outliers (fast and slow)

Guess distribution
Delayed startup distribution
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Outliers Mixture Model

• Two new parameters
• p probability of diffusion process
• g probability of fast guess (distributes density
  over response options)

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Outliers EWMA

• Some outliers are fast guesses
• Exponentially weighted moving average
• From statistical quality control
• Detects deviation from chance performance

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Outliers EWMA

• EWMA algorithm
• sort RTs (from all conditions) choices
• choose constants L1.5, l.01
• compute zi lxi(1-l)z(i-1) and UCLi
• if zi lt UCLi ? fast guess
• otherwise ? lower limit of diffusion process
  reached
• delete observations below this lower limit

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Handling outliers

• Mixture model approach
• Exponentially weighted moving average
• Both can be combined
• (resulting in lower g)

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Reducing parameters

• Reduce processing load
• On the fitting algorithm
• On the substantive researcher (interpretation!)
• More efficient fitting

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Reducing parameters

• A normal parameter set (C conditions)

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Reducing parameters

• Absolute fixing
• E.g. Range of nondecision time st 0

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Reducing parameters

• Relative fixing
• E.g. Starting point zi ½boundary separation ai

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Reducing parameters

• Design fixing
• Describe diffusion parameter vectors as products
  of design matrices D and design parameter vectors
  ?

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Reducing parameters

• E.g. Nondecision times Ter(1), , Ter(C) equal

! New notation Design parameter vector
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Reducing parameters

• E.g. Nondecision times Ter(1), , Ter(C) equal

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Reducing parameters

• E.g. Linear increase in drift rate according to
  a known covariate E

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Reducing parameters

• Across-condition effect on drift rate only, no
  effects on other parameters

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Reducing parameters

• Advantages
• Less parameters to worry about (Last example 8C
  parameters instead of 9C)
• Impose regression models on diffusion model
  parameters
• Test hypotheses by comparing models

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Fitting the model

• Grouped Maximum Likelihood
• Fixed bins

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Fitting the model

• Choose a parameter set
• Compute expected proportions Paxi
• Condition a
• Response x
• Bin i

Upper bound of bin i
Lower bound of bin i
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Fitting the model

• Calculate observed frequency Oaxi
• Compute deviance ?P
• Minimize ? over all P

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Fitting the model

• Optimization problems
• Computationally intensive
• Local minima
• Strategies
• Good starting guess (EZDIFF)
• Several Nelder-Mead simplex runs
• Jump from suspicious points

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The DMA Toolbox

• Efficient and accurate
• Easy to use
• Fast
• Freely downloadable (sometime soon)
• http//ppw.kuleuven.be/okp/dmatoolbox/

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Data panel
Settings panels
Queue panel
Tools panel
Status
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Statistical inference

• Compare parameters (Wald test / d method)
• Compare nested models (likelihood ratio)
• Compare nonnested models (information criteria)

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Statistical inference

• Compare nested models

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Simulations Outlier treatment

• Added outliers (2.5 fast, 2.5 slow)
• Disabled/enabled outlier treatment

Relative bias ()
p -1 g -90
Standard errors (1000)
p 6 g 90
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Simulations Power analysis

• Generate 1000 data sets
• N 250 in each of 4 conditions
• Drift rate varies over conditions (e.g. 0.00
  0.10 0.20 0.30)
• Evaluate significance of difference (??)
• No-effects model
• Model allowing drift rate changes
• Significant in gt99 of data sets (a 10-6)

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Simulations Selectivity

• Generate 1000 data sets
• N 250 in each of 4 conditions
• All parameters constant over conditions
• Evaluate significance of difference (??)
• No-effects model
• Model allowing drift rate changes
• Acceptance rate should criterion a

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Future plans

• Immediate future
• Get DMAT out of beta-testing
• Work out details regarding GML
• Not-so-immediate future
• Increase flexibility (e.g. nonrandom fluctuation
  in parameters)
• Bayesian

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Fin

• Suggestions?
• Comments?
• Questions?