Fitting the Ratcliff Diffusion Model: classical and Bayesian approaches

1
Fitting the Ratcliff Diffusion Model classical
and Bayesian approaches

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

2
Overview

• The Ratcliff Diffusion Model
• Substantive restrictions in the diffusion model
• The DMA Toolbox for MATLAB
• An example application
• A Bayesian implementation

3
The Ratcliff Diffusion Model

• Simultaneous analysis of reaction time and
  accuracy (or binary) data
• Wide applicability good fit to real data
• Basic idea sequential sampling
• Interesting parameters
• a Speed-accuracy trade-off
• z Participant bias
• v Quality of the stimulus
• Ter Nondecision time
• 3 trial-to-trial variability parameters

Ratcliff (1978)
4
The Ratcliff Diffusion Model
Ter
v
v
Evidence
a
z
0.0
0.125
0.250
0.375
0.500
0.625
0.750
Time (sec)
5
Fitting the RDM

• No analytical ML estimators
• High-dimensional numerical optimization
• Discretization of RT distribution

6
Fitting the RDM

• More computationallytractable

Expected proportion in bin b
Over conditions, responses and bins
Multinomial log-likelihood
7
Fitting the RDM

• Still computationally intense

8
Goal of our research

• Develop flexible methods for diffusion model
  analysis
• Handle outliers and guesses
• Implement substantive restrictions on parameters
  across conditions
• Make diffusion modeling easier by publishing
  software

9
Matrix notation

• To formalize constraints, assume the below
  equality

All the drift rates (for all conditions) in one
vector
Parameters to estimate
Restrictions you impose
10
Matrix notation

• Many types of restrictions are possible

Allow z to take any combination of values
Keep Ter constant across conditions
11
Matrix notation

• Many types of restrictions are possible

v depends on covariate E ( linear regression)
Allow a to assume only two values ( ANOVA)
12
Analysis strategy

• Data RT and accuracy
• Fit Ratcliff Diffusion Model
• Compare models of increasing complexity that
  capture across-condition changes
• If models are nested

13
The DMA Toolbox

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

14
An example

• Brightness discrimination
• 25 levels of brightness
• Accuracy vs. speed instruction
• 0 Dark1 Bright"

Ratcliff Rouder (1998)
15
Design matrices

• Model 1 Allow a to change with ACC/SPE
  instruction and v to linearly evolve with
  stimulus strength
• Model 2 Allow v to follow a polynomial
  regression ( sigmoidal shape)
• Model 3 Let v take any value
• Model 4 Allow all parameters to vary freely

16
Model fits
17
Model fits

• Polynomial regression fits pattern pretty well
• ACC/SPE instruction has marked influence on v

18
Model recovery

• Empirical and theoretical CDFs

ACC, medium bright
SPE, very dark
19
More complex models?

• Design matrices are easy, but limited
• Even moderately complex designs seem haphazard
  and ad hoc
• (but we like complex designs)

20
Bayesian implementation

• Much more computation
• Fewer computational problems
• More flexibility
• Easy to compute statistics

21
Easy to compute statistics

• Posterior distribution of speed/accuracy trade-off

22
Bayesian implementation

• Much more computation
• Fewer computational problems
• More flexibility
• Easy to compute statistics
• Complex functional forms

23
Complex functional forms

• E.g., fit a Weibull function to drift rates

24
Complex functional forms
25
Complex functional forms
26
Bayesian implementation

• Much more computation
• Fewer computational problems
• More flexibility
• Easy to compute statistics
• Complex functional forms
• Hierarchical extensions

27
Bayesian implementation

• Much more computation
• Fewer computational problems
• More flexibility
• Easy to compute statistics
• Complex functional forms
• Hierarchical extensions
• Latent classes

28
Latent classes

• Assume that a data set contains outliers, which
  come from a certain distribution
• Assign each individual trial to a component
  distribution
• Diffusion model
• Outlier
• Estimate probability that a given data point is
  an outlier

29
(No Transcript)
30
Bayesian implementation

• Much more computation
• Fewer computational problems
• More flexibility
• Easy to compute statistics
• Complex functional forms
• Hierarchical extensions
• Latent classes
• State-switching regimes (hypothetical example)

31
State-switching
32
Future work

• Apply state-switching
• Apply simple hierarchical extension
• Write more specialized software
• Pushing the limits of WinBUGS now
• Need to work out full conditionals
• Apply Bayesian model selection
• Choose between models with or without outlier
  components