Applying the Wiener diffusion
process as a psychometric
measurement model
Joachim Vandekerckhove and Francis Tuerlinckx
Research Group Quantitative Psychology, K.U.Leuven
Overview
•
•
•
•
•
•
An example problem
The diffusion model
Cognitive psychometrics
Random effects diffusion models
Explanatory diffusion models
Conclusions
An example problem
• Speeded category verification
task
– Participants evaluate item category
memberships
P
Block
1
Mammals
1
Mammals
1
Mammals
1
Mammals
…
…
Item
RT
Answer
An example problem
• Speeded category verification
task
– Participants evaluate item category
memberships
P
Block
Item
RT
Answer
1
Mammals
Dog
523
“Yes”
1
Mammals
1
Mammals
1
Mammals
…
…
An example problem
• Speeded category verification
task
– Participants evaluate item category
memberships
P
Block
Item
RT
Answer
1
Mammals
Dog
523
“Yes”
1
Mammals
Cat
475
“Yes”
1
Mammals
1
Mammals
…
…
An example problem
• Speeded category verification
task
– Participants evaluate item category
memberships
P
Block
Item
RT
Answer
1
Mammals
Dog
523
“Yes”
1
Mammals
Cat
475
“Yes”
1
Mammals
Spider
657
“No”
1
Mammals
…
…
An example problem
• Speeded category verification
task
– Participants evaluate item category
memberships
P
Block
Item
RT
Answer
1
Mammals
Dog
523
“Yes”
1
Mammals
Cat
475
“Yes”
1
Mammals
Spider
657
“No”
1
Mammals
Wombat
723
“No”
…
…
…
…
…
An example problem
• Speeded category verification
task
– Participants evaluate item category
memberships
– Measure both RT and response
An example problem
• Speeded category verification
task
– Participants evaluate item category
memberships
– Measure both RT and response
– Each participant evaluates each
item exactly once
– Expectation: “Typical” members
(e.g., Dog, Cat) are easier
An example problem
• The problem
– Standard assumptions violated
• Bivariate data (RT and binary response) do not
conform to the assumptions made by standard
models (e.g., normality)
– Different sources of variability
• RT and response are partly determined by both
participants (ability) and items (difficulty)
An example problem
• The problem
– Standard assumptions violated
• Typical problem in mathematical psychology
• Approach: use process models
An example problem
• The problem
– Standard assumptions violated
• Typical problem in mathematical psychology
• Approach: use process models
– Different sources of variability
• Typical problem in psychometrics
• Approach: hierarchical models (multilevel models;
mixed models; e.g., crossed random effects of
persons and items)
Diffusion model
• Wiener diffusion model
– Process model for choice RT
– Predicts RT and binary choice simultaneously
– Principle: Accumulation of information
x
pij

, t pij  Wiener a pij , b pij , pij , d pij

(For persons p, conditions i, and trials j.)
τ
d
a
z = ba
0.0
0.125
0.250
0.375
0.500
0.625
0.750
time
Diffusion model
• Many associated problems
– Technical issues
• Parameter estimation / Model comparison
– Substantive issues
• Difficult to combine information across participants
– Problem if many participants with few data each
– Problem if items are presented only once (e.g., words)
• Unlikely that parameters are constant in time (i.e.,
unexplained variability)
• Almost completely descriptive – differences over
persons/trials/conditions cannot be explained
Cognitive psychometrics
• Use cognitive models as measurement
model
• Try to explain differences
– between trials, manipulations and persons
– by regressing the parameters on covariates
Cognitive psychometrics
• Most common measurement model:
Gaussian
– Normal linear model (linear regression,
ANOVA):
y
N  , 2
pi
Indexes p for
persons, i for
conditions

pi

 pi = b0  b1 xi  b 2 z p 
– But often not a realistic model
– Unsuited for choice RT
Cognitive psychometrics
• Common measurement model in
psychometrics: Logistic
– Two-parameter logistic model (item response
theory):
Measurement
level describes
the data
Regression
component
explains
differences
y pi
Bernoulli  pi 

 pi = 1  e
 pi

1
 pi = b 0  b1 xi  b 2 z p 
Transform the
parameter(s) to
a linear scale
Adding random effects
• Not all data points come from the same
distribution
• Differences between participants/items/…
exist, but causes unknown
p  y pij  =

 Meas d


 p d  = 1
pij

pij
, Θ pij  p d pij  d d pij
Adding random effects
• Case of the diffusion model’s drift rate
p  x pij , t pij  =

 Wiener a
pij

, b pij , pij , d pij  N d pij |  pi ,

x
pij
d pij
, t pij  Wiener a pij , b pij , pij , d pij 

N  pi , pi2

2
pi
 dd
pij
Adding random effects
• Ratcliff diffusion model
x

Wiener a pij , b pij , pij , d pij
pij , t pij 
b pij U  b , b

p
 pij U  ,

p

N  pi ,
2
p

p

p
d pij




Measurement level
(Wiener process)
Trial-to-trial variability in
bias
Trial-to-trial variability in
nondecision time
Trial-to-trial variability in
information uptake rate
Adding random effects
• Crossed random effects diffusion model
x
pij
d pij
, t pij  Wiener a pij , b pij , pij , d pij 

N  pi , p2

 pi =  p  i
p

N 0,  
2

and i

N  ,  
2

Adding random effects
• Addition of random effects
– Allows for excess variability
• Due to item differences
• Due to person differences
– Allows to build “levels of randomness”
– Importantly, can be accomplished with the
diffusion model
– Only feasible in a Bayesian statistical
framework
Applying to data
• Crossed random effects diffusion model
p
i

N 
N 0,  
2
T
Pop. distr. of
item easiness
(distractors)
item easiness
(targets)
person aptitude

,
2
T

or i
Mean

N  D , 
2
D
Stdev
 D = 0.21   D = 0.11
T = 0.37  T = 0.12
  = 0.04

Explanatory modeling
• Previous models were descriptive
– Didn’t use covariates
– Mixed models merely quantify variability
• Use external factors as predictors to
– analyze the data
– explain the differences in parameter values
(i.e., reduce unexplained variance)
Explanatory modeling
• Variability in choice RT due to
– Inherent (stochastic) variability in sampling
– Trial-to-trial differences
– Participant effects
– Participant’s group membership
– Item effects
– Item type
– Combination of the above
–…
Explanatory modeling
• Use basic “building blocks” for modeling
– Random/Fixed effects
– Person/Item side
– Hierarchical/Crossed
– Use covariates
(continuous/categorical/binary)
Explanatory modeling
x
pij

, t pij  Wiener a pij , b pij , pij , d pij

explaining variability in drift rate
d pij

N  pi , p2

 pi =  p  i
p
i

N z
N 1 z p1  2 z p 2 
0 xi 0  z 1 xi1 
,  2
,  2


Applying to data
d pij

N  pi , p2

 pi =  p  i
p
i

N z
N 0,  
2
z 0 = 0.9698
z 1 = 0.0659

 z 1Typicalityi1 ,  
2
0

  = 0.0673
R 2 = 0.7178
Conclusions
• Category verification data
– Variance in person aptitude small (0.04)
relative to variance in item easiness (≈ 0.11)
– Item easiness correlates with typicality
Conclusions
• More results (not discussed)
– Other parameters besides drift rate may be
analyzed
• e.g., encoding time is negatively correlated with
word length (at ±7ms/letter)
– Results hold across semantic categories (not
just for mammals)
General conclusion
• Hierarchical diffusion models
– combine a realistic process model for choice
and reaction time with random effects and
explanatory covariates
– allow to analyze complex data sets in a
statistically (and substantively) principled
fashion with relative ease
Future work
• Efficient software for fitting hierarchical
diffusion models
• Model selection and evaluation methods
Thank you
• Questions, comments, suggestions
welcome