Analysis of RT distributions
with R
Emil Ratko-Dehnert
WS 2010/ 2011
Session 10 – 25.01.2011
Last time...
• RT distributions in the field
– Convolution
– Ex-Gaussian, Ex-Wald, Gamma, Weibull
– Comparing functional fits
• Bootstrapping
• Creating functions in R
2
Ex-Gauss distribution
Ex-Gauss
• Very good fits to RT data
0.10
0.05
response processes)
0.00
gauss -> residual perceptual/
y5
• Correspondance to mental
0.15
0.20
and a gaussian distribution
processes (exp -> decision,
exg(mu=0, sigma=1.5, nu=0.5)
exg(mu=0, sigma=1.5, nu=1)
exg(mu=0, sigma=1.5, nu=2)
exg(mu=0, sigma=1.5, nu=5)
exg(mu=0, sigma=1.5, nu=9)
0.25
• Convolution of an exponential
0.30
Manipulating exponential parameter
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-2
0
2
4
xx
6
8
10
3
Ex-Wald distribution
Ex-Wald
• Is the convolution of a Wald and an exponential distribution
• Represents decision and response components as a
diffusion process (Schwarz, 2001)
• Neurally plausible (single cell recordings) + parameters can
be interpreted psychologically
• Very good fits to RT data and highly successful in modelling
RTs in various cognitive fields
4
Diffusion Process
Ex-Wald
Information space
Respond „A“
A
Mean drift ν
Evidence
time
B
drift rate ~N(ν,η)
Boundary separation
z
Respond „B“
5
Gamma distribution
• Series of exponential
0.5
Gamma densities
distributions
• Suitable for three-stage
exponential models
0.1
0.2
y1
0.0
• Above average fits
0.3
processes, β = reflects
processes
gamma(shape = 1, scale = 2)
gamma(shape = 2, scale = 2)
gamma(shape = 3, scale = 2)
gamma(shape = 5, scale = 1)
gamma(shape = 9, scale = 0.5)
0.4
• α = average scale of
approximate number of
Gamma
0
2
4
6
xx
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10
12
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Weibull Distribution
• Like a series of races the
weibull(shape = 1, scale = 1)
weibull(shape = 1, scale = 2)
weibull(shape = 1.5, scale = 1)
weibull(shape = 2, scale = 2)
weibull(shape = 2, scale = 3)
weibull(shape = 3.6, scale = 3)
0.8
an asymptotic description
1.0
Weibull densities
weibull distribution renders
0.4
y1
0.6
of their minima
• γ should lie between 1 (exp.)
Weibull
appropriate for processes
which can be modelled as
0.0
• Decent functional fits,
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and 3.6 (gauss.)
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1
2
3
4
5
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7
xx
races
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Palmer et al. (2009)
• Compared functional fits for three different search
tasks (feature, conjunction, spatial config.)
• H0: fit to normal distribution
• All proposed distributions could reject H0, but not
equally well
1: Ex-Gauss, 2: Ex-Wald, 3: Gamma, 4:Weibull
8
II
FUNCTIONAL FORMS OF RANDOM
VARIABLES
9
So far...
• We looked at densities and (cumulative) distribution
functions for analysis of RTs
• As all densities for RTs are unimodal and
rightskewed they can be inappropriate for analysis
• Similarly all CDF are sigmoidal, so they might not be
adequate to compare
10
RTs in ms
11
1.0
Survivor function of Ex-Gauss distributions
z5
0.4
0.2
0.0
Survivor function
0.6
0.8
exg(mu=0, sigma=1.5, nu=0.5)
exg(mu=0, sigma=1.5, nu=1)
exg(mu=0, sigma=1.5, nu=2)
exg(mu=0, sigma=1.5, nu=5)
exg(mu=0, sigma=1.5, nu=9)
0
5
10
xx
• The survivor function F(t) is the probability
that the lifetime of an object is at least t
• In oder words:
the probability that failure occurs after t
F (t )  P( RT  t )  1  F (t )
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15
1.0
0.8
0.2
0.4
1.0
0.6
Survivor function of Ex-Gauss distributions
exg(mu=0, sigma=1.5, nu=0.5)
exg(mu=0, sigma=1.5, nu=1)
exg(mu=0, sigma=1.5, nu=2)
exg(mu=0, sigma=1.5, nu=5)
exg(mu=0, sigma=1.5, nu=9)
0.0
0.8
exg(mu=0, sigma=1.5, nu=0.5)
exg(mu=0, sigma=1.5, nu=1)
exg(mu=0, sigma=1.5, nu=2)
exg(mu=0, sigma=1.5, nu=5)
exg(mu=0, sigma=1.5, nu=9)
5
10
15
0.6
xx
0.0
0.2
0.4
z5
0
0
5
10
15 13
0.2
0.4
y1
exg(mu=0, sigma=1, nu=1)
exg(mu=0, sigma=0.5, nu=1)
exg(mu=0, sigma=1, nu=1)
exg(mu=0, sigma=0.5, nu=1)
0.0
Hazard function
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0.8
1.0
Comparing Ex-Gauss CDF and hazard function
-2
0
2
4
xx
• The hazard function h(t) gives the likelihood that
an event will occur in the next small interval dt
in time, given that it has not occured before that
point in time
• Thus, it is the conditional probability:
f (t )
h(t )  lim P(t  RT  t  dt | RT  t ) / dt 
dt  0
F (t )
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6
0.2
0.4
y1
exg(mu=0, sigma=1, nu=1)
exg(mu=0, sigma=0.5, nu=1)
exg(mu=0, sigma=1, nu=1)
exg(mu=0, sigma=0.5, nu=1)
0.0
Connection to survivor function
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0.8
1.0
Comparing Ex-Gauss CDF and hazard function
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0
2
4
xx
• When F(t) is differentiable, the hazard
function can be expressed as a function of the
survivor function F(t):
d
h(t )   ln F (t )
dt
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6
Ex: h(t) of Ex-Gauss and Wald
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0.6
0.8
1.0
Comparing Ex-Gauss CDF and hazard function
0.2
0.4
y1
exg(mu=0, sigma=1, nu=1)
exg(mu=0, sigma=0.5, nu=1)
0.0
exg(mu=0, sigma=1, nu=1)
exg(mu=0, sigma=0.5, nu=1)
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0
2
4
6
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0.6
0.4
y1
exg(mu=0, sigma=1, nu=1)
exg(mu=0, sigma=0.5, nu=1)
0.2
Cumulative Hazard Function
0.8
1.0
Comparing Ex-Gauss CDF and hazard function
0.0
exg(mu=0, sigma=1, nu=1)
exg(mu=0, sigma=0.5, nu=1)
• Accumulted hazard over time
H (t ) 

t
0
-2
0
2
4
xx
h(u) du
• Is an alternative (but equivalent) representation of the hazard h(t)
• cf. Density <-> Distribution
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6
0.5
1.0
1.5
haz exg(mu=0, sigma=1, nu=1)
haz exg(mu=0, sigma=0.5, nu=1)
cum haz exg(mu=0, sigma=1, nu=1)
cum haz exg(mu=0, sigma=0.5, nu=1)
0.0
h1
2.0
2.5
3.0
Hazard and cumulative Hazard for Ex-Gauss
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-5
0
5
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0.2
0.4
y1
exg(mu=0, sigma=1, nu=1)
exg(mu=0, sigma=0.5, nu=1)
exg(mu=0, sigma=1, nu=1)
exg(mu=0, sigma=0.5, nu=1)
0.0
Literature
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0.8
1.0
Comparing Ex-Gauss CDF and hazard function
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0
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4
xx
• Ashby, Tein &
Balakrishnan, 1993
• Maddox, Ashby &
Gottlob, 1998
• Bloxom, 1984
• Thomas, 1971
• Colonius, 1988
• Burbeck & Luce, 1982
• Luce, 1986
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6
III
ESTIMATION THEORY
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Next steps
• Theoretical analysis of distributions and their
discrimination is important but in research
practice another aspect also paramount
• „good“ estimation of densities, distribution and
hazard functions are the first step to analyse RT
data
22
AND NOW TO
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