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Piéron's Law is a psychophysical regularity in signal detection tasks that states that mean response times decrease as a power function of stimulus intensity. In this article, we extend Piéron's Law to perceptual two-choice decision-making tasks, and demonstrate that the law holds as the discriminability between two competing choices is manipulated, even though the stimulus intensity remains constant. This result is consistent with predictions from a Bayesian ideal observer model. The model assumes that in order to respond optimally in a two-choice decision-making task, participants continually update the posterior probability of each response alternative, until the probability of one alternative crosses a criterion value. In addition to predictions for two-choice decision-making tasks, we extend the ideal observer model to predict Piéron's Law in signal detection tasks. We conclude that Piéron's Law is a general phenomenon that may be caused by optimality constraints.
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ORIGINAL RESEARCH ARTICLE
published: 02 January 2012
doi: 10.3389/fnins.2011.00143
Piérons law and optimal behavior in perceptual
decision-making
Leendert van Maanen*, Raoul P. P. P. Grasman, Birte U. Forstmann and Eric-Jan Wagenmakers
University of Amsterdam, Amsterdam, Netherlands
Edited by:
Michael Platt, Duke University, USA
Reviewed by:
Benjamin Hayden, Duke University
Medical Center, USA
Willem Huijbers, Harvard Medical
School, USA
*Correspondence:
Leendert van Maanen, Cognitive
Science Center Amsterdam, Plantage
Muidergracht 24, 1018 TV
Amsterdam, Netherlands.
e-mail: l.vanmaanen@uva.nl
Piérons Law is a psychophysical regularity in signal detection tasks that states that mean
response times decrease as a power function of stimulus intensity. In this article, we extend
Piérons Law to perceptual two-choice decision-making tasks, and demonstrate that the
law holds as the discriminability between two competing choices is manipulated, even
though the stimulus intensity remains constant. This result is consistent with predictions
from a Bayesian ideal observer model. The model assumes that in order to respond opti-
mally in a two-choice decision-making task, participants continually update the posterior
probability of each response alternative, until the probability of one alternative crosses a
criterion value. In addition to predictions for two-choice decision-making tasks, we extend
the ideal observer model to predict Piérons Law in signal detection tasks. We conclude
that Piérons Law is a general phenomenon that may be caused by optimality constraints.
Keywords: Piéron’s law, decision-making, random-dot motion paradigm, Bayesian ideal observer
INTRODUCTION
Cognitive science features several psychophysical laws. These laws
are not only inherently interesting, but often they also bring
together phenomena that seem unrelated at first sight. One exam-
ple of a psychophysical law is Piérons Law. Piérons Law states
that mean response times (MRT) decrease as a power law with
increasing stimulus intensity I :
MRT = αI
β
+ γ. (1)
Here α and β are scaling parameters that determine the slope
of the function and γ is an intercept (Luce, 1986). The specific
parameter values differ between application domains.
Originally, Piérons Law was formulated as an effect of stimu-
lus intensity (Piéron, 1914). For example, when participants were
instructed to press a button as soon as a light was switched on,
MRTs were found to follow a power law decrease with increas-
ing luminance of the light. Over the last century, the law has
been reported in many different domains, including brightness
detection (Piéron, 1914), tone detection (Chocholle, 1940), taste
detection of dissolved substances (Bonnet et al., 1999), odor detec-
tion (Overbosch et al., 1989), heat detection (Banks, 1973), and the
go/no-go task (Jaskowski and Sobieralska, 2004).
In recent years, Piérons Law has been found to hold in two-
alternative forced choice (2AFC) tasks as well (Pins and Bonnet,
1996; Palmer et al., 2005; Stafford et al., 2011). Similar to the initial
studies that reported Piérons Law, MRTs were found to decrease
as a power law with increasing stimulus intensity, even though the
task was not a signal detection task but instead a choice task. For
example, Stafford et al. (2011) demonstrate that reaction times in
the Stroop color naming task scale according to a power law with
the luminance of the color dimension.
These studies raise the question whether Piérons Law describes
a specific relation between stimulus intensity and MRT or whether
Piérons Law is related to the more general notion of discriminabil-
ity in (perceptual) decision-making. Thus, the typical power law
decrease in MRT is not only observed with increasing stimulus
intensity, but also when a decision becomes increasingly easy. One
reason for arguing in favor of this hypothesis is that the typi-
cal stimulus detection task that is associated with Piérons Law
can be thought of as a 2AFC task. The decision that is required
is that between providing a response or withholding a response.
The stimulus intensity can now be thought of as a factor on the
decision difficulty, because a low intensity stimulus discriminates
poorly between the two response alternatives (respond or withhold
a response).
In this paper we hypothesize that Piérons Law extends to the
more general notion of stimulus discriminability. We first show
data that support that MRTs in a 2AFC task decrease as a power
law with increasing discriminability of the choice alternatives. In
particular, we show that the relation between MRT and stimu-
lus discriminability is better supported by a power function than
by an exponential function. This comparison quantifies the like-
lihood that stimulus discriminability indeed leads to Piéron-like
behavior. The experimental result supports the idea that Piérons
Law is more broadly applicable than only in stimulus detection
paradigms. Next, we discuss how Piéron-like behavior may be a
consequence of optimal decision-making. That is, we present a
Bayesian ideal observer model (e.g., Brown et al., 2009) and show
that it predicts that MRT decreases as the power of the discrim-
inability of the choices increases. Finally, we extend the Bayesian
ideal observer model to include stimulus detection behavior, and
discuss under what assumptions optimal stimulus detection would
lead to Piérons Law.
In the process of studying Piérons Law in a 2AFC task, we also
make a methodological point: Although never explicitly formu-
lated in this way, Piérons Law relates stimulus intensity to mean
detection time, rather than mean response time. To account for the
www.frontiersin.org January 2012 | Volume 5 | Article 143 | 1
van Maanen et al. Piérons law in RDM
non-detection related components of the response time, an inter-
cept parameter is added (γ in Equation 1). To study the relation
between stimulus discriminability and decision time in the 2AFC
paradigm
1
, we separately estimate the non-decision time using a
more elaborate choice-response time model (the Linear Ballistic
Accumulator model, Brown and Heathcote, 2008). For compari-
son, we also include the analysis without this additional procedure,
and show how this leads to different – implausible – results.
EXPERIMENT 1: STIMULUS DISCRIMINABILITY IN THE RDM
TASK
A 2AFC task that is perfectly suited to address the question at
hand is the random-dot motion task (RDM, Ball and Sekuler,
1982; Britten et al., 1992; Salzman and Newsome, 1994; Shadlen
and Newsome, 2001; Churchland et al., 2008; Forstmann et al.,
2008, 2010a,b; Niwa and Ditterich, 2008; Ho et al., 2009; Mulder
et al., 2010; Van Maanen et al., 2011). In this task participants are
required to indicate the apparent direction of motion of a cloud of
dots that is presented on a computer screen. Typically, a percentage
of the dots moves in a designated direction (the target direction),
while the remaining dots move randomly. The choice alternatives
are presented on a fixed distance from the center of the dot cloud,
and their similarity can be manipulated by changing the angular
distance between the alternatives, as in Figure 1.
In Experiment 1, participants were asked to make a choice
between two alternative directions of motion. We manipulated
the angular distance between the response alternatives to opera-
tionalize choice difficulty. Response alternatives that were in close
spatial proximity were hypothesized to be hard to discriminate.
Response alternatives that were not in close spatial proximity were
hypothesized to be easy to discriminate (Figure 1). Note that in
this experiment we do not manipulate stimulus intensity, as the
dot cloud remains the same in all conditions. Therefore, evidence
for a power curve in this experiment would support the hypothesis
that Piérons Law is a general relation between MRT and stimulus
discriminability.
1
We use decision time to refer to the time required to make a choice in 2AFC tasks,
and detection time to refer to the time required to detect a stimulus in a stimulus
detection task.
FIGURE 1 | RDM stimulus display. A proportion of the dots move in a
target direction indicated by the arrows, while the remaining dots move
randomly. The discriminability of the t arget is manipulated by changing the
angular distance (d ) between the alternatives.
METHODS
Participants
Six students (three female, age range 18–20) from the University
of Amsterdam participated for course credit. All had normal or
corrected-to-normal vision.
Random-dot motion stimulus
To create the moving-dot kinematogram we used the Variable
Coherence Random-Dot Motion (VCRDM) library for Psych-
toolbox in Matlab (Brainard, 1997)
2
. The appearance of motion
in VCRDM is created by controlling the location of a subset of
dots for three frames in a row. That is, when the second frame
is drawn, the location of a subset of dots will be recomputed
to align with the target direction. The location of the remaining
dots is randomly assigned. The size of the subset –often referred
to as the coherence level is under the experimenter’s control.
We set the coherence at 25%. Pilot studies with several levels of
coherence indicated that at a coherence of 25% participants per-
formed above chance for the hardest conditions, whereas they
performed below ceiling at the easiest conditions. Each dot con-
sisted of three by three pixels, and the initial locations of each dot
sequence were uniformly distributed in an aperture of 5 visual
degrees diameter.
Design and procedure
Participants were instructed to indicate the apparent direction
of motion of the random-dot kinematogram. The two-choice
alternatives were represented by two circles (one blue, one yel-
low) that were located at 5 visual degrees from the center of
the aperture. If the direction of motion was toward the blue
alternative, the participants were required to press “z”; if the
direction of motion was toward the yellow alternative they were
required to press “m.” The location of the alternatives was ran-
domized over the top half of the imaginary circle. However, the
blue alternative always remained to the left of the yellow alter-
native. Therefore, the “z” and “m response mapping was always
congruent with the target positions on the screen. The angular
distances between the alternatives were exponentially distributed,
to maximize the probability of diverging model fits. Consecu-
tively, the angular distances were 11.5˚, 16˚, 22.5˚, 32˚, 45˚, 64˚, or
90˚. The presentation order of the different angular distances was
pseudo-randomized in such a way that there were never more than
two consecutive trials with the same angular distance between the
alternatives.
After a short training session with 14 trials (2 for each angle) the
experiment was presented in seven blocks of 210 trials each. After
each block, the participant could take a short break. Participants
received feedback on the accuracy of their response: A screen stat-
ing “Correct!” or “Incorrect!” remained visible for 400 ms. At the
beginning of each trial, a red fixation dot was presented together
with the alternatives. After 500 ms, the RDM stimulus was pre-
sented which remained on screen until the participant made a
response. If the response was faster than 200 ms a feedback screen
appeared that stated Te snel!” (too fast). If the participant did not
respond within 2000 ms, the feedback “Incorrect” was given.
2
The library can be downloaded from http://www.shadlen.org/Code/VCRDM.
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van Maanen et al. Piérons law in RDM
Analyses
Based on the hypothesis that MRT and difficulty have a power
relationship, we fitted a three-parameter power function to the
data:
MRT = αd
β
+ γ. (2)
Note that Equation (2) is almost identical to Piérons Law
(Equation 1), except that the variable indicating stimulus intensity
(I) is replaced with a variable indicating angular distance (d). In
addition, we fitted a three-parameter exponential function to the
data to study if another functional relationship could account for
the observed effects:
MRT = αe
βd
+ γ. (3)
We fitted both functions using standard simplex optimization
routines (Nelder and Mead, 1965). This was repeated 10,000 times
with randomized initial values to avoid local minima. To deter-
mine goodness of fit, we computed the correlation between the
MRT for each angle and both models’ predictions for each angle.
In addition we computed Bayesian Information Criterion (BIC,
Schwarz, 1978; Raftery, 1996) values for each model to obtain the
evidence ratio. The evidence ratio for the power function over the
exponential function is computed according to e
1
2
(BIC
pow
BIC
exp
)
(Wagenmakers and Farrell, 2004). This expression quantifies how
many times more likely the data are to have occurred under the
power function than under the exponential function, given their
respective BIC values.
In addition to these simple model fits, we fit a more complex
model in which the intercept parameter (γ) is fixed for all condi-
tions. This method provides more constraint on the parameter that
we are least interested in (γ), allowing for a clearer interpretation
of the non-linear model fits that describe the choice behavior. We
first estimate the time required for non-decision related processes
using the full RT distribution and the error rate and then use these
estimates to fix γ in the power and exponential curves.
Linear ballistic accumulator model
To estimate γ, we fit the linear ballistic accumulator (LBA, Brown
and Heathcote,2008) model to the data (Figure 2). The LBA model
assumes that a decision is made by the accumulation of evidence
for a particular option until a decision threshold has been reached.
In the LBA model, the decision threshold b is a free parameter. The
starting point of the accumulation is drawn from a uniform dis-
tribution [0, A], with A as a free parameter. The speed of the
accumulation of each alternative i is controlled by a specific drift
parameter v
i
(and typically a common drift variance parameter
s). Because each response alternative is represented by a sepa-
rate accumulator, the accumulator that reaches the threshold the
quickest determines the response, and the time required to reach
the threshold is the decision time. Crucially, the LBA model also
has a parameter that quantifies the amount of time required for
peripheral, non-decision processes (that is, the intercept γ).
In LBA and related response-time models, stimulus differences
are often modeled by allowing drift rate to vary (e.g., Brown and
Heathcote, 2008; Wagenmakers et al., 2008; Ho et al., 2009; Va n
Maanen et al., 2009; Van Maanen and Van Rijn, 2010). Therefore,
in the first LBA model that we fit to the data (i.e., Model 1), we
allowed the drift rate of the model to vary over the different angu-
lar distances, with all other parameters fixed. In addition to drift
rate, Model 2 also allows the non-decision time to vary over the
angles. Although there is no theoretical reason for this assumption,
it is important to assess that in the data the non-decision time does
not vary over angles, so that we can estimate a single value for γ.
Models 3 and 4 are identical to Models 1 and 2 but in addition
allow the drift rate variance to vary over angles (Churchland et al.,
2011). Parameter values were optimized using quantile maximum
likelihood estimation (Heathcote et al., 2002), using the 0.1, 0.3,
0.5, 0.7, and 0.9 quantiles of the RT distributions for correct and
error responses.
RESULTS AND DISCUSSION
We excluded trials in which the participants failed to respond in
time (over 2000 ms, 3.3% of the trials) as well as trials in which the
participants responded too fast (faster than 200 ms, 0.9%). Table 1
presents the mean accuracy per condition; Figure 3 presents MRT
of the correct responses, for each participant separately. The results
show that MRT decreases with the angular distance, whereas accu-
racy increases. This is consistent with the idea that smaller angular
distances are more difficult and would therefore lead to more
errors and slower correct responses. Participant 3 displays behav-
ior that does not clearly follow this result, although the smallest
angular distance yields slower responses than the widest angle. We
could not find a reason why this participant displays behavior that
does not follow our hypotheses. Therefore, there is no reason to
FIGURE 2 | LBA model. The accumulator representing response alternative
1 is on average faster than the competing accumulator. The model therefore
predicts that response alternative 1 will be selected on the majority of
trials. See text for details.
Table 1 | Average accuracy scores per angular distance (and average
within-subjects standard error of the mean, Loftus and Masson, 1994).
Angle Mean accuracy (SEM)
11.5 0.67 (0.032)
16 0.71 (0.030)
22.5 0.77 (0.027)
32 0.82 (0.025)
45 0.85 (0.023)
64 0.89 (0.020)
90 0.88 (0.021)
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van Maanen et al. Piérons law in RDM
FIGURE 3 | Mean response times (MRT) as a function of angular distance. Each plot represents the data from one participant. (Error bars represent
within-subjects standard error of the mean, Loftus and Masson, 1994). The best fitting power function and the best fitting exponential function are overlaid.
Table 2 | Correlation coefficient (ρ), BIC values, and evidence ratios
(Wagenmakers and Farrell, 2004) for the three-parameter power
function and the exponential function for each participant.
Participant ρ BIC Evidence ratio
Exp Pow Exp Pow
1 0.98 0.97 64.22 61.66 0.28
2 0.96 0.95 54.56 53.08 0.48
3 0.74 0.71 45.52 44.88 0.66
4 0.98 0.98 58.47 55.21 0.19
5 0.98 0.97 56.40 54.47 0.38
6 0.99 0.99 64.90 62.46 0.29
exclude this participant from the analyses. However, it should be
noted that the results do not hinge on this participant, and would
stand even if we would exclude this participant.
Table 2 presents the goodness of fit of the power and expo-
nential functions presented in Equations (2) and (3), as well as
evidence ratios. The fact that the evidence ratios are between 0
and 1 indicates that there is slightly more evidence in favor of the
exponential function than the power function.
It is misleading to accept these model results at face value,
because the parameter estimates cannot be interpreted in psy-
chological terms. A plausible interpretation of these functions
should be that each response consists of a decision and additional
processes such as stimulus encoding and response execution (cf.
Sternberg, 1969; Luce, 1986). The additional processes are cap-
tured by the intercept γ, whereas the actual decision process is
Table 3 | Best fitting parameter values for the three-parameter power
and exponential functions.
Participant Exponential Power
αβγ(ms) αβγ(ms)
1 0.16 0.04 540 0.51 0.18 320
2 0.23 0.05 600 0.91 0.68 550
3 1038 0.82 810 3722 4.41 810
4 0.26 0.03 810 1.31 0.08 0
5 0.28 0.05 620 0.84 0.47 510
6 0.28 0.05 570 0.94 0.58 500
captured by the remaining time course. This interpretation entails
that γ should be in a reasonable range of 200 to 500 ms. How-
ever, γ ranges here from 0 to 810 ms (Tab l e 3 ). While a negative
value for this parameter is impossible because it would suggest
a negative duration
3
,avalueofγ =0 ms suggests that no time
was required for processes unrelated to the decision. On the other
hand, a value of γ =810 ms is higher than some of the observed
response times (and even higher than the MRT for large angles),
which also makes it impossible to interpret γ as non-decision time.
Thus, the under constraint of these non-linear models leads to
implausible parameter estimates, and therefore the results lack
psychological credibility. Because of the unexpected behavior of
3
We constrained the simplex routine to only allow positive values. Without this
constraint, the best fitting models included values of γ < 0 ms.
Frontiers in Neuroscience | Decision Neuroscience January 2012 | Volume 5 | Article 143 | 4
van Maanen et al. Piérons law in RDM
Participant 3, the best fitting parameters take different values than
for the remaining participants.
For these reasons, we take a more elaborate approach in fitting
the power and exponential functions by first fitting an LBA model
to the data and then estimating the parameters of the non-linear
functions on the decision time. The LBA model that best balanced
model fit with the number of free parameters was Model 1, the
model in which only drift rate varied over angles. The average BIC
value for Model 1 over participants was 7092, which was consid-
erably less than any of the other models (7149, 8276, and 7307,
respectively)
4
. Thus, the BIC results indicate that only drift rate
variations were required to account for the data and no additional
parameters were needed.
For Model 1, the best fitting parameter values are presented
in Table 4
5
. As expected, the LBA model captures the increase
in mean drift rate with increasing angular distance. The estimated
non-decision time parameters (γ) seem to be in a reasonable range
of 270–500 ms.
A power function fits better
The LBA model allowed a more constrained estimation of the
non-decision time parameter γ. We can now estimate power and
exponential curves using this parameter estimate to assess whether
the data follows a Piéron-like pattern. We fitted Equations (2) and
(3) to the data of Experiment 1, allowing two free parameters (α
and β), and one fixed parameter γ. Again, we fit both functions
using standard simplex optimization routines (Nelder and Mead,
1965). The best fitting exponential and power curves are presented
in Figure 3 (the best fitting parameters are presented in Ta ble 6 ).
The results show that the power function was the best fit-
ting model for all participants. Tab le 5 presents the goodness of
fit for the exponential and power functions presented in Equa-
tions (3) and (2). The power function is consistently the better
model, as indicated by the lower BIC values and higher correla-
tions. Although the data of Participants 3 and 4 do not clearly
distinguish between the models, the power function clearly is the
better model for the remaining participants. In terms of the evi-
dence ratio, the power function is 15 to 415 times more likely
than the exponential function for these participants. This suggests
that an increase in angular distance yields a power law decrease
in MRT, and not an exponential law. The model comparison of
the psychologically constrained models thus shows that behavior
4
These BIC values are higher than the BIC values obtained with the power and
exponential functions because here we fit the quantiles instead of MRT.
5
The model fit itself is presented in the Appendix.
in 2AFC tasks is Piéron-like when the discriminability of choice
alternatives is manipulated.
BAYESIAN IDEAL OBSERVER
In detection tasks, Piérons Law relates MRT to stimulus intensity
by means of a power function (Equation 1). Experiment 1 showed
that MRT and choice difficulty in a 2AFC task are also related by
means of a power function. This result suggests that Piérons Law
may generalize to choice behavior in which the discriminability of
the correct alternative relative to the incorrect alternative is manip-
ulated, instead of the intensity of the stimulus. To understand
better why participants in Experiment 1 behave in accordance to
Piérons Law, we developed a Bayesian ideal observer model. This
model assumes that participants make the optimal choice, given
the uncertainty of the task (for example, a noisy stimulus).
METHODS
The RDM stimulus consists of a set of dots each moving in a
particular direction. A proportion of dots move in the same direc-
tion whereas the remaining dots move in random directions. In
order to make a correct decision, the observer needs to decide
whether a certain amount of evidence for a particular response
alternative outweighs the evidence for other response alternatives.
If the observer performs this task on average in the minimum time
required for a particular error rate, the observer is said to be opti-
mal (Bogacz et al., 2006; Brown et al., 2009). Optimal behavior
is achieved if the observer computes for each response alternative
the posterior probability that it is the target based on the evidence
observed so far (Baum and Veeravalli, 1994):
P(H
i
|D) =
P(D|H
i
)P(H
i
)
j
P(D|H
j
)P(H
j
)
,
with H
i
the hypothesis that motion direction i generated the
RDM stimulus and x
1
, ..., x
t
D the observed motion directions
over time. Here we assume that the prior probabilities for each
alternative are equal and hence can be ignored:
P(H
i
|D) =
P(D|H
i
)
j
P(D|H
j
)
.
On the basis of new incoming evidence, the model continually
updates the posterior probability of each response alternative,until
the probability of one of the alternatives crosses a preset response
criterion θ:
P(D|H
i
)
j
P(D|H
j
)
θ.
Table 4 | Best fitting LBA model parameters for the data of Experiment 1.
Participant γ (ms) Abs v
11.5
v
16
v
22.5
v
32
v
45
v
64
v
90
1 416 0.28 0.28 0.38 0.61 0.65 0.80 0.86 0.97 1.04 1.01
2 389 0.50 0.50 0.54 0.92 1.03 1.12 1.19 1.32 1.41 1.30
3 306 0.90 0.90 0.59 0.66 0.84 0.71 0.93 0.79 1.05 1.03
4 504 0.53 0.57 0.32 0.57 0.59 0.64 0.69 0.81 0.89 0.96
5 271 0.60 0.60 0.31 0.61 0.56 0.59 0.73 0.76 0.86 0.86
6 402 0.43 0.43 0.41 0.74 0.84 1.06 1.09 1.18 1.29 1.33
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van Maanen et al. Piérons law in RDM
Table 5 | Correlation coefficient (ρ) and BIC values for the
two-parameter power function and the exponential function for each
participant.
Participant ρ BIC Evidence ratio
Exp Pow Exp Pow
1 0.94 0.97 58.73 64.28 15.97
2 0.85 0.94 48.30 53.78 15.49
3 0.59 0.63 45.47 46.16 1.41
4 0.96 0.97 55.69 55.71 1.01
5 0.90 0.97 47.97 56.37 66.82
6 0.92 0.99 50.97 63.03 415.14
Table 6 | Best fitting parameter values for the two-parameter power
and exponential functions.
Participant Exponential Power
αβ αβ
1 0.24 0.0076 0.45 0.27
2 0.33 0.0061 0.59 0.23
3 0.56 0.0018 0.67 0.072
4 0.52 0.0054 0.81 0.19
5 0.51 0.0049 0.81 0.19
6 0.33 0.0086 0.70 0.31
Note that the intercept is obtained through LBA model fitting (see Table 4).
The time needed to reach that particular criterion θ determines
the decision time (DT). Because in the current context we only
discuss two-choice decisions, this equation reduces to a sequential
probability ratio test (SPRT, Wald, 1947)
P
(
D|H
1
)
P
(
D|H
2
)
= θ
, (4)
in which alternative 1 is chosen if θ
θ/(1 θ) and alternative 2
is chosen if θ
(1 θ)/θ (Baum and Veeravalli, 1994).
RESULTS AND DISCUSSION
Piéron’s law in the ideal observer model for 2AFC tasks
We assume here that the evidence for the correct response alter-
native is represented by a Gaussian distribution with a mean μ
i
representing the direction of motion. The variance of the distrib-
ution σ
2
may represent individual differences in perceptual ability,
but will remain constant in our simulations. Thus, at each time
step the model samples from the Gaussian distribution:
f
(
x|μ, σ
)
=
1
2πσ
2
e
(xμ
i
)
2
/2σ
2
(5)
until the decision threshold has been reached
6
.
6
Because of the circular arrangement of response alternatives in the RDM task a
slightly more accurate sampling distribution would be the von Mises distribution,
Without loss of generality we assume that the correct response is
always alternative 1. The evidence for alternative 1 at any time step
is computed from the Gaussian distribution function (Equation
5). Substituting Equation (5) in Equation (4) and taking the log
shows that an evidence sample S
t
for any alternative is proportional
to the distance between the alternatives:
S
t
= log
1
2πσ
2
exp
(x μ
1
)
2
/2σ
2
1
2πσ
2
exp
(x μ
2
)
2
/2σ
2
(
μ
1
μ
2
)
2
(6)
Following this computation of the evidence at each time step
the ideal observer model predicts that decision times decrease as a
power curve with the angular distance between the alternatives, as
shown by Stafford and Gurney, 2004): Suppose that at each time
step, the posterior probability of either alternative is increased
with the average evidence sample E{S
t
}. Then a decision is made
as soon as Σ
DT
t=1
E{ S
t
} log θ
, with a decision time of approx-
imately DT. Therefore DT ·E{S
t
} log θ
, which means that the
mean decision time (MDT) is
MDT = C · E
{
S
t
}
1
(7)
with C =log θ
. Because S
t
(μ
1
μ
2
)
2
, Equation (7) entails
Piérons Law (Equation 1) with the intercept γ =0.The value of
the scaling parameters α and β depends on the threshold and the
variance in the assumed Gaussian distributions.
To corroborate this result, we explored the model’s behavior
as a function of angular distance by running a 2AFC simulation.
That is, the model had two response options and we manipulated
the angular distance from 11.5˚ to 90˚. In addition, we modeled a
range of response criteria (from 0.6 to 0.9). The variance of the
sampling distribution was kept constant at σ
2
=4. The result of
these simulations are presented in Figure 4. Each data point in
Figure 4 was estimated using 10,000 Monte Carlo samples. The
model’s behavior exhibits Piérons Law: when transformed to a
log-log-scale the relation between angular distance and MDT is
approximately linear. This pattern is observed independent of the
response criterion (θ
) that is being adopted.
These simulations are in line with the results from Experiment
1, and together the model and experiment support the view that
Piérons Law is a consequence of optimal behavior, at least in
perceptual 2AFC tasks. To extend these results to the traditional
field of application of stimulus detection, we demonstrate how the
ideal observer model predicts Piérons Law in stimulus detection
experiments.
Piéron’s law in the ideal observer model for stimulus detection
In stimulus detection experiments, the intensity of a stimulus is
manipulated and participants are required to indicate the pres-
ence or absence of the stimulus. An ideal observer (ideal stimulus
detector) weighs the likelihood of the presence of the stimulus (i.e.,
which can be thought of as a Gaussian distribution wrapped around a circle. Because
this distribution leads to similar results we chose to use the more generally applicable
Gaussian.
Frontiers in Neuroscience | Decision Neuroscience January 2012 | Volume 5 | Article 143 | 6
van Maanen et al. Piérons law in RDM
FIGURE 4 | Ideal observer model behavior as a function of the angular
distance between alternatives, for four different response criterion
values. Left: Mean decision time (MDT) vs angular distance. Middle: log MDT
vs log Angle. Right: Proportion correct vs Angle. θ
: response criterion value.
P(D|H
1
) in Equation 4) against the likelihood of the absence of
the stimulus (i.e., P(D|H
2
)). Because any evidence supporting the
presence (or absence) of the stimulus contributes equal evidence
against the absence (or presence) of the stimulus, Equation (4) can
be written as
P
(
D|H
1
)
1 P
(
D|H
1
)
= θ
. (8)
Piérons Law emerges from an ideal observer model for stimulus
detection if the likelihood of the presence (or absence) of the stim-
ulus at each time step scales linearly with the stimulus intensity.
In that case, MDT will decrease with stimulus intensity accord-
ing to a power function (Equation 7) which will result in Piérons
Law. Figure 5 presents the results of a simulation of this process.
On each time step a value is drawn from a Gaussian distribution
with stimulus intensity as the mean and σ =0.005. The stimulus
intensities ranged from 0.01 to 0.05. Equation (8) was computed
until the posterior probability exceeded threshold (θ
), and the
number of time steps required was recorded. Each data point in
Figure 5 was estimated using 10,000 Monte Carlo samples. Again,
the linearity in log-log space illustrates that the non-linear relation
in the MRT data follows a power curve.
GENERAL DISCUSSION AND CONCLUSION
Piérons Law has previously only been studied in experiments in
which stimulus intensity was manipulated. Here, we hypothesized
that Piérons Law may be a special case of a more general rela-
tionship between choice difficulty and mean response time. To
support this conjecture, we performed an experiment in which the
difficulty of choice in a random-dot motion paradigm was manip-
ulated by adjusting the angular distance between two response
alternatives. We found a power law relation between angular
distance (discriminability) and mean response time, supporting
our hypothesis of a general mechanism behind Piérons Law. A
Bayesian ideal observer model showed that participants perform-
ing a 2AFC task may respond in a Piéron-like manner because
it is the optimal way of minimizing overall response times (for a
fixed error rate). The ideal observer model can also be extended
to stimulus detection behavior, showing that Piérons Law reflects
optimal detection behavior under varying stimulus intensities.
FIGURE 5 | Ideal observer model behavior as a function of stimulus
intensity, for five different response criterion values. Left: Mean decision
time (MDT) vs stimulus intensity. Right: log MDT vs log Intensity. θ
:
response criterion value.
One particular aspect of our study deserves some additional
consideration. We conclude that the decline of MRT with choice
difficulty in Experiment 1 can be best described by a power func-
tion. An important step toward this conclusion was to determine
which part of the observed response time we wanted to explain.
As we have argued, directly fitting the models to the observed
data does not provide enough constraint to make an appropri-
ate model selection inference. Therefore, we first determined the
decision time by subtracting an estimate for the non-decision
time from the RTs. The non-decision time was estimated using
the LBA model of choice-response time (Brown and Heathcote,
2008).
The validity of this approach rests on the extent to which it
is reasonable to exclude non-decision time and focus on decision
time. Because the exponential and power functions also implicitly
focus on decision time, we believe that in this study, our approach
is plausible. After subtracting non-decision time the evidence in
favor of the power function was overwhelming, which was not
the case for the models that were directly fitted to MRT. In some
instances, the traditional approach may lead to an accurate esti-
mate of the non-decision time parameter. In those cases, a more
elaborate two-step fitting approach may not appear to be neces-
sary. However, even in such cases we recommend the two-step
fitting approach.
www.frontiersin.org January 2012 | Volume 5 | Article 143 | 7
van Maanen et al. Piérons law in RDM
It should be noted that although we used a specific processing
model to obtain appropriate non-decision time estimates (the LBA
model), the same results could have been obtained with any other
model that takes the variance in the RT distribution into account
(e.g., Donkin et al., 2011). Where the LBA model is a full process
model that attempts to describe many aspects of decision-making,
we could have also used simpler process models (e.g., the EZ dif-
fusion model, Wagenmakers et al., 2007), or even any descriptive
model of the RT distribution (Matzke and Wagenmakers, 2009).
In summary, we have shown that MRT in a 2AFC task decreases
with stimulus discriminability according to a power function. In
addition, we provided a Bayesian ideal observer analysis of both
2AFC tasks and stimulus detection tasks. This analysis showed that
optimal behavior in these tasks follows a power function when
stimulus discriminability is manipulated. These results support
the view that Piérons Law originates from optimal information
processing.
AUTHOR NOTE
Leendert van Maanen, Raoul Grasman, Birte Forstmann, and
Eric-Jan Wagenmakers, Department of Psychology, University of
Amsterdam. Correspondence concerning this article should be
addressed to Leendert van Maanen,Cognitive Science CenterAms-
terdam,University of Amsterdam, Plantage Muidergracht 24, 1018
TV Amsterdam, the Netherlands, e-mail: l.vanmaanen@uva.nl.
We thank Helen Steingröver for data collection. This research
was supported by VENI and VIDI grants from the Netherlands
Organization for Scientific Research.
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Conflict of Interest Statement: The
authors declare that the research was
conducted in the absence of any com-
mercial or financial relationships that
could be construed as a potential con-
flict of interest.
Received: 20 September 2011; accepted:
12 December 2011; published online: 02
January 2012.
Citation: van Maanen L, Grasman
RPPP, Forstmann BU and Wagenmak-
ers E-J (2012) Piérons law and opti-
mal behavior in perceptual decision-
making. Front. Neurosci. 5:143. doi:
10.3389/fnins.2011.00143
This article was submitted to Frontiers
in Decision Neuroscience, a specialty of
Frontiers in Neuroscience.
Copyright © 2012 van Maanen, Gras-
man, Forstmann and Wagenmakers. This
is an open-access article distributed under
the terms of the Creative Commons Attri-
bution Non Commercial License, which
permits non-commercial us e, distribu-
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are credited.
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van Maanen et al. Piérons law in RDM
APPENDIX
Figures A1A6 present the fit of the Linear Ballistic Accumulator (LBA) model to the data of Experiment 1. Each figure
represents the data (black circles) and model fit (red lines) for one participant. The top line represents the cumulative RT dis-
tribution of correct responses, the bottom line represents the cumulative RT distribution of incorrect responses. Both lines are
weighted to the probability of an incorrect response. Each graph represents the different angular distances (d) used in Experiment 1.
FIGURE A1 | LBA model fit for Participant 1.
Frontiers in Neuroscience | Decision Neuroscience January 2012 | Volume 5 | Article 143 | 10
van Maanen et al. Piérons law in RDM
FIGURE A2 | LBA model fit for Participant 2.
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van Maanen et al. Piérons law in RDM
FIGURE A3 | LBA model fit for Participant 3.
Frontiers in Neuroscience | Decision Neuroscience January 2012 | Volume 5 | Article 143 | 12
van Maanen et al. Piérons law in RDM
FIGURE A4 | LBA model fit for Participant 4.
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van Maanen et al. Piérons law in RDM
FIGURE A5 | LBA model fit for Participant 5.
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van Maanen et al. Piérons law in RDM
FIGURE A6 | LBA model fit for Participant 6.
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