EZ2-package: EZ2 diffusion modeling of Response Time and Accuracy

In EZ2: Modeling Human Response Times and Accuracy using the EZ/EZ2 Diffusion Model

Description

EZ2 fits a simplified version of the diffusion model for human and monkey response times and accuracy to the means and variances of the observables. Ratcliff's diffusion model specifies the information accumulation process in the brain in a two-alternative forced choice perceptual decision as a drift diffusion that conforms to the stochastic differential equation

dZ(t) = ν dt + sdW(t),

where ν specifies the accumulation speed (which is a function of perceptual quality of the stimulus as well as the quality of the stimulus' stored representation in memory), and s is the noisiness of the accumulation process. The process is assumed to start at Z(0) = z (z is often called the “starting point”). Each alternative is associated with a direction in which the process Z(t) develops. A decision can be made only if sufficient information is accumulated towards either of the alternatives. That is, the model assumes that there are thresholds associated with each of the two alternatives, which cause the brain to make the decision in favor of the reached threshold. For the alternative that is associated with the downward direction, the threshold is conventionally set at 0. For the alternative that is associated with the upward direction the threshold is conventionally called a (the “boundary separation”). The starting point z lies in between the lower and upper threshold (0 < z < a) and is interpreted as a bias parameter: If one alternative is for some reason favored over the other (e.g., because of relative frequency of occurence), the starting point is closer to that alternative. The time it takes Z to reach either of these thresholds is considered the Decision time. (In the Stochastic Calculus literature this is also called the exit time of the process Z starting at Z(0)=z from the interval (0,a).) The Response Time for a given decision is the sum of the Decision time, stimulus encoding time, and motor execution time. The latter two times are lumped into a Non-decision time parameter often called T_er. The upper threshold alternative is most commonly identified with the correct alternative for a given stimulus, while the lower threshold alternative is identified with the incorrect alternative. Hence, there are ‘correct’ response times which have a mean and variance, and there are ‘incorrect’ response times which also have a mean and variance. These means and variances are predicted by the diffusion model. Furthermore, the diffusion model predicts the percentage of error-responses.

In Ratcliff's diffusion model, holding all things equal across different trials, the starting point z, the information speed ν, and the non-decision time T_{er} still have some variability. This variability is captured in extra parameters that specify the across trials variances of these parameters. The EZ and EZ2 models assume that, holding all things equal across different trials, all parameters are constant. Hence, EZ and EZ2 leave out these extra variance parameters. The difference between EZ and EZ2 is that the EZ model assumes that z = a / 2, which is appropriate in many experiments, while EZ2 does not make this assumption.

This package contains functions to compute the response time means and variances of the decision (exit) times and proportions of error, given the parameter values of the EZ(2) model. The means and variances can be computed either conditionally on wether the response was (in)correct or by pooling across correct and incorrect response. Furthermore, the package provides utility functions for fitting data. The original intent was to provide method of moment estimators, but the package also supports least squares fitting of extensive models (including models that allow for contaminant RTs) to RTs from multiple experimental conditions.

Details

Package:	EZ2
Type:	Package
Version:	1.0
Date:	2007-08-31
License:	GPL version 2 (or later)

need a data frame containing (at least) the moments (RT means, RT variances, and proportions correct) that you would like to model.

Specify for each used moment a formula of the form vrt1 ~ EZ2.vrt(v1, z, a).

Supply the model with the data and a starting point for all unknown parameters to EZ2 or EZ2batch to calculate method of moment estimators or least squares estimators of the unknown parameters in the model (that is, v1, z, and a in vrt1 ~ EZ2.vrt(v1, z, a)).

Your model can be complex and extensive; for instance

vrt1 ~ EZ2.vrt(v1, z, a) + p0*(maxRT-minRT)^2/12+p0*(1-p0)*(EZ2.mrt(v1, z, a)-(maxRT+minRT)/2)^2

is equaly valid.

Author(s)

Raoul P. P. P. Grasman Maintainer: Raoul Grasman <[e-mail]>

References

Grasman, R. P. P., Wagenmakers, E.-J., & van der Maas, H. L. J. (2007). On the mean and variance of response times under the diffusion model with an application to parameter estimation, J. Math. Psych. 53: 55–68.

Examples

## create some data (theoretical values, not simulated) Needless to say, in reality 
## you would like to fit real data!
A = seq(.08,.13,len=6)
X2 = data.frame(A=A)
X2$vrt0 = sapply(A, function(a) EZ2.vrt(.1,.05,a))
X2$pe0 = sapply(A, function(a) EZ2.pe(.1,.05,a))
X2$vrt1 = sapply(A, function(a) EZ2.vrt(.2,a-.05,a))
X2$pe1 = sapply(A, function(a) EZ2.pe(.2,a-.05,a))

X2 = as.data.frame(X2)          # now pretend that X2 is the data frame that 
                                # you may have computed from real data

## fit an EZ2 model on each row
#  method 1:
EZ2batch(c(v0=.11,v1=.21,z=.05,a=.09), 
 vrt0 ~ EZ2.vrt(v0,z,a), 
  pe0 ~ EZ2.pe(v0,z,a), 
 vrt1 ~ EZ2.vrt(v1,a-z,a), 
  pe1 ~ EZ2.pe(v1, a-z, a), data=X2)

# method 2 (eventually less typing):
mdl <- list( vrt0 ~ EZ2.vrt(v0,z,a), 
              pe0 ~ EZ2.pe(v0,z,a), 
             vrt1 ~ EZ2.vrt(v1,a-z,a), 
              pe1 ~ EZ2.pe(v1, a-z, a)
           )
EZ2batch(c(v0=.11,v1=.21,z=.05,a=.09), mdl, data=X2)

EZ2 documentation built on May 2, 2019, 6:20 p.m.