PAM: Piecewise Attention Model

Description

The PAM (aka dual-process model) is an evidence accumulation model developed to study cognition in conflict tasks like the Eriksen flanker task. It is similar to the SSP, but instead of a gradual narrowing of attention, target selection is discrete. Its total drift rate is $$v(x,t) = 2*a_{outer}*p_{outer} + 2*a_{inner}*p_{inner} + a_{target}*p_{target},$$ where $a_{inner}$ and $a_{outter}$ are 0 if $t >= t_s$ and 1 otherwise. The PAM otherwise maintains the parameters of the SDDM.

Usage

dPAM(rt, resp, phi, x_res = "default", t_res = "default")
pPAM(rt, resp, phi, x_res = "default", t_res = "default")
rPAM(n, phi, dt = 1e-05)

Value

For the density a list of PDF values, log-PDF values, and the sum of the log-PDFs, for the distribution function a list of of CDF values, log-CDF values, and the sum of the log-CDFs, and for the random sampler a list of response times (rt) and response thresholds (resp).

Arguments

rt

vector of response times

resp

vector of responses ("upper" and "lower")

phi

parameter vector in the following order:

Non-decision time ($t_{nd}$). Time for non-decision processes such as stimulus encoding and response execution. Total decision time t is the sum of the decision and non-decision times.
Relative start ($w$). Sets the start point of accumulation as a ratio of the two decision thresholds. Related to the absolute start z point via equation $z = b_l + w*(b_u - b_l)$.
Perceptual input strength of outer units ($p_{outer}$).
Perceptual input strength of inner units ($p_{inner}$).
Perceptual input strength of target ($p_{target}$).
Target selection time ($t_s$).
Noise scale ($\sigma$). Model noise scale parameter.
Decision thresholds ($b$). Sets the location of each decision threshold. The upper threshold $b_u$ is above 0 and the lower threshold $b_l$ is below 0 such that $b_u = -b_l = b$. The threshold separation $a = 2b$.
Contamination ($g$). Sets the strength of the contamination process. Contamination process is a uniform distribution $f_c(t)$ where $f_c(t) = 1/(g_u-g_l)$ if $g_l <= t <= g_u$ and $f_c(t) = 0$ if $t < g_l$ or $t > g_u$. It is combined with PDF $f_i(t)$ to give the final combined distribution $f_{i,c}(t) = g*f_c(t) + (1-g)*f_i(t)$, which is then output by the program. If $g = 0$, it just outputs $f_i(t)$.
Lower bound of contamination distribution ($g_l$). See parameter $g$.
Upper bound of contamination distribution ($g_u$). See parameter $g$.

x_res

spatial/evidence resolution

t_res

time resolution

n

number of samples

dt

step size of time. We recommend 0.00001 (1e-5)

Author

Raphael Hartmann & Matthew Murrow

References

White, C. N., Ratcliff, R., & Starns, J. J. (2011). Diffusion models of the flanker task: Discrete versus gradual attentional selection. Cognitive Psychology, 63(4), 210-238.

Examples

Run this code

# Probability density function
dPAM(rt = c(1.2, 0.6, 0.4), resp = c("upper", "lower", "lower"),
     phi = c(0.25, 0.5, -0.3, -0.3, 0.3, 0.25, 1.0, 0.5, 0.0, 0.0, 1.0))

# Cumulative distribution function
pPAM(rt = c(1.2, 0.6, 0.4), resp = c("upper", "lower", "lower"),
     phi = c(0.25, 0.5, -0.3, -0.3, 0.3, 0.25, 1.0, 0.5, 0.0, 0.0, 1.0))

# Random sampling
rPAM(n = 100, phi = c(0.25, 0.5, -0.3, -0.3, 0.3, 0.25, 1.0, 0.5, 0.0, 0.0, 1.0))

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