WDSTP: Weibull Dual-Stage Two-Phase Model of Selective Attention

Description

A continuous approximation of the Dual-Stage Two-Phase model of conflict tasks. The Dual-Stage Two-Phase model assumes that choice in conflict tasks involves two processes: a decision process and a target selection process. Unlike the CDSTP, the target selection process here is a Weibull cumulative distribution function. The decision process is an SDDM but with drift rate $$v(x,t) = (1 - w(t))*(\mu_t + c*\mu_{nt}) + w(t)*\mu_2,$$ where $w(t) = 0$ before target selection and $w(t) = 1$ after target selection. A full derivation of this model is in the ream publication.

Usage

dWDSTP(rt, resp, phi, x_res = "default", t_res = "default")
pWDSTP(rt, resp, phi, x_res = "default", t_res = "default")
rWDSTP(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 $z = b_l + w*(b_u - b_l)$.
Relative start of the target selection process ($w_{ts}$). Sets the start point of accumulation for the target selection process as a ratio of the two decision thresholds. Related to the absolute start $z_{ts}$ point via equation $z_{ts} = b_{lts} + w_ts*(b_{uts} – b_{lts})$.
Target stimulus strength ($\mu_t$).
Congruence parameter ($c$). Set experiment congruency. In congruent condition $c = 1$, in incongruent condition $c = -1$, and in neutral condition $c = 0$.
Non-target stimulus strength ($\mu_{nt}$).
Drift rate following target selection i.e. stage 2 ($\mu_2$).
Scale parameter for Weibull function ($\lambda$).
Shape parameter for Weibull function ($\kappa$).
Noise scale ($\sigma$). Model scaling 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 $a = 2b$.
Target selection decision thresholds ($b_{ts}$). Sets the location of each decision threshold for the target selection process. The upper threshold $b_{uts}$ is above 0 and the lower threshold $b_{lts}$ is below 0 such that $b_{uts} = -b_{lts} = b_{ts}$. The threshold separation $a_{ts} = 2b_{ts}$.
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

Hübner, R., Steinhauser, M., & Lehle, C. (2010). A dual-stage two-phase model of selective attention. Psychological Review, 117(3), 759-784.

Examples

Run this code

# Probability density function
dWDSTP(rt = c(1.2, 0.6, 0.4), resp = c("upper", "lower", "lower"),
       phi = c(0.3, 0.5, 1.0, 1.0, -1.0, -0.5, 8.0, 1.0, 1.0, 1.0, 2.0, 1.3, 0.0, 0.0, 1.0))

# Cumulative distribution function
pWDSTP(rt = c(1.2, 0.6, 0.4), resp = c("upper", "lower", "lower"),
       phi = c(0.3, 0.5, 1.0, 1.0, -1.0, -0.5, 8.0, 1.0, 1.0, 1.0, 2.0, 1.3, 0.0, 0.0, 1.0))

# Random sampling
rWDSTP(n = 100, phi = c(0.3, 0.5, 1.0, 1.0, -1.0, -0.5, 8.0, 1.0, 1.0, 1.0, 2.0, 1.3,
                        0.0, 0.0, 1.0))

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