Pleskac and Busemeyer's 2DSD Model for Decision Confidence

Likelihood function and random number generator for a generalization of the 2DSD Model presented by Pleskac & Busemeyer (2010). It includes following parameters: DDM parameters: a (threshold separation), z (starting point; relative), v (drift rate), t0 (non-decision time/ response time constant), d (differences in speed of response execution), sv (inter-trial-variability of drift), st0 (inter-trial-variability of non-decisional components), sz (inter-trial-variability of relative starting point), s (diffusion constant).

Usage

d2DSD(rt, response = "upper", th1, th2, a, v, t0 = 0, z = 0.5, d = 0,
  sz = 0, sv = 0, st0 = 0, tau = 1, lambda = 0, s = 1,
  simult_conf = FALSE, precision = 6, z_absolute = FALSE,
  stop_on_error = TRUE, stop_on_zero = FALSE)

r2DSD(n, a, v, t0 = 0, z = 0.5, d = 0, sz = 0, sv = 0, st0 = 0,
  tau = 1, lambda = 0, s = 1, delta = 0.01, maxrt = 15,
  simult_conf = FALSE, z_absolute = FALSE, stop_on_error = TRUE)

Arguments

rt: a vector of RTs. Or for convenience also a data.frame with columns rt and response.
response: character vector, indicating the decision, i.e. which boundary was met first. Possible values are c("upper", "lower") (possibly abbreviated) and "upper" being the default. Alternatively, a numeric vector with values 1=lower and 2=upper or -1=lower and 1=upper, respectively. For convenience, response is converted via as.numeric also allowing factors. Ignored if the first argument is a data.frame.
th1: together with th2: scalars or numerical vectors giving the lower and upper bound of the interval, in which the accumulator should end at the time of the confidence judgment (i.e. at time rt+tau). Only values with th2>=th1 are accepted.
th2: (see th1)
a: threshold separation. Amount of information that is considered for a decision. Large values indicate a conservative decisional style. Typical range: 0.5 < a < 2
v: drift rate. Average slope of the information accumulation process. The drift gives information about the speed and direction of the accumulation of information. Large (absolute) values of drift indicate a good performance. If received information supports the response linked to the upper threshold the sign will be positive and vice versa. Typical range: -5 < v < 5
t0: non-decision time or response time constant (in seconds). Lower bound for the duration of all non-decisional processes (encoding and response execution). Typical range: 0.1 < t0 < 0.5. Default is 0.
z: (by default relative) starting point. Indicator of an a priori bias in decision making. When the relative starting point z deviates from 0.5, the amount of information necessary for a decision differs between response alternatives. Default is 0.5 (i.e., no bias).
d: differences in speed of response execution (in seconds). Positive values indicate that response execution is faster for responses linked to the upper threshold than for responses linked to the lower threshold. Typical range: -0.1 < d < 0.1. Default is 0.
sz: inter-trial-variability of starting point. Range of a uniform distribution with mean z describing the distribution of actual starting points from specific trials. Values different from 0 can predict fast errors (but can slow computation considerably). Typical range: 0 < sz < 0.2. Default is 0. (Given in relative range i.e. bounded by 2*min(z, 1-z))
sv: inter-trial-variability of drift rate. Standard deviation of a normal distribution with mean v describing the distribution of actual drift rates from specific trials. Values different from 0 can predict slow errors. Typical range: 0 < sv < 2. Default is 0.
st0: inter-trial-variability of non-decisional components. Range of a uniform distribution with mean t0 + st0/2 describing the distribution of actual t0 values across trials. Accounts for response times below t0. Reduces skew of predicted RT distributions. Values different from 0 can slow computation considerably. Typical range: 0 < st0 < 0.2. Default is 0.
tau: post-decisional accumulation time. The length of the time period after the decision was made until the confidence judgment is made. Range: tau>0. Default: tau=1.
lambda: power for judgment time in the division of the confidence measure by the judgment time (Default: 0, i.e. no division which is the version of 2DSD proposed by Pleskac and Busemeyer)
s: diffusion constant. Standard deviation of the random noise of the diffusion process (i.e., within-trial variability), scales a, v, sv, and th's. Needs to be fixed to a constant in most applications. Default is 1. Note that the default used by Ratcliff and in other applications is often 0.1.
simult_conf: logical. Whether in the experiment confidence was reported simultaneously with the decision, as then decision and confidence judgment are assumed to have happened subsequent before response and computations are different, when there is an observable interjudgment time (then simult_conf should be FALSE).
precision: numerical scalar value. Precision of calculation. Determines the the stepsize of integration w.r.t. z and t0. Represents the number of decimals precisely computet on average. Default is 6.
z_absolute: logical. Determines whether z is treated as absolute start point (TRUE) or relative (FALSE; default) to a.
stop_on_error: Should the diffusion functions return 0 if the parameters values are outside the allowed range (= FALSE) or produce an error in this case (= TRUE).
stop_on_zero: Should the computation of densities stop as soon as a density value of 0 occurs. This may save a lot of time if the function is used for a likelihood function. Default: FALSE
n: integer. The number of samples generated.
delta: numeric. Discretization step size for simulations in the stochastic process
maxrt: numeric. Maximum decision time returned. If the simulation of the stochastic process exceeds a decision time of maxrt, the response will be set to 0 and the maxrt will be returned as rt.

Value

d2DSD gives the density/likelihood/probability of the diffusion process producing a decision of response at time rt and a confidence judgment corresponding to the interval [ th1, th2]. The value will be a numeric vector of the same length as rt.

r2DSD returns a data.frame with three columns and n rows. Column names are rt (response time), response (-1 (lower) or 1 (upper), indicating which bound was hit), and conf (the value of the confidence measure; not discretized!).

The distribution parameters (as well as response, tau, th1 and th2) are recycled to the length of the result. In other words, the functions are completely vectorized for all parameters and even the response boundary.

Details

For confidence: tau (post-decisional accumulation time), lambda the exponent of judgment time for the division by judgment time in the confidence measure, th1 and th2 (lower and upper thresholds for confidence interval).

Note that the parameterization or defaults of non-decision time variability st0 and diffusion constant s differ from what is often found in the literature.

The drift diffusion model (DDM; Ratcliff and McKoon, 2008) is a mathematical model for two-choice discrimination tasks. It is based on the assumption that information is accumulated continuously until one of two decision thresholds is hit. For introduction see Ratcliff and McKoon (2008).

The 2DSD is an extension of the DDM to explain confidence judgments based on the preceding decision. It assumes a post decisional period where the process continues the accumulation of information. At the end of the period a confidence judgment (i.e. a judgment of the probability that the decision was correct) is made based on the state of the process. Here, we use a given interval, given by th1 and th2, assuming that the data is given with discrete judgments and pre-processed, s.t. these discrete ratings are translated to the respective intervals. The 2DSD Model was proposed by Pleskac and Busemeyer (2010).

All functions are fully vectorized across all parameters as well as the response to match the length or rt (i.e., the output is always of length equal to rt). This allows for trial wise parameters for each model parameter.

For convenience, the function allows that the first argument is a data.frame containing the information of the first and second argument in two columns (i.e., rt and response). Other columns (as well as passing response separately argument) will be ignored.

Note

The parameterization of the non-decisional components, t0 and st0, differs from the parameterization sometimes used in the literature. In the present case t0 is the lower bound of the uniform distribution of length st0, but not its midpoint. The parameterization employed here is in line with the functions in the rtdists package.

The default diffusion constant s is 1 and not 0.1 as in most applications of Roger Ratcliff and others. Usually s is not specified as the other parameters: a, v, and sv, may be scaled to produce the same distributions (as is done in the code).

The function code is basically an extension of the ddiffusion function from the package rtdists for the Ratcliff diffusion model.

References

Pleskac, T. J., & Busemeyer, J. R. (2010). Two-Stage Dynamic Signal Detection: A Theory of Choice, Decision Time, and Confidence, Psychological Review, 117(3), 864-901. doi:10.1037/a0019737

Ratcliff, R., & McKoon, G. (2008). The diffusion decision model: Theory and data for two-choice decision tasks. Neural Computation, 20(4), 873-922.

Author

For the original rtdists package: Underlying C code by Jochen Voss and Andreas Voss. Porting and R wrapping by Matthew Gretton, Andrew Heathcote, Scott Brown, and Henrik Singmann. qdiffusion by Henrik Singmann. For the d2DSD function the C code was extended by Sebastian Hellmann.

Examples

# Plot rt distribution ignoring confidence
curve(d2DSD(x, "upper", -Inf, Inf, tau=1, a=2, v=0.4, sz=0.2, sv=0.9), xlim=c(0, 2), lty=2)
curve(d2DSD(x, "lower", -Inf, Inf, tau=1, a=2, v=0.4, sz=0.2, sv=0.9), col="red", lty=2, add=TRUE)
curve(d2DSD(x, "upper", -Inf, Inf, tau=1, a=2, v=0.4),add=TRUE)
curve(d2DSD(x, "lower", -Inf, Inf, tau=1, a=2, v=0.4), col="red", add=TRUE)

# Generate a random sample
dfu <- r2DSD(5000, a=2,v=0.5,t0=0,z=0.5,d=0,sz=0,sv=0, st0=0,  tau=1, s=1)
# Same RT distribution but upper and lower responses changed
dfl <- r2DSD(50, a=2,v=-0.5,t0=0,z=0.5,d=0,sz=0,sv=0, st0=0,  tau=1, s=1)
head(dfu)
#>     rt response       conf
#> 1 1.42        1  0.2387984
#> 2 0.81        1  0.9773506
#> 3 0.76        1  1.1594624
#> 4 1.03        1  2.2139801
#> 5 2.53        1  1.6380412
#> 6 0.21       -1 -0.5891229

d2DSD(dfu, th1=-Inf, th2=Inf, a=2, v=.5)[1:5]
#> [1] 0.18807364 0.43036375 0.46031878 0.31945417 0.04162406
# Scaling diffusion parameters leads do same density values
s <- 2
d2DSD(dfu, th1=-Inf, th2=Inf, a=2*s, v=.5*s, s=2)[1:5]
#> [1] 0.18807364 0.43036375 0.46031878 0.31945417 0.04162406
if (requireNamespace("ggplot2", quietly = TRUE)) {
  require(ggplot2)
  ggplot(dfu, aes(x=rt, y=conf))+
    stat_density_2d(aes(fill = after_stat(density)), geom = "raster", contour = FALSE) +
    facet_wrap(~response)
}

boxplot(conf~response, data=dfu)


# Restricting to specific confidence region
dfu <- dfu[dfu$conf >0 & dfu$conf <1,]
d2DSD(dfu, th1=0, th2=1, a=2, v=0.5)[1:5]
#> [1] 0.04546310 0.10403197 0.17687802 0.04932451 0.04948609

# If lower confidence threshold is higher than the upper, the function throws an error,
# except when stop_on_error is FALSE
d2DSD(dfu[1:5,], th1=1, th2=0, a=2, v=0.5, stop_on_error = FALSE)
#> error: invalid parameter combination th1 = 1, th2 = 0
#> error: invalid parameter combination th1 = 1, th2 = 0
#> [1] 0 0 0 0 0