dsv_dfddm {fddm} | R Documentation |
Partial Derivative of 5-parameter DDM PDF with respect to sv (inter-trial variability in the drift rate)
Description
Partial Derivative of the density function for the 5-parameter variant of
the Ratcliff diffusion decision model (DDM) with respect to w, the
inter-trial variability in the drift rate. This variant contains the
following parameters:
v
(drift rate), a
(threshold separation),
t0
(non-decision time/response time constant), w
(relative starting point), sv
(inter-trial variability of drift),
and sigma
(diffusion coefficient of underlying Wiener process).
Usage
dsv_dfddm(
rt,
response,
v,
a,
t0,
w = 0.5,
sv = 0,
sigma = 1,
sl_thresh = 0.355,
err_tol = 1e-06
)
Arguments
rt |
A vector of response times (in seconds). If a response time is
non-positve, then its density will evaluate to |
response |
Binary response(s) that correspond(s) to either the "lower" or "upper" threshold. This model parameter can either be a singular value or a vector. The value(s) in 'response' can be of the following data types:
|
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, then the sign will be positive; similarly a negative
value indicates that the received information supports the response linked
to the lower threshold. Allowed range: |
a |
Threshold separation. Amount of information that is considered for
a decision. Large values indicate a conservative decisional style. Allowed
range: |
t0 |
Non-decision time or response time constant (in seconds). Lower
bound for the duration of all non-decisional processes (encoding and
response execution). If this value is greater than |
w |
Relative starting point. Indicator of an a priori bias in decision
making. When the relative starting point |
sv |
Inter-trial-variability of drift rate. Standard deviation of a
normal distribution with mean |
sigma |
Diffusion coefficient of the underlying Wiener process. Allowed
range:
. |
sl_thresh |
Threshold for deciding when to use the "small-time" variant
or the "large-time" variant. If the "effective response time" is greater
than |
err_tol |
Allowed error tolerance of the overall calculation. Since the
partial derivative of the density function contains one infinite sum, this
parameter defines the precision of the approximation to that infinite sum.
If the provided error tolerance is less than |
Details
All of the model inputs and parameters (rt
, response
,
v
, a
, t0
, w
, sv
, sigma
) can be
input as a single value or as a vector of values. If input as a vector of
values, then the standard R
input wrapping will occur.
sv
- Both the "small-time" and "large-time" variants of the density
function have two further variants: one with a constant drift rate v
(i.e., sv
= 0
), and one with a variable drift rate v
(i.e., sv
> 0
). The details of the differences between these
two density functions can be found in our associated paper. To use the
density function with a constant drift rate, leave the parameter sv
to its default value of sv = 0
, as this will indicate no drift to the
function. To use the density function with a variable drift rate, set the
parameter sv
to some non-negative value, i.e., sv
> 0
.
sigma
- The default value of this parameter is 1
because it
only scales the parameters v
, a
, and sv
, as shown
above. However, other formulations of the DDM may set sigma = 0.1
(see Ratcliff (1978), the fourth reference), so care must be taken when
comparing the results of different formulations.
Value
A vector containing the partial derivatives of the DDM PDF with
precision err_tol
whose length matches that of the longest input
parameter (usually rt
).
References
Navarro, D. J., & Fuss, I. G. (2009). Fast and accurate calculations for first-passage times in Wiener diffusion models. Journal of Mathematical Psychology, 53(4), 222-230.
Gondan, M., Blurton, S. P., & Kesselmeier, M. (2014). Even faster and even more accurate first-passage time densities and distributions for the Wiener diffusion model. Journal of Mathematical Psychology, 60, 20-22.
Blurton, S. P., Kesselmeier, M., & Gondan, M. (2017). The first-passage time distribution for the diffusion model with variable drift. Journal of Mathematical Psychology, 76, 7-12.
Hartmann, R., Klauer, K. C. (2021). Partial derivatives for the first-passage time distribution in Wiener diffusion models. Journal of Mathematical Psychology, 103, 102550.
Ratcliff, R. (1978). A theory of memory retrieval. Psychological review, 85(2), 59.
Examples
# derivative with respect to rt (response time)
dt_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
sv = 0.4, err_tol = 1e-6)
# derivative with respect to t0 (non-decision time)
dt0_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
sv = 0.4, err_tol = 1e-6)
# derivative with respect to a (threshold separation)
da_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
sv = 0.4, err_tol = 1e-6)
# derivative with respect to v (drift rate)
dv_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
sv = 0.4, err_tol = 1e-6)
# derivative with respect to w (relative starting point)
dw_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
sv = 0.4, err_tol = 1e-6)
# derivative with respect to sv (inter-trial variability in the drift rate)
dsv_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
sv = 0.4, err_tol = 1e-6)
#-------- Log Versions --------#
# # derivative with respect to rt (response time)
# dt_log_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
# sv = 0.4, err_tol = 1e-6)
# # derivative with respect to t0 (non-decision time)
# dt0_log_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
# sv = 0.4, err_tol = 1e-6)
# # derivative with respect to a (threshold separation)
# da_log_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
# sv = 0.4, err_tol = 1e-6)
# # derivative with respect to v (drift rate)
# dv_log_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
# sv = 0.4, err_tol = 1e-6)
# # derivative with respect to w (relative starting point)
# dw_log_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
# sv = 0.4, err_tol = 1e-6)
# # derivative with respect to sv (inter-trial variability in the drift rate)
# dsv_log_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
# sv = 0.4, err_tol = 1e-6)