Introduction
In fixed interval schedules (FI), subjects receive reinforcement only
for the first response that occurs after a specific amount of time has
elapsed (the fixed interval). Responses happening before the interval
ends are recorded but have no consequence. Fixed-interval schedules are
known to produce characteristic response patterns. These patterns
typically involve an initial pause at the beginning of each interval,
followed by a period of increasing responding that reaches a high and
steady rate until reinforcement is delivered.
Curvature analysis provides a quantitative measure to characterize
how responding deviates from an ideal increasingly steady rate. This
information is valuable for understanding how different factors, such as
drugs or manipulations of the reinforcement schedule itself, can
influence response patterning. The curvature index can reveal deviations
from a steady responding rate and help identify how different factors
influence response patterning.
Here we introduce two functions for calculating the curvature index,
one based on numerical integration and another using Fry’s derivation.
The first, named curv_index_int takes the following
parameters:
cr a numeric vector representing the cumulative
response data.t a numeric vector representing time points
corresponding to the x-axis in a cumulative response distribution.
The second, named curv_index_fry takes the following
parameters:
cr a numeric vector of cumulative response.time_in a numeric vector of the response times.if_val a numeric value for the base of the triangle
(e.g., FI value or the last time value).n the number of sub-intervals to divide the area under
the curve by (4 by default).
Both functions return a numeric value for the curvature index.
Example
First let’s load an example data set from an experimental trial and
create the data points for the cumulative response vector:
data("r_times")
r_times <- r_times[r_times < 60] # This example is a Peak Procedure trial, so we keep only the responses up to the FI value of 60s.
head(r_times, n = 30)
## [1] 28.1 40.7 44.2 44.4 44.7 45.0 45.4 47.9 48.1 48.3 48.6 48.8 49.8 50.2 50.7
## [16] 51.2 51.4 51.7 51.9 52.7 53.0 53.5 53.7 53.9 54.1 54.3 54.9 55.3 55.5 55.7
cr <- seq_along(r_times) # The Cumulative Response from 1 to n data points.
cr
## [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
## [26] 26 27 28 29 30 31 32 33 34 35 36 37
Now let’s calculate the curvature index using both methods for
comparison purposes:
int_index <- curv_index_int(cr, r_times)
fry_index <- curv_index_fry(cr, r_times, 60)
Note that for the curv_index_fry function
we omitted the n parameter, using the default subdivision
value of 4.
## [1] "Numerical integration index: 0.501803743737452"
## [1] "Fry's integration index: 0.655405405405405"
