Consider what happens when the target is clockwise in Experiment 4’s uncued condition. Let
c represent that clockwise target and let
d represent the distracter. Thus the stimulus in the target position can be described
c+
n1 and the stimulus in the distracter position can be described
d+
n2. Finally, assume that the observer responds incorrectly when
where
η represents a sample of the internal noise and the templates
wa and
wc, conform to
7. To determine whether
f(
x) = Max{
x,c} or
f(
x) = Sgn(
x)|
x|
p is a better representation, assume for the time being that
Given the difference between
c and
d,
wct(
c +
n1) >
wct (
d +
n2) and
wat(
d +
n1) >
wat (
c +
n2) for virtually any noise samples
n1 and
n2. Because
f is non-decreasing, we can simplify
12 to say that the observer will respond incorrectly on clockwise trials when
These last two equations can be combined to yield
If
p were much greater than 1, we would not expect to find any effect of
n2 on response accuracy. However, the results of Experiment 4 indicate that incorrect responses occur not only when
ctn1 (and thus
wctn1) is negative, but also when
atn2 (and thus
watn2) is positive; thus
p must not be much greater than 1. In fact, when
p is exactly 1, and
wa and
wc are constrained to be rotated versions of
d, simulation indicates that
wctc−
watc is maximized when
wa and
wc are rotated ±28 degrees, a value that conforms to the result of Experiment 4 (see main text).