Quantity discrimination in Clark's nutcrackers
Supplement
Methods
Model
Humans and non-human animals may represent numbers on a logarithmic scale (see Pica,
Lemer, Izard, & Dehaene, 2004). Under these conditions psychometric principles such as
Weber’s law may apply, that is, the ability to indicate that one stimulus is different from another
is proportional to the magnitude of the differences between the stimuli. Psychometric principles
may account for the ability of an observer to indicate which of two numbers of things (n1 or n2)
is larger during a simultaneous two-choice object task like the one used in the current study. A
model reported by Pica and colleagues (2004) based on psychophysical principles estimates the
probability of a correct response, given a choice between two present quantities (n1 and n2). The
model assumes that the internal representation of two quantities is not precise, and occasionally
n1 can be confused with n2, with errors being distributed along a Gaussian distribution for each
representation.
P(comparison) = ∫
0
𝑥+𝐴𝑏𝑠(𝑛1−𝑛2) 2
+∞ −1/2( 𝑤√𝑛12 +𝑛22 )
𝑒
√2𝜋 𝑤 √𝑛12 + 𝑛22
1
𝐴𝑏𝑠(𝑛1 − 𝑛2)
𝑑𝑥 = 𝑒𝑟𝑓𝑐 (
)
2
√2 𝑤 √𝑛12 + 𝑛22
The Gaussian distributions for the internal representations of n1 and n2 are then added to form
another Gaussian distribution with a mean of n2-n1 and a standard deviation of 𝑤√𝑛12 + 𝑛22 to
derive the probability of making an error in the comparison. The internal Weber fraction (w)
indicates the quality of the internal representation.
Results
1
Quantity discrimination in Clark's nutcrackers
Table S1. Number of correct responses (choosing the larger quantity) for each bird during for
each condition.
Condition Sony
Puck
Starr
Betsy
Susan
Total
Total
Effect
number
%
size
correct
correct
number
number
number
number
number
correct
correct
correct
correct
correct
out of 8
out of 8
out of 8
out of 8
out of 8
1v2
8
7
8
7
4
34
85
0.770
1v3
6
7
5
6
5
29
72.5
0.836
1v4
8
7
8
7
8
38
95
0.978
1v5
7
8
8
6
7
36
90
0.950
2v3
6
7
6
6
3
28
70
0.598
2
out of 40
Quantity discrimination in Clark's nutcrackers
2v4
7
8
4
7
5
31
77.5
0.687
2v5
6
7
6
3
7
29
72.5
0.612
3v4
5
6
7
5
2
25
62.5
0.395
3v5
8
7
4
7
6
32
80
0.746
3v8
7
8
4
6
7
32
80
0.746
4v5
6
7
3
5
4
25
62.5
0.413
4v6
8
7
7
7
7
36
90
0.981
4v8
4
6
7
7
7
31
77.5
0.766
5v6
3
7
6
5
7
28
70
0.547
6v7
6
7
7
6
5
31
77.5
0.881
3
Quantity discrimination in Clark's nutcrackers
6v8
5
5
7
5
6
28
70
0.816
6v9
7
8
5
6
5
31
77.5
0.784
6v12
6
4
6
6
3
25
62.5
0.447
8v9
5
4
6
4
4
23
57.5
0.429
8v10
5
8
4
5
7
29
72.5
0.612
8v16
7
7
6
6
7
33
82.5
0.958
Total %
77.4
84.5
73.8
72.6
69.0
correct
4
Quantity discrimination in Clark's nutcrackers
Table S2. Number of responses for each bird for each condition of the Number, Volume, and
Area comparison tests. W = whole pine nuts, H = half pine nuts, S = small array of nuts (~ 5mm
spacing), L = large array of nuts (~ 3cm spacing).
Sony
Puck
Starr
Betsy
Susan
number
number
number
number
number
larger
larger
larger
larger
larger
out of 8
out of 8
out of 8
out of 8
out of 8
Total
Total
Effect
number
%
size
-0.545
larger
out of 40
Number controls
1Wv2H
4
1
2
4
3
14
35
2Wv4H
3
1
3
2
3
12
30
0.784
3Wv6H
5
1
4
2
4
16
40
-0.326
Total %
50
12.5
37.5
33.3
41.7
larger
number
Volume controls
1Wv1H
7
8
7
7
8
37
92.5
0.980
2Wv2H
8
8
7
8
5
7
38
95
0.978
Quantity discrimination in Clark's nutcrackers
3Wv3H
7
8
6
8
7
36
90
0.938
Total %
91.7
100
83.3
95.8
91.7
larger
volume
Area controls
2Sv2L
2
2
3
3
2
12
30
-0.840
3Sv3L
3
3
3
3
6
18
45
-0.206
4Sv4L
5
3
3
6
4
21
52.5
0.108
Total %
41.7
33.3
37.5
50
Larger
area
6
50