To compare the generalization among the four groups, we compare difference thresholds: a color distance beyond which novel colors are treated significantly different from T+, or in other words a distance at which the chick discriminates between T+ and novel colors. We adopt the threshold criterion from standard psychophysical methods to determine sensory discrimination thresholds. Both types of thresholds are determined by uncertainty. Whereas in sensory discrimination, performance is limited by actual noise in the sensory neurons or by stimulus-inherent variations in appearance, in the present study performance is limited by uncertainty about the presence of a reward (Lynn,
2010). In the case of sensory discrimination, the threshold describes the minimal physical distance necessary to yield a just-noticeable (perceptual) difference. In terms of discrimination thresholds, “just noticeable” was defined as the difference that is detected some proportion of the time. Likewise for generalization, this means that the stimulus is treated differently (is discriminated from T+) some proportion of the time. In psychophysics, this proportion, known as the threshold criterion, is usually a performance halfway between chance level and 100% T+ choices, and is used to interpolate the threshold from a model fitted to the data. In a two-alternative forced-choice task in which the subject is asked to choose the T+ over a simultaneously presented comparison stimulus, the chance level is 50%. Therefore the threshold would be a distance from T+ at which T+ is chosen in 75% of all presentations. In a simultaneous four-alternative forced-choice task, the chance level would be 25%, so the threshold criterion would be 62.5%. However, this is only true if the three comparison stimuli are
identical. Here, we compare four
different colors simultaneously, which all have a different choice probability that has to be considered (
Equation 1). Since our model converts choice frequencies into probabilities we can apply a two-alternative forced-choice threshold criterion, despite using a multiple-choice paradigm to generate the data by asking for the relative probability
Pr(T+) of T+ compared to any novel color with a measurement value
X:
with
P(T
+) as the probability of T+ and
P(
X) as the probability of a color at a discriminable distance
X. For colors very close to T+,
Pr(T
+) is around 50% and increases with increasing discriminability of novel colors from T+ until it reaches 100% for colors that are very different from T+. Therefore, we can interpolate the difference threshold using a criterion of
Pr(T
+) = 75%. If we assume a Gaussian or a Laplace likelihood function (
Equations 2 and
3), and disregard normalization,
Equation 6 can be written as
for a Gaussian likelihood function and
for a Laplace likelihood function. As becomes obvious from
Equations 7 and
8, the thresholds can be directly calculated from the estimated parameters (
σ or
a). Assuming a threshold criterion of 0.75 (75%), solving for
X reveals a linear relationship between the parameter and the threshold, with
Xthreshold = 1.482
σ for a Gaussian likelihood function and
Xthreshold = 1.099
a for a Laplace likelihood function. This threshold interpolation is performed for each parameter value in the posterior distribution, which was obtained using the Metropolis–Hastings algorithm, as described in the previous section (
Equation 5). This way a probability distribution of thresholds with mean and upper and lower 95% confidence limits of the mean can be obtained as summary statistics and a measure of confidence to compare generalization between conditions.