All analyses were performed using the open source statistical programming language R (
http://www.r-project.org/, in the public domain)
15 in the RStudio environment.
18 FoS curves were constructed for each of the three tested locations by using a maximum likelihood estimation method to fit the following function:
\begin{eqnarray*}\Psi \left( {x,t} \right) = fp + \left( {1 - fp - fn} \right) \times \left[ {1 - G\left( {x,t,s} \right)} \right]\end{eqnarray*}
where
fp is the false positive rate defining the lower asymptote of Ψ,
fn is the false negative rate defining the upper asymptote of Ψ, and
G(
x,t,s) is the value at
x of a cumulative Gaussian function with mean
t and standard deviation
s. The mean (
t), standard deviation (
s), and upper asymptote (
fn) were free parameters in the fitting procedure. The lower asymptote (
fp) was set at 0.5 for the 2IFC task as per the recommendations of Wichmann and Hill
19 (
Fig. 2; Fig.
3), whereas for the yes-no task,
fp was set at the false positive response rate, estimated as the proportion of false positive catch trials for which a response was detected (i.e. the lower asymptote is set to 0 if no catch trial responses are detected). The difference in lower asymptote position for the 2 tasks is inherent to the nature of the tasks: choosing between 2 alternatives results in a chance level of 50% of picking correctly, whereas for a yes-no task an observer may not detect any stimuli, resulting in a possible 0% detection rate. The FoS curve shown in
Figure 1A is for the yes-no task.
FoS curves were used to quantify sensitivity (intensity value corresponding to 0.5 seen for yes-no task and 0.75 correct for 2IFC task), response variability (standard deviation of the fit), and maximum response rate (upper asymptote of the FoS curve). In the absence of criterion effects in the yes-no task, sensitivity and response variability quantified this way are mathematically equivalent between yes-no and 2IFC FoS curves despite the difference in scaling of the FoS curve. Consequently, we assume that within-participant, within-location differences in sensitivity and response variability between the two tasks result from criterion bias in the yes-no task. Because an unseen stimulus has a 50% probability of a correct response in the 2IFC task, false negative rates (fn) are expected to differ by a factor of 2 between the 2 methods, such that maximum response rates (MR = 1-fn) are expected to be related by the function 1-MRyes-no = 1–2MR2IFC. Note that maximum response rates were the upper asymptote of the fitted functions and were not constrained to the intensity range of the perimeter, so some maximum response rates were inferred from stimulus intensities below 0 dB.
Comparisons were performed using linear mixed models, accounting for within-subject effects. Models of the form x ∼ 1 + (1|participant) in which x denotes the parameter of interest, 1 denotes the intercept representing the fixed effect of mean paired difference between the 2IFC and yes-no tasks, and (1|participant) represents random effects of participant were compared with null models without the fixed effect. Residuals were checked for all models and found to have approximately Gaussian distributions. Models were compared by χ2 likelihood ratio test, with P < 0.017 being considered statistically significant after accounting for familywise error rate by Bonferroni correction. This approach is analogous to paired t-tests while accounting for within-subject effects.