Psychometric functions from each experiment were fit using
psignifit 4 to estimate threshold and slope parameters via a Bayesian numerical integration method (
Schütt, Harmeling, Macke, & Wichmann, 2016). A cumulative Gaussian was used as the underlying function, with contrast values expressed in decibel (dB) units (where
\(C_{dB} = 20\log_{10}(C_{\%})\)). We converted the slope estimates (σ parameters from the fitted Gaussians) to equivalent Weibull β values using the approximation β = 10.3/σ. Threshold was defined as the target contrast corresponding to an accuracy of 75% correct.
The two-stage model of
Meese, Georgeson, and Baker (2006) was fit to the threshold data from Experiments 1 and 3 using a simplex algorithm to minimize the error between the model and data. We normalized the thresholds and pedestal contrasts to the appropriate monocular threshold for each stimulus type. The model is defined by a series of equations:
\begin{equation}
{\mathrm{Stage1}_L = \frac{C_L^m}{S + C_L + \omega C_R},}
\end{equation}
\begin{equation}
{\mathrm{Stage1}_R = \frac{C_R^m}{S + C_R + \omega C_L},}
\end{equation}
\begin{equation}
{\mathrm{binsum} = \mathrm{Stage1}_L + \mathrm{Stage1}_R,}
\end{equation}
\begin{equation}
{\mathrm{Stage2} = \frac{\mathrm{binsum}^p}{Z + \mathrm{binsum}^q},}
\end{equation}
where
CL and
CR are the (normalized) contrasts displayed to the left and right eyes, and
m,
S, ω,
p,
q, and
Z are free parameters in the model. A further free parameter,
k, is used to convert the model outputs to either d-prime or threshold values (note that in the original model specification [
Meese, Georgeson, & Baker, 2006], this parameter was called σ, but we use the
k symbol here to avoid confusion with the standard deviation of the cumulative Gaussian used when fitting the psychometric functions to estimate thresholds). Thresholds are defined by iteratively adjusting the target contrast until the following equality is satisfied:
\begin{equation}
{\mathrm{Stage2}_{\mathrm{target+pedestal}} - \mathrm{Stage2}_{\mathrm{pedestal}} = k,}
\end{equation}
and
d′ (d-prime) for a single target level is defined as
\begin{equation}
{d^{\prime } = \frac{\mathrm{Stage2}_{\mathrm{target+pedestal}} - \mathrm{Stage2}_{\mathrm{pedestal}}}{k/\tau },}
\end{equation}
where the parameter τ reflects the value of
d′ at detection threshold. Defining threshold as the 75% correct point on the psychometric function (as here) yields a value of
\(\tau = \Phi ^{-1}(0.75)\sqrt{2}\) = 0.954 (where Φ
−1 is the inverse cumulative normal density function). The combined denominator term (
k/τ) therefore represents internal additive noise in the model.
We ran the simplex algorithm from 100 random starting vectors for each data set and chose the solution for each data set that gave the smallest root mean squared error (RMSE) between the model and data.
We also implemented a Bayesian hierarchical version of the model using the Stan probabilistic programming language (
Carpenter et al., 2017). This used a binomial generator function to model the proportion correct data at each target level and was fit simultaneously to all participants for a given experiment, but separately for each chromatic condition of Experiment 1, and the flickering disc data from Experiment 3 (i.e., four fits in total, as for the simplex fitting). Prior distributions for the parameters
p,
q,
m, and ω were Gaussian, with means determined from published values (see first row of
Table 1). Priors for parameters
S,
Z, and
k were uniform. This modeling primarily focuses on examining posterior parameter distributions, rather than a model comparison approach. We generated over 1 million posterior samples for the model and retained 10% of them for plotting.
Finally, we adapted the two-stage model to include parallel pathways to process achromatic, L-M, and S-(L+M) stimuli that mutually suppress each other. We added additional suppressive terms at the first (monocular) stage of the model, for example:
\begin{eqnarray}
{}^{AC}\mathrm{Stage1}_L = \frac{AC_L^m}{S + AC_L + \omega _A AC_R + \omega _R RG_R + \omega _B BY_R},\nonumber\\
\end{eqnarray}
where
AC represents the achromatic contrast,
RG represents the L-M contrast,
BY represents the S-(L+M) contrast, and ω
A, ω
R, and ω
B are the accompanying weights of interocular suppression. There is an equivalent expression for the right eye and for each of the two isoluminant chromatic pathways. To simplify the model and avoid free parameters that are poorly constrained by the data, we fixed several parameters (
p,
q,
m,
S,
k, and ω for the within-pathway suppression) at the values from the fits from Experiment 1 (
Table 1, lower rows). This left nine free parameters: a
Z parameter for each mechanism and six cross-mechanism weights of interocular suppression. These parameters were again estimated within a Bayesian hierarchical framework, using the binomial proportion correct data from Experiment 2. Note that the model as specified does not currently include monocular suppression between different pathways, as we did not collect any data for these conditions. Previous work (e.g.,
Chen et al., 2000;
Mullen & Losada, 1994) has measured such interactions, and they could in principle be incorporated into the denominator of either Stage 1 or Stage 2 in the model.