The 1D mixture models split errors into three components (
Bays et al., 2009) (
Figure 1). Throughout the article, we shall focus on this three-component model (the misbinding model), although the approach is the same for related models such as the two-component mixture model (without misbinding;
Zhang & Luck, 2008), and all such models have been adapted for 2D. The misbinding model has three sources of errors: imprecision, misbinding, and random guessing:
\begin{eqnarray}P\left( {\hat \theta {\rm{\;}}} \right){\rm{\;}} &=& {\rm{\;}}\alpha {\phi _\kappa }\left( {\hat \theta {\rm{\;}} - {\rm{\;}}\theta } \right) + {\rm{\;}}\beta \frac{1}{m}\mathop \sum \limits_i^m {\phi _\kappa }\left( {\hat \theta {\rm{\;}} - {\rm{\;}}{\varphi _i}} \right) \nonumber\\
&&+ {\rm{\;}}\gamma \frac{1}{{2\pi }},\end{eqnarray}
where
\(P( {\hat \theta \;} )\) is the probability of finding a response orientation
\(\hat \theta \), θ is the orientation of the target stimulus, φ
κ is the von Mises distribution (circular analogue of the Gaussian distribution), ϕ
i is the orientation of the nontarget stimulus
i,
m is the number of nontarget stimuli, and guessing is uniform over the entire circle (2π). The parameters α, β, γ, and κ control the proportion of target responding, nontarget responding, guessing, and the concentration of the von Mises distribution, respectively. The spread of the distributions of target and nontarget responses is assumed to be the same. As α, β, and γ must sum to 1, α is not included as a free parameter in the fitting. The three free parameters (β, γ, κ) are estimated using maximum likelihood methods.
To adapt the model for 2D data, several changes are needed. First, 2D coordinates are used in place of angles, which means that the von Mises distribution is replaced with a bivariate Gaussian distribution ψ
σ with standard deviation
σ and zero covariance. Second, a distribution for the random guesses must be chosen. For simplicity, we begin by assuming that they are drawn from a uniform distribution over the entire area of the screen (
A). This gives the following response density function:
\begin{eqnarray}P\;\left( {{\bf{\hat \theta }}{\rm{\;}}} \right) &=& {\rm{\;\alpha \;}}{{\rm{\psi }}_{\rm{\sigma }}}\left( {{\bf{\hat \theta }}\; - {\bf{\theta }}} \right) + {\rm{\;\beta }}\frac{1}{m}\mathop \sum \limits_i^m {{\rm{\psi }}_{\rm{\sigma }}}\left( {{\bf{\hat \theta }}{\rm{\;}} - {\rm{\;}}{\varphi _i}} \right){\rm{\;}} \nonumber\\
&&+ {\rm{\;\gamma }}\frac{1}{A},\end{eqnarray}
where here,
θ and
ϕ are vectors indicating locations on the screen, and again, α, β, and γ sum to 1, which together with σ yield three free parameters.
The model will operate in any spatial units, as long as A is changed to reflect this. For the examples provided here, we will use 1366*768 as the screen area and pixels as units. The dimensions of the screen will dictate the scale of the errors in the task (i.e., on a 40 cm × 30 cm screen, the errors will be under 50 cm) and thus the scale of the imprecision parameter σ. The models take in a dimensions argument, which sets the screen dimensions for simulations and fitting. In other words, it implicitly sets the units for the model. Any appropriate scaling values are possible (e.g., 960*540 pixels, 46*26 cm, 30*17 visual degrees). This allows you to convert data collected on different screen sizes into standard units (e.g., visual degrees or percentage of screen size), for example, to correct for any influence that different screen sizes may have on the data. A rectangular screen is assumed, so if a different shape screen is used (e.g., a circular window on a screen), the model will need to be modified to account for this. Different dimensions can be supplied for simulating and fitting the data (e.g., if the stimuli may not appear too close to the edges of the screen but possible response locations are not so constrained). This can also be achieved by using the method described in the “Effect of stimulus separation on recovery” section below.
Previous 1D models used a circular space, meaning that responses wrapped around the domain, without any edges. In contrast, the 2D spatial tasks involve a finite space defined by the screen or screen region. When simulating the model, any responses that fall outside of the screen when drawn from the normal distributions are replaced by new samples drawn from the same distribution.
We use the framework for 1D models provided in MemToolbox (
Suchow et al., 2013; MemToolbox.org) to provide a new toolbox (MemToolbox2D;
Grogan et al., 2019;
https://doi.org/10.5281/zenodo.3752705) for modeling 2D data that integrates with the existing 1D MemToolbox. New functions were created to run the fitting, display plots, correct for biases, and simulate the models. Thus, MemToolbox2D will not interfere with the functions of MemToolbox.