We quantified observers’ sensitivity to the target filters in natural images in a generalized linear model (GLM) framework (see
Figures 3 and
4). We describe the most pertinent aspects of the framework below, but for the impatient reader, we note that these equations accumulate to the fitglme() function in MATLAB or, equivalently, the lmer() function in R with the lme4 package (
Bates, Mächler, Bolker, & Walker, 2015).
In a standard single-interval detection paradigm in which a target is either present or absent, sensitivity,
d′, is calculated as
\begin{equation}d^{\prime} = \phi H - \phi F\end{equation}
where ϕ is the normal integral function,
H is the proportion of hits, and
F is the proportion of false alarms, under the assumption of equal variance. An observer's criterion (or bias),
c, is calculated as
\begin{eqnarray}
c = \frac{1}{2}\left( {\phi H + \phi F} \right)\quad
\end{eqnarray}
In a GLM,
d′ and
c (bias) are computed as predictor weights β
1 and β
0, respectively, that are passed through a probit link function, which is the normal integral function:
\begin{equation}{\eta _{\left[ i \right]}} = {\beta _0} + {\beta _1}{S_{01\left[ i \right]}}\end{equation}
\begin{equation}p\left( {presen{t_i}} \right) = \;\phi {\eta _i}\end{equation}
where η is the sum of weighted linear predictors and
S01 is the absence or presence of the signal (i.e., 0 or 1, respectively) on the
ith trial. By fitting such a probit model, estimates of the predictor weights β
1 and β
0 are identical to
d′ and
c, respectively, as calculated in
Equation 13 and
Equation 14. Whereas these equations fully specify sensitivity and bias in a single interval present/absent judgment task, some small modifications are needed to quantify sensitivity in a two-alternative forced-choice task (2AFC) as in our experiment. First,
S01 denotes whether the target filter(s) appeared in the left or right spatial interval, defined as −.5 or .5, respectively. Similarly, observers’ reports (i.e., “target appeared in the left or right interval”) were defined as 0 and 1, respectively. Finally, in a 2AFC, observers have two opportunities to detect the target—once per spatial interval—and so raw
d′ will be greater than in a single-interval detection design. Therefore, sensitivity (but not bias) must be scaled by
\(\frac{1}{{\sqrt 2 }}\) (
Macmillan & Creelman, 2004):
\begin{equation}d_{2AFC}^{\prime} = \;\frac{1}{{\sqrt 2 }}{\beta _1} = \;\frac{1}{{\sqrt 2 }}\left( {\phi H - \phi F} \right)\end{equation}
Importantly, we can extend
Equation 15 to quantify sensitivity to any number of other predictors,
xω:
\begin{equation}{\eta _{\left[ i \right]}} = {\beta _0} + {\beta _1}{S_{01\left[ i \right]}} + \ldots + \;{\beta _\omega }{x_{\omega \left[ i \right]}}\end{equation}
Consider, for example, the influence of filter amplitude (α) on an observer's sensitivity:
\begin{equation}{\eta _{\left[ i \right]}} = {\beta _0} + {\beta _1}{\alpha _{\left[ i \right]}}{S_{01\left[ i \right]}}\end{equation}
Note that filter amplitude is entered into the model as an interaction with target location because the model's predicted outcome is a spatial report; target amplitude alone can only predict a change in bias. In preliminary model fits, we found that such bias was not significantly different from zero and thus included only interactive terms to facilitate interpretability of the standard bias term, β0. We selected other model predictors according to the model that produced the lowest Akaike information criterion (AIC; see below).
Finally, we implemented this model as a multilevel GLM (GLMM) to partially pool coefficient estimates across observers (
Gelman & Hill, 2007). By using a GLMM, we model each observer's predictor weights as having come from a population distribution with mean µ and variance σ
2:
\begin{eqnarray}
{\eta _{\left[ i \right]}} = {\beta _{0,j}} + \;{\beta _{1,j}}{\alpha _{j\left[ i \right]}}{S_{01,j\left[ i \right]}} + \ldots \qquad
\end{eqnarray}
where
\begin{eqnarray}
{\beta _{\omega ,j}} = \;{\mu _\omega } + {\varepsilon _{\omega ,j}}\;|\;\sigma _\omega ^2\qquad
\end{eqnarray}
Here, ε
ω,j is the offset for each predictor ω and observer
j, relative to the parameter's mean µ, contingent on the parameters’ estimated population variance σ
2. The partial pooling of observers’ data in a GLMM results in more extreme values being pulled toward the population mean estimate. Note that in our experiment, however, such pooling is relatively minor due to the large number of trials, and therefore high precision, of each observer's estimated performance, as well as the relatively small number of observers. Because images were drawn randomly from trial to trial from a pool of tens of thousands of images, we did not expect many, if any, repeats of each image. We therefore did not model the background images as a random effect, but we note that such a design could be chosen in future to estimate the variance associated with each tested background.
We entered into the model the factors target amplitude, number of filters, and target-background alignment, which, as noted above, were each entered as an interaction with the spatial interval of the target. In hindsight, our inclusion of the condition in which target amplitude was 0 was unnecessary. For all such trials, therefore, we set all predictors to have a value of 0 so they were omitted from model calculations. The model fit was improved by including nonlinear terms by raising amplitude and number of Gabors to the exponents 0.5 and 2, respectively. We further tested all combinations of interactions, but none improved the model fit as assessed by the AIC.