Paying attention to a stimulus affords it many behavioral advantages, but whether attention also changes its subjective appearance is controversial. K. A. Schneider and M. Komlos (2008) demonstrated that the results of previous studies suggesting that attention increased perceived contrast could also be explained by a biased decision mechanism. This bias could be neutralized by altering the methodology to ask subjects whether two stimuli were equal in contrast or not rather than which had the higher contrast. K. Anton-Erxleben, J. Abrams, and M. Carrasco (2010) claimed that, even using this equality judgment, attention could still be shown to increase perceived contrast. In this reply, we analyze their data and conclude that the effects that they reported resulted from fitting symmetric functions that poorly characterized the individual subject data, which exhibited significant asymmetries between the high- and low-contrast tails. The strength of the effect attributed to attentional enhancement in each subject was strongly correlated with this skew. By refitting the data with a response model that included a non-zero asymptotic response in the low-contrast regime, we show that the reported attentional effects are better explained as changes in subjective criteria. Thus, the conclusion of Schneider and Komlos that attention biases the decision mechanism but does not alter appearance is still valid and is in fact supported by the data from Anton-Erxleben et al.

*σ*) of 1°. Since FWHM = 2

*σ,*an FWHM of 1° gives

*σ*≈ 0.42°; therefore, the stimuli in Anton-Erxleben et al. were less than half as large as those in Schneider and Komlos. Stimulus visibility strongly depends on stimulus size. For example, for 4-cpd gratings (used in both studies; Schneider and Komlos, 2008 also used 2-cpd stimuli), decreasing the radius of a stimulus from 1° to 0.5°, comparable to the difference between the stimuli in Anton-Erxleben et al. and Schneider and Komlos, reduces contrast sensitivity by a factor of approximately 2.5 for vertical gratings or 1.75 for oblique gratings (Díaz-Otero, Caballero, Lorenzo, & Sigüenza, 1995).

*x*) ≡

*φ*(

*u*)

*du*and

*φ*(

*x*) ≡

*τ*is the subjective equality criterion (subjects report that the two targets have equal contrast if the absolute difference in perceived contrast is less than

*τ*), Δ

*c*is the actual difference in contrast between the two target stimuli,

*α*is the hypothesized attentional boost in the perceived contrast of the cued target, and

*σ*

^{2}is the variance of the perceived contrast difference;

*h*is a scale factor, and

*γ*is a skew parameter (Figure 2); and

*α*= 0) plus a non-zero asymptote or “guessing rate”

*g*at low contrast, applied with a threshold

*μ*

_{ g }and slope

*σ*

_{ g }. In fitting this third function, the

*σ, μ*

_{ g, }and

*σ*

_{ g }parameters were linked across conditions, so that only the subjective criteria

*τ*and

*g*were free to independently vary among the conditions; there were nine free parameters in total to fit the data from the three conditions for each subject.

*R*

^{2}was computed over all subjects and conditions as

*R*

^{2}≡ 1 −

_{err}=

*y*

_{ ijk }−

*f*

_{ ijk })

^{2}for the

*N*

_{sub}subjects,

*N*

_{exp}conditions,

*N*

_{c}contrast levels, data

*y,*and model function

*f,*and the total sum of squares SS

_{tot}=

*y*

_{ ijk }−

_{ ij })

^{2}, where

_{ ij }=

*y*

_{ ijk }. The root mean square error (RMSE) was calculated as RMSE =

*f*

_{1}, Equation 1) to the mean data across subjects, as shown in Figures 3 (overlapping) and 4 (separated). Directly comparing the three conditions in Figure 3, the most prominent feature is that the subjects relaxed their subjective equality criteria during the neutral condition compared to the other two conditions, such that they were more likely to report the targets having equal contrast, increasing the response amplitude. Another feature to note in this figure is that the peaks of the fitted functions in all three conditions are shifted to the left, toward the low-contrast test stimuli, with the peak of the fit to the “test cued” data shifted to a lower contrast than was the peak of the fit to the “standard cued” data. The location of the peak of the fitted function is governed by the hypothesized attentional contrast boosting parameter (

*α*), and therefore, the significant difference in

*α*between the “test cued” and “standard cued” conditions was interpreted by Anton-Erxleben et al. (2010) as supporting their hypothesis that attention enhances perceived contrast.

*c*) test stimuli and differ only for the low-contrast (negative Δ

*c*) test stimuli. Subjects were more likely to report that a test and standard stimulus had equal contrasts when the test stimulus had lower contrast than the standard, compared to when the absolute difference was the same but the standard stimulus had the lower contrast. This asymmetry between the high- and low-contrast data means that fitting a symmetric function, such as the equality judgment function (

*f*

_{1}, Equation 1) used here, or the scaled Gaussian used in Anton-Erxleben et al. (2010), results in large systematic errors, as can be seen in the residuals in Figure 4. It is also important to note that the test stimulus contrast that subjects were most likely to report equal in contrast to the standard was in fact the standard contrast (Δ

*c*= 0); however, the maximum of the fitted function is shifted to the left of this, such that the maximum of the data is underestimated by the function, as the fit attempts to accommodate the significant tail in the data at low contrast. This appears not only in the mean data but also in the individual subject data. A single typical subject is shown in the first row of Figure 5, all of the subjects and the fitted function parameters are shown in Supplementary Figure S1, and the mean residuals across subjects from their individual fits are shown in Figure 6. Our interpretation of these data is that subjects perceived the veridical contrast, independent of the cuing condition, but that their response data deviates from the response model for the low-contrast test stimuli due to the addition of some non-zero asymptotic phenomenon.

*f*

_{2}, Equation 2; Figure 2). As seen in the single subject in Figure 5 (second row), all of the subjects in Supplementary Figure S2, and the mean residuals across subjects in Figure 6, the addition of a skew parameter improves the fits considerably, eliminating the systematic errors. Unlike the equality judgment function that was derived from signal theory (see Schneider & Komlos, 2008), the scaled skew normal function is merely descriptive and does not provide insight into the response mechanism. However, comparing the skew (

*γ*) from these fits to the hypothesized attentional effect (

*α*) from the equality judgment fits, as shown in Figure 7, we see a very strong correlation (

*r*= 0.54,

*p*= 0.0039) between the two—the stronger the skew in the data, the more the optimal fitted function must be pulled to the left to accommodate the low-contrast asymmetry and, therefore, the stronger the deviation of the fitted symmetrical function away from veridical contrast perception.

*f*

_{2}, Equation 2) fit the data very well, it is difficult to interpret since it is merely a descriptive function rather than a mechanistic model. Because the process that caused the skew is unknown, it is impossible to determine whether or not the attentional cues altered perceived contrast. To rectify this, we fit the data to a third function whose parameters can be directly interpreted. Because the low-contrast data appeared in many subjects to asymptote to a non-zero value, and given the reduced visibility of the stimuli used in Anton-Erxleben et al. (2010) compared to those used in Schneider and Komlos (2008; see Methods section), we hypothesized that the asymmetries between the low-and high-contrast data likely resulted from a noise process occurring in the low-contrast regime. Although the low-contrast stimuli might be visible, there might be a threshold below which the subjects might lack sufficient information to properly perform the equality judgment and might, therefore, resort to a default behavior of responding that the two targets had equal contrast with probability

*g*. The third function we fit incorporated this low-contrast noise into the equality judgment function (

*f*

_{2}, Equation 3). For this third set of fits, to strongly test the hypothesis posited in Schneider and Komlos that attentional cues affect subjective decision criteria but do not alter perceived contrast, we forced

*α*= 0, i.e., no attentional effect upon perceived contrast was allowed, and also assumed that the underlying distribution of perceived contrast was identical among the three conditions (

*σ*was linked among the conditions).

*f*

_{3}, Equation 3), which assumes that only the subjective criteria

*τ*(the equality threshold) and

*g*varied among the attentional cue conditions, had

*R*

^{2}= 0.94 and RMSE = 0.16 and fit the data better (smaller residuals and reduced systematic error) than the simple equality judgment function (

*f*

_{1}, Equation 1), with

*R*

^{2}= 0.92 and RMSE = 0.19, that permits the attentional cues to alter objective parameters. Both models used nine free parameters for each subject across the three experimental conditions and fit the data worse than the scaled skew normal function (

*f*

_{2}, Equation 2), which used twelve free parameters and had

*R*

^{2}= 0.95 and RMSE = 0.14.

*f*

_{3}, Equation 3) that assumed a constant distribution of perceived contrast across attentional cuing conditions was able to explain the subjects' responses, with differences among the conditions only requiring changes in the subjective equality threshold criteria and the low-contrast asymptotic behavior. Therefore, we conclude that the data reported by Anton-Erxleben et al. were in fact consistent with Schneider and Komlos' (2008) hypothesis that attention alters subjective decision criteria but not perceived contrast.

- Anton-Erxleben et al. (2010) discuss at length the significant correlation they observe between the amplitude and standard deviation parameters in the scaled Gaussian function they used to fit the equality judgment data. The equality judgment response function (
*f*_{1}, Equation 1) used in Schneider and Komlos (2008), which was derived from a signal theory model, contains no explicit amplitude parameter. Instead, the independent parameters are*τ,*the subjective equality criterion, and*σ,*the standard deviation. As noted in Schneider and Komlos, the contrast at which the response function is maximum is independent of*τ*. The response amplitude at this contrast is a function of*τ*and*σ,*namely Φ(*τ*/*σ*) − Φ(−*τ*/*σ*), and it is, therefore, not surprising that this amplitude is correlated with*σ*. Anton-Erxleben et al. are, thus, incorrect in stating that the independence of the equality judgment parameters is not justified, as claimed by Schneider and Komlos. This issue also affects their discussion of Valsecchi, Vescovi, and Turatto (2010). - Anton-Erxleben et al. (2010) criticize the equality judgment response model (
*f*_{1}, Equation 1) in Schneider and Komlos (2008) because “it allows for ill-defined, plateau-like peaks, which renders PSE estimation unreliable.” However, these plateau-like peaks do occur in subjects' data when*τ*/*σ*is large (e.g., see Subjects S5 and S7 in Supplementary Figure S1), which is the reason the response model was originally developed in Schneider and Bavelier (2003). The PSE can be reliably determined even when*τ*/*σ*is large as long as a sufficiently large range of test stimuli contrasts is used to sample the slopes on either side of the plateau. The scaled Gaussian function used by Anton-Erxleben et al. is a reasonable approximation to the equality judgment function when*τ*/*σ*is small.

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