How well can observers selectively attend only to dots that are lighter or darker than the background when all dot intensities are present? Observers estimated centroids of briefly flashed, sparse clouds of 8 or 16 dots, ranging in intensity from dark black to bright white on a gray background. Attention instructions were to equally weight: (i) dots brighter than the background, assigning zero weight to others; (ii) dots darker than the background, assigning zero weight to others; (iii) all dots. For each observer, a quantitative estimate of the operational attention filter (the weight exerted in the centroid estimates as a function of dot intensity) was derived for each attention instruction in each dot condition. Attended dots typically have 4× the weights of unattended dots. Whereas observers performed remarkably well in estimating centroids and achieving the three required attention filters, they achieved higher accuracy when equally weighing all dots than when selectively attending to dots of only one contrast polarity. Although their attention filters are similar, individual observers use significantly different parameters in their centroid computations. The complete model of performance enables perceptual measurements of observers' attention filters for shades of gray that are as accurate as physical measurements of color filters.

*My experience is what I agree to attend to*. Only those items which I

*notice*shape my mind—without selective interest, experience is an utter chaos.”

*spatial filter*. Short-term memory cannot hold more than 4 or 5 letters for recall. When 3 × 3 or 3 × 4 letter arrays (i.e., arrays of 3 rows with either 3 or 4 letters in a row) are flashed briefly, the process of spatial selective attention determines which one of the three rows of letters will ultimately gain access to the limited-capacity short-term memory.

^{2}and white dots had a luminance of 68.1 cd/m

^{2}. The mean gray background had a luminance of 34.2 cd/m

^{2}.

^{2}, and each display consisted entirely of this value aside from a sparse cloud of dots. In any given display, there were either 8 or 16 dots. The dots in a given 8-dot display had Weber contrasts of −1, −0.75 −0.5, −0.25, 0.25, 0.5, 0.75, and 1 (Figure 1). In 16-dot stimuli, there were two dots of each of these eight Weber contrasts (Figure 2). The locations of the dots in a given display were randomly drawn from a circular bivariate Gaussian density with a standard deviation of 60 pixels (1.70 deg).

*N*= 8, 16, with each of the three attention instructions

*A*= Attend-to-Dark, Attend-to-Light, Attend-to-All. We refer to the condition in which

*N*dots were displayed and the attention instruction was

*A*as condition (

*N*,

*A*). The raw data from condition (

*N*,

*A*) consists of the

*x*- and

*y*-locations,

*X*

_{ N,A }(

*t*) and

*Y*

_{ N,A }(

*t*) of the observer's response for each trial

*t*. Although these raw centroid-estimation data are interesting in and of themselves, it is of even greater interest to interpret the data in terms of the process of selective attention that enables observers, in their centroid computation, to give differential weights to attended dots and to ignore unattended dots. The process of selective attention is encapsulated in a model in which filters, determined by attention instructions, selectively weight the dots according to the dots' Weber contrasts, and these weights are what is incorporated into the centroid computation. The model has filters; observers have selective attention. Because of the intimate connection between the two, we will speak of the “observers' filters” even though the attention filter is a model construct.

*not*the case for the model introduced below.

^{1}Our focus in this study is on the attentional selection of input information, not on how the brain computes centroids. We seek to determine which sorts of attention filters people can achieve and to describe these filters quantitatively. The point of our model is to enable a more accurate characterization of attention filters across different attention conditions (Attend-to-Dark, Attend-to-Light, Attend-to-All) and different numbers of stimulus dots (8 vs. 16) by taking into account the effects of imperfect centroid computations and individual differences.

- The attention filter used by the observer may differ from the ideal (target) filter.
- The centroid computation may be imperfect. We model the imperfection by a single parameter, a bias to weight peripheral dots in the cloud differently from more central dots.
- The resulting centroid computation may be corrupted by internal response noise.

*systematic*errors in observers' responses whereas factor (3) leads to

*random*errors in observers' responses. The model enables us to measure variations in each of these three factors for each observer in each attention condition and in each different numbers-of-dots condition.

*F*

_{ N,A }to each dot in the internally represented stimulus. This process replaces each dot in the stimulus by a point with a weight that depends on the dot's Weber contrast. Specifically, if a dot has Weber contrast

*c*, then this dot gets replaced by a point with weight

*F*

_{ N,A }(

*c*). It is crucial to realize that although the observer may be trying to achieve a particular target attention filter (e.g., to weight all dots darker than the background equally while giving all dots lighter than the background weight 0), in practice the attention filter

*F*

_{ N,A }that he/she achieves always deviates from this target filter. This deviation of the attention filter from the target filter is one important source of systematic error.

*N*,

*A*), we extract an unbiased estimate of the standard deviation of the observer's internal response noise.

*N*,

*A*), we first derive the attention filter and distance-distortion parameter that minimize the sum of squared deviations (across trials) of the model-predicted centroids from the observer's centroid settings. We then derive an unbiased estimate of the standard deviation of the observer's internal response noise

^{2}from residual sum of squared deviations of predicted from observed centroids (see 1 for details).

*N*= 8 or 16 and

*A*= Attend-to-Dark, Attend-to-Light, or Attend-to-All, we write

*F*

_{ N,A }for the attention filter used by the observer in condition (

*N*,

*A*). That is, for any Weber contrast

*c*= −1, −3/4, …, 1, the attention filter

*F*

_{ N,A }(

*c*) gives the weight assigned to dots of contrast

*c*in centroid estimates in condition (

*N*,

*A*). Target filters for each attention condition (

*A*) are depicted in Figure 6.

*z*represent a dot in the display, let

*c*(

*z*) be the Weber contrast of

*z*, and let

*d*(

*z*) be the Euclidean distance (a positive real number) of

*z*from the centroid of the

*F*

_{ N,A }-filtered dot cloud. The weight

*w*(

*z*) of a dot

*z*on trial

*t*of condition (

*N*,

*A*) is the product of two factors, an attention filter applied to the dot's contrast and a distance correction function applied to the dot's distance from the centroid of the

*F*

_{ N,A }-filtered dot cloud:

*R*

_{ N,A }(

*d*(z)) is a one-parameter, monotonic function (given by Equation A3 in 1) that can either be a decreasing function of

*d*(

*z*) (in which case the centroid computation is “robust”) or an increasing function of

*d*(

*z*) (in which case the centroid computation is “anti-robust”).

*x*- and

*y*-locations of the response on trial

*t*in condition (

*N*,

*A*) are given by

*z*presented on trial

*t*,

*w*(

*z*) is given by Equation 1, and

*x*(

*z*) and

*y*(

*z*) are the

*x*-location and

*y*-location of

*z*, respectively. The estimated standard deviations

*σ*

_{ x }and

*σ*

_{ y }of noise

_{ x }and noise

_{ y }did not differ systematically and so are subsumed in a single parameter

*σ*,

*σ*= ((

*σ*

_{ x }

^{2}+

*σ*

_{ y }

^{2})/2)

^{1/2}, the standard deviation of the symmetrically distributed internal response noise.

*F*

_{ N,A }(0) = 0 (i.e., implying that dots that match the background (invisible dots) exert zero weight in the centroid), and we constrain

*F*

_{ N,A }so that

*c*= −1, −0.75, …, 1 used in each display. Under these constraints, for each condition (

*N*,

*A*), the filter

*F*

_{ N,A }, and distance correction function

*R*

_{ N,A }are chosen to produce predicted response locations

_{ N,A }(

*t*) and

_{ N,A }(

*t*) that minimize the sum of squared distances of the trial-by-trial observed responses. That is, we minimize

*t*in condition (

*N*,

*A*).

*c*= −1, −0.75, …, 1 (excluding 0 to which the attention filter assigns the weight 0). The absolute values of the attention filter weights are constrained to sum to 1.0 making the total number of degrees of freedom absorbed by the attention filter equal to 7. The eighth degree of freedom is contributed by the free parameter

*α*

_{ N,A }controlling the form of the distance correction function

*R*

_{ N,A }(Equation A3 in 1). The parameter

*σ*giving the standard deviation of the internal response noise is derived from the deviation of the model fit from the observed responses; thus it does not constrain the fit of the model to the data.

*RMSFE*

_{ N,A }(see Equation A7 in 1), which reflects the

*systematic deviation*of the attention filter

^{3}achieved by the observer in condition (

*N*,

*A*) from the target attention filter for condition (

*N*,

*A*). Second, we compute an unbiased estimate of the standard deviation

*σ*of the

*internal response noise*(see Equation 5)

*.*

*N*,

*A*), Table 1 gives the mean distance of responses made by all observers in condition (

*N*,

*A*) from the centroids of (1) the dark dots, (2) the light dots, and (3) all dots in the display. These results provide preliminary evidence that observers can adjust their performance to match varying attentional demands. Note first that in the Attend-to-Dark condition (for both 8- and 16-dot displays), responses are closer to the centroid of the Dark dots than they are to the centroid of the Light dots or to the centroid of All dots. Moreover, as might be expected, the mean distance of the response is greater from the centroid of the Light dots than it is from the centroid of All dots. An analogous pattern holds for responses made in the Attend-to-Light conditions. Interestingly, the most accurate responses (those that come closest to their actual target centroids) are those made in the Attend-to-All conditions (for both 8 and 16 dots).

Mean distance of response from centroid of dark dots | Mean distance of response from centroid of light dots | Mean distance of response from centroid of all dots | |
---|---|---|---|

Attend-to-Dark (8) | 24.2 | 47.1 | 32.1 |

Attend-to-Light (8) | 44.4 | 23.9 | 33.6 |

Attend-to-All (8) | 25.4 | 26.8 | 17.8 |

Attend-to-Dark (16) | 23.8 | 37.2 | 26.6 |

Attend-to-Light (16) | 38.0 | 23.7 | 26.6 |

Attend-to-All (16) | 25.4 | 24.5 | 18.1 |

*α*, and the standard deviation

*σ*of internal response noise that best predict the observers' performances. An attention filter was computed for each observer in each condition. The attention filters, averaged over the 11 observers, for each of the 6 conditions, are shown in Figure 7. Figure 7A plots the best fitting attention filter (average of all observers)

*F*

_{ N,Attend-to-All}when observers attend to all dots for

*N*= 8 and for

*N*= 16. Also shown for reference is the target attention filter

*F*

_{Attend-to-all}

^{Target}. Error bars give 95% confidence intervals. Figures 7B and 7C show the corresponding plots for the Attend-to-Dark and Attend-to-Light conditions.

*F*

_{ N,Attend-to-All}filter (for each of

*N*= 8 and 16) is strikingly flat across all non-zero contrasts, showing only a slight drop in sensitivity for the two Weber contrasts closest to 0. Evidently, observers are able to weight all dots nearly equally, despite the differences in Weber contrast between dots.

*F*

_{8,Attend-to-Light}and

*F*

_{16,Attend-to-Light}, show a high degree of selectivity for positive vs. negative Weber contrasts; the Attend-to-Dark filters,

*F*

_{8,Attend-to-Dark}and

*F*

_{16,Attend-to-Dark}, are equally selective. Nevertheless, the four selective filters,

*A*=

*Attend-to-Light*,

*Attend-to-Dark*for

*N*= 8 and 16, show systematic deviations from their target forms. The weights given to unattended contrasts—negative Weber contrasts by

*F*

_{ N,Attend-to-Light}and positive contrasts by

*F*

_{ N,Attend-to-Dark}—are all significantly greater than 0; observers cannot completely ignore unattended dots. To characterize just how good the selective attention filters are, we consider just the three most extreme contrasts at each end of the range and omit the two contrasts closest to zero, which are less discriminable. The ratio of the average weights for these three attended to the three unattended contrasts ranges from 3.60:1 to 4.33:1. That is, among the 6 most discriminable of the 8 dot contrasts, the attention filter gives attended contrasts typically a 4:1 weight advantage over unattended contrasts. Not perfect selection, but surprisingly good.

*RMSFE*

_{ N,A }values (across all observers) for

*N*= 8, 16 and

*A*=

*Attend-to-All*,

*Attend-to-Dark*, and

*Attend-to-Light*are given in Table 2, and indeed the mean value of

*RMSFE*

_{8,A }is lower than that for

*RMSFE*

_{16,A }for all three attention instructions. A repeated measures ANOVA reveals a significant main effect for cloud size (

*F*(1, 10) = 9.601,

*p*= 0.011), indicating that observers are slightly but significantly better at matching their attention filter to the target filter when displays contain only eight versus sixteen dots.

Attention instructions | Mean RMSFE | Standard deviation of RMSFE | ||
---|---|---|---|---|

8-Dot clouds | 16-Dot clouds | 8-Dot clouds | 16-Dot clouds | |

Attend-to-Dark | 0.189 | 0.209 | 0.105 | 0.076 |

Attend-to-Light | 0.165 | 0.205 | 0.053 | 0.053 |

Attend-to-All | 0.058 | 0.075 | 0.020 | 0.025 |

Mean | 0.137 | 0.163 | 0.059 | 0.051 |

*N*,

*A*), the internal response noise standard deviation, which we assume is common to each of the horizontal and vertical components of the internal response noise, is estimated by

*t*, 1 ≤

*t*≤ 200 in condition (

*N*,

*A*). In this equation, the number of degrees of freedom in the model is 8,

*X*

_{ N,A }(

*t*) and

*Y*

_{ N,A }(

*t*) give the

*x*- and

*y*-coordinates of the observer's response on trial

*t*in condition (

*N*,

*A*), and

_{ N,A }(

*t*) and

_{ N,A }(

*t*) give the

*x*- and

*y*-locations predicted by the model.

_{ N,A }across

*N*or

*A*can plausibly be assumed to reflect differences in the cost incurred in adapting to these task variations.

_{ N,A }of the standard deviation of random response error given by Equation 5 for all conditions (

*N*,

*A*). There are several trends to note in Table 3. First,

_{ N,A }is lower for

*N*= 8 versus

*N*= 16 dots for each of the three attention instructions. Second, for each of

*N*= 8 and

*N*= 16,

_{ N,Attend-to-All}is lower than

_{ N,Attend-to-Dark}, and similarly

_{ N,Attend-to-Dark}is lower than

_{ N,Attend-to-Light}. A repeated measures ANOVA confirms that all of these differences are significant: for the effect due to dot number,

*F*(1, 10) = 10.61,

*p*= 0.009. In addition, within-subject contrasts reveal that the difference in noise between the

*Attend-to-All*vs. the

*Attend-to-Dark*conditions is significant (

*F*(1, 10) = 5.831,

*p*= 0.036), as is the difference in noise between the

*Attend-to-Dark*vs. the

*Attend-to-Light*conditions (

*F*(1, 10) = 5.380,

*p*= 0.043). As one might expect given these two results, the difference in noise between the

*Attend-to-All*vs. the

*Attend-to-Light*conditions is also highly significant (

*F*(1, 10) = 20.820,

*p*= 0.001).

Attention instructions | Unbiased estimates σ ^ _{ N,A } of the standard deviation of internal response noise | Standard deviation of σ ^ _{ N,A } | ||
---|---|---|---|---|

N = 8 | N = 16 | N = 8 | N = 16 | |

Attend-to-Dark | 15.89 | 17.31 | 4.43 | 3.61 |

Attend-to-Light | 17.38 | 19.13 | 4.84 | 4.13 |

Attend-to-All | 14.36 | 15.19 | 2.71 | 3.56 |

Mean | 15.87 | 17.21 |

*N*= 8 and

*N*= 16 conditions, then the averaging performed in the 16-dot conditions would be expected to yield response errors substantially smaller than those observed in the 8-dot conditions. Specifically, we should expect the values of

_{8,A }to be approximately equal to

_{16,A }. As observed above, we find on the contrary that

_{8,A }is slightly but significantly

*smaller*than

_{16,A }. The simplest account of this finding is that the dominant source of noise compromising performance is late noise, e.g., motor noise or random, perceptual mislocalization of the (noiselessly computed) centroid prior to response production. Assuming this is true, the current results imply that increasing the number of dots from 8 to 16 produces a slight but significant increase in this late noise.

*Attend-to-Light*and

*Attend-to-Dark*conditions using 16 dots, only 8 dots are supposed to contribute to the target centroid. The same is true of the

*Attend-to-All*condition using 8 dots. Thus the difference between

_{16,Attend-to-Light}and

_{16,Attend-to-Dark}provides a measure of the cost in additional internal response noise incurred in attempting to ignore the irrelevant dots in the

*Attend-to-Light*and

*Attend-to-Dark*conditions. As might be expected, given the results noted above, the paired comparison

*t*-test results in Table 4 reveal that

_{16,Attend-to-Light}and

_{16,Attend-to-Dark}, suggesting that the presence of the to-be-ignored dots and the task of ignoring them in the

*Attend-to-Light*and

*Attend-to-Dark*conditions introduces increased levels of internal response noise into the centroid calculation process.

Conditions compared | t | df | Significance (two-tailed) |
---|---|---|---|

Attend-to-All (8)–Attend-to-Dark (16) | 4.011 | 10 | p = 0.0012 |

Attend-to-All (8)–Attend-to-Light (16) | 4.669 | 10 | p = 0.0004 |

*α*

_{ N,A }, in the model to characterize distance distortions. This elaborated model enables us to determine whether our observers use computations that are robust or anti-robust, i.e., to test the hypothesis that observers differentially weighted peripheral versus central elements in the cloud (for computational details, see 1).

*α*, we use a two-pass fitting procedure: First, we fit an attention filter under the assumption that the observer is using a center of gravity computation free of distance distortion. This yields an estimated centroid for each display from which we derive a distance-from-centroid

*d*(

*z*) for every dot

*z*in each display. In Pass 2, we fit a refined model to the same data in which the weight exerted by each dot becomes a product of two elements: (1) the weight attributed by the attention filter (based on the dot's contrast—as in Pass 1) and (2) a function

*R*

_{ N,A }(

*d*(

*z*)) of the dot

*z*'s distance-from-centroid

*d*(

*z*) acquired in Pass 1.

*R*

_{ N,A }can be either a decreasing function of distance, making the weighting a “robust” computation, or an increasing function of distance, yielding an “anti-robust” computation (see Figure 8 and 1 for details).

*F*-tests (

*df*numerator = 1,

*df*denominator = 192) assessing whether the parameter

*α*

_{ N,A }significantly improved the model fit (the model's estimate of the centroid location for each individual stimulus) to the data (the observer's estimate of the centroid) for each observer in each condition. For each observer, the top row gives the estimated value of

*α*

_{ N,A }, which can take values between −1 and 1; negative values indicate centroid estimators that tend toward robustness; positive values indicate estimators that tend toward anti-robustness. The second row gives the

*F*-value, and the third row gives the corresponding

*p*-value. Cells that appear in bold give conditions in which the centroid computation was robust at the

*p*< 0.05 level of significance, and cells that appear in italic give conditions in which the centroid computation was anti-robust. (It should be noted that whether or not the parameter

*α*

_{ N,A }improves the model fit significantly in a given instance depends on other factors in addition to value of

*α*

_{ N,A }itself; thus, for example, it might happen for some observer that 0 <

*α*

_{ A 1,N 1 }<

*α*

_{ A 2,N 2 }, yet

*α*

_{ A 1,N 1 }improves the fit significantly in condition (

*N*

_{1},

*A*

_{1}), whereas

*α*

_{ A 2,N 2 }does not produce a significant improvement in the fit in condition (

*N*

_{2},

*A*

_{2}).)

Observer | Attend-to-Dark | Attend-to-Light | Attend-to-All | Measure | |||
---|---|---|---|---|---|---|---|

8-Dot Cloud | 16-Dot Cloud | 8-Dot Cloud | 16-Dot Cloud | 8-Dot Cloud | 16-Dot Cloud | ||

1 | −0.568^{a} | 0.912 | 0.958 | −0.956 | 0.920 | −0.979 | α |

0.000^{b} | 2.740 | 22.200 | 0.660 | 9.080 | 3.110 | F | |

0.991^{c} | 0.100 | 0.000 | 0.418 | 0.003 | 0.080 | P | |

2 | 0.903 | 0.896 | 0.919 | −0.997 | −0.997 | −0.968 | α |

1.310 | 1.210 | 1.740 | 4.520 | 7.610 | 1.050 | F | |

0.254 | 0.272 | 0.189 | 0.035 | 0.006 | 0.308 | P | |

3 | −0.980 | −0.985 | 0.841 | −0.971 | −0.979 | −0.977 | α |

7.620 | 8.570 | 0.340 | 1.200 | 6.960 | 4.650 | F | |

0.006 | 0.004 | 0.563 | 0.274 | 0.009 | 0.032 | P | |

4 | 0.994 | −0.972 | 0.890 | −0.799 | 0.995 | 0.993 | α |

15.280 | 1.050 | 0.600 | 0.000 | 11.940 | 12.660 | F | |

0.000 | 0.308 | 0.441 | 0.985 | 0.001 | 0.001 | P | |

5 | 0.921 | 0.965 | 0.953 | 0.992 | 0.989 | 0.967 | α |

1.300 | 18.520 | 13.160 | 14.140 | 95.560 | 28.590 | F | |

0.256 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | P | |

6 | −0.971 | −0.981 | −0.978 | −0.977 | −0.982 | −0.997 | α |

1.320 | 5.560 | 4.200 | 2.080 | 10.190 | 10.030 | F | |

0.251 | 0.019 | 0.042 | 0.151 | 0.002 | 0.002 | P | |

7 | 0.966 | 0.994 | 0.995 | 0.994 | 0.993 | 0.992 | α |

14.290 | 11.070 | 9.890 | 3.870 | 26.370 | 22.550 | F | |

0.000 | 0.001 | 0.002 | 0.051 | 0.000 | 0.000 | P | |

8 | −0.978 | −0.984 | −0.960 | −0.979 | 0.996 | 0.993 | α |

10.670 | 16.210 | 0.960 | 2.690 | 5.450 | 12.620 | F | |

0.001 | 0.000 | 0.329 | 0.103 | 0.021 | 0.001 | P | |

9 | −0.962 | −0.975 | −0.973 | 0.779 | 0.995 | 0.922 | α |

3.390 | 6.580 | 6.110 | 0.090 | 6.620 | 5.230 | F | |

0.067 | 0.011 | 0.014 | 0.771 | 0.011 | 0.023 | P | |

10 | 0.960 | −0.942 | −0.999 | 0.873 | 0.909 | −0.980 | α |

16.930 | 0.050 | 2.570 | 0.180 | 1.370 | 3.610 | F | |

0.000 | 0.816 | 0.110 | 0.669 | 0.244 | 0.059 | P | |

11 | 0.824 | −0.999 | 0.818 | −0.958 | 0.940 | 0.953 | α |

0.470 | 1.370 | 0.280 | 0.520 | 20.060 | 43.640 | F | |

0.496 | 0.244 | 0.596 | 0.472 | 0.000 | 0.000 | P |

^{a}Negative values of *α* indicate robustness (overvaluing central versus peripheral dots); positive values indicate anti-robustness (overvaluing peripheral dots in the centroid computation). Statistically significant (*p* < 0.05) negative *α* values (robust) are shown and appear in bold; statistically significant positive *α* values (anti-robust) appear in italic; *α* values not statistically different from 0.0 are black.

^{b}Degrees of freedom for *F*: *df* numerator = 1, *df* denominator = 192.

^{c}Probability that *α* has not improved the model, i.e., level of statistical significance.

*p*< 0.05, whereas only 14 showed robust computations with

*p*< 0.05. Note in addition that individual observers tended to be consistent in direction of their distance-distortion bias: each row of Table 5 tends to contain exclusively bold cells or italic cells. Only two observers show a mixture both. Figure 8 shows the best fitting distance correction functions for each of the observers in each condition (

*N*,

*A*). See 1 for the definition of

*R*

_{ N,A }. The dashed blue curve is proportional in height to the expected number of dots at each distance in the condition corresponding the given panel; it indicates the distance values where

*α*most influences the centroid computation.

*F*

^{+}and

*F*

^{−}, differentially sensitive to dots of different Weber contrasts. Under this hypothesis, each of these two filters operates in a strictly bottom-up fashion, i.e., the sensitivity of each of these two filters is immutable and completely beyond the reach of attention control. One filter

*F*

^{+}is strongly activated (approximately equally well) by dots of positive Weber contrast but only slightly activated by dots of negative Weber contrast. This is the filter used under the “Attend-to-Light” instruction. The second filter

*F*

^{−}is strongly activated (approximately equally well) by dots of negative Weber contrast but only slightly by dots of positive Weber contrast and used under “Attend-to-Dark” condition. The observer synthesizes the filter to be used in “Attend-to-All” by assigning maximal gain to both filters. The resulting attention filter would have approximately equal sensitivity to dots of all contrasts, positive and negative (Figure 7a).

*N*,

*Attend-to-All*), (

*N*,

*Attend-to-Dark*), and (

*N*,

*Attend-to-Light*) separately for each observer and each of

*N*= 8 and

*N*= 16. The analysis compared four nested models that we will refer to as Model

_{0}, Model

_{1}, Model

_{2}, and Model

_{3}. Each model assumed a fixed distance correction function

*R*

_{ N,A }across all three attention conditions (i.e.,

*α*

_{ N,Attend-to-Dark}=

*α*

_{ N,Attend-to-Light}=

*α*

_{ N,Attend-to-Allι }=

*α*).

_{0}has no free parameters. This model assumes that the subject computes a true centroid (no distance distortion) giving equal weights to all dots (this would be the optimal single filter to use if you had to pick one filter to deal with all three attention conditions).

_{1}still assumes that the subject has to use the same filter for all three attention conditions but allows this filter to vary freely and also allows distance distortion (

*α*is free to roam). Thus, Model

_{1}has 8 free parameters.

_{2}allows the three attention filters to be drawn from a 2-dimensional (instead of a 1-dimensional) subspace of functions. This allows for two arbitrary linearly independent filters; any third filter must be a combination of the first two. In addition to letting

*α*to vary, this model has 7 free parameters for

*F*

_{ N,Attend-to-Dark}, 7 for

*F*

_{ N,Attend-to-Light}, and one additional parameter

*θ*used to generate

*F*

_{ N,Attend-to-All}= cos(

*θ*) ×

*F*

_{ N,Attend-to-Dark}+ sin(

*θ*) ×

*F*

_{ N,Attend-to-Light}. This gives a total of 16 free parameters.

_{3}places no constraints on the three attention filters. This model has 22 free parameters:

*α*and 7 degrees of freedom for each of the three attention filters.

Observer | N | Percent of variance above Model_{0} accounted for by Model_{1} (first filter dimension) | Percent of additional variance accounted for by Model_{2} (second filter dimension) | Percent of additional variance accounted for by Model_{3} (third filter dimension) | p-value for F-test^{a} assessing significance of Model_{3} vs. Model_{2} (third vs. second filter dimension) |
---|---|---|---|---|---|

1 | 8 | 1.9% | 49.6% | 1.3% | 0.018 |

16 | 7.5% | 24.4% | 3.3% | 0.000 | |

2 | 8 | 4.7% | 27.3% | 0.7% | 0.382 |

16 | 1.6% | 15.6% | 1.2% | 0.213 | |

3 | 8 | 10.4% | 25.1% | 0.7% | 0.406 |

16 | 7.0% | 20.2% | 1.3% | 0.106 | |

4 | 8 | 9.0% | 3.1% | 1.6% | 0.094 |

16 | 5.8% | 2.6% | 1.1% | 0.330 | |

5 | 8 | 5.5% | 20.4% | 1.8% | 0.024 |

16 | 6.1% | 12.1% | 2.7% | 0.003 | |

6 | 8 | 3.9% | 21.8% | 0.9% | 0.289 |

16 | 6.6% | 12.1% | 1.6% | 0.079 | |

7 | 8 | 9.8% | 11.1% | 1.6% | 0.070 |

16 | 6.7% | 10.3% | 1.0% | 0.314 | |

8 | 8 | 2.7% | 44.8% | 1.2% | 0.039 |

16 | 2.5% | 29.8% | 1.1% | 0.101 | |

9 | 8 | 2.4% | 51.3% | 1.1% | 0.026 |

16 | 1.1% | 33.8% | 0.5% | 0.566 | |

10 | 8 | 3.7% | 21.0% | 0.4% | 0.771 |

16 | 3.8% | 15.4% | 0.7% | 0.528 | |

11 | 8 | 2.6% | 50.5% | 0.8% | 0.126 |

16 | 2.7% | 23.1% | 2.9% | 0.001 | |

Mean | 8 | 5.1% | 29.6% | 1.1% | 0.204 |

16 | 4.7% | 18.1% | 1.6% | 0.204 |

*α*produces a statistically significant improvement in the fit of the model to the data. So, although the model may seem complicated, all the elements are essential and statistically justified.

*F*

^{−}and

*F*

^{+}were the only human preattentive filters sensitive to dots of different Weber contrasts, then it would follow that the only attention filters people could achieve would be combinations of

*F*

^{−}and

*F*

^{+}. Of course, if human vision had other filters selectively sensitive to dots of different Weber contrasts, then observers should be able to achieve filters outside the plane of filters spanned by

*F*

^{−}and

*F*

^{+}. Research is under way to address this question.

*Attend-to-All*condition than it is in either of the 16-dot,

*Attend-to-Dark*conditions even though eight dots are counted in all of these conditions.

*A*projects only weakly into this space of achievable filters, then we expect the filter combination used by the observer to correlate poorly with the target filter, leading to a high value of filter error

*RMSFE*

_{ N,A }.

*Attend-to-Light*and

*Attend-to-Dark*conditions than in the 8-dot

*Attend-to-All*condition? A model in which some fraction of stimulus dots was lost, i.e., not represented perceptually, and in which the fraction of lost dots increased with the number of stimulus dots, would seem to have more internal noise for 16 than 8 presented dots. In other respects, the class of dot-loss model is very difficult to distinguish from the class of dot-weighting models that we consider here. Additional studies are required to resolve these complex issues.

*N*= 8 and

*N*= 16 in each of the three attention instructions (

*A*= Attend-to-Dark,

*A*= Attend-to-Light, and

*A*= Attend-to-All). We will refer to the condition in which

*N*dots were displayed and the attention instruction was

*A*as condition (

*N*,

*A*). The data from condition (

*N*,

*A*) consist of the

*x*- and

*y*-locations,

*X*

_{ N,A }(

*t*) and

*Y*

_{ N,A }(

*t*), of the observer's response across all trials

*t*. It is assumed that on a given trial the observer's response (the location he/she clicks on) is an estimate of the centroid of the dots in the display following the application of an attention filter differentially sensitive to different dot Weber contrasts.

*F*

_{ N,A }be the attention filter used by the observer in condition (

*N*,

*A*). That is, for any Weber contrast

*c*,

*F*

_{ N,A }(

*c*) gives the weight assigned to a dot

*z*of contrast

*c*in the observer's centroid estimates in condition (

*N*,

*A*). In this case, on trial

*t*of condition (

*N*,

*A*) the

*x*- and

*y*-locations of the actual centroid of the attention-filtered display are

*z*presented on trial

*t*, and

*x*(

*z*),

*y*(

*z*), and

*c*(

*z*) are the

*x*-location,

*y*-location, and Weber contrast of

*z*, respectively.

*x*- and

*y*-locations,

*X*

_{ N,A }(

*t*) and

*Y*

_{ N,A }(

*t*), of the observer's centroid estimates to deviate from

_{ N,A }(

*t*) and

_{ N,A }(

*t*) for two reasons; first, the responses will be degraded by noise. Additionally, we expect that observers may not combine information about disparate dots precisely according to Equation A1. We anticipate that the weight given a particular dot

*z*may deviate from Equation A1 depending on the distance of

*z*from the true centroid. Specifically, we assume that the

*x*- and

*y*-locations of the observer's response on trial

*t*in condition (

*N*,

*A*) are given by Equation 2 where the weight

*w*(

*z*) assigned to a given dot

*z*on trial

*t*is

*R*

_{ N,A }as a function intended to capture biases in the weighting of central versus peripheral points in the dot cloud. Specifically, we estimate a parameter

*α*

_{ N,A }to determine the distance correction factor

*R*

_{ N,A }as follows:

*R*

_{ N,A }is defined for −1 <

*α*

_{ N,A }< 1. For

*α*

_{ N,A }< 0,

*R*

_{ N,A }is a decreasing function of the distance of a dot from (

_{ N,A }(

*t*),

_{ N,A }(

*t*)); thus, in this case, the bias introduced by

*R*

_{ N,A }yields a “robust” estimator of the centroid—i.e., an estimator that gives most weight to dots near the middle of the cloud and is relatively immune to dots located in the periphery. The reverse is true if

*α*

_{ N,A }> 0; in this case,

*R*

_{ N,A }is an increasing function of distance, yielding an “anti-robust” estimator of the centroid that gives more weight to peripheral than to central dots in the cloud. In actuality, for

*α*

_{ N,A }in a large neighborhood around 0,

*R*

_{ N,A }is essentially constant across all dot distances in any of our displays; specifically, for

*α*

_{ N,A }greater than around 0.025 and less than around 0.975, dots contribute to the centroid computation with weights that are roughly independent of their distance (in pixels) from (

_{ N,A }(

*t*),

_{ N,A }(

*t*)); that is, the computation is approximately a standard centroid (as given by Equation A1).

*F*

_{ N,A }(0) = 0 and constrain

*F*

_{ N,A }to satisfy Equation 3. Under these constraints, for each condition (

*N*,

*A*), the filter

*F*

_{ N,A }, and the value of

*α*

_{ N,A }are chosen to minimize the sum of squared distances of the responses estimated by the model (

_{ N,A }(

*t*) and

_{ N,A }(

*t*)) from the observed responses. That is, we minimize Equation 4.

*α*

_{ N,A }= 0 for each condition (

*N*,

*A*), and in addition, for any dot contrast

*c*,

*F*

_{ N,A }'s given in Equations A4, A5, and A6). We use the following statistic to estimate the root-mean-square filter error that compromises performance in a given condition:

*c*= −1, −0.75, …, 1.

*α*

_{ N,A }= 0 and

*RMSE*

_{ N,A }= 0, he/she might still deviate significantly trial by trial from the target centroid if his/her responses were contaminated by high levels of noise. In assessing the cost of imposing an attention filter, one must take into account not only systematic error reflected by

*α*

_{ N,A }and

*RMSFE*

_{ N,A }but also such random response error. We use the statistic

_{ N,A }given by Equation 5 to estimate the contribution of random noise to observers' responses in a given condition (

*N*,

*A*).

_{ N,A }is an unbiased estimate of the standard deviation of the noise component of the observer's responses.

^{2}The “deviations of predicted from observed centroids,” which are here modeled as internal response noise, consist of two components: “true internal noise” and “model prediction error.” These cannot be separated in the present experiments. Experiments using a double-pass procedure (Drew, Chubb, & Sperling, 2009) are underway to determine the relative contributions of these two factors to deviations of predicted from observed centroids.