Attention as a serial contiguous spotlight has met challenges from parallel-noisy models and evidence that attention can split across multiple locations. To assess this question, this study compared the dynamics at cued and uncued locations during a cueing task through response classification. Four observers performed a yes/no contrast discrimination of Gabors appearing at two locations, with simultaneous cues appearing at one location with 80% validity. Stimuli were presented for 272 ms in ‘contrast noise’; the peak contrasts of the Gabors varied randomly across 12 intervals of the stimulus duration. Response classification yielded a ‘classification number’ for each location (cued and uncued) and interval, giving two classification number functions (cued and uncued) across the stimulus duration. Serial models predict delays in the uncued functions, whereas parallel models do not. No evidence for delays at uncued locations was found, with no significant differences for the amplitude-matched cued and uncued functions, or for the functions' peak times after smoothing with third-degree polynomials. Also, the relative integrals between the cued and uncued functions were fit well to a parallel-noisy weighted likelihood model assuming a linear summation of responses across intervals. Thus, a parallel description of attention seemed to best explain the results in this study.

*directly*imply a limited-capacity serial attentional model.

*Journal of Vision*introduced by Eckstein & Ahumada, 2002). Several studies have also assessed temporal information use with response classification, sometimes known as classification movies (e. g., Caspi, Beutter, & Eckstein, 2004; Ghose, 2006; Ludwig, Gilchrist, McSorley, & Baddeley, 2005; Neri & Heeger, 2002; Shimozaki, Chen, Abbey, & Eckstein, 2007).

^{rd}-degree polynomials. The serial model predicts a difference in the peak times, whereas the parallel model does not. Also, the results were compared to a representative parallel-noisy model for the cueing task, the weighted likelihood model. First, the behavioral response rates (hits, misses, corrections, and false alarms) were compared to the weighted likelihood model. Second, the integrals of the cued and uncued functions were compared to the predictions from an extension of the weighted likelihood model. In addition to the assumptions of the weighted likelihood model, the extended model assumed a linear combination of information use across intervals, weighted by the smoothed classification number functions from the cued location. This comparison of integrals was performed as a method to assess the total amount of information use throughout the stimulus duration, both for the human observers and the weighted likelihood model.

^{2}), simultaneously and continuously displayed throughout the stimulus duration. A dark fixation cross was visible continuously in the center of the display (length = 0.5°, width = 4.2′, luminance = 0.77 cd/m

^{2}), as well as four dark fiduciary ‘tick marks’ near each possible signal location (1.25° from the center of the stimulus) to reduce location uncertainty (length = 0.5°, width = 4.2′, luminance = 0.77 cd/m

^{2}). Also, two copies of the signal and nonsignal pedestal Gabors were visible continuously 6.5° (signal) and 3.5° (pedestal) above the central fixation point as standard references for the observer.

^{2}, and all stimuli were achromatic (gray, CIE coordinates, x = .284, y = .309).

*d*′ and bias (criterion) that accounts for decisions over multiple locations (e.g., Eckstein et al., 2000; Palmer, 1995; Palmer et al., 1993; Palmer et al., 2000; Verghese, 2001), as it falls under the class of ‘parallel-noisy’ models that predict set size effects (decreasing performance with increasing set size or number of items) based on parallel processing over increasingly noisier decisions. An important aspect of the weighted likelihood model is that it predicts a cueing effect when the cue validity is used optimally. As this is a parallel model, this implies that cueing effects, by themselves, do not indicate a serial effect of attention.

*n,*the weighted likelihood model starts with an input variable for each location

*j*(cued or uncued),

*x*

_{ j,n }. This input variable is assumed to be Gaussian-distributed, and assumed to be the sum of the internal representation of the mean contrast for that location and trial (

*α*

_{ j,n }), plus an error term (

*ɛ*

_{ j,n }), such that

_{1}) for each location j. First, the likelihood of signal presence (

*l*

_{ j,n }) is calculated, which is the prior probability of the input variables at all locations, given signal presence at that location (S

_{1,j,n }). For this task with two locations, the input variables comprise a two-element vector

**x**

_{ n }, where

*σ*= 1) for

*x,*with a mean of zero for the signal absent distribution and a mean of

*d*′ for the signal present distribution, we can define a probability density function g(

*x*) for both signal present and signal absent distributions, where

_{0}= the state of signal absence, and S

_{1}= the state of signal presence. For p(

**x**

_{ n }∣S

_{1,j,n }), we must consider the joint probability of signal presence at location

*j*and signal absence at location ≠

*j*. Therefore,

_{1,j,n }∣

**x**

_{ n }) by weighting the likelihoods by the internal estimate of the prior probability of signal presence at that location (p(S

_{1,j }) =

*w*

_{ j }), such that

**x**

_{ n }) is not included in this formulation. Ideally the weights would represent the cue validity (i.e.,

*w*

_{ cued }= 0.80,

*w*

_{ uncued }= 0.20); however, the human observer may diverge from these ideal weightings. Thus, the estimate of the human observer weights serves as an assessment of how well (or optimally) the observer could use the information provided by the cue validity.

_{0,n }∣

**x**

_{ n }), we must consider the probability of signal absence at the cued location and the uncued location; therefore,

*crit*); if the ratio is larger than the criterion, the model responds ‘yes,’ and if the ratio is below the criterion, the model responds ‘no.’ Thus, we may predict the behavioral response rates for the model for a given

*d*′, weight and criterion from the following cumulative probabilities Pr over all trials

*n*:

*d*′, criterion, and weight (Shimozaki et al., 2003). Then the best fits of these predicted response rates to the response rates of the human observers were found. The parts of Figure 5 in red indicate those parameters that were free to vary in the fits with the observer. These were the weights for the cued and uncued locations (

*w*

_{ j }), the criterion (

*crit*), and the internal representation of the total noise (

*ɛ*

_{ j,n }). The last item (

*ɛ*

_{ j,n }) was expressed as the overall

*d*′ for the observer. Also, fits were performed with a fixed optimal weight of 0.80 and

*d*′ and the criterion as free parameters.

*x*

_{ j,n }is the result of a cross-correlation (linear operator) of an ideal template with the stimulus; in white Gaussian spatial noise, the ideal template is the same as the signal. The weight for the likelihood for the cued location (

*w*

_{ cued }) is the probability of cue appearance at the cued location, or the cue validity (0.80). The criterion is the ratio of the probabilities for target presence and target absence, which was 1.0 for this study. Finally, there would be no internal noise, and therefore

*ɛ*

_{ j,n }=

*ɛ*

_{ j,n,external }.

*crit*) = 0 represents unbiased decisions.

*a*) was calculated as the mean of the added contrast noise (

*ɛ*

_{ external }) across all trials (

*N*), taken separately for each interval (

*i*), each location (

*j,*where

*j*=

*cued*or

*uncued*), and each outcome (

*k,*where

*k*=

*false alarm*or

*correct rejection*), as follows:

*T*

^{2}statistic (Harris, 1985), which is the multivariate generalization of the univariate

*t*statistic, and analogous to the

*t*statistic has two forms, one-sample and two-sample (see 1). The one-sample Hotelling

*T*

^{2}statistic with a diagonal covariance matrix

**K**(i.e., all zero correlations) is distributed as a

*χ*

^{2}statistic, and it was found that the covariance matrices for the sample classification number functions were nearly diagonal (see 1). Therefore, for differences between the sample classification number functions and hypothesized models (with

*q*= number of free parameters), the one-sample Hotelling

*T*

^{2}statistics were evaluated as

*χ*

^{2}statistics for the model, and

*df*=

*p*−

*q*.

*χ*

^{2}results indicated these cueing effects were highly significant.

*χ*

^{2}goodness of fit statistics. As mentioned earlier, the weighted likelihood model is a parallel model of performance that also gives an estimate of the observed cue validity effect relative to an optimal Bayesian decision rule. Also, it provides a measure of

*d*′ and bias that accounts for multiple locations (Eckstein et al., 2000; Palmer, 1995; Palmer et al., 1993, 2000; Verghese, 2001). Table 1a gives the weighted likelihood fits with the cued location weight (

*w*

_{ cued }) as a free parameter, with the optimal cued weight equaling the actual cue validity of 0.80 (and therefore 1.00 − 0.80 = 0.20 at the uncued location). The free parameters (

*d*′,

*w*

_{ cued }, log(criterion)) equal the total degrees of freedom (3, for each trial type); thus, no

*χ*

^{2}statistics are presented. As shown in Table 1a, these estimated weights for each observer indeed were near the optimal weight of 0.80, except for FB, whose weight was less than optimal (0.67). For

*w*

_{ unued }/

*w*

_{ cued }, the optimal weight of 0.80 corresponds to a ratio of 0.250 = (1.0 −

*w*

_{ cued })/

*w*

_{ cued }= (1.0 − 0.8)/0.8.

Table 1a | ||||||
---|---|---|---|---|---|---|

Observer | d′ | Weight, cued location (w _{ cued }) | log(criterion) | w _{ uncued }/w _{ cued } | ||

AW | 1.50 | 0.81 | −0.18 | 0.235 | ||

FB | 1.60 | 0.67 | 0.05 | 0.492 | ||

LO | 1.92 | 0.81 | −0.42 | 0.235 | ||

SS | 1.61 | 0.80 | −0.13 | 0.250 | ||

Table 1b | ||||||

Observer | d′ | Weight, cued location (w _{ cued }) | log(criterion) | χ ^{2}(1) | p-value | Sig. |

AW | 1.50 | 0.80 | −0.17 | 1.141 | 0.2854 | |

FB | 1.60 | 0.80 | 0.01 | 60.504 | 0.0000 | * |

LO | 1.92 | 0.80 | −0.41 | 0.437 | 0.5085 | |

SS | 1.61 | 0.80 | −0.13 | 0.291 | 0.5895 |

*w*

_{ cued }) of 0.80. This gave a fit with two free parameters (

*d*′, criterion) and thus one degree of freedom for the

*χ*

^{2}statistic (3 total

*df*− 2). As might be expected from Table 1a, the fits to the optimally weighted model were not significantly different (indicating nearly optimal weighting) for all observers except for FB. Thus, the observers' performances (except FB) were fit well to a parallel model with an optimal weighting of cued information.

*d*'s were relatively consistent across observers, ranging from 1.50 to 1.92, as would be expected from the relatively consistent results for proportion correct (Table B1a, column 1). Generally across observers the log-criteria were slightly negative, indicating a slight bias to respond ‘yes.’

*T*

^{2}statistics (Table C2) confirmed that all functions (across intervals) were significantly different from zero (one-sample Hotelling

*T*

^{2}, Table C2a), and that all the cued functions were significantly greater than the uncued functions (two-sample Hotelling

*T*

^{2}, Table C2b).

*t*-tests, corrected for multiple comparisons across the 12 intervals with the Simes-Hochberg method; this method is similar to the Bonferroni correction, but is considered more appropriate given the conservative characteristics of the Bonferroni correction (Hochberg, 1988; Simes, 1986). Significant differences from zero are indicated in Figure 7 as filled circles, and are also summarized in Table C1. A significant difference from zero would indicate a significant use of information for that interval at the given location. As shown in Figure 7, there was evidence of significant information use for both cued and uncued locations throughout the stimulus duration. The first interval with significant information use might be considered an estimate of the latency for this task. By this definition, the latency for the cued location was the first interval (22.7 ms) for all observers except FB, who had a latency of the second interval (45.3 ms). These results were consistent with the results from Shimozaki et al. (2007), which found evidence of the first use of information at cued locations at 37.5 ms (2 observers) to 75 ms (1 observer). For the uncued location, these estimates of latency were slightly later, the first interval (22.7 ms) for AW and SS, the second interval (45.3 ms) for LO, and the third interval (68.0 ms) for FB.

*t*-tests, adjusted for multiple corrections with the Simes-Hochberg method. These results are also summarized in Table C1. For all observers, the cued and uncued functions differed significantly by the 2

^{nd}interval (45.3 ms); for all observers except FB, the cued and uncued functions differed significantly interval-by-interval throughout most of the stimulus duration (AW: intervals 2–7,9; LO: intervals 2–9; SS: intervals 2–9,11,12). FB had fewer intervals with significant cued vs. uncued differences (intervals 2, 5, and 6), related to the higher amplitude of her uncued function relative to her cued function.

*T*

^{2}statistics (Table 2). As expected from the interval-by-interval comparisons of the cued vs. uncued functions, FB had the largest scale factor of 0.57. The cued and the scaled uncued functions could not be distinguished statistically from each other, except for SS. For SS, a posthoc analysis was performed comparing the cued and uncued functions for each interval using independent two-sample

*t*-tests, corrected for multiple corrections by the Simes-Hochberg method. It was found that the two functions differed only in the 7

^{th}interval at 158.7 ms (

*t*(19622) = 3.155, p(uncorrected) = 0.0016). Thus, for three observers the comparison of the cued and the scaled uncued functions found no evidence of a difference, which does not favor a strong serial hypothesis. Also, the results of SS do not appear to suggest a shift in the functions in time, and thus does not appear to be congruent with the predictions from a serial model. Rather, there appears to be a ‘dip’ in the uncued function at an intermediate stimulus duration. This dip differs from the other observers, and there appears to be no obvious explanation.

*t*) from quadratic (2

^{nd}) to quartic (4

^{th}) degree were considered, such that

*c*

_{4}= 0 for the 3

^{rd}-degree polynomial, and

*c*

_{3}=

*c*

_{4}= 0 for the 2

^{nd}-degree polynomial. Fits were evaluated with one-sample Hotelling

*T*

^{2}statistics, evaluated as

*χ*

^{2}'s and with degrees of freedom = 12 − number of free parameters. It was found that no 4th-degree polynomial significantly improved the fit compared to the 3rd-degree polynomial (Hotelling

*T*

^{2}difference = 0.04 to 2.94;

*p*= 0.6548 to 0.0863). Thus, only the 3

^{rd}-degree polynomial fits are presented, with the following format:

^{rd}-degree polynomials, and Table D1 in 4 summarizes the one-sample Hotelling

*T*

^{2}statistics, evaluated as

*χ*

^{2}'s and with degrees of freedom = 12 − 4 = 8. Also, Table D1 lists the fitted coefficients for the fits (

*c*

_{0}to

*c*

_{3}), with time (

*t*) in seconds. Except for the uncued data for SS, no significant differences were found between the data and the fitted polynomial models, indicating relatively good fits. As mentioned earlier, the data for SS at the uncued location had a ‘dip’ at the 7

^{th}interval (158.7 ms), which led to the poor fits to both the 3rd-degree and 4th-degree polynomial functions (4th-degree fit: Hotelling

*T*

^{2}= 19.89, degrees of freedom = 7,

*p*= 0.0058).

^{rd}-degree polynomial fits, and then perturbing those values assuming Gaussian distributions with the same standard errors as the observed classification numbers. These resampled classification number functions were then refit to new 3

^{rd}-degree polynomials with

*χ*

^{2}goodness-of-fits, and the peak times were found for the refit functions; standard errors were computed across iterations (50,000 for each peak time). The peak times (from the fitted polynomials) and the estimated standard errors are presented in Table 3. Also, Table 3 summarizes the comparisons of the cued and uncued peak times for each observer, assessed as independent two-sample

*t*-tests. The cued peak times tended to be slightly less than the uncued peak times. However, the estimated standard errors for the uncued peak times were larger than those for the cued peak times; this was due to the lower-amplitude (i.e., ‘flatter’) uncued functions, leading to more uncertainty in the estimated peak times. This was particularly true for AW, who had a distinctly flat function for the uncued location, and thus a rather large estimated standard error (72.6 ms). The result is that no significant differences were found between the peak times of the cued and uncued functions for any observer, arguing against a strong serial attentional hypothesis.

^{rd}-degree polynomials), with the standard errors calculated with the same method as the peak times (resampling based on the classification number standard errors and refitting with 3

^{rd}-degree polynomials with

*χ*

^{2}goodness-of-fits over 50,000 iterations). The cued function integrals were significantly greater than the uncued function integrals, as assessed with independent two-sample

*t*-tests, with ratios (uncued/cued) ranging from 0.396 (for LO) to 0.545 (for FB).

Observer | Integral | Std. Err. | t | p-value | Sig. | Ratio, uncued/cued | |
---|---|---|---|---|---|---|---|

AW | cued | 0.687 | 0.029 | 10.283 | 0.0000 | * | 0.396 |

uncued | 0.272 | 0.029 | |||||

FB | cued | 0.583 | 0.031 | 6.082 | 0.0000 | * | 0.545 |

uncued | 0.318 | 0.031 | |||||

LO | cued | 0.921 | 0.029 | 12.481 | 0.0000 | * | 0.438 |

uncued | 0.403 | 0.029 | |||||

SS | cued | 1.003 | 0.029 | 12.605 | 0.0000 | * | 0.478 |

uncued | 0.479 | 0.030 |

*w*

_{ uncued }/

*w*

_{ cued }, from Table 1a) optimally should be (1.0 − 0.8)/0.8 = 0.25, and ranged from 0.235 (for AW and LO) to 0.492 (for FB). However, the scaling factors equalizing amplitudes across the cued and uncued classification number functions ranged from 0.33 (for AW) to 0.57 (for FB); also, the relative integrals from Table 4 ranged from 0.396 (for LO) to 0.545 (for FB). Thus, the general order across observers seem to correspond across these measures, with AW having the lowest and FB having the highest values; however, the relative weightings predicted by the weighted likelihood model seem lower than those expected from both the scaling factors and the relative integrals. However, we would not necessarily expect to find a linear relationship between the weights predicted from weighted likelihood and the characteristics of the classification number functions. One issue is that behavioral cueing effects (assessed as valid hit rate − invalid hit rate) predicted by the weighted likelihood model depends upon

*d*′, with small cueing effects predicted at both low and high

*d*'s (Shimozaki et al., 2003). Also, there would be differences in cueing effects based upon different criteria in the weighted likelihood model; consider the case in which the observer is completely biased to always respond ‘yes’ or ‘no’, which clearly would lead to no observable cueing effects. For these reasons, a final analysis was undertaken to predict the relative relationship between the weights from the weighted likelihood model and cued and uncued classification number functions, including the relative integral values for the cued and uncued functions, with an extension of the weighted likelihood model presented earlier (Figure 5).

*x*

_{ j,n }for a given location

*j, uncued*or

*cued,*and trial

*n*) is the result of the sum of weighted responses (

*y*

_{ i,j,n }) across intervals (

*i*), with the total number of intervals =

*H*= 12. Each unweighted response is the internal representation for the contrast for a particular location

*j*and trial n (

*α*

_{ j,n }), added to the internal representation of the total noise for that location

*j,*trial

*n,*and interval

*i*(

*ɛ*

_{ i,j,n }), with

*i*.

*i*at the cued location, as given by the 3

^{rd}-degree polynomial fits described above (

*a*

_{ i,cued }). Thus, the weighted internal response

*j*and trial

*n*is the sum of the weighted responses across the intervals, or

_{ j,cued }) were determined by the observers' classification numbers for the cued location, as derived from the 3

^{rd}-degree polynomial fits. The internal representation of the total noise (

*ɛ*

_{ i,j,n }) was determined so that the sensitivity for the extended model (

*d*′) matched that of the observers' fits to the weighted likelihood model presented earlier (Table 1a). The weights for the cued and uncued locations (

*w*

_{ j }) and the criteria (

*crit*) were also determined by the observers' fits to the weighted likelihood model. Thus, the extended model was completely determined and had no free parameters. Also, the observers' use of information at the uncued location was not represented, except as the difference in weights at the cued and uncued locations for the fitted weighted likelihood model (

*w*

_{ j }). The predicted classification number functions from the extended model were found with Monte Carlo simulations with 50,000 trials for each observer.

^{rd}-degree polynomials. For these analyses, the cued functions for the extended model and the polynomials were matched to give the best fits. Figure 11 presents the same 3

^{rd}-degree polynomial fits shown in Figure 8, with cued in blue and uncued in red, and presents the extended model fits in gray. The amplitudes for the uncued functions were lower than those for the cued locations. This is expected from the difference in weights at the cued and uncued locations in the extended model, and is the representation of the cueing effect for the classification number functions. Note that this difference is predicted from a parallel-noisy model, analogous to cueing effects for behavioral responses (Table 1; Eckstein et al., 2002; Shimozaki et al., 2003). This finding highlights the point that cueing effects in cueing tasks do not necessarily indicate a serial, limited capacity attentional mechanism.

*T*

^{2}tests with the covariance matrices from the original data. The fits were good, which was not necessarily surprising, as the interval weights for the extended model did come from the cued location functions. However, it suggests that any nonlinear transformation of the extended model (i.e., the calculation of the likelihoods) did not have much relative effect from interval to interval on the resulting classification number functions.

^{rd}-degree polynomial fits (also shown in Figure 11). Table 5 gives the overall fits for the uncued functions, assessed as two-sample Hotelling

*T*

^{2}tests with the covariance matrices from the original data. There were no significant differences for any observer, and the fits were good for all observers except SS, whose predicted uncued function was somewhat less than the 3

^{rd}-degree polynomial fits.

Observer | 3rd -degree polynomial uncued integral | Std. Err. | Extended weighted likelihood uncued integral | Std. Err. | t | p-value | Sig. |
---|---|---|---|---|---|---|---|

AW | 0.272 | 0.029 | 0.260 | 0.036 | 0.248 | 0.8040 | |

FB | 0.318 | 0.031 | 0.404 | 0.038 | 1.765 | 0.0775 | |

LO | 0.403 | 0.029 | 0.362 | 0.039 | 0.841 | 0.4001 | |

SS | 0.479 | 0.030 | 0.379 | 0.038 | 2.102 | 0.0355 | * |

^{rd}-degree polynomial fits (also presented in Table 4). The standard errors for the integrals from the extended model were found with the same basic method as before. The predicted values for the classification numbers from the extended model were resampled with the standard errors from the original data; then these values were fitted with 3

^{rd}-degree polynomials with

*χ*

^{2}goodness-of-fits over 50,000 iterations. Independent two-sample

*t*-tests found no significant differences between the extended model and the original 3

^{rd}-degree polynomial fits, except for SS, which found that the extended model predictions were significantly smaller. The significant results for SS, however, likely were affected by the ‘dip’ at 158.7 ms that led to a poor fit to the original 3

^{rd}-degree polynomial (Table D1 and Figure 9). In summary, over both sets of results (Hotelling

*T*

^{2}'s and integrals) the extended weighted likelihood model of weighted responses across intervals gave relatively good predictions for the uncued classification number functions, and therefore the relative differences between the cued and uncued functions. In other words, the total information use throughout the stimulus duration at the cued and uncued locations were predicted reasonably well with the extended weighted likelihood model.

*d*'s) and faster rises to the asymptotes (suggesting speeded processing). While the current study did not manipulate response times, inferences and comparisons can be made to the deadline procedure of Carrasco et al. (Carrasco et al., 2004, 2006; Carrasco & McElree, 2001). Both findings of increased performance and speeded processing for the cued locations in these studies were represented in the classification number functions. First, increased performance was represented by the increased amplitudes of the cued functions. Second, speeded processing (as defined by Carrasco et al.) was represented by the greater initial rise of the cued functions, concurrent with the greater amplitudes. Note, however, that the same aspects were predicted well by the extended weighted likelihood model, a parallel-noisy model (Figure 11). The reason is that the weighted likelihood model considers both cued and uncued locations in its decision, compared to separate analyses at each location, as done by Carrasco et al. (Carrasco et al., 2004, 2006; Carrasco & McElree, 2001). Thus, the weighted likelihood model assumes the same sensitivity at each location, but with the differential (larger) weighting at the cued location. To repeat a point made earlier, the predicted differences between the cued and uncued classification number functions made by the extended weighted likelihood model are analogous to the differences in behavioral performance (expressed as valid hit rate − invalid hit rate) predicted by the weighted likelihood model (Table 1; Eckstein et al., 2002; Shimozaki et al., 2003). They are also analogous to the results found for spatial response classification (classification images) at cued and uncued locations in Eckstein et al. (2002). In that study, they found that the cued classification images had greater amplitudes then the uncued classification images; however, these differences were predicted by the weighted likelihood model, and scaling the uncued classification images to match the amplitudes of the cued classification images resulted in no difference in shape, again as predicted by the weighted likelihood model.

^{th}-degree polynomial fits did not significantly improve the fits given by the 3

^{rd}-degree polynomials, as would be anticipated with a two-peak function predicted by the discrete sampling model within the stimulus duration for the current study. However, it should be noted that the periodic sampling in VanRullen et al. (2007) only occurred for more difficult discriminations, and parallel processing was found in easier discriminations. Also, their assessment was over a longer period of time (1 second), which may induce a more periodic sampling behavior than the shorter duration of the current study (272 ms).

^{rd}-degree polynomials, fits were attempted with log-Gaussian and Poisson functions. These were attempted as representative functions with positive skew, which seemed to describe the basic characteristic of the classification number functions. Also, the log-Gaussian function was employed previously to fit equivalent classification number functions in saccadic decisions (Ludwig et al., 2005). The fits to the log-Gaussian and Poisson functions were relatively poor, particularly for the cued functions. For the log-Gaussian functions, the differences in the quality of fits, compared to Ludwig et al. (2005), could be due to several factors. The most obvious is the saccadic vs. attentional information use, but others include static conditions vs. temporal noise, slightly different assessments of fits (correlations instead of Hotelling

*T*

^{2}'s), different signals (Gaussians vs. Gabors), the use of precues, and longer stimulus durations (500 ms and 1000 ms vs. 272 ms).

^{rd}-degree polynomial functions) were not significantly different from each other. Thus, these classification number functions do not suggest a difference between the cued and uncued temporal dynamics in the form of a delay at the uncued location, and therefore did not indicate a serial attentional mechanism. There was a difference in the amplitudes and integrals of the classification number functions, with greater amplitudes and integrals for the cued locations. These differences in the classification number functions, however, could be predicted by a parallel-noisy model based upon the weighted likelihood model that assumed a linear integration of information across intervals.

*T*

^{2}statistics

*T*

^{2}statistic (Harris, 1985). The Hotelling

*T*

^{2}is the multivariate generalization of the univariate

*t*statistic; analogous to the

*t*statistic, the Hotelling

*T*

^{2}has two forms, one-sample and two-sample. The one-sample Hotelling

*T*

^{2}is appropriate for comparisons of a sample (multivariate) vector against a fixed or known population vector, such as differences of a sample vector from a hypothesized model or from zero. It is calculated as

*T*

^{2}is the Hotelling

*T*

^{2}statistic,

*N*= sample size,

**a**= the sample vector (the classification number function),

**a**

_{ 0 }= the population (model) vector, [

**a**−

**a**

_{ 0 }]

^{ t }is the transpose of [

**a**−

**a**

_{ 0 }], and

**K**

^{−1}is the inverse of the covariance matrix of the sample vector. For tests of significance,

*T*

^{2}may be transformed into an

*F*statistic by the following:

*p*= length of the vector [

**a**−

**a**

_{ 0 }], degrees of freedom (

*df*) for the numerator (

*df*

_{ numerator }) =

*p,*and

*df*

_{ denominator }=

*N*−

*p*.

*T*

^{2}is appropriate for comparisons of two sample vectors (classification number functions), and is calculated as

*N*

_{1}= the sample size for the first sample vector,

*N*

_{2}= the sample size for the second sample vector,

**a**

_{ 1 }= the first sample vector, and

**a**

_{ 2 }= the second sample vector. The covariance matrix

**K**is computed as the pooled variances/covariances across

**a**

_{ 1 }and

**a**

_{ 2 }. The corresponding

*F*statistic is computed as

*df*

_{ numerator }=

*p,*and

*df*

_{ denominator }=

*N*

_{1}+

*N*

_{2}−

*p*− 1.

*T*

^{2}statistic with a diagonal covariance matrix

**K**(i.e., all zero correlations), the

*T*

^{2}statistic may be described as

*H*= the number of intervals = 12. This is distributed as a

*χ*

^{2}statistic for differences between the sample classification number function (

**a**) and a hypothesized model (

**a**

_{ 0 }). It was found that the correlations across intervals for the classification number functions were nearly zero (mean across observers = −0.002, with a range from −0.053 to 0.034), and not significantly different from zero for any observer. Across observers, the single-sample

*t*values for differences from zero ranged from 1.037 to 1.604, giving

*p*-values ranging from 0.3302 to 0.1092. Thus, for comparisons of hypothesized models (with

*q*= number of free parameters) against the sample classification number functions, the one-sample Hotelling

*T*

^{2}statistics were evaluated as

*χ*

^{2}statistics for the model, and

*df*=

*p*−

*q*.

Table B1a | ||||
---|---|---|---|---|

Observer | Proportion correct | Valid hit rate | Invalid hit rate | False alarm rate |

AW | 0.7921 | 0.8048 | 0.6185 | 0.2906 |

FB | 0.7508 | 0.7537 | 0.6531 | 0.2307 |

LO | 0.7372 | 0.8768 | 0.7404 | 0.2635 |

SS | 0.7543 | 0.8037 | 0.6323 | 0.2588 |

Table B1b | ||||

Observer | Cueing (valid hit − invalid hit) | χ ^{2}(1) | p-value | Sig. |

AW | 0.1863 | 414.2 | 0.0000 | * |

FB | 0.1006 | 118.5 | 0.0000 | * |

LO | 0.1364 | 293.7 | 0.0000 | * |

SS | 0.1714 | 354.3 | 0.0000 | * |

AW (n = 9779) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Time (ms) | Cued | Diff. from zero | Uncued | Diff. from zero | ||||||

Class. Number | SE | t | p | Sig. | Class. Number | SE | t | p | Sig. | |

22.7 | 0.916 | 0.351 | 2.606 | 0.0092 | * | 1.207 | 0.353 | 3.421 | 0.0006 | * |

45.3 | 3.193 | 0.343 | 9.314 | 0.0000 | * | 0.957 | 0.348 | 2.749 | 0.0060 | * |

68.0 | 4.428 | 0.343 | 12.916 | 0.0000 | * | 0.686 | 0.353 | 1.940 | 0.0523 | |

90.7 | 3.480 | 0.347 | 10.035 | 0.0000 | * | 0.986 | 0.346 | 2.852 | 0.0043 | * |

113.3 | 4.011 | 0.346 | 11.608 | 0.0000 | * | 1.277 | 0.350 | 3.649 | 0.0003 | * |

136.0 | 3.911 | 0.350 | 11.166 | 0.0000 | * | 1.540 | 0.353 | 4.362 | 0.0000 | * |

158.7 | 2.955 | 0.350 | 8.442 | 0.0000 | * | 0.958 | 0.344 | 2.788 | 0.0053 | * |

181.3 | 2.540 | 0.347 | 7.326 | 0.0000 | * | 1.309 | 0.347 | 3.771 | 0.0002 | * |

204.0 | 2.384 | 0.356 | 6.705 | 0.0000 | * | 0.850 | 0.339 | 2.507 | 0.0122 | |

226.7 | 1.108 | 0.344 | 3.217 | 0.0013 | * | 0.746 | 0.348 | 2.146 | 0.0318 | |

249.3 | 1.138 | 0.347 | 3.277 | 0.0011 | * | 0.706 | 0.350 | 2.018 | 0.0436 | |

272.0 | 1.261 | 0.350 | 3.603 | 0.0003 | * | 0.283 | 0.346 | 0.817 | 0.4140 | |

Time (ms) | Cued vs. Uncued | Scaled Uncued scale factor = 0.33 | ||||||||

t | p | Sig. | Class. Number | SE | ||||||

22.7 | −0.585 | 0.5584 | 3.658 | 1.069 | ||||||

45.3 | 4.576 | 0.0000 | * | 2.900 | 1.055 | |||||

68.0 | 7.599 | 0.0000 | * | 2.079 | 1.071 | |||||

90.7 | 5.093 | 0.0000 | * | 2.988 | 1.048 | |||||

113.3 | 5.561 | 0.0000 | * | 3.868 | 1.060 | |||||

136.0 | 4.768 | 0.0000 | * | 4.666 | 1.070 | |||||

158.7 | 4.072 | 0.0000 | * | 2.903 | 1.041 | |||||

181.3 | 2.508 | 0.0121 | 3.968 | 1.052 | ||||||

204.0 | 3.123 | 0.0018 | * | 2.576 | 1.028 | |||||

226.7 | 0.739 | 0.4597 | 2.261 | 1.053 | ||||||

249.3 | 0.876 | 0.3809 | 2.140 | 1.060 | ||||||

272.0 | 1.987 | 0.0469 | 0.857 | 1.049 | ||||||

FB (n = 9779) | ||||||||||

Time (ms) | Cued | Diff. from zero | Uncued | Diff. from zero | ||||||

Class. Number | SE | t | p | Sig. | Class. Number | SE | t | p | Sig. | |

22.7 | 0.210 | 0.378 | 0.556 | 0.5783 | −0.173 | 0.381 | 0.455 | 0.6490 | ||

45.3 | 2.077 | 0.371 | 5.595 | 0.0000 | * | 0.261 | 0.375 | 0.696 | 0.4862 | |

68.0 | 2.322 | 0.372 | 6.240 | 0.0000 | * | 1.538 | 0.381 | 4.031 | 0.0001 | * |

90.7 | 3.098 | 0.374 | 8.273 | 0.0000 | * | 1.966 | 0.372 | 5.291 | 0.0000 | * |

113.3 | 3.345 | 0.373 | 8.970 | 0.0000 | * | 1.513 | 0.378 | 4.005 | 0.0001 | * |

136.0 | 3.813 | 0.377 | 10.102 | 0.0000 | * | 2.057 | 0.379 | 5.422 | 0.0000 | * |

158.7 | 2.523 | 0.378 | 6.680 | 0.0000 | * | 1.338 | 0.370 | 3.613 | 0.0003 | * |

181.3 | 2.203 | 0.373 | 5.900 | 0.0000 | * | 2.140 | 0.374 | 5.717 | 0.0000 | * |

204.0 | 1.897 | 0.383 | 4.951 | 0.0000 | * | 1.152 | 0.365 | 3.157 | 0.0016 | * |

226.7 | 1.945 | 0.371 | 5.242 | 0.0000 | * | 1.327 | 0.375 | 3.541 | 0.0004 | * |

249.3 | 2.238 | 0.375 | 5.975 | 0.0000 | * | 1.085 | 0.378 | 2.872 | 0.0041 | * |

272.0 | 1.742 | 0.376 | 4.634 | 0.0000 | * | 1.016 | 0.372 | 2.731 | 0.0063 | * |

Time (ms) | Cued vs. Uncued | Scaled Uncued scale factor = 0.57 | ||||||||

t | p | Sig. | Class. Number | SE | ||||||

22.7 | 0.715 | 0.4748 | −0.304 | 0.669 | ||||||

45.3 | 3.443 | 0.0006 | * | 0.458 | 0.657 | |||||

68.0 | 1.472 | 0.1410 | 2.697 | 0.669 | ||||||

90.7 | 2.144 | 0.0320 | 3.450 | 0.652 | ||||||

113.3 | 3.452 | 0.0006 | * | 2.654 | 0.663 | |||||

136.0 | 3.281 | 0.0010 | * | 3.609 | 0.666 | |||||

158.7 | 2.242 | 0.0249 | 2.347 | 0.649 | ||||||

181.3 | 0.120 | 0.9047 | 3.754 | 0.657 | ||||||

204.0 | 1.407 | 0.1594 | 2.022 | 0.640 | ||||||

226.7 | 1.173 | 0.2406 | 2.327 | 0.657 | ||||||

249.3 | 2.167 | 0.0302 | 1.904 | 0.663 | ||||||

272.0 | 1.372 | 0.1701 | 1.783 | 0.653 | ||||||

LO (n = 9804) | ||||||||||

Time (ms) | Cued | Diff. from zero | Uncued | Diff. from zero | ||||||

Class. Number | SE | t | p | Sig. | Class. Number | SE | t | p | Sig. | |

22.7 | 0.837 | 0.362 | 2.313 | 0.0207 | * | 0.789 | 0.363 | 2.173 | 0.0298 | |

45.3 | 3.648 | 0.353 | 10.325 | 0.0000 | * | 1.691 | 0.358 | 4.730 | 0.0000 | * |

68.0 | 5.373 | 0.352 | 15.263 | 0.0000 | * | 1.313 | 0.364 | 3.601 | 0.0003 | * |

90.7 | 4.912 | 0.356 | 13.814 | 0.0000 | * | 2.205 | 0.355 | 6.217 | 0.0000 | * |

113.3 | 5.130 | 0.354 | 14.475 | 0.0000 | * | 1.842 | 0.360 | 5.120 | 0.0000 | * |

136.0 | 5.307 | 0.359 | 14.798 | 0.0000 | * | 2.548 | 0.363 | 7.021 | 0.0000 | * |

158.7 | 4.196 | 0.359 | 11.699 | 0.0000 | * | 1.802 | 0.354 | 5.086 | 0.0000 | * |

181.3 | 4.240 | 0.355 | 11.957 | 0.0000 | * | 2.202 | 0.357 | 6.168 | 0.0000 | * |

204.0 | 3.276 | 0.366 | 8.961 | 0.0000 | * | 1.052 | 0.349 | 3.010 | 0.0026 | * |

226.7 | 1.972 | 0.355 | 5.552 | 0.0000 | * | 0.870 | 0.358 | 2.432 | 0.0150 | * |

249.3 | 1.880 | 0.358 | 5.258 | 0.0000 | * | 1.044 | 0.361 | 2.893 | 0.0038 | * |

272.0 | 1.087 | 0.360 | 3.023 | 0.0025 | * | 0.691 | 0.356 | 1.941 | 0.0523 | |

Time (ms) | Cued vs. Uncued | Scaled Uncued scale factor = 0.42 | ||||||||

t | p | Sig. | Class. Number | SE | ||||||

22.7 | 0.093 | 0.9259 | 1.879 | 0.865 | ||||||

45.3 | 3.893 | 0.0001 | * | 4.027 | 0.851 | |||||

68.0 | 8.013 | 0.0000 | * | 3.125 | 0.868 | |||||

90.7 | 5.391 | 0.0000 | * | 5.249 | 0.844 | |||||

113.3 | 6.511 | 0.0000 | * | 4.386 | 0.857 | |||||

136.0 | 5.406 | 0.0000 | * | 6.068 | 0.864 | |||||

158.7 | 4.750 | 0.0000 | * | 4.289 | 0.843 | |||||

181.3 | 4.051 | 0.0001 | * | 5.242 | 0.850 | |||||

204.0 | 4.399 | 0.0000 | * | 2.504 | 0.832 | |||||

226.7 | 2.185 | 0.0289 | 2.072 | 0.852 | ||||||

249.3 | 1.647 | 0.0996 | 2.485 | 0.859 | ||||||

272.0 | 0.783 | 0.4336 | 1.644 | 0.847 | ||||||

SS (n = 9832) | ||||||||||

Time (ms) | Cued | Diff. from zero | Uncued | Diff. from zero | ||||||

Class. Number | SE | t | p | Sig. | Class. Number | SE | t | p | Sig. | |

22.7 | 1.273 | 0.357 | 3.568 | 0.0004 | * | 1.305 | 0.365 | 3.578 | 0.0003 | * |

45.3 | 3.457 | 0.357 | 9.674 | 0.0000 | * | 1.299 | 0.355 | 3.663 | 0.0002 | * |

68.0 | 4.609 | 0.353 | 13.042 | 0.0000 | * | 1.894 | 0.364 | 5.196 | 0.0000 | * |

90.7 | 4.855 | 0.354 | 13.702 | 0.0000 | * | 2.305 | 0.360 | 6.411 | 0.0000 | * |

113.3 | 4.640 | 0.360 | 12.905 | 0.0000 | * | 2.690 | 0.365 | 7.365 | 0.0000 | * |

136.0 | 5.700 | 0.358 | 15.913 | 0.0000 | * | 2.930 | 0.367 | 7.979 | 0.0000 | * |

158.7 | 5.055 | 0.355 | 14.232 | 0.0000 | * | 1.073 | 0.362 | 2.961 | 0.0031 | * |

181.3 | 3.936 | 0.354 | 11.114 | 0.0000 | * | 1.959 | 0.358 | 5.478 | 0.0000 | * |

204.0 | 3.645 | 0.361 | 10.103 | 0.0000 | * | 1.702 | 0.353 | 4.816 | 0.0000 | * |

226.7 | 3.283 | 0.360 | 9.124 | 0.0000 | * | 2.267 | 0.358 | 6.329 | 0.0000 | * |

249.3 | 3.266 | 0.360 | 9.061 | 0.0000 | * | 1.289 | 0.357 | 3.615 | 0.0003 | * |

272.0 | 1.744 | 0.360 | 4.842 | 0.0000 | * | 0.307 | 0.357 | 0.860 | 0.3895 | |

Time (ms) | Cued vs. Uncued | Scaled Uncued scale factor = 0.46 | ||||||||

t | p | Sig. | Class. Number | SE | ||||||

22.7 | −0.063 | 0.9499 | 2.836 | 0.793 | ||||||

45.3 | 4.286 | 0.0000 | * | 2.824 | 0.771 | |||||

68.0 | 5.349 | 0.0000 | * | 4.117 | 0.792 | |||||

90.7 | 5.049 | 0.0000 | * | 5.012 | 0.782 | |||||

113.3 | 3.805 | 0.0001 | * | 5.848 | 0.794 | |||||

136.0 | 5.399 | 0.0000 | * | 6.371 | 0.798 | |||||

158.7 | 7.850 | 0.0000 | * | 2.332 | 0.788 | |||||

181.3 | 3.929 | 0.0001 | * | 4.258 | 0.777 | |||||

204.0 | 3.845 | 0.0001 | * | 3.701 | 0.769 | |||||

226.7 | 2.000 | 0.0455 | 4.929 | 0.779 | ||||||

249.3 | 3.899 | 0.0001 | * | 2.802 | 0.775 | |||||

272.0 | 2.834 | 0.0046 | * | 0.668 | 0.776 |

Table C2a | ||||||
---|---|---|---|---|---|---|

Observer | Hotelling T ^{2} | df _{2} | F(12,df _{2}) | p-value | Sig. | |

AW | cued | 815.36 | 9767 | 67.870 | 0.0000 | * |

uncued | 63.46 | 9767 | 5.283 | 0.0000 | * | |

FB | cued | 514.48 | 9767 | 42.825 | 0.0000 | * |

uncued | 107.52 | 9767 | 8.950 | 0.0000 | * | |

LO | cued | 1379.74 | 9792 | 114.849 | 0.0000 | * |

uncued | 153.05 | 9792 | 12.740 | 0.0000 | * | |

SS | cued | 1483.28 | 9820 | 123.468 | 0.0000 | * |

uncued | 203.08 | 9820 | 16.904 | 0.0000 | * | |

Table C2b | ||||||

Observer | Hotelling T ^{2} | df _{2} | F(12,df _{2}) | p-value | Sig. | |

AW | 146.04 | 19545 | 12.163 | 0.0000 | * | |

FB | 43.31 | 19545 | 3.607 | 0.0000 | * | |

LO | 184.23 | 19595 | 15.344 | 0.0000 | * | |

SS | 177.81 | 19651 | 14.809 | 0.0000 | * |

^{rd}-degree polynomials of the form

Observer | Hotelling T ^{2} | df | p-value | Sig. | c0 | c1 | c2 | c3 | |
---|---|---|---|---|---|---|---|---|---|

AW | cued | 12.78 | 8 | 0.1196 | −1.19 | 125.63 | −884.1 | 1674.4 | |

uncued | 3.82 | 8 | 0.8730 | 1.17 | −6.83 | 85.6 | −268.7 | ||

FB | cued | 8.27 | 8 | 0.4076 | −1.83 | 108.45 | −707.8 | 1327.5 | |

uncued | 7.32 | 8 | 0.5025 | −1.56 | 63.99 | −368.3 | 618.0 | ||

LO | cued | 13.19 | 8 | 0.1055 | −1.79 | 153.95 | −983.0 | 1684.5 | |

uncued | 9.22 | 8 | 0.3241 | −0.04 | 43.16 | −255.3 | 389.5 | ||

SS | cued | 13.60 | 8 | 0.0928 | −0.77 | 115.11 | −664.7 | 1032.3 | |

uncued | 21.12 | 8 | 0.0068 | * | 0.65 | 25.85 | −103.0 | 27.0 |

*t*):

*c*

_{0}equals the y-offset,

*c*

_{1}equals the peak amplitude,

*c*

_{2}determines the peak time (the ‘location’), and

*c*

_{3}equals the spread or ‘scale.’ The second function was a Poisson, defined as a function of interval (

*i*):

*c*

_{0}equals the y-offset,

*c*

_{1}determines the peak amplitude, and

*c*

_{2}is the mean and the variance.

*T*

^{2}statistics evaluated as

*χ*

^{2}'s and with degrees of freedom = 12 − number of free parameters. Thus, the log-Gaussian fits had 12 − 4 = 8 degrees of freedom, and the Poisson fits had 12 − 3 = 9 degrees of freedom. Table D2 also summarizes the free parameters for the functions, in terms of milliseconds for the log-Gaussian fits, and in terms of interval (1 to 12) for the Poisson fits. Across both functions the fits for the cued functions were generally poor, with significant differences found for all but one fit (FB, log-Gaussian). The fits for the uncued functions tended to be better, with no significant differences found across observers, except for SS.

Table D2a | |||||||||
---|---|---|---|---|---|---|---|---|---|

Observer | Hotelling T ^{2} | df | p-value | Sig. | c0 | c1 | c2 | c3 | |

AW | cued | 28.16 | 8 | 0.0004 | * | 0.44 | 5.25 | 61 | 0.90 |

uncued | 13.40 | 8 | 0.0989 | 0.99 | 1.34 | 92 | −0.43 | ||

FB | cued | 11.09 | 8 | 0.1965 | 0.23 | 2.96 | 76 | 1.03 | |

uncued | 6.12 | 8 | 0.6335 | −0.23 | 2.14 | 92 | 0.93 | ||

LO | cued | 18.28 | 8 | 0.0192 | * | 0.62 | 3.75 | 53 | 0.89 |

uncued | 5.55 | 8 | 0.6970 | 0.78 | 0.65 | 106 | −0.28 | ||

SS | cued | 23.72 | 8 | 0.0026 | * | 1.12 | 4.18 | 71 | 0.93 |

uncued | 23.24 | 8 | 0.0031 | * | 1.14 | 1.50 | 83 | −0.49 | |

Table D2b | |||||||||

Observer | Hotelling T ^{2} | df | p-value | Sig. | c0 | c1 | c2 | ||

AW | cued | 37.19 | 9 | 0.0000 | * | 1.39 | 25.31 | 5.45 | |

uncued | 9.21 | 9 | 0.4181 | 0.75 | 9.27 | 5.88 | |||

FB | cued | 20.85 | 9 | 0.0133 | * | 1.20 | 13.18 | 5.96 | |

uncued | 15.27 | 9 | 0.0838 | 0.44 | 10.11 | 6.81 | |||

LO | cued | 24.20 | 9 | 0.0040 | * | 1.13 | 17.78 | 4.99 | |

uncued | 6.21 | 9 | 0.7187 | 0.64 | 3.92 | 6.13 | |||

SS | cued | 29.08 | 9 | 0.0006 | * | 2.03 | 21.30 | 6.03 | |

uncued | 23.68 | 9 | 0.0048 | * | 1.01 | 9.02 | 5.74 |