In the Posner cueing paradigm, observers’ performance in detecting a target is typically better in trials in which the target is present at the cued location than in trials in which the target appears at the uncued location. This effect can be explained in terms of a Bayesian observer where visual attention simply weights the information differently at the cued (attended) and uncued (unattended) locations without a change in the quality of processing at each location. Alternatively, it could also be explained in terms of visual attention changing the shape of the perceptual filter at the cued location. In this study, we use the classification image technique to compare the human perceptual filters at the cued and uncued locations in a contrast discrimination task. We did not find statistically significant differences between the shapes of the inferred perceptual filters across the two locations, nor did the observed differences account for the measured cueing effects in human observers. Instead, we found a difference in the magnitude of the classification images, supporting the idea that visual attention changes the weighting of information at the cued and uncued location, but does not change the quality of processing at each individual location.

_{c}) by the prior probability of the target being present in that location (precue validity). The likelihood from the uncued location is weighted (w

_{u}) by its corresponding prior probability of target presence (1 minus precue validity). The result is an overall likelihood of the filter responses given target presence across the two locations. The Bayesian observer then calculates an overall likelihood of the data given target absence. Finally, the model computes a ratio of the likelihoods and makes a decision by comparing the likelihood ratio to a decision criterion or threshold. Figure 1 shows a schematic of the Bayesian observer for a task in which the signal is a Gaussian “contrast increment” embedded in white Gaussian noise. summarizes the mathematical expressions describing the Bayesian observer for the Posner paradigm.

_{u}= 0 in Figure 1). For the more general case where the cue is valid a certain percent of the time (cue validity = 80%), the Bayesian observer simply gives more weight to evidence (or information) arising from the cued location.

_{ideal filter}/E

_{human filter}). This measure is known as the efficiency of the perceptual filter. For simple linear tasks in white Gaussian noise, the efficiency can be directly calculated by computing the squared correlation (match) between the perceptual filter and the optimal filter (which is the signal). However, when the external noise does not have equal power in all the frequencies (nonwhite noise), then the degree of match between the perceptual filter and the signal is not the sole factor determining the performance of the filter. In these cases, the optimal filter does not match the signal. For tasks such as the Posner paradigm where the decision is a nonlinear function of the data, no simple calculations are available and Monte Carlo simulations and/or numerical approximations are required to compute the task performance associated to a perceptual filter (Nolte & Jaarsma, 1967).

_{c}and w

_{u}) in our Bayesian model (see Figure 1 and ).

_{c}and w

_{u}in Figure 1). Because of the nonlinear stage in the Bayesian observer in the Posner paradigm, the mathematical relationship between the weights used in the model for the likelihood for each location and the ratio of magnitudes of the obtained classification images is not easily derived analytically. We therefore performed extensive Monte Carlo simulations with the Bayesian observer with two Gaussian perceptual filters to empirically measure the relationship between these two. Figure 5 shows the ratio of magnitudes of the classification images and input weights used in the model (see “Methods” for technical details about fitting routine used)3. This relationship can potentially be used to infer the underlying weights used by the human observers for the cued and uncued locations from the obtained human classification images.

^{2}). The observers then pressed one of two keys on a computer keyboard to select their decision (signal present or signal absent). Feedback about the correct decision was provided, but no feedback about the signal location was given.

^{2}statistic is a generalization of the univariate t statistic to multivariate vectors, and can be used to test for differences between a sample multivariate vector and a population vector or between two-sample multivariate vectors. We used one-sample and two-sample Hotelling T

^{2}statistics (Harris, 1985) to do hypothesis testing of the radial averages of the classification images. The Hotelling T

^{2}statistic is where x is a vector containing the observed radial average of the classification image, and x

_{0}is either a population or a hypothesized radial average classification image. K

^{−1}is the inverse of the covariance matrix that contains the sample variance of each of the elements of the radial average classification images, and the sample covariance between them. To test for significance, the T

^{2}statistic can be transformed to an F statistic using the following relationship: where p is the number of dependent variables (number of vector elements in the radial average of the classification images), and N is the number of observations (number of false alarm trials for our case). The obtained F statistic can be compared to an F

_{critical}with p degrees of freedom for the numerator and N-p degrees of freedom for the denominator.

^{2}, which is given by the following expression: where x

_{1}and x

_{2}are vectors containing the observed radial averages of the two classification images; N

_{1}and N

_{2}refer to the number of observations for the two classification images. For the two-sample test, a pooled covariance K is computed combining the sum of square deviations and sum of squared products from both samples. To test for significance, the two-sample T

^{2}statistic can be transformed to an F statistic using the following relationship: where p, N

_{1}, and N

_{2}are defined before. The obtained F

_{statistic}can be compared to an F

_{critical}with p degrees of freedom for the numerator and N

_{1}+N

_{2−}p − 1 degrees of freedom for the denominator.

Observer | Hit rate (valid trials) | Hit rate (invalid trials) | False alarm rate (all trials) | Cueing effect (HR_{v}–HR_{iv}) |
---|---|---|---|---|

O.C. | 0.824 | 0.716 | 0.235 | 0.108 |

K.F. | 0.845 | 0.655 | 0.194 | 0.190 |

K.C. | 0.890 | 0.729 | 0.270 | 0.160 |

A.H. | 0.880 | 0.649 | 0.227 | 0.231 |

^{2}statistic was used to test whether the radial averages of the classification images were significantly different from a hypothesized null classification image (vector of zeros). All radial averages of the classification images were significantly different (

*p*< .01) from the null classification image. The two sample Hotelling T

^{2}statistic showed that the differences between the classification images at the cued and uncued locations were statistically significant for all four observers (

*p*< .001).

_{1}and K

_{2}) and one standard deviation for each of the two Gaussians (σ

_{1}and σ

_{2}). DOG is given by p ]Table 2 shows the χ

^{2}best-fit parameters for the radial averages for the cued and uncued locations for all four human observers. The table also includes a χ

^{2}goodness of fit for each of the fits.

Observer | Perceptual filter at the cued location | Perceptual filter at the uncued location | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

K_{1} | K_{2} | σ_{1} | σ_{2} | χ^{1} | w_{1} | K_{1} | K_{2} | σ_{1} | σ_{2} | χ^{2} | w_{2} | |

O.C. | 1.56 | 0.57 | 4.9 | 7.3 | 21.09 | 0.76 | 0.60 | 0.11 | 4.3 | 9.3 | 7.598 | 0.24 |

K.F. | 2.14 | 0.79 | 5.1 | 8.2 | 19.07 | 0.84 | 0.46 | 0.08 | 5.3 | 14.0 | 25.07 | 0.16 |

K.C. | 1.2 | 0.16 | 4.3 | 11.8 | 39.25 | 0.80 | 0.65 | 0.11 | 4.3 | 8.9 | 18.8 | 0.20 |

A.H. | 1.7 | 0.51 | 4.9 | 8.4 | 33.83 | 0.88 | 0.76 | 0.5 | 7.3 | 6.4 | 20.71 | 0.12 |

^{2}were calculated (see “Methods”) to statistically compare the shape of the radial average of the perceptual filters for the cued and uncued locations. We found no statistically significant difference between radial averages for the cued and the scaled uncued locations (

*p*> .01).4

Observer | Equal weighting | |||
---|---|---|---|---|

Hit rate (valid trials) | Hit rate (invalid trials) | False alarm rate (all trials) | Cueing effect (HR_{v}–HR_{iv}) | |

O.C. | 0.939 | 0.919 | 0.053 | 0.02 |

K.F. | 0.923 | 0.943 | 0.059 | −0.02 |

K.C. | 0.930 | 0.929 | 0.071 | 0.001 |

A.H. | 0.930 | 0.969 | 0.050 | −0.039 |

Observer | Equal weighting | |||
---|---|---|---|---|

Hit Rate (valid trials) | Hit Rate (invalid trials) | False alarm rate (all trials) | Cueing effect (HR_{v}–HR_{iv}) | |

O.C. | 0.819 | 0.799 | 0.231 | 0.02 |

K.F. | 0.821 | 0.824 | 0.233 | −0.03 |

K.C. | 0.892 | 0.883 | 0.264 | 0.009 |

A.H. | 0.880 | 0.924 | 0.229 | −0.043 |

_{c}and w

_{u}in Figure 1) and the ratio of the magnitude of the classification images obtained for the model (e.g., Figure 5). From this lookup table we could then infer the weights used by the observers from the ratio of the magnitude of the human classification images. The simulations for the Bayesian model were performed by injecting internal noise and adjusting the criterion in order to match the false alarm rates observed in humans. The procedure was done separately for each human observer. The weights inferred for the cued location were: 0.76 (O.C.), 0.84 (K.F.), 0.8 (K.C.), and 0.88 (A.H.).

^{2}In the perceptual template model (PTM) model, attention changes what is referred to as the external noise exclusion, which is identical to what traditionally is known as the sampling efficiency in the linear template model (Burgess, Wagner, Jennings, & Barlow, 1981).

^{4}A potential problem is that the two sample Hotelling T

^{2}assumes equal covariance. This is clearly not true, at least for observers A.H. and K.F., where the covariance for the uncued location was scaled by a constant, resulting in higher variance than for the cued location. Ito and Schull (1964) have shown that the Hotelling T

^{2}statistic is robust to violations of the equal covariance when N is large. We believe our case, N of approximately 1,625, to be sufficiently large.

_{c}(x.y) and F

_{u}(x,y) and are normalized to have unit length. The image at the cued and uncued locations is given by g

_{c,i}(x,y) and g

_{u,i}(x,y). The first subscript refers to the locations (“c” for cued and “u” for uncued), whereas the second subscript refers to the i

^{th}trial.

_{c,i}(x,y) and n

_{u,i}(x,y) are the external image noise samples at the cued and uncued locations, which are independently sampled.

_{c,i}and λ

_{u,i}) to the stimuli in the i

^{th}trial is given by where ε

_{c,i}and ε

_{u,i}is a random scalar corresponding to internal noise, which is independently sampled for each trial and location (cued and uncued) from a Gaussian distribution with standard deviation σ

_{int}.

_{c,i}and λ

_{u,i}) given that the signal is present at the cued location, L(λ

_{c},λ

_{u}|s

_{c},n

_{u}), and a likelihood of the responses given that the signal is present at the uncued location L(λ

_{c},λ

_{u}|n

_{c},s

_{u}). The model then computes an overall likelihood of the responses given that the signal is present by weighting the individual likelihood from each location by a weight (w

_{c}and w

_{u}):

_{c},λ

_{u}|n

_{c},n

_{u}). Finally, the Bayesian model computes the ratio of the likelihood for signal presence and signal absence:

_{ratio}) to a decision threshold or criterion:

_{ratio}> threshold, then respond “signal present,”; otherwise respond “signal absent.”

_{u}and d′

_{c}are defined as the mean response of the perceptual filter to the signal present location minus the response to the signal absent location divided by the standard deviation of the response (including the effects of external and internal noise): where, <λ

_{c},s> is the expected value of these responses of the perceptual filter at the cued location when the signal is present; <λ

_{c},n> is the expected value of the response of the perceptual filter at the cued location when the signal is absent; σ

_{λc}is the standard deviation of the response due to external noise; and, σ

_{int}is the standard deviation of the additive internal noise. Similarly, d′

_{u}is given by

*th*is the decision criteria.

^{2}test in multivariate analysis of variance when variance-covariance matrices are not equal. Biometrika, 51, 71–82.