Search performance for a target tilted in a known direction among vertical distractors is well explained by signal detection theory models. Typically these models use a maximum-of-outputs rule (Max rule) to predict search performance. The Max rule bases its decision on the largest response from a set of independent noisy detectors. When the target is tilted in either direction from the reference orientation and the task is to identify the sign of tilt, the loss of performance with set size is much greater than predicted by the Max rule. Here we varied the target tilt and measured psychometric functions for identifying the direction of tilt from vertical. Measurements were made at different set sizes in the presence of various levels of orientation jitter. The orientation jitter was set at multiples of the estimated internal noise, which was invariant across set sizes and measurement techniques. We then compared the data to the predictions of two models: a Summation model that integrates both signal and noise from local detectors and a Signed-Max model that first picks the maxima on both sides of vertical and then chooses the value with the highest absolute deviation from the reference. Although the function relating thresholds to set size had a slope consistent with both the Signed-Max and the Summation models, the shape of individual psychometric functions was in the most crucial conditions better predicted by the Signed-Max model, which chooses the largest tilt while keeping track of the direction of tilt.

*identification*task because the observer has to identify the sign of the target orientation with respect to a mean, as opposed to the standard odd-man-out search tasks where the target varies along a single direction.

*n*locations in the 2 intervals and chooses the interval that evokes the greatest response in this class of detectors. Therefore, a correct decision will be made if the maximum output comes from any of the locations in the signal interval. If we assume that the response of the tilted detector is a sample from the probability density function,

*f*(

*r*) for a vertical stimulus and

*f*(

*r-kϑ*) for a tilted stimulus, then the probability of a correct response is given by where

*r*is the response from a CW-oriented detector,

*ϑ*is the orientation of the target,

*k*is a sensitivity parameter that scales the orientation, and

*F(r)*is the probability distribution We will refer to Equation 1 as the standard Max rule. Here we assume that

*f*(

*x*) has a Gaussian distribution.

*difference detector*. This subtraction operation produces distractor responses that have a zero mean, CW responses that have a positive mean value, and CCW responses that have a negative mean. Note that while the responses of the two individual detectors have a standard deviation equal to 1, the difference distribution has a standard deviation that is larger by a factor of The distributions in the bottom panel of Figure 2 represent the output of this difference operation.

*r*generated by the target or the distractors would be the largest, in absolute magnitude, if all the other samples produced responses between −

*r*and

*r*.

*r*is the response from an individual detector,

*ϑ*is the orientation difference,

*f*(

*r*) is the probability density function for responses,

*k*a sensitivity parameter, and

*n*is the uncertainty parameter. The Signed-Max model is a variant of the standard Max model that takes absolute values and keeps track of their sign. The equation is the sum of two terms. The first part of the equation calculates the probability that the response to a CW target (a sample from the distribution is larger in magnitude than the response to vertical distractors (

*n*-1 samples from the distribution . Note that these responses are samples from the difference distribution of CW and CCW responses. The second part calculates the probability that one of the

*n*-1 distractor responses has the largest positive value, and the response from the target and the other distractor elements is smaller in magnitude. The integration limits in the first and second parts of the equation go from −

*r*to +

*r*. This is because any value

*r*generated by the target or distractor will be the largest in absolute magnitude if all of the other responses lie between −

*r*and

*r*. The integration limits of the entire expression go from 0 to ∞ because only positive responses are correct for a clockwise target.1

*n*-1 vertical distractors has a mean equal to

*kϑ*, where

*ϑ*is the CW target angle and the variance is

*nσ*

^{2}. The probability that a sample from such a distribution has a value greater than 0 (CW) is where and .

*k*is a parameter that defines signal-to-noise ratio. In Equation 3, the factor

*k/n*modulates the signal-to-noise ratio. Because

*k*and

*n*cannot be independently evaluated in this formulation, we set

*k*equal to the total variance and calculated the best-fitting value for the parameter

*n*.

*k*represents a scalar that relates orientation to detector response. The parameter

*n*represents uncertainty, or the number of detectors (locations) monitored. Uncertainty will affect the Sum model by increasing the variance of the summed distribution, and will affect the Signed-Max model by increasing the units to be monitored. We used an iterative procedure to find the best-fitting values of

*k*and

*n*for a given set of data. In the Signed-Max model, we left both

*k*and

*n*free to vary, whereas we effectively had only one free parameter for the Sum model. Because both

*k*and

*n*modulate the signal-to-noise ratio in the Sum model, we set

*k*equal to the inverse of the internal noise plus the external noise, and allowed

*n*to vary. It is of interest to note that the values of

*k*returned by the Signed-Max fit, where it varied freely, closely matched the fixed value of

*k*in the Sum model.

^{2}. Pilot data indicated that crowding effects were under control. Thresholds obtained with two elements were similar whether the elements were on opposite sides of fixation or were separated by 22.5° (the angular separation for 16 elements).

*dimensional*noise (e.g, Verghese & Stone, 1995) is characterized by variability within the dimension of interest (orientation), rather than the standard pixel contrast-modulated noise used in similar studies. It has the advantage of directly affecting orientation detectors responsible for the task, and not requiring any assumption about the way contrast is related to orientation. Different amounts of jitter were used in separate sessions. We decided not to use the QUEST procedure (Watson & Pelli, 1983) as in the Baldassi and Burr (2000) study, but rather a fixed set of angles that spanned the whole range of performance from chance to perfect, interleaved within a session. While adaptive procedures concentrate trials around the inferred threshold value, we wanted the same number of trials over the entire psychometric function. Orientation jitter was introduced by setting the standard deviation of the orientation distribution of both target and distractors to a multiple of the internal orientation noise estimate (threshold for 1 element). The mean of the distractor distribution was 0 (vertical), and the mean of the target distribution was ± target angle, with 50% probability of being CW or CCW. Target and distractors were independently drawn from these noisy orientation distributions. In the no-noise condition, the added noise was 0, so target and distractors were displayed at the mean of their distributions.

*k*value was used to determine the equivalent internal orientation noise, a fixed parameter in the model fits to each observer’s data. As an alternative measure to estimate the equivalent internal orientation noise for that task, which is a key parameter of the study, observer S.B. measured additional orientation thresholds at set size 1 at different orientation noise levels.

*σ*

_{int}is internal variability and

*σ*

_{ext}the orientation jitter that we added. The numerator is the d’ value for 82% correct. The text box in Figure 5 reports the actual estimate of

*σ*

_{int,}or equivalent internal orientation noise, for one observer. This value is almost identical to the estimated standard deviation of the psychometric function for set size 1 with no noise. More importantly, it was similar to the equivalent noise values estimated by reanalyzing thresholds measured across set sizes ranging from 1 to 16 using pixel contrast-modulated noise (Baldassi & Burr, 2000). The latter set of data showed that the internal noise does not change with set size, suggesting that the dominating source of internal noise arises locally from each element, rather than globally, following the integration process (as in Morgan et al., 1998). This provides empirical support for the common first stage of the two models stated in the Models section. The value of

*k*estimated from internal noise was consistent with the estimate of

*k*from the Signed-Max model fits to the data. Therefore, we have converging evidence from various sources to justify the use of this noise estimate in the fits of the Sum model. It is reasonable to use the standard deviation at the set size 1 with no noise as a fixed parameter because this value seems to be independent of both the set size and the added external noise.

^{2}values for each model fit for each separate condition (subplot in Figure 6). As previously stated in more detail, we expect the two models to differ from each other in the way they fit the whole psychometric function across the conditions we explored. In particular, a different signature should be represented by diverging slopes of the fits as the set size increases. Indeed, this is what we observed for both our subjects. When the set size is small, 2 and 4, the two models yield similar trends to the fits, with virtually overlapping functions. When instead the set size increases up to 16, the overall picture is that psychometric functions have increasing slope consistent with the Signed-Max model, whereas the Sum model predicts shifts along the abscissa without changing slope. This is true for both observers, although it is more evident in V.A.’s data.

^{2}values for the Signed-Max and the Sum models for the two observers.

S.B. | V.A. | ||||||
---|---|---|---|---|---|---|---|

SetSize | Noise | SignedMax | Sum | Difference | SignedMax | Sum | Difference |

0 | 0.764 | 0.689 | −0.075 | 0.933 | 0.816 | −0.117 | |

0.5 | 0.421 | 0.369 | −0.052 | 2.48 | 1.984 | −0.496 | |

2 | 1 | 1.433 | 1.433 | −0.183 | 4.869 | 6.313 | 1.444 |

2 | 3.136 | 2.617 | −0.519 | 2.433 | 2.167 | −0.266 | |

4 | 4.104 | 3.644 | −0.46 | 1.056 | 0.939 | −0.117 | |

0 | 2.328 | 2 | −0.328 | 0.947 | 1.97 | 1.023 | |

0.5 | 1.298 | 1.383 | 0.085 | 1.623 | 1.298 | 1 | |

4 | 1 | 1.528 | 1.273 | 2 | 0.495 | 0.412 | 4 |

2 | 0.51 | 0.566 | 0.056 | 0.569 | 3.9 | 3.331 | |

4 | 3.533 | 3.029 | 0.5 | 2.428 | 2.017 | −0.728 | |

0 | 0.983 | 1.98 | 0.997 | 1.371 | 16 | 14.629 | |

0.5 | 0.368 | 1.3 | 0.932 | 3.588 | 2.86 | 4 | |

8 | 1 | 1.854 | 3.544 | 1.69 | 0.615 | 6.82 | 6.205 |

2 | 1.173 | 66.2 | 65.027 | 19.3 | 484.286 | 464.986 | |

4 | 1.031 | 0.99 | 1 | 1.611 | 1.818 | 0.207 | |

0 | 0.327 | 23.125 | 22.798 | 2.13 | 3.529 | 1.399 | |

0.5 | 0.286 | 1.711 | 1.425 | 1.03 | 670 | 668.97 | |

1 | 1.211 | 1.663 | 0.452 | 0.697 | 38.286 | 37.589 | |

2 | 1.393 | 1.875 | 0.482 | 2.827 | 43.25 | 40.423 | |

4 | 0.166 | 2.7 | 2.534 | 1.234 | 3.783 | 2.549 |

Weighted χ^{2} values obtained by fitting Equations 2 and 3 (Signed-Max and Sum models, respectively) to the psychometric functions obtained by the two observers, S.B. and V.A. . The difference column reports the difference between the Sum and the Signed-Max models; negative values indicate the advantage of the Sum (italic) and positive values of the Signed-Max (bold) model.

*n*averaged across noise conditions corresponds to the actual set size for observer S.B. for both the Signed-Max and the Sum models. Observer V.A.’s average estimate of

*n*reflects the number of elements in the display at low set sizes, but exceeds that number for the two larger set sizes used. While the uncertainty increases with set size, this increase is not proportional to the displayed set size. So it cannot simply be explained by an intrinsic uncertainty factor. This naïve observer exhibits non-optimal behavior only at larger set sizes.

*n*) and the gain (

*k*) parameters to vary allowed us to quantitatively compare the outcomes of the two models (the gain parameter

*k*was free only in the Signed-Max model).

*n*=set size independent random variables to generate predictions of the Max model. Even though it is computationally equivalent to our account, an absolute Max is not satisfying conceptually. In our account, each direction of tilt away from vertical activates two mirrored detectors whose activity is monitored by the observer and labeled. Actually, a standard Max rule takes place on either side and then the two maxima are compared, similar to the Max rule applied to 2IFC tasks, where a max is assumed to be extracted in each interval, and the two maxima are compared. We think that this labeling takes place in tasks such as location search and

*m*-alternative forced choice (e.g., 10AFC, Solomon & Morgan, 2001). Here the outputs are monitored in the location rather than the orientation domain, and the location producing the highest response is eventually chosen. The equivalent of Morgan and colleagues’ (1998) absolute Max in the positional context would be to collapse all the positional information onto a single abstract space and take the Max. If such a strategy were used in an

*m*AFC task, then the Max response would have to be remapped back onto the original space. We think this is not biologically plausible, nor economical, and assume that such tasks are instead accomplished by labeled detectors. The model sketched in Figure 1 is biologically plausible and can be extended to other tasks, once the nature of the task and the behavior of front-end filters are taken into account.

^{1}The Signed-Max model defined above produces the same fit to the data as Equation A3 in Carrasco et al., 2000. We chose a different exposition than Carrasco et al. to reflect biological plausibility, that is, cortical detectors do not usually produce modulations below baseline.