Necessary condition under constant

and

P(miss) =P(“no” | TP)

=_P(examining and rejecting 𝑙 distractors | TP)

+_{P(examining and rejecting 𝑙 − 1 distractors and the target | TP)} = 𝑘−1 𝑙
𝑘 𝑙
· (1 − 𝑝1)𝑙+ 𝑘−1 𝑙−1
𝑘 𝑙
· (1 − 𝑝1)𝑙−1·𝑝2 = (𝑘 − 1)! (𝑘 − 1 − 𝑙)!𝑙! · (𝑘 − 𝑙)!𝑙! 𝑘! · (1 − 𝑝1)𝑙 + (𝑘 − 1)! ((𝑘 − 1) − (𝑙 − 1))!(𝑙 − 1)! · (𝑘 − 𝑙)!𝑙! 𝑘! · (1 − 𝑝1)𝑙−1·𝑝2 = 𝑘 − 𝑙 𝑘 · (1 − 𝑝1)𝑙+ 𝑙 𝑘 · (1 − 𝑝1)𝑙−1·𝑝2, (4.4) respectively.

4.3.1

Necessary condition under constant

𝒑

and

𝒑

Assume that 𝑝1 and 𝑝2 are invariant of 𝑘. The question of how the counting

threshold 𝑙 is determined arises. One might easily think of several simple options, for example, a constant threshold or a fixed ratio of 𝑙 to the set size 𝑘 etc. Nevertheless, a second thought disproves their appropriateness of being the counting threshold. Consider first the case in which the search is terminated after examining a fixed proportion of display items. Let 𝑟 = 𝑘𝑙 =const., then

P(miss) = 𝑘 − 𝑟 𝑘 𝑘 · (1 − 𝑝1)𝑟 𝑘 + 𝑟 𝑘 𝑘 · (1 − 𝑝1)𝑟 𝑘−1·𝑝2 = (_{1 − 𝑟)(1 − 𝑝1})𝑟 𝑘 +𝑟(1 − 𝑝₁)𝑟 𝑘−1·𝑝₂

4.3. REVISING ASSUMPTION 3: QUITTING BY COUNTING 83

It implies that P(miss) should be decreasing for increasing 𝑘 if 𝑝1 > 0, and

constant if 𝑝1=0, because 𝑟, 𝑝1and 𝑝2in the equation are assumed to be constant.

This prediction is inconsistent with Requirement a). Now consider the case in which 𝑙 is constant, i.e., remains the same for all 𝑘. We can assess the monotonicity ofP(miss) as a function of 𝑘 by calculating its derivative:

d d𝑘P(miss) = 𝑘 − (𝑘 − 𝑙) 𝑘2 (1 − 𝑝1) 𝑙 ₋ 𝑙 𝑘2(1 − 𝑝1) 𝑙−1_·𝑝 2 = 𝑙 𝑘2(1 − 𝑝1) 𝑙−1_{(1 − 𝑝} 1−𝑝2) > 0, if 𝑝1+𝑝2 < 1.

Thus, Requirement a) can be met with the constraint 𝑝1+𝑝2 < 1, which is not

quite restrictive. However, this inference is valid only under the condition 𝑙 ≤ 𝑘. For the case 𝑘 < 𝑙, we should assume that simply all items will be examined, i.e, an exhaustive search. Considering both cases together, the mere idea of aborting the search after examining a fixed number of items is inconsistent with the empirical data: If the fixed threshold 𝑙 is larger than all set size levels realized in the experiment, decreasing miss rates should be observed; if some of the realized set size levels exceed 𝑙, i.e., 𝑙 lies between two set size levels, the miss rates should first decrease then increase. This means that 𝑙 must be smaller than all set sizes to be able to reproduce a pattern of constantly increasing miss rates. Since many studies include small set sizes (less than 10), the case that 𝑙 is smaller than all set size levels realized in the studies is very implausible. Besides, the concept of a counting threshold that is invariant of the set size itself does not seem convincing.

Although these two options have been disproved, the discussion is not in vain. It provides a hint for finding out conditions that 𝑙 needs to satisfy so that the miss probability is increasing. An observation is that assuming 𝑙 is constant, the ratio 𝑘𝑙 decreases as 𝑘 increases. One might speculate that decreasing 𝑘𝑙 could

be a necessary condition. A closer look at Equation (4.4) for the miss probability provides a rational argument for this speculation. The term 𝑘−𝑙𝑘 · (1 − 𝑝1)𝑙 is the

probability of the case where the searcher misses the target because he or she did not look at it at all; the term 𝑘𝑙 · (1 − 𝑝1)𝑙−1·𝑝2is the probability of the case

where although the target is looked at, the searcher failed to identify it as such. The ratio of the former to the latter is thus 𝑘−𝑙𝑙 ·

1−𝑝1

alarm rate and miss rate are usually very low (under 10%), 𝑝1and 𝑝2must be

very small (≤ 0.05). This means that the fraction 1−𝑝1

𝑝₂ should be a large value

(≥ 19). Consequently, the ratio 𝑘−𝑙𝑙 · 1−𝑝1

𝑝₂ can only fall below 1 if 𝑘−𝑙

𝑙 is smaller

than about 0.05. In other words, the case where the target is not included in the searched set makes up the dominating source of miss errors, unless the search is almost exhaustive (i.e., more than 95% of the items must be examined).

From this observation, we can infer that the option of the increasing ratio

𝑙

𝑘 with 𝑘 should be excluded. If 𝑘𝑙 was increasing with 𝑘, then it would be

the only increasing factor in Equation (4.4). If 𝑘𝑙 is below 0.95,P(miss) would

be decreasing with increasing 𝑘 because the dominating part 𝑘−𝑙𝑘 · (1 − 𝑝1)𝑙 is

decreasing. If 𝑘𝑙 is above 0.95,P(miss) would hardly be increasing with increasing

𝑘 because the monotonic behavior of the factor (1 − 𝑝1)𝑙(and (1 − 𝑝1)𝑙−1) is faster

than a linear increase3 and the influence of the only increasing factor 𝑘𝑙 is very

limited (more slowly than linear, from 0.95 to 1 at the most, and then shrunk by the small 𝑝2). Even if this influence was strong enough (for specific value

configuration of the variables) to compensate all other decreasing factors, the increase in miss probability would be minuscule. In other words, under the assumption of increasing 𝑘𝑙, we would not be able to observe such monotonic

and substantial increases in empirical miss rates. Any model that predicts an increasing proportion of the searched subset with increasing set size would fail to predict the empirical pattern in Requirement a).

On the other hand, if we assume 𝑘𝑙 to be decreasing with increasing 𝑘, as one

might speculate from the previous discussion, increasing miss probability can be a natural consequence. If 𝑘𝑙 decreases with increasing 𝑘 so fast that even 𝑙

decreases, then itself would be the only decreasing factor in Equation (4.4). Even if it starts from 1, it would fall below 0.95 very soon. Due to the dominance of 𝑘−𝑙𝑘 · (1 − 𝑝1)𝑙, its influence can be easily compensated and overtaken by the

exponential increase of the factor (1 − 𝑝1)𝑙−1. Nevertheless, searching even less

items when there are more appears less plausible than searching a fixed number of items regardless of set size. If 𝑙 remains increasing as 𝑘 increases, then all

3_{Note that for very small 𝑝}

1, the increase of (1 − 𝑝1)𝑙is approximately linear, if 𝑙

𝑘 is constant

and 𝑙 is relatively small. Under the condition of increasing 𝑘𝑙, 𝑙 would increase even faster as 𝑘

4.3. REVISING ASSUMPTION 3: QUITTING BY COUNTING 85

factors in Equation (4.4) would be decreasing except 𝑘−𝑙𝑘 . Because 𝑘𝑙 is decreasing,

the factor (1 − 𝑝1)𝑙−1 decreases more slowly than it does if 𝑘𝑙 is constant (as

discussed above), i.e., more slowly than linear. This means that the increasing behavior of 𝑘−𝑙𝑘 can be of similar magnitude as the decreasing behavior of all

other factors so that it determines the monotonic behavior ofP(miss) eventually.

In other words, depending on the properties of 𝑘𝑙, the miss probability can be

increasing. This is the only relation that can fulfill Requirement a).

Explained in plain language without reference to formulas, the critical part of the reasoning above is to consider the relative impact of the two situations where a miss error occurs: The target was not included in the searched subset in the first place, although all examined items are rejected correctly; the target was included in the searched subset but rejected, like the other examined items. If the occurrence of a miss error requires an erroneous processing, i.e., a misidentification of the target, the miss rate cannot rise when there are more items in the display. Saying “no” implies that all distractors, once attended to, must be rejected correctly. Rejecting all of them correctly becomes harder when more distractors are attended to. Because there is only one target on each target-present trial of a standard visual search task, including it in a random sample — and subsequently misidentifying it — also becomes harder when there are more items in the display. Therefore, the extra occurrences of miss errors for larger set sizes must result largely from the omission of the target rather than its misidentification. To ensure a sufficiently large relative contribution of the omission, the proportion of the items selected to examine must become smaller as the set size gets larger. Only this condition can predict increasing miss rates as the set size increases.

If this is the case, what is the mechanism behind it? Can we describe the relation between the proportion of items to search and the set size more precisely than “it seems to be negative?” It may come to one’s mind that the purpose of looking at stimuli is to collect information to make a decision. What about viewing the problem from the perspective of “how the observer decides which response option to choose based on her search experience?”