Acciones variables sin sismo

Analyzing the monotonicity of the predicted P(false alarm) and P(miss) as

functions of set size 𝑘 enables a quick evaluation of the applicability of the assumptions from a mathematical point of view. Despite its apparent simplicity, this rudimentary model consisting of Assumptions 1 to 3 is not able to predict the three patterns observed in empirical data.

• Requirement a) cannot be met because it follows from Equation (4.2) that

P(miss) is a decreasing function in 𝑘.

• Requirement b) cannot be met because it follows from Equation (4.1) that

P(false alarm) is an increasing function in 𝑘.

• Since

P(miss) −P(false alarm) = (1 − 𝑝1)𝑘−1𝑝2− (1 − (1 − 𝑝1)𝑘)

= (_{1 − 𝑝1})𝑘−1_{(1 + 𝑝2}−𝑝_{1) − 1,}

Requirement c) can be met if and only if 𝑝2−𝑝1 > _(1−𝑝1

1)𝑘−1 − 1. As the empirical error rates are usually very low (less than 0.1), 𝑝1and 𝑝2would

be estimated at very small values according to the model. Consequently, for every fixed 𝑘, the set of 𝑝1and 𝑝2fulfilling this inequality is sufficiently

large. However, for fixed 𝑝1and 𝑝2which fulfill the inequality, if we let

𝑘 increase, the inequality will not hold once 𝑘 exceeds a certain value, because _(1−𝑝1

1)𝑘−1 gets larger as 𝑘 increases. This means that for constant 𝑝_{1and 𝑝2, the model predicts that the false alarm probability will always} outpace the miss probability as 𝑘 increases and thus exceed it when 𝑘 gets large enough. This is associated with the behavior that as 𝑘 increases, the predicted discrepancy between miss probability and false alarm probability, i.e., (1 − 𝑝1)𝑘−1(1 + 𝑝2−𝑝1) − 1, declines because (1 − 𝑝1)𝑘−1gets smaller. Both

are inconsistent with the empirical observation that miss rate dominates false alarm rate with increasing ascendancy as 𝑘 increases.

The inconsistency between the implications of Equations (4.1) and (4.2) and Requirement a) and b) motivated Zenger and Fahle (1997) to introduce further assumptions of strategy use by observers. Their approach is to relax the constraint of constant 𝑝1and 𝑝2, assuming that observers shift their discrimination

thresholds so that a certain quantity (can be understood as “risk”) is minimized. They proposed that this quantity can be the probability of misclassification (the sum ofP(“𝑇” | 𝐷) and P(“𝐷” | 𝑇))) weighted by the incidence probability of

distractors and the target 2𝑘−1_2𝑘 P(“𝑇” | 𝐷) + _2𝑘1P(“𝐷” | 𝑇), referred to as “Strategy

2”, or the expected overall error rate (P(miss) +P(false alarm)), referred to as

“Strategy 3”. Since 𝑝1exerts a larger influence on both quantities than 𝑝2does as 𝑘

increases (reflected in the factor 2𝑘−1_2𝑘 and the power 𝑘), both strategies require 𝑝1

to become smaller. Consequently, both strategies predict a threshold shift in the more conservative direction and 𝑝2has to become larger according to SDT. In this

way, the assumptions of strategy use in Zenger and Fahle (1997) are essentially equivalent to a relaxation of Assumption 2 in my rudimentary model, such that 𝑝_{1decreases whereas 𝑝2increases with increasing 𝑘. The authors claimed that} the modified model with this sort of threshold adaptation (“Strategy 2” and “Strategy 3”) alone is sufficient to reproduce the patterns of increasing miss rates and relatively flat false alarm rates. They further conclude that we should refrain from attributing errors to multiple sources such as premature termination, as per Occam’s razor.

Zenger and Fahle (1997) doubtlessly provided an original approach to model- ing error rates in visual search. They showed via simulation that their model

4.1. REQUIREMENTS ON A ERROR RATE MODEL AND BASIC

ASSUMPTIONS 73

with threshold adaptation could predict increasing miss rates and relatively flat false alarm rates. However, their results appear insufficient to forgo taking other sources of errors into account when modeling. First of all, there are two systematic differences between the empirical data and model predictions. For the purpose of illustration, Figure 3 in Zenger and Fahle (1997) is reprinted here as Figure 4.2. It plots the miss and false alarm rated predicted by their model for set sizes between 1 and 12, assuming the application of “Strategy 1”, “Strategy 2” and “Strategy 3.”

The first systematic difference is that the growth of the empirical miss rate speeds up with increasing set size and does not appear to be limited in the data, whereas it slows down and is limited in their model. In other words, the data show miss rates as a convex, increasing function of set size, whereas their model predicts this relation as a concave, increasing function. Although the authors considered this as a consequence of second-order effects, neglecting it may conceal important information carried by the data because the convex pattern appears consistently in visual search data whereas the concavity seems to be an intrinsic property of their model.

The second systematic difference is that the empirical false alarm rates do not show any upward trend but rather occasionally a slight downward trend with increasing 𝑘, whereas the false alarm rates predicted by their model always ascend slightly despite of a generally flat pattern. These differences become more profound when the value of the error rate becomes larger (reflecting more difficult search tasks). Another notable point is that the set size in the empirical data used in Zenger and Fahle (1997) consists of merely three levels with relatively small distance (2, 8, 14). This limits the inspection of patterns in the data and may undermine or even conceal critical discrepancies between the data and model predictions.

The analysis above demonstrates the limitations of the sort of models that assume classification errors (or strictly speaking, genuine processing errors) to be the only source of errors in visual search. Now one could argue that these limitations might result from other assumptions of the models. Nevertheless, in addition to the mathematical arguments discussed above, there are rationality arguments as well as empirical evidence for a substantial role of incomplete

Figure 4.2: Error rates predicted by the model of Zenger and Fahle (1997) for set sizes between 1 and 12, assuming the application of “Strategy 1”, “Strategy 2” and “Strategy 3.” Reprinted from “Missed targets are more frequent than false alarms: A model for error rates in visual search,” by B. Zenger and M. Fahle, 1997, Journal of Experimental Psychology: Human Perception and Performance, 23(6), p. 1787. Copyright 1997 by the American Psychological Association.

search. In the next section, I will elaborate the issue of how observers decide when to terminate a search.