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Why Does the Base Rate Appear to Be Ignored? the Equiprobability Hypothesis

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Why Does the Base Rate Appear to Be Ignored? the Equiprobability Hypothesis Psychonomic Bulletin & Review 2009, 16 (6), 1065-1070 doi:10.3758/PBR.16.6.1065 Why does the base rate appear to be ignored? The equiprobability hypothesis MASASI HATTORI Ritsumeikan University, Kyoto, Japan AND YUTAKA NISHIDA Osaka University, Osaka, Japan The base rate fallacy has been considered to result from people’s tendency to ignore the base rates given in tasks. In the present article, we note a particular, common structure of the tasks (the imbalanced probability structure) in which the fallacy is often observed. The equiprobability hypothesis explains the mechanism that produces the fallacy. This hypothesis predicts that task material that overrides people’s default equiprobability assumption can facilitate normative Bayesian inferences. The results of our two experiments strongly supported this prediction, and none of the alternative theories considered could explain the results. Since a seminal study by Kahneman and Tversky (1973), P(H ) P(H |D) P(D|H ) r . (1) a vast number of studies have been concerned with the P(D) controversial topic of base rate neglect (for reviews, see, Here, P(D) is expressed as e.g., Barbey & Sloman, 2007a; Koehler, 1996). People are supposed to be insensitive to base rate information when P(D) P(D|H )P(H ) P(D|H )P(H ) they engage in a task such as the following, modified from .8 r .01 .096 r .99. (2) Eddy (1982, pp. 251–254; underlines and brackets will be Therefore, the correct answer for the task is explained later): P(H |D) .8 r .01 y .078. (3) Recently, an incurable disease called X syndrome has .8 r .01 .096 r .99 begun to be reported. X syndrome is known to show symptoms similar to a cold. Suppose you are a doc- From Equation 1, the low base rate, P(H), lowers the pos- tor working in a hospital. You are expected to make terior probability, P(H |D). However, modal responses in a judgment about whether or not a patient is infected typical experiments are about 80% (see, e.g., Bar-Hillel, by X syndrome based on the following information. 1980), the same rate as the true positive rate. This pervasive and robust phenomenon has been called base rate neglect The prevalence of X syndrome is 1%. A patient who (or base rate fallacy), because it was believed to demon- is infected with X syndrome has an 80% chance of strate people’s insensitivity to the base rate information. testing positive [having a cough]. However a patient In considering whether and why the base rate is ne- who is not infected with the syndrome has a 9.6% glected, we should ask what base rate “neglect” means. chance of testing positive [having a cough]. Now As long as a person makes a response, some value must a patient tests positive [has a cough]. What is the be allocated—at least implicitly—to the base rate, and it is chance that the patient is actually infected with X not in this sense ignored. As is obvious from Equation 1, syndrome? Please answer intuitively. % if we suppose P(H |D) P(D|H), which is a common answer, we can derive P(H) P(D). This is exactly what (Correct Answer: 7.8%) base rate neglect implies. If participants assume that the This is called a base rate task (or a BR task, hereafter). probability of “testing positive” and the probability of Let P(H) be the base rate (prevalence) of having the dis- “having X syndrome” are equal, they will give the true ease (i.e., the hypothesis) and P(D) be the probability that positive rate as an answer for a posterior probability. In a person tests positive (i.e., data). This is a Bayesian in- our view, base rates are not ignored, but people make a ference task to derive the posterior probability of X syn- default equiprobability assumption, P(H) P(D). Thus, drome, P(H |D), from the true positive (detectability) rate, we propose that participants do not neglect the base rate, P(D|H). There is a relationship between the true positive but that they infer that the posterior and the true positive rate and the posterior probability, as follows: rates are almost equal by postulating near equality in the M. Hattori, [email protected] 1065 © 2009 The Psychonomic Society, Inc. 1066 HATTORI AND NISHIDA ABB1 [X syndrome] 3 [Positive Test] H (1%) [Fever] D (10%) D (10.3%) D (1%) [X syndrome] H (1%) H (1%) 0.92% 0.08% 0.92% B2 H (1%) D (2%) 0.2% 0.2% 0.8% 0.8% 9.5% 9.2% 0.84%0.16% 1.84% Figure 1. Probabilistic task structures expressed by Euler circles (with correspondence between probability and area size). (A) A task used in Experiment 1. (B1, B2, B3) Tasks used in the F1, F2, and F10 conditions, respectively, in Experiment 2. marginal probabilities of the two target events that are fo- reasoning, even in a typical BR task. To encourage partici- cused on in the task. pants to give up their equiprobability assumptions and to BR tasks share some characteristics that we regard as grasp the task structure, we adopted a technique to utilize essential in leading people to the fallacy. The most im- their knowledge about the statistical structure of the envi- portant feature, we believe, is the fact that the marginal ronment (see, e.g., McKenzie, 2006). Hypothesis H was probabilities, P(H) and P(D), differ greatly. Figure 1A “being infected with X syndrome” (a fictitious disease), (a special Euler diagram) shows the set-size relation- and data D was “having a cough” as a symptom of the ship between H and D of the aforementioned task. This disease. X syndrome (H) is a comparatively rare disease is drawn to establish correspondence between each sub- [i.e., the base rate, P(H), is low, as in typical BR tasks] and set probability and each division area of the subsets. Fig- is likely to cause a cough [i.e., P(D|H) is high]. However, ure 1A shows that (1) the area of P(D) is large in compari- having a cough (D) is a common matter in everyday life, son with that of P(H); therefore, (2) although the subset so the IPS [i.e., P(H) P(D)] can be easily recognized. H&D occupies a large part of set H, it occupies a much This is because alternative causes for a cough—including smaller part of set D. Consequently, P(D|H) is large, but a cold and dust—can be easily brought to mind, and this P(H |D) is small. This is the common distinctive structural would promote the recognition that even if the patient has feature in the BR tasks and seems to be closely related to a cough, he or she is not necessarily infected by X syn- the task’s difficulty. The structure with a great difference drome [i.e., P(H |D) is low]. To probe participants’ mental in the marginal probabilities of two target events is called representations of the task with IPS, we introduced tasks an imbalanced probability structure (IPS). called “Est-D” and “Est-HD.” In our view, the BR tasks are difficult in nature because they have an IPS, which conflicts with people’s implicit Method assumption that the two target sets are almost equal in Participants. A total of 36 undergraduate students from Ritsu- size. In other words, we commit an error in the tasks not meikan University participated in the experiment as unpaid volun- because the probabilistic inference is intrinsically dif- teers. Equal numbers (n 18) of participants were randomly as- signed to the cough and PT conditions, described below. ficult, but because two processes—one to recognize the BR task. The tasks were shown at the beginning of the present task structure and another to infer the structure on the article. In the positive test (PT) condition (i.e., control), the words basis of the equiprobability assumption—compete with “testing positive” (underlined) were used, whereas in the cough con- each other. According to the equiprobability hypothesis, dition, the words “having a cough” (in square brackets) were used we predict that (1) people’s output responses will roughly instead. conform to the normative Bayesian solution if the default Est-D/HD: Tasks for estimating P(D) and P(H&D). A sample answer sheet for each task is shown in the Appendix. In each task, equiprobability assumption is dismissed somehow and the total number of people who suffer from X syndrome (H) was set the IPS of the task is recognized (Experiment 1), and that as a reference point for estimation, and participants were asked to (2) people infer more normatively as a task structure be- graphically estimate the size of subsets: The total number of people comes closer to the assumed one (Experiment 2). who have a cough (or who have tested positive, D) was estimated in the Est-D task; and the total number of people who suffer from EXPERIMENT 1 X syndrome and who also have a cough (or also have tested posi- tive) was estimated in the Est-HD task. To avoid biasing participants Facilitation of Bayesian Inference toward any particular size (i.e., toward greater or smaller than H), three options were prepared (see the Appendix). In Experiment 1, we examined the prediction that the Procedure. Participants were tested either individually or in recognition of the IPS will facilitate participants’ Bayesian groups of 2. The tasks were printed in a booklet, with each task on a EQUIPROBABILITY AND BASE RATE FALLACIES 1067 Cough .3 .2 .1 0 Positive Test .3 Proportion of Participants .2 .1 0 0 102030405060708090100 Responses (%) Figure 2.
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