Noncausal Base Rates
A noncausal base rate describes how common something is without offering an obvious causal explanation of the case. The studies discussed here found that responses to these problems differed from responses to causal base-rate problems, raising questions about how participants interpreted them.
THE IDEA
NONCAUSAL BASE RATES
The statistical reasoning problems used in the experiments in which we obtained positive correlations with cognitive ability (Stanovich & West, 1998c) have involved causal aggregate information, analogous to the causal base rates discussed by Ajzen (1977) and Bar-Hillel (1980, 1990) – that is, base rates that had a causal relationship to the criterion behavior. Noncausal base-rate problems – those involving base rates with no obvious causal relationship to the criterion behavior – have had a much more controversial history in the research literature (Bar-Hillel, 1990; Birnbaum, 1983; Cohen, 1981, 1982, 1986; Cosmides & Tooby, 1996; Gigerenzer, 1991b, 1993, 1996; Gigerenzer & Hoffrage, 1995; Kahneman & Tversky, 1996; Koehler, 1996; Kyburg, 1983; Levi, 1983; Macchi, 1995).
In several experiments, we examined some of the noncausal base-rate problems that are notorious for provoking philosophical dispute. One was an AIDS testing problem modeled on Casscells, Schoenberger, and Grayboys (1978):
Imagine that AIDS occurs in 1 in every 1000 people. Imagine also there is a test to diagnose the disease that always gives a positive result when a person has AIDS. Finally, imagine that the test has a false-positive rate of 5%. This means that the test wrongly indicates that AIDS is present in 5% of the cases where the person does not have AIDS. Imagine that we choose a person randomly, administer the test, and that it yields a positive result (indicates that the person has AIDS). What is the probability that the individual actually has AIDS, assuming that we know nothing else about the individual’s personal or medical history?
The Bayesian posterior probability for this problem is slightly less than .02. In several analyses and replications (see Stanovich, 1999; Stanovich & West, 1998c) in which we classified responses of less than 10% as Bayesian, responses of more than 90% as indicating strong reliance on indicant information, and responses between 10% and 90% as intermediate, we have found that subjects giving the indicant response were higher in cognitive ability than those giving the Bayesian response. In addition, when tested on causal base-rate problems (e.g., Fong et al., 1986), the greatest base-rate usage was displayed by the group highly reliant on the indicant information in the AIDS problem. The subjects giving the Bayesian answer on the AIDS problem were least reliant on the aggregate information in the causal statistical reasoning problems.
A similar violation of the expectation of positive manifold was observed on the notorious cab problem (see Bar-Hillel, 1980; Lyon & Slovic, 1976; Tversky & Kahneman, 1982):
A cab was involved in a hit-and-run accident at night. Two cab companies, the Green and the Blue, operate in the city in which the accident occurred. You are given the following facts: 85% of the cabs in the city are Green and 15% are Blue. A witness identified the cab as Blue. The court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each of the two colors 80% of the time. What is the probability that the cab involved in the accident was Blue?
Bayes’ rule yields .41 as the posterior probability of the cab being Blue. Thus, responses over 70% were classified as reliant on indicant information, responses between 30% and 70% as Bayesian, and responses less than 30% as reliant on base-rate information (these labels are to be taken only as pointers to response classes – we do not mean to imply that responses labelled as Bayesian necessarily imply Bayesian reasoning). Again, it was found that subjects giving the indicant response were higher in cognitive ability and need for cognition than those giving the Bayesian or base-rate response (Stanovich & West, 1998c, 1999). Finally, both the cab problem and the AIDS problem were subjected to the second of Slovic and Tversky’s (1974) methods of operationalizing the understanding/acceptance principle – presenting respondents with arguments explicating the traditional normative interpretation (Stanovich & West, 1999). On neither problem was there a strong tendency for responses to move in the Bayesian direction subsequent to explication.
The results from both of these problems indicate that the noncausal base-rate problems display patterns of individual differences quite unlike those shown on the causal aggregate problems (Stanovich & West, 1998c). This is particularly interesting considering that it is precisely such noncausal base-rate problems like the AIDS and cab problem that have been the focus of the most intense debate in the literature (Cohen, 1979, 1981, 1982, 1986; Koehler, 1996; Kyburg, 1983; Levi, 1983). Several authors argue that a rote application of the Bayesian formula to these problems is unwarranted because noncausal base rates of the AIDS-problem type lack relevance and reference-class specificity. Finally, our results might also suggest that the “Bayesian” participants on the AIDS problem might not actually be arriving at their response through anything resembling Bayesian processing, because on causal reasoning problems these respondents were less likely to rely on the aggregate information.
When to use it
Evaluating medical test results when disease prevalence is low.
Assessing eyewitness testimony in the context of base rates (e.g., rare events).
Examining how people weigh eyewitness information and population frequencies in legal judgments.
Checking whether an aggregate describes the relevant reference class for an individual case, rather than accepting or rejecting it solely because it lacks a causal explanation.
FROM THEORY TO EVERYDAY LIFE
In practice
AIDS Testing Problem
In this hypothetical problem, AIDS occurs in 1 in 1000 people. The test detects every affected person but has a false-positive rate of 5%. For a randomly selected person with a positive result, the Bayesian probability of having the disease is about 2% (slightly less than .02). These are stipulated problem assumptions, not clinical test specifications.
Cab Problem
Suppose 85% of the city’s cabs are Green and 15% are Blue. A witness identifies the cab as Blue and correctly distinguishes each color 80% of the time. Under the problem’s assumption that fleet proportions supply the relevant prior, the Bayesian probability that the cab was Blue is about 41%.
A MOMENT TO REFLECT
Put it to work.
Think of a real-world scenario where you encountered a noncausal base rate, such as a medical test with a low disease prevalence. How did you reason about the probability?
Why might people with higher cognitive ability sometimes rely more on indicant information than on noncausal base rates?
In the hypothetical AIDS problem, how would you explain combining the test result with disease prevalence rather than treating either piece of information as sufficient on its own?
Often confused with
References
- Stanovich & West, 1998c — Stanovich, K. E. & West, R. F. (1998)
- Stanovich & West, 1999 — Stanovich, K. E. & West, R. F. (1999)
- Ajzen, 1977 — Ajzen, I. (1977)
- Bar-Hillel, 1980 — Bar-Hillel, M. (1980)
- Bar-Hillel, 1990 — Bar-Hillel, M. (1990)
- Casscells, Schoenberger, & Grayboys, 1978 — Casscells, W., Schoenberger, A., & Grayboys, T. (1978)
- Lyon & Slovic, 1976 — Lyon, D. & Slovic, P. (1976)
- Tversky & Kahneman, 1982 — Tversky, A. & Kahneman, D. (1982)
- Slovic & Tversky, 1974 — Slovic, P. & Tversky, A. (1974)
- Fong et al., 1986 — Fong, G. T. et al. (1986)
- Stanovich, 1999 — Stanovich, K. E. (1999)
CONTINUE THE CONVERSATION
Source texts
- Heuristics and Biases The Psychology of Intuitive Judgment (Thomas Gilovich, Dale Griffin, Daniel Kahneman)