Ignoring statistics (particularly probability) in heuristic inductive thinking-
Dr. Munish Alagh, PDF-IIM(A).
Inductive Thinking involves, generalising from the particular to the general.
Heuristics, involve cognitive short-cuts to make a decision easily.
Heuristics people use in inductive reasoning tasks often do not respect the required statistical principles. People consequently overlook statistical variables such as sample size, correlation, and base rate when they solve inductive reasoning problems
Statistical Problems and Nonstatistical Heuristics
As we have seen, people often solve inductive problems by use of a variety of intuitive heuristics—rapid and more or less automatic judgmental rules of thumb. These include the representativeness heuristic (Kahneman & Tversky, 1972, 1973), the availability heuristic (Tversky & Kahneman, 1973), and the anchoring heuristic (Tversky & Kahneman, 1974). In problems where these heuristics diverge from the correct statistical approach, people commit serious errors of inference. The following heuristics, the biases they lead to and the statistical principles that are ignored therein are discussed :
Ø Adjustment and Anchoring.
According to Kahneman and Tversky (1974) there are three types of probabilistic questions with which people are concerned.
What is the probability that object A belongs to class B?
What is the probability that event A originates from process B?
What is the probability that process B will generate event A?
People answer such questions by relying on the representativeness heuristic according to which probabilities are evaluated by the degree to which A is representative of B, that is, by the degree to which A resembles B. For example, when A is highly representative of B, the probability that A originates from B is judged to be high. On the other hand, if A is not similar to B, the probability that A originates from B is judged to be low.
There is a type of research on problems of a particular type which has shown that people order the occupations by probability and by similarity in exactly the same way. They consider that a person, Steve, whose probability that he is a librarian, for example, is assessed by the degree to which he is representative of, or similar to, the stereotype of a librarian. This is known as the representativeness heuristic.
Infact people who are asked to assess probability are not stumped, because they do not try to judge probability as statisticians and philosophers use the word. A question about probability or likelihood activates a mental shotgun, evoking answers to easier questions. Judging probability by representativeness has important virtues: the intuitive impressions that it produces are often-indeed, usually-more accurate than chance guesses would be.
This approach to the judgement of probability however leads to serious errors, because similarity, or representativeness, is not influenced by several factors that should affect judgments of probability:
Insensitivity to prior probability of outcomes:
One of the factors that have no effect on representativeness but should have a major effect on probability is the prior probability, or base-rate frequency, of the outcomes. In case of Steve, for example, the fact that there are many more farmers than librarians in the population should enter into any reasonable estimate of the probability that Steve is a librarian rather than a farmer. Considerations of base-rate frequency, however, do not affect the similarity if people evaluate probability by representativeness, therefore, prior probabilities will be neglected. Certain differing prior probabilities were given for two professions to subjects, in two different cases, also the personality description of several individuals, allegedly sampled at random from a group of 100 professionals, including both the occupations were given. The subjects were asked to assess, for each description, the probability that it belonged to one of the occupations. The odds that any particular description belongs to any one of the professions should be higher when the prior probability of that particular occupation is more. However subjects in the two conditions produced essentially the same probability judgements. Apparently, subjects evaluated the likelihood that a particular description belonged to a particular occupation, from the two, by the degree to which the description was representative of the two stereotypes, with little or no regard for the prior probabilities of the categories.
The subjects used prior probabilities correctly when they had no other information. However, prior probabilities were effectively ignored when a description was introduced, even when this description was totally uninformative. Evidently, people respond differently when given no evidence and when given worthless evidence. When no specific evidence is given, prior probabilities are ignored.
Nisbett and Borgida (1975),quoted in The reference stated above showed that consensus information, that is, base rate information about the behaviour of a sample of people in a given situation, often has little effect on subjects attributions about the causes of a particular target individual's behavior. When told that most people behaved in the same way as the target, subjects shift little or not at all in the direction of assuming that it was situational forces, rather than the target's personal dispositions or traits, that explain the target's behavior.
It is noticed that subjects use prior probabilities correctly when they have no other information. However, prior probabilities are effectively ignored when a description is introduced, even when this description is totally uninformative. Evidently, people respond differently when given no evidence and when given worthless evidence. When no specific evidence is given, prior probabilities are ignored.
Insensitivity to sample size:
To evaluate the probability of obtaining a particular result in a sample drawn from a specified population, people typically apply the representativeness heuristic. That is, they assess the likelihood of a sample result by the similarity of this result to the corresponding parameter The similarity of a sample statistic to a population parameter does not depend on the size of the sample. Consequently, if probabilities are assessed by representativeness, then the judged probability of a sample statistic will be essentially independent of sample size. Indeed, when subjects assessed the distributions of the sample results for samples of various sizes, they produced identical distributions . A similar insensitivity to sample size has been reported in judgments of posterior probability, that is, of the probability that a sample has been drawn from one population rather than from another. Here again, intuitive judgments are dominated by the sample proportion and are essentially unaffected by the size of the sample, which plays a crucial role in the determination of the actual posterior odds . In addition, intuitive estimates of posterior odds are far less extreme than the correct values. The underestimation of the impact of evidence has been observed repeatedly in problems of this type. It has been labeled "conservatism."
Misconceptions of chance:
People expect that a sequence of events generated by a random process will represent the essential characteristics of that process even when the sequence is short. Thus, people expect that the essential characteristics of the process will be represented, not only globally in the entire sequence, but also locally in each of its parts. A locally representative sequence, how-ever, deviates systematically from chance expectation: it contains too many alternations and too few runs. Another consequence of the belief in local representativeness is the well-known gambler's fallacy. Chance is commonly viewed as a self-correcting process in which a deviation in one direction induces a deviation in the opposite direction to restore the equilibrium. In fact, deviations are not "corrected" as a chance process unfolds, they are merely diluted. Misconceptions of chance are not limited to naive subjects. A study of the statistical intuitions of experienced research psychologists revealed a lingering belief in what may be called the "law of small numbers," according to which even small samples are highly representative of the populations from which they are drawn. The responses of these investigators reflected the expectation that a valid hypothesis about a population will be represented by a statistically significant result in a sample with little regard for its size. As a consequence, the researchers put too much faith in the results of small samples and grossly overestimated the replicability of such results. In the actual conduct of research, this bias leads to the selection of samples of inadequate size and to overinterpretation of findings.
Insensitivity to predictability: People are sometimes called upon to make such numerical predictions as the future value of a stock, the demand for a commodity, or the outcome of a football game. Such predictions are often made by representativeness. The degree to which the description is favorable is unaffected by the reliability of that description or by the degree to which it permits accurate prediction. Hence, if people predict solely in terms of the favorableness of the description, their predictions will be insensitive to the reliability of the evidence and to the expected accuracy of the prediction demonstrated that intuitive predictions violate this rule, and that subjects show little or no regard for considerations of predictability 
That is, the prediction of a remote criterion was identical to the evaluation of the information on which the prediction was based The students who made these predictions were undoubtedly aware of the limited predictability never-theless, their predictions were as ex-treme as their evaluations.
This mode of judgment violates the normative statistical theory in which the extremeness and the range of predictions are controlled by considerations of predictability. When predictability is nil, the same prediction should be made in all cases If predictability is perfect, of course, the values predicted will match the actual values and the range of predic-tions will equal the range of outcomes. In general, the higher the predictability, the wider the range of predicted values. Several studies of numerical prediction have
The illusion of validity:
As we have seen, people often predict by selecting the outcome (for example, an occupation) that is most representative of the input (for example, the description of a person). The confidence they have in their prediction depends primarily on the degree of representativeness (that is, on the quality of the match between the selected outcome and the input) with little or no regard for the factors that limit predictive accuracy The unwarranted confidence which is produced by a good fit between the predicted outcome and the input information may be called the illusion of validity. This illusion persists even when the judge is aware of the factors that limit the accuracy of his predictions.
The internal consistency of a pattern of inputs is a major determinant of one's confidence in predictions based on these inputs Highly consistent patterns are most often observed when the input variables are highly redundant or correlated. Hence, people tend to have great con-fidence in predictions based on redundant input variables. However, an elementary result in the statistics of correlation asserts that, given input variables of stated validity, a prediction based on several such inputs can achieve higher accuracy when they are independent of each other than when they are redundant or correlated. Thus, redundancy among inputs decreases accuracy even as it increases confidence, and people are often confident in predictions that are quite likely to be off the mark
Regression to the mean:
Regression to the mean- involves moving closer to the average than the earlier value of the variable observed. Also regression to the mean has an explanation, but does not have a cause.
An important principle of skill training: rewards for improved performance work better than punishment of mistakes. This proposition is supported by much evidence from research.
Regression to the mean, involves that poor performance is typically followed by improvement and good performance by deterioration, without any help from either praise or punishment.
The feedback to which life exposes us is perverse. Because we tend to be nice to other people when they please us and nasty when they do not, we are statistically punished for being nice and rewarded for being nasty.
Regression does not have a causal explanation. Regression effects are ubiquitous, and so are misguided casual stories to explain them. The point to remember is that the change from the first to the second occurrence does not need a causal explanation. It is a mathematically inevitable consequence of the fact that luck played a role in the outcome of the first occurence.
Regression inevitably occurs when the correlation between two measures is less than perfect.
The correlation coefficient between two measures, which varies between 0 and 1, is a measure of the relative weight of the factors they share.
Correlation and regression are not two concepts-they are different perspectives on the same concept. The general rule is straightforward but has surprising consequences: whenever the correlation between two scores is imperfect, there will be regression to the mean.
Our mind is strongly biased toward causal explanations and does not deal well with “mere statistics.” When our attention is called to an event, associative memory will look for its cause, more precisely, activation will automatically spread to any cause that is already stored in memory. Causal explanations will be evoked when regression is detected, but they will be wrong because the truth is that regression to the mean has an explanation but does not have a cause.
Regression effects are a common source of trouble in research, and experienced scientists develop a healthy fear of the trap of unwarranted causal inference.
Statistics can be used, but is often not used in intuitive thinking:
Even when judgments are based on the representativeness heuristic, there may be an underlying stratum of probabilistic thinking. In many of the problems studied by Kahneman and Tversky, people probably conceive of the underlying process as random, but they lack a means of making use of their intuitions about randomness and they fall back on representativeness.
Adjustment and Anchoring:
Biases in the evaluation of compound events are particularly significant in the context of planning. The successful completion of an undertaking, such as the development of a new product, typically has a conjunctive character: for the undertaking to succeed, each of a series of events must occur. Even when each of these events is very likely, the overall probability of success can be quite low if the number of events is large. The general tendency to overestimate the probability of conjunctive events leads to unwarranted optimism in the evaluation of the likelihood that a plan will succeed or that a project will be completed on time. Conversely, disjunctive structures are typically encountered in the evaluation of risks. A complex system, such as a nuclear reactor or a human body, will malfunction if any of its essential components fails. Even when the likelihood of failure in each component is slight, the probability of an overall failure can be high if many components are involved. Because of anchoring, people will tend to underestimate the probabilities of failure in complex systems.
The subjects state overly narrow confidence intervals which reflect more certainty than is justified by their knowledge about the assessed quantities.
Anchoring in the assessment of subjective probability distributions.: the subjects state overly narrow confidence intervals which reflect more certainty than is justified by their knowledge about the assessed quantities
it is natural to begin by thinking about one's best estimate of the parameter and to adjust this value upward. If this adjustment like most others is insufficient, then the upper value of the distribution will not be sufficiently extreme. A similar anchoring effect will occur in the selection of the lower value of the distribution, which is presumably obtained by adjusting one's best estimate downward. Consequently, the confidence interval between the lower and upper values of the distribution will be too narrow, and the assessed probability distribution will be too tight.
Availability which is discussed above, is affected by various factors which are not related to actual frequency. If the availability heuristic is applied, then such factors will affect the perceived frequency of classes and the subjective probability of events. Consequently, not only does the use of the availability heuristic leads to systematic biases, there are also effects on the statistical picture which is pictured by us as a result.
“Errors” in probabilistic reasoning are in fact not violations of probability
Most so-called “errors” in probabilistic reasoning are in fact not violations of probability theory. Examples of such “errors” include overconfidence bias, conjunction fallacy, and base-rate neglect.
Over-confidence bias-systematic discrepancy between confidence and relative frequency is termed “overconfidence.”
Has probability theory been violated if one’s degree of belief (confidence) in a single event (i.e., that a particular answer is correct) is different from the relative frequency of correct answers one generates in the long run? The answer is “no.” It is in fact not a violation according to several interpretations of probability. According to the frequentists, probability theory is about frequencies, not about single events. To compare the two means comparing apples with oranges. According to subjectivists a discrepancy between confidence and relative frequency is not a “bias,” albeit for diff erent reasons. For a subjectivist, probability is about single events, but rationality is identified with the internal consistency of subjective probabilities. So, in conclusion, a discrepancy between confidence in single events and relative frequencies in the long run is not an error or a violation of probability theory from many experts’ points of view. It only looks so from a narrow interpretation of probability that blurs the distinction between single events and frequencies fundamental to probability theory. 
The original demonstration of the “Conjunction fallacy” was with problems involving matching a description of a lady, with a) her profession and b) her profession and an activity she was involved in. Subjects were asked which of two alternatives was more probable. Tversky and Kahneman, however, argued that the “correct” answer is a), because the probability of a conjunction of two events, such as b), can never be greater than that of one of its constituents. They explained this “fallacy” as induced by the representativeness heuristic. They assumed that judgments were based on the match (similarity, representativeness) between the description of the lady and the
two alternatives. That is, since the lady was described based on her activity and b)
contains her activity people believe that b)is more probable.
Is the “conjunction fallacy” a violation of probability theory, as has been claimed in the literature? Has a person who chooses b) as the more probable alternative violated probability theory? Again, the answer is “no.” Choosing b) is not a violation of probability theory, and for the same reason given above. For a frequentist, this problem has nothing to do with probability theory. Subjects are asked for the probability of a single event (that the lady has a particular profession), not for frequencies. Note that problems which are claimed to demonstrate the “conjunction fallacy” are structurally slightly different from “confidence” problems. In the former, subjective probabilities ( a) or b)) are compared with one another; in the latter, they are compared with frequencies. To summarize the normative issue, what is called the “conjunction fallacy” is a violation of some subjective theories of probability. It is not, however, a violation of the major view of probability, the frequentist conception.
The base-rate fallacy
The example is from Casscells, Schoenberger, and Grayboys (1978, p. 999) and presented by Tversky and Kahneman (1982, p. 154) to demonstrate the generality of the phenomenon:
If a test to detect a disease whose prevalence is 1/1000 has a false positive rate of 5%, what is the chance that a person found to have a positive result actually has the disease, assuming you know nothing about the person’s symptoms or signs?
Sixty students and staff at
answered this medical diagnosis problem. Almost half of them judged the
probability that the person actually had the disease to be 0.95 (modal answer),
the average answer was 0.56, and only 18% of participants responded 0.02. The
latter is what the authors believed to be the correct answer. Note the enormous
variability in judgments. Harvard
Little has been achieved in explaining how people make these judgments and why the judgments are so strikingly variable.
But do statistics and probability give one and only one “correct” answer to that problem?
The answer is again “no.” And for the same reason, as the reader will already guess. As in the case of confidence and conjunction judgments, subjects were asked for the probability of a single event, that is, that
“a person found to have a positive result actually has the disease.” If the mind is an intuitive statistician of the frequentist school, such a question has no necessary connection to probability theory.
A more serious difficulty is that the problem does not specify whether or not the person was randomly drawn from the population to which the base rate refers.
Statistical principles are not learned from everyday experience because the relevant in-stances are not coded appropriately.
The lack of an appropriate code also explains why people usually do not detect the biases in their judgments of probability.
The inherently subjective nature of probability has led many students to the belief that coherence, or internal consistency, is the only valid criterion by which judged probabilities should be evaluated. From the standpoint of the formal theory of subjective probability, any set of internally consistent probability judgments is as good as any other. This criterion is not entirely satisfactory, because an internally consistent set of subjective probabilities can be incompatible with other beliefs held by the individual. Consider a person whose subjective probabilities for all possible outcomes of a coin-tossing game reflect the gambler's fallacy. That is, his estimate of the probability of tails on a particular toss increases with the number of consecutive heads that preceded that toss. The judgments of such a person could be internally consistent and therefore acceptable as adequate subjective probabilities according to the criterion of the formal theory. These probabilities, however, are incompatible with the generally held belief that a coin has no memory and is therefore incapable of generating sequential dependencies. For judged probabilities to be considered adequate, or rational, in-ternal consistency is not enough. The judgments must be compatible with the entire web of beliefs held by the individual. Unfortunately, there can be no simple formal procedure for assessing the compatibility of a set of probability judgments with the judge's total system of beliefs.
 Nisbett, Richard E., Krantz, David H., Jepson, Christopher. And Kunda, Z.(1983, p. 339)
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