Holy smokes! The "'hot-hand fallacy' fallacy"! - Cultural Cognition of Health

It’ super-duper easy to demonstrate that individuals of low to moderate Numeracy — an information-processing disposition that consists in the capacity & motivation to engage in quantitative reasoning — are prone to all manner of biases–like “denominator neglect,” “confirmation bias,” “covariance [non]detection,” the “conjunction fallacy,” etc.

It’s harder, but not impossible, to show that individuals high in Numeracy are more prone to biased reasoning under particular conditions.

In one such study, Ellen Peters and her colleagues did an experiment in which subjects evaluated the attractiveness of proposed wagers.

For one group of subjects, the proposed wager involved outcomes of a positive sum & nothing, with respective probabilities adding to 1.

For another group, the proposed wager had a slightly lower positive expected value and proposed outcomes were a positive sum & a negative sum (again with respective probabilities adding to 1).

Because the second wager had a lower expected value, and added “loss aversion” to boot, one might have expected subjects to view the first as more attractive.

But in fact subjects low in Numeracy ranked the two comparable in attractiveness.  Maybe they couldn’t do the math to figure out the EVs.

But the real surprise was that among subjects high in Numeracy, the second wager– the one that coupled a potential gain and a potential loss– was rated as being substantially more attractive than the first — the one that coupled a potential gain with a potential outcome of zero and a higher expected EV.

This result, which is hard to make sense of if we assume that people generally prefer to maximize their wealth, fit Peters et al.’s hypothesis that the cognitive proficiency associated with high Numeracy guides decisionmaking through its influence in calibrating affective perceptions.

Because those high in Numeracy literally feel the significance of quantitative information, the necessity of doing the computations necessary to evaluate the second wager, Peters et al. surmised, would generate a more intense experience of positive affect for them than would the process of evaluating the first wager, the positive expected value of which can be seen without doing any math at all.  Lacking the same sort of emotional connection to quantitative information, the subjects low in Numeracy wouldn’t perceive much difference between the two wagers.

But can we find real-world examples of biases in quantitative information-processing distinctive to individuals high in Numeracy?  Being able to is important not only to show that the Peters et. al result has “practical” significance but also show that it is valid .  Their account of what they expected to and did find hangs together, but as always there are alternative explanations for their results.  We’d have more reason to credit the explanation they gave– that high Numeracy can actually cause individuals to make mistakes in quantitative reasoning that low Numeracy ones wouldn’t — in the real world.

That way of thinking is an instance of the principle of convergent validity : because we can never be “certain” that the inference we are drawing from an empirical finding isn’t an artifact of some peculiarity of the study design, the corroboration of that finding by an empirical study using different methods — ones not subject to whatever potential defect diminished our confidence in the first — will supply us with more reason to treat the first finding as valid.

Indeed, the confidence enhancement will be reciprocal: because there will always be some alternative explanation for the findings associated with the second method, too, the concordance of the results reached via those means with the results generated by whatever method informed the first study gives us more reason to credit the inference we are drawing from the second.

Okay, so  now we have some realllllllly cool “real world” evidence of the distinctive vulnerability of high Numeracy types to a certain form of quantitative-reasoning bias.

It comes in a paper, the existence of which I was alerted to in the blog of stats legend  (& former Freud expert ) Andrew Gelman, that examines the probability that we’ll observe the immediate recurrence of an outcome if we examine some sequence of binary outcomes generated by a process in which the outcomes are independent of one another– e.g., of getting “heads” again after one getting “heads” rather than “tails” in the previous flip of a fair coin.

We all know that if the events are independent, then obviously the probability of the previous event recurring is exactly the same as the probability that it would occur in the first place.

So if someone flipped a coin 100 times, & we then examined her meticulously recorded results, we’d discover the probability that she got “heads” after any particular flip of “heads” was 0.50, the same as it would be had she gotten “tails” in the previous flip.

Indeed, only real dummies don’t get this!  The idea that the probability of independent events is influenced by the occurrence of past events is one of the mistakes that those low to moderate Numeracy dolts make!

They (i.e., most people) think that if a string of “heads” comes up in a “fair” coin toss (we shouldn’t care if the coin is fair; but that’s another stats legend /former Freud expert Andrew Gelman blog post), then the probability we’ll observe “heads” on the next toss goes down, and the probability that we’ll observe “tails” goes up. Not!

Only a true moron , then, would think that if we looked at a past series of coin flips, the probability of a “heads” after a “heads” would be lower than the probability of a “heads” after a “tail”! Ha ha ha ha ha! I want to play that dope in poker! Ha ha ha!