![]() A very large difference score, however, tells us that both, numerator and denominator are near the endpoints of their scales, but at opposite ends. If the ratio is very large, the denominator is probably very small. Second – and relatedly – the size of the ratio lets us estimate the size of the denominator. In contrast, difference scores make do with a modest and symmetrical distribution around 0, where the maximum is X max – Y max. This asymmetry yields highly skewed distributions. While 1.0 is the midpoint, lowering the numerator cannot make the ratio negative, whereas lowering the denominator can move the ratio towards infinity. First, ratios are bounded by 0 at the floor, but they have no ceiling. Yet, ratios and difference scores differ in important ways. Either way, they probably thought that taking into account the way a comparison group is being perceived can only improve measurement and prediction. McCauley and others later moved from ratios to difference scores without much comment. Elegant as Bayes’ method is, it is not a good description of how people perceive the typicality of various traits in social groups. In other words, Bayes’ theorem demands the calculation of a ratio of conditional probabilities so that a person can be classified as Japanese or Luxembourgian given their differential probabilities of suicide. In Bayes’ theorem, the ratio of the probability that a Japanese person will die by suicide, p(S|J), divided by the probability that a Luxembourgian will die by suicide, p(S|L), is equal to the ratio of posterior classification, i.e., the probability that a suicide is Japanese, p(J|S), over the probability that a suicide is Luxembourgian, p(L|S), if multiplied by the ratio of the prior probability that a person is Japanese, p(J), over the prior probability that a person is Luxembourgian, p(L). ![]() This means that beliefs can be expressed probabilistically and that a set of beliefs is – or at least should be – consistent in Bayes’ way. Why did McCauley & Stitt think diagnostic ratios are superior? They started from the premise – a prior belief you might say – that all cognition, and hence social cognition, is Bayesian. ![]() We can see this even in McCauley & Stitt’s own data. Simple percentage estimates for a group are more highly correlated with trait typicality ratings than are diagnostic ratios. Sure enough, they found that diagnostic ratios are correlated with typicality ratings (‘How typical is committing suicide of the Japanese?’), but in a sustained decade-long quest, my colleagues and I showed that the numerator (% Japanese) does all the work, whereas the denominator (% Luxembourgians) degrades the measure instead of sharpen it (reviewed in Krueger, 2008). McCauley & Stitt argued that the diagnostic ratio is a better and truer measure of stereotyping than the good old-fashioned percentage value obtained for the Japanese. ![]() ![]() According to McCauley & Stitt, this perception differential makes suicide stereotypical of the Japanese and counter-stereotypical of the Luxembourgians and it ought to be expressed as a diagnostic ratio here 3/1. Let’s say the perceived prevalence of suicide in Japan is 3%, whereas it is 1% in Luxembourg. They have – thank god – a low rate of suicide, but this rate may be – and may be perceived to be – a bit larger than in the rest of the world, or in your own country if it is not Japan. I remember like it was the day before yesterday, when a classmate in graduate school summarized an article by McCauley and Stitt (1978), which purported to show that social stereotypes are Bayesian, that is, that they are relative. ![]()
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