Chance News 14
The [Supreme] Court concluded that mental health professionals' predictions were "not always wrong...only most of the time."
Two rows and three columns:
| Treatment Deaths per 1000 women
No mammography screening 4 Mammography screening
Consequently, there is "a 25 percent relative risk reduction." He would prefer focusing on the difference in the number of deaths which yields the more revealing and perhaps more honest statement: "The absolute risk reduction is 4 minus 3, that is, 1 out of 1000 women (which corresponds to .1 percent)." However, "Counting on their clients' innumeracy, organizations that want to impress upon clients the benefits of treatment generally report them in terms of relative risk reduction...applicants [for grants] often feel compelled to report relative risk reductions because they sound more impressive." Although he did not use this example, one's relative "risk" of winning the lottery is infinitely greater if one buys a ticket, yet one's absolute "risk" of winning has hardly improved at all. Most of his numerical examples are typified by his discussion of the picture given below which indicates the superiority of dealing with counts. Note that "H" represents having the disease and "D" represents a diagnosis having the symptom as seen by testing positive. Characteristically, there is a large number in the population who do not have the disease and because of the possibility of a wrong classification, the number of false positives (99) outweighs the number of true positives (8) resulting in P(disease| symptom) being much lower (8/(8+99)) than P(symptom| disease) (.8). This type of result, low probability of disease given symptom, is true even when ".8" is replaced by a number much closer to 1 provided there are many more who do not have the disease.
Here is an example he did not consider but it also illustrates the superiority of dealing with counts. Instead of two populations--diseased and healthy--which are greatly different in size, consider Boys and Girls and the desire to predict gender based on some simple test. Assume that 50% of births are Boys so that P(Boy) = P(Girl) = 1/2. A simple, inexpensive, non-invasive gender-testing procedure indicates that it is "perfect" for boys, P(Test Boy| Boy) = 1, implying P(Test Girl| Boy) = 0. Unfortunately, this simple, inexpensive, non-invasive gender-testing procedure for girls is a "coin toss," P(Test Girl| Girl) = P(Test Boy| Girl) = 1/2. Application of Bayes theorem yields what seems to be a strange inversion, P(Boy| Test Boy) = 2/3 and P(Girl| Test Girl) = 1. That is, somehow, "perfection" switched from Boy to Girl. The test is perfect in "confirming" that a Boy is a Boy and has a 50% error rate in confirming that a Girl is a Girl. The test is perfect in "predicting" that a person who tests as a girl is in fact a girl but has 33% error rate in predicting that a person who tests as a Boy is in fact a Boy. Thus, the term perfect is ambiguous. Perfection in confirmation, i.e., the test conditional on the gender, does not mean perfection in prediction, i.e., the gender conditional on the test.
Some of the puzzlement disappears if we deal with counts; the table below is equivalent to Gigerenzer's "tree" diagram. Assume 50 Boys and 50 Girls to start with. Every one of the 50 Boys will test as a Boy--none of the Boys test as a Girl; of the 50 Girls, 25 will test as a Boy and 25 will test as a girl. Therefore, P(Girl| Test Girl) = 1. One is tempted to to explain the switch by using the lingo of medical testing: false positives, false negatives, sensitivity, specificity, positive predictive value, negative predictive value. However, one hesitates to designate either gender as diseased even though the mathematics is the same.
Test Boy Test Girl Total
Boy 50 0 50
Girl 25 25 50
Gigerenzer rightly concludes that the language of statistics is not natural for most individuals. Perhaps the puzzlement in this specific example is at least partly due to the natural language known as English. Boys, Girls, Test Boys and Test Girls are too confusing. . Replace "Boy" by "Norwegian" and "Girl" by "German" and assume that there are as many Norwegians as Germans. Let every Norwegian be "Blond," so that P(Blond| Norwegian) = 1 and only half the Germans are Blond. Thus, P(German| Not Blond) =1; the switch, P(German| Not Blond) = P(Blond| Norwegian) = 1, is rather obvious. Is the this situation easier to understand because of the linguistics--hair color and ethnicity are easily distinct as Test Boy and Boy are not?
Discussion Questions 1. Gigerenzer has a chapter entitled, "(Un)Informed Consent." Based on your experience, what do you imagine the chapter contains? 2. A drawing of two tables (that is, physical tables on which things are placed) appears on page 10. He claims the tables (due to Roger Shepard) are identical in size and shape. After staring at them in disbelief of the claim, how would you verify the contention? 3. Physicians sometime make the following type of statement:"Never mind the statistics, I treat every patient as an individual." Defend this assertion. Criticize this assertion. 4. The physicist, Lord Rutherford, is reputed to have said, " If your experiment, needs statistics you ought to have done a better experiment." Defend and criticize Lord Rutherford. 5. Assume an asymptomatic woman has a mammogram which looks suspicious and then a biopsy which is negative. Would she be grateful for the clean bill of health or would she become an advocate who opposes (mass) screening? Suppose instead we assume a man has a suspiciously high PSA and the painful multiple biopsies (6-12 "sticks") are all negative. Would he be grateful for the clean bill of health or would he become an advocate who opposes (mass) screening? 6. Calculated Risks also deals with the risk to the physician making a recommendation and a diagnosis. Discuss why in our present-day litigious society the risks to the physician (who may or may not recommend a test or may or may not make a diagnosis) are not symmetrical. Along these lines, who are the vested interests involved in maintaining screening and testing? 7. Revisit the Boy/Girl scenario but now the test always says Boy regardless of gender, P(Test Boy| Boy) = P(Test Boy| Girl) = 1. Complete the table for this version. Obviously, this test has the advantage of being extremely simple, cost-free and non-invasive. Use either the Probability Format or the Frequency Format to comment on the statistical worthiness of this test.