## Info

30 9,970

10,000

I constructed this table by first subdividing the total of 10,000 individuals into 30 with colorectal cancer and 9,970 without colorectal cancer because 0.3% of 10,000 = 30 and 10,000 - 30 = 9,970. For those with colorectal cancer, half would be expected to have a positive occult-stool test and half would be expected to have a negative test based on Table 3.1, producing the values of 15 in the first numerical column here. For those without colorectal cancer, 3% would be expected to have a positive occult-stool test based on the upper-right cell in Table 3.1, and 3% of 9,970 = 299. Summing across the first row gives 314 positive tests among 10,000 people. Only 15 of these positive tests (5% of 314) would occur in people who actually have colorectal cancer.

I constructed this table by first subdividing the total of 10,000 individuals into 30 with colorectal cancer and 9,970 without colorectal cancer because 0.3% of 10,000 = 30 and 10,000 - 30 = 9,970. For those with colorectal cancer, half would be expected to have a positive occult-stool test and half would be expected to have a negative test based on Table 3.1, producing the values of 15 in the first numerical column here. For those without colorectal cancer, 3% would be expected to have a positive occult-stool test based on the upper-right cell in Table 3.1, and 3% of 9,970 = 299. Summing across the first row gives 314 positive tests among 10,000 people. Only 15 of these positive tests (5% of 314) would occur in people who actually have colorectal cancer.

of positive test results is 314, of which only 15 are actually associated with cancer. That is, only about 5% of people with a positive occult-stool test would act ually have colorectal cancer, and you might be fort unate enough to be in the 95% without cancer, despite a positive test. Even though the likelihood of a false positive result for an indiv idual without colorectal cancer is relatively small (3%) compared to the likelihood of a correct positive result for an individual with cancer (50%), the number of people without cancer is so much larger than the number with cancer that most positive test result s occur in people w ithout cancer.

Now let's return to scent identification by dogs. The results of Schoon's research are summarized in Table 3.3, which has the same format as Table 3.1. The crit ical question is similar to that for diagnostic testing for colorectal cancer: if a dog identifies a suspect as hav ing an odor matching that from a crime scene, what is the probability that the suspect is guilty? Just as in the cancer example, we need more informat ion than the values shown in Table 3.3 to answer this question. In the cancer example, the additional information was the frequency of colorectal cancer in the population. The parallel information for the scent-ident ificat ion example would be the number of potential suspect s for the crime. If we assume that only one person committed the crime and that there are 10 possible suspect s, the proport ion of suspect s that is guilty is 1/10, or 10%, just as the proportion of people with colorectal cancer is 30/10,000, or 0.3%. Unfortunately, for most crimes it's not very clear how many potential suspects there are. However, imagine one of those classic murder mysteries on an estate in the English countryside. The owner, an eccentric bachelor, has 10 servants (the gardener, the butler, the cook , etc.). He invites 10 guests for a weekend of hunting. On Saturday evening, the owner is discovered murdered. A handkerchief with no identifying marks has been left on the floor of the library where the owner's body is found. The local constable brings his trained dog to match the scent of the handkerchief to that of one of the suspect s. It's obvious that one of the 10 servants or 10 guests committed the dastardly deed, so the calculat ions can be made just as in the cancer example (to keep things reasonably simple, we'll assume that

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