The Language of Argument

2 5 8

C HaP Te r 1 1 ■ C h a n c e s

Tables like these work by dividing the population into groups. We already

learned some names for these groups:

Hypothesis (h) Not Hypothesis (~h)

Evidence (e) True Positives False positives

Not Evidence (~e) False Negatives True Negatives

Population

False positives are sometimes also called false alarms, and false negatives are

sometimes called misses. A little more terminology is also common:

Pr(h) 5 base rate or prevalence or prior probability

Pr(h|e) 5 solution or posterior probability

Pr(e|h) 5 sensitivity of the test

Pr(~e|~h) 5 specificity of the test

1 – Pr(e|h) 5 1 – sensitivity 5 false negative rate

1 – Pr(~e|~h) 5 1 – specificity 5 false positive rate

You don’t need to use these terms in order to calculate the probabilities, but

it is useful to learn them so that you will be able to understand people who

discuss these issues.

One of the most important lessons of Bayes’s theorem is that the base

rate has big effects. To see how much it matters, let’s recalculate the solution

[Pr(h|e)] in our colon cancer example for different values of the base rate

[Pr(h)] using the same test with the same sensitivity (Pr(e|h) 5 0.9) and spe-

cificity [Pr(~e|~h) 5 0.97]:

If Pr(h) 5 0.003, then Pr(h|e) 5 0.083

If Pr(h) 5 0.03, then Pr(h|e) 5 0.48

If Pr(h) 5 0.3, then Pr(h|e) 5 0.93

Construct tables to confirm these calculations of Pr(h|e) for base rates of 0.03

and 0.3.

exercise Vi

These calculations show that a positive test result for a given test means

a lot more when the base rate is high than when it is low. Thus, if doctors

use the specified test as a screening test in the general population, and if

the rate of colon cancer in that general population is only 0.003, then a

positive test result by itself does not show that the patient has cancer. In

contrast, if doctors use the specified test as a diagnostic test only for people

with certain symptoms, and if the rate of colon cancer among people with

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