Last time in Uncertainty Wednesday, I announced that we would look at the PSA Test Example using absolute numbers instead of probabilities. Now you may recall that we defined probability as how likely something is to happen. And in introducing the PSA Test Example, I provided the probabilities for the elementary events for a 50-year old male as follows:
P({AL}) = P(healthy *and* low PSA) = 0.907179
P({AH}) = P(healthy *and* high PSA) = 0.089721
P({BL}) = P(cancer *and* low PSA) = 0.001519
P({BH}) = P(cancer *and* high PSA) = 0.001581
How do we go from these to absolute numbers? What we want to know is what happens if we look at a group of 10,000 50-year old males assuming that these probabilities apply equally to ever male in the group. Put differently we are assuming that one male in the group having cancer does not make it more or less likely that another does and one receiving a high signal does not make it more or less likely that another does. It is important to recognize that this embeds many assumptions, such that this type of cancer is not contagious and that the test equipment in use is working properly each time!
With this assumption, we can simply apply the probabilities as fractions to the total population. And with a bit of rounding we get the following counts which I will denote using N as in “number of”
N({AL}) = N(healthy *and* low PSA) = 9072
N({AH}) = N(healthy *and* high PSA) = 897
N({BL}) = N(cancer *and* low PSA) = 15
N({BH}) = N(cancer *and* high PSA) = 16
You can easily verify that this adds up to 10,000.
Now let’s revisit the question about the conditional probability that you have cancer (B) after receiving a high PSA level signal (H). We can see that in total there are N(H) = 897 + 16 = 913 people who receive an H signal. Of these only N({BH}) = 16 have cancer. So the likelihood of having cancer conditional on a high PSA level is
P(B | H) = N({BH}) / N(H) = 16 / 913 = 0.0175
or 1.75% which is pretty much the same as the 1.73% we found previously (the difference is the result of rounding when we went to absolute numbers).
Many people find it much more intuitive to think about absolute numbers. You should try out using the absolute numbers to answer the other question which was “How likely is it that you do have cancer (A) even though your PSA level was low (L)?” (left as an exercise for the reader – I love saying that!).
Keep in mind that given the assumption of the probabilities applying equally to each member of the group, using absolute numbers is really just a scaling of the probabilities (in our case by a factor of 10,000). To answer the conditional questions we wind up forming fractions in which both the numerator and the denominator were scaled by the same factor and hence the factor immediately cancels back out. You should convince yourself of this by using 100,000 or 1,000,000 as the size of the group.
Next week we will wrap up this example by looking at two measures, called sensitivity and specificity, that are widely used to assess the quality of medical tests (meaning how strong a signal is the test producing).
