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STA100 STA100 Statistical Methods Lecture 8
Text Sections: 4.2, 4.
Here’s an example which is very important (and will feature prominently in homework 3). It concerns what we do when we have to make a decision but don’t have enough information to be sure about what we are doing. This comes up in life all the time, but here we’ll restrict our attention to a simple test.
A quick idea first: recall from last time
We’ll use this extensively in this lecture. Just notice that we can multiply both sides of our equation to solve for the famous multiplication formula.
Imperfect Diagnostic Tests
Suppose you wish to test high school athletes for the use of steroids. Unfortunately, tests are imperfect. There’s always a chance that you will accuse someone unjustly or let a guilty party off without penalty. Speaking a little more technically, any real world test for a condition (like “this individual used steroids” or “this person did the crime”) may produce false positives or false negatives.
In our case a false positive corresponds to a “clean" athlete who tests positive for steroid use. A false negative corresponds to an athlete who actually did use steroids testing “clean”.
A true positive is, of course, when the test indicates steroid use for someone who did use steroids. What is a false negative?
We'll need two definitions: The true positive rate of a test is called its sensitivity. The true negative rate of a test is called its specificity.
How would you measure these things? For example, on a quick “Google” right now I found that, in an article discussing a new flu diagnostic kit, (Quach, Newby, Daoust, Rubin, & McDonald,
There were 85 positive results detected by QuickVue, for a sensitivity of 79.2% (42 of 53) and a specificity of 82.6% (204 of 247), with a positive predictive value (PPV) of 49.4% (42 of 85) and a negative predictive value (NPV) of 94.9% (204 of 215).
I’ve reconstructed this in a table below:
Flu Present Flu Not Present Test Positive 42 43 85 Test Negative 11 204 215 53 247 300
So, how to get sensitivity? This is just a conditional probability. We’ve defined sensitivity as the true positive rate, i.e. “how often do we find it when it is there?” Thus recalling our definition of conditional probability from last lecture:
We write
Looking at the table we have
Working the same way, we’ve defined specificity as the true negative rate, i.e. “how often do we exclude it (reject it) it when it is not there?” Again substituting into our definition of conditional probability:
We write
Note that people who use steroids come in two types: those who get caught (Region 4) and those who do not (Region 2). Also, people who test positive also come in two types: those who actually did use steroids and got caught, and those who did not use steroids but are “found guilty” anyway. Remember that tests are imperfect and guilty people are sometimes set free and innocent people are sometimes unjustly sent to prison. It’s just a fact of life.
Since our question states: “You test an athlete and find that the test indicates steroid use. What is the probability that the athlete has, in fact, used steroids?” we are being asked for a conditional probability. We would like to know
The top of our fraction (numerator) is just the size of Region 4. The bottom (denominator) is the size of Region 3 together with Region 4. To figure out these probabilities/sizes we use our given information. We have been told that the rate of steroid use is 2%, so the sizes of Regions 2 and Region 4 together give us 0.02. What does the sensitivity tell us?
Writing this out as
Gives us
Or
That is, the size of Region 4 is 0.019.
We have immediately (watch the dominos start to fall) that, since Region 4 together with Region 2 have size 0.02, the size of Region 2 is 0.001.
Now work with the specificity.
Written in the form of the multiplication rule:
Now is the size of Region 1 while the other probabilities are known to us, so
Thus the size of Region 1 is. We still have to figure out the size of Region 3. Remember that all four Regions together add up to 1, so
And so finally the size of Region 3 is 0.0784. Summing up: Region 1 size is 0. Region 2 size is 0. Region 3 size is 0. Region 4 size is 0.
We are done once we substitute into our equation from before
So, the athlete standing before you who tested positive probably (better than 80% chance) DID NOT use steroids. If you find this surprising, it’s not that the test is bad (these are decent enough numbers), it’s just that testing for a rare condition is difficult!!!
