Misued? Percent Information

[carey]

Nice example of percentage issues and issues interpreting false positives and negatives!

If the 25% increase in positives occurs only in the false positive group, then not only will the number of false negatives remain the same, but the number of false positives will increase! The following example demonstrates this.

Below is an initial distribution of 100 test results: 20 positive and 80 negative. While we also assume 20 people have cancer and 80 have no cancer, the people with and without cancer doesn't correspond exactly with the test results because we also assume false positives and false negatives.

Code:


                 | Positive   | Negative  Total
Cancer           |      12    |      8       20
No Cancer        |       8    |     72       80
       Total            20          80

Now increase positive tests by 25% as required in the description, i.e., from 20 to 25, but draw them all from negative tests of people who have no cancer as follows.

Code:


                 | Positive   | Negative  Total
Cancer           |      12    |      8       20
No Cancer        |      13    |     67       80
       Total            25          75

The condition of the problem has been satisfied (an increase of 25% in positive tests) but the false positives went from 8 to 13 and the false negatives remained the same at 8.

Note:
In going from table 1 to table 2, the column Total distribution remains invariant (20 and 80) since that's the distribution of cancer and no cancer patients we assumed. However, the row Total can redistribute (e.g., from 20 80 to 25 75) as it represents the distribution of test results, as long as the total equals 100.