Misued? Percent Information

[SlideRule]

from an astute source:

       We recently heard a case of misusing percent on a local radio station. The
announcer was describing a process for detecting a specific type of cancer. Medical
researchers had become concerned that there was a significant number of "false
negatives" resulting from the detection process previously used; that is, on occasion
a person for whom the detecting process had identified no cancer would suddenly
have symptoms of the cancer, leading researchers to suspect that the cancer had
actually been present at the time of the original negative test. To address this
problem, researchers developed a more comprehensive detection process. This new
detection process appeared to be more successful, since the number of positive tests
(apparent detection of cancer) increased by 25%.
       So far, there is nothing wrong with the announcer's description. But he then
said something that got our "mathematical attention". Specifically, he said, "in
other words, 25% of the people who had negative tests with the previous detection
process actually had this specific cancer." Is this statement correct?

BEST!
SlideRule

[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.

[Albert Chan]

(05-23-2024 10:56 PM)SlideRule Wrote:  "in other words, 25% of the people who had negative tests with the previous detection
process actually had this specific cancer." Is this statement correct?

Technically, it is possible, but it is un-related to the new test.

carey's number, true cancer rate = 20%

Code:


                 | Positive   | Negative  Total
Cancer           |       0    |     20       20
No Cancer        |      20    |     60       80
       Total            20          80

False positive = 20/20 = 100%      // extremely bad test!
False negative = 20/80 = 25%

[ttw]

Remember that 98.6° of all statistics are made up on the spot.

[SlideRule]

… Is this statement correct?
      To examine this statement let us make a preliminary assumption about the prevalence of this type of cancer. Let us suppose that the previous detection process yields positive results for 4% of the population tested; the new process would detect 25% more than this. A 25% increase would raise this 4% of the population testing positive to 5% testing positive. How many people would have negative tests with each process? (Recall that a negative test is one in which no cancer is detected.)
      The original detection process identified 96% as having negative results. The new improved process identified 95% as having negative results. If this were applied to a typical group of 100 people, one person of the original 96 found to be negative by the original process is now identified by the new process as apparently having this specific cancer. This one person represents 1.04% of the 96, not 25% as the announcer thought.
      We performed this computation with the assumption that 4% tested positive using the original process. …
…
      The radio announcer would have been correct if 50% of the population tested positive for this cancer using the original process. This is extremely unlikely. The actual incidence of cancer is probably nearer the small values of x …

BEST!
SlideRule

source: MISUSING MEDICAL PERCENTS
David R. Duncan and Bonnie H. Litwiller
University of Northern Iowa
Cedar Falls, IA 50614