Goosing denominators before calculating FH prevalence

In the 2nd Danish report, “Patients” are mathematically derived by inflating key denominators before calculation.

Analogy(Skip to this page to dive right into the material. For a more gradual approach please begin here.)

Analogy of denominator-goosing

The authors goosed the denominator of a key metric, lowering the fraction, which, after division, inflates the outcome.  This could appear complicated (at first), so we will use a simple analogy that shares the same mathematical structure. Imagine that a school administrator is evaluating a 4H class which is tending animals. He has some numbers, but he will have to derive the others.

Crucially, he knows that each student has the same animals and in the same proportion as the other students. 

The administrator is asked to answer the questions:

  1. How many total living things are in the whole class?
  2. How many total mammals are there?
  3. How many birds?

He is given enough in the above table to carry the known proportion of each student over to the entire class.

How many living things are in the whole class?

Using the sample student’s numbers, he knows that the proportion of Geese, Cows, Sheep, and Pigs to total Living Things is 6 ÷ 12, or .5.  Turning his attention to the whole class, he sees 60 such Geese, Cows, Sheep and Pigs. With an eye on the established proportion of 50%, he can divide 60 by .5 and end with 120 total living things.

How many mammals?

Example of inflated denominators

He knows that for one student the proportion of Cows, Sheep, and Pigs to total Mammals is 4÷10, or .4.  The class has 40 such Cows, Sheep and Pigs, so he can divide 40 by .4 and end with 100 total Mammals. Subtracting 40 from 100, he calculates 60 “other mammals.”

Essentially, what he is after when he calculates the fraction 4/10th is, “How many other mammals are thereDividing the known mammals by 4/10th will give us the total number of mammals, and from that we can derive through subtraction the number for the “Other mammals.”

Important note to be kept in mind for later chapters: this methodology depends upon symmetry of proportion between the sample student and the class of 10 students. Without a sufficient degree of symmetry, the methodology fails. For example, if the sample student has 2 cows, but the actual class of 10 students has 30 cows, and not the 20 required by symmetry, then the result will be skewed. We will derive our 50% proportion from the sample student, but then after carrying this proportion over to the whole class, instead of dividing 60 by .5, we will divide 70 by .5. With mammals and birds combined at the outset, we calculate 140 instead of 120.

Additionally, good math is reversible, these numbers would not be. Let’s call this 140 our “top-down” total.

Without symmetry, if I first separate the birds from the mammals, I reach a “bottom-up” total of 145 animals.

• Birds: 20 ÷ 1 = 20
• Mammals: 50 ÷ .4 = 125

If I reverse the “top-down” total of 140, by the established 50% proportion, I end with 70. With the “bottom-up” total of 145, I end with 72.5

Presuming a symmetry that does not exist, the math leaks. The results are unreliable.

How many birds?

He does the same for the birds.  There are 2 geese. There are no other birds to speak of. Thus, there is no “fraction” of geese-to-total-birds to calculate or work with. Since we had a step where the mammals were divided by .4, as a formality we might also divide the number of geese by 1, but otherwise it is a meaningless exercise: 20 ÷ 1 = 20.

An Error

What if I first conflate the word, “geese,” with the ambiguity in the word, “animals,” and then pass them all off as “mammals”? I can now use the total of the whole group (which includes the birds) for my denominator. So instead of deriving a proportion of mammals where Cows, Sheep, and Pigs are divided by 10 Mammals, they are divided by 12 “Animals.”  Below, we will put the correct math on the left side of the table and the goosed math on the right.  Note how the denominator on the right is inflated due to the inclusion of the geese in total animals, increasing it to 12.

Analogy of goosed math with farm animals

How many mammals if I goose the denominator? I get 20 more mammals on paper than actually exist in the barns. I goosed the denominator by counting the named mammals for the numerator, but taking advantage of the ambiguity of the word “Animal” I include the geese in the denominator. This leverages the outcome. And how many birds will I have if I put a pig in the denominator? … or if I put all the other mammals in the denominator? Why not?

Correct Math and Goosed Math
Correct and goosed denominators

Again, essentially, what we are after when we calculate the fraction to be used for division is, How many other birds are there? The division by the fraction will get us the total number of birds, from that we can derive the missing number.  But here, we don’t have a right to the question, because we already have all the birds from the beginning. There are no others. For example, if I say, “Here is 1/4th of a pie;” and then ask, “Now what am I missing?” 1 minus 1/4th = I’m missing 3/4th.  But if I say, “Here is a whole pie,” and then ask, “What am I missing?”  The answer is, “Nothing.” There is no remainder or missing slice: we already have the whole pie.

We get 100 more geese on paper than exist in the barns. And we can include them under the ambiguous term, “animals.”

  • In all, goosing each denominator while calculating mammals and birds separately, we can double the population of animals from 120 to 240, without having to find any new animals in the real world.