Good math is reversible; these numbers are not.

As we said earlier, the methodology requires symmetry of proportion between the spectrum and the general population.[1]  Without that, the results will be unstable. And the results below are indeed unstable.  Add this instability to the inflation of key denominators, and the problem is even more conspicuous.  When the report teases out the FH and FDB and calculates them individually “from scratch,” the numbers are not consistent with the report’s total when FH and FDB are combined at the outset of calculation.  We can see this clearly by looking at the study’s results. If the prevalence of FH LDLR + FDB APOB is supposed to be 1:217, then out of a population of 1,000,000 we would expect to find 4,608 mutation carriers.  But if we accept the breakdown then the LDLR portion of 1:395 gives us 2,532 out of 1,000,000 and the APOB portion coming in at a rate of 1 for every 118 gives us 8,475. Adding the two constituents together we get 11,007, which yields a prevalence rate of 1:91, next to the 1:217 prevalence rate found in the results, which gave us 4,608 … so which is it?

LDLR and APOB results are not parallel

The predominance of LDLR over APOB probands in the spectrum actually inverts and suddenly APOB is more prevalent than the LDLR hits in the 98,000 sample.  This is obvious when comparing the study’s population results to the spectrum probands used to derive the ratio. (It is also obvious when comparing the results to the probands in Brusgaard’s paper.)

APOB and LDLR prevalence reversal

Note that, in the results(above left), the reports’ FDB APOB prevalence rate of 1:118 is higher than the FH LDLR rate of 1:395. Again, per million, the FDB APOB’s 1:118 yields 8,475 mutation carriers and the FH LDLR’s 1:395 yields 2,532.  This is upside down. It is well established in the industry that the LDLR mutations, as a whole, are more prevalent than the disease-causing APOB mutations. And the authors have already shown in the 2nd report Supplement that APOB R3500Q/W make up only 13% of the total probands found (FH LDLR + FDB APOB) (above right). But then, as we see, in the results, APOB mutations suddenly outnumber the LDLR by over 200%. 

Chart of FDB APOB and FH LDLR  prevalence irregularity

For their ADH (FH + FDB) prevalence, the whole business of carrying over the .387 fraction of Top4 probands rests on the unstated assumption that there is a symmetry of proportion between the spectrum and the Copenhagen sample population. We can only expect a roughly accurate result if the proportions between the two sets of numbers are roughly parallel.

Although inflating the denominator accounts for much of the distortion, even before this, we can see the inversion in the raw data. In the 2nd report’s Supplementary, Table 2, we see the number of FDB APOB to FH LDLR is 19 APOB for 36 Top3 LDLR in the spectrum, but this predominance of LDLR flips when the report presents the molecular hits from the 98,000 Copenhagen residents. We now have 111 APOB for 63 Top3 LDLR.   There were almost twice as many Top3 LDLR as APOB in the spectrum, but then when we get to the molecular hits in the population study, we see the opposite: the Top3 LDLR are outnumbered by twice as many APOB.  This is a view of the raw data before denominators were inflated, so we have more than one problem with this inversion of LDLR predominance. 

Where symmetry is necessary asymmetry is present

The assumed symmetry of proportion between the authors’ reported spectrum and their reported population is just not there. But we can’t yet disagree on the size of the gap between estimates.  First, we must deal with the outright reversal in the established predominance of LDLR over APOB … just as we wouldn’t disagree about the difference between the size of an elephant compared to that of a mouse until we first confronted the suspicious statement that the mouse was bigger than the elephant.  Such a reversal in FH predominance over FDB might apply to a geographical region subject to founder effect, but I have found no expert opinion stating that FDB prevalence outnumbers FH prevalence in a sufficiently diverse population. It is universally agreed that FDB is the rarer of the two.

The 2nd report, Table 1, contradicts established prevalence ratios of FDB to FH, which are roughly 2 for every 1. The raw data here shows 111 FDB (APOB) for 63 TOP3 FH (LDLR), a predominance of APOB over the Top3, and possibly an equivalence of total FH (LDLR) to total FDB (APOB) in the general population. It is curious that if this reversal or even parity between the two classes of mutations is correct, then it would have been a brilliant discovery. Why didn’t the authors lead in with this story?

We can at least say one thing with confidence: the method of using a proportion of probands in the spectrum to calculate prevalence in the general population is unstable.  We should have seen this coming.  A proband to members of a family is like a hub to the spokes of the wheel. Different wheels have different ratios of spokes to hubs, and different cultures within a single nation have different family sizes. Can I really assess a portion of hubs relative to total hubs in a small town and then use this proportion on a fraction of spokes in a large city in order to tally up all spokes in that city? 

How to accurately determine that ratio is worthy of further debate.  Since, however, we are reconciling the authors’ own reports, we will later continue to use their .387 fraction. Readers are invited to substitute their own estimates in the equations. Even a wide range of estimates will show that the larger problem is with the method used and not just the ratio chosen. For example, suppose that I ask, “What is the ratio of black to white pearls?”… while referring to a necklace which includes white plastic beads.  There may be this or that proposed ratio, which needs to be resolved.  But the bigger problem is the existence of plastic beads where one expected pearls. Likewise, the issue of the ratio of Top4 mutation carriers to Ex-Top4 needs to be resolved. But we mustn’t take our eyes off of the bigger issue: False Positives are included in results where one expected genuine mutation carriers. We will demonstrate this swap later in the core deduction, beginning here.  Keeping this in mind for later, let’s continue our evaluation of the leveraged denominators.


[1] See this page.