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Measuring Haplotype Rarity

Wheaton Method

In Kelly Wheaton’s interpretation of Casey’s method, the percentage of men who do hold the modal value is subtracted from the percentage who do not hold the individual’s particular value.

For example, if the “Casey score” is 98 (i.e., only 2% have that value) and 33% have the modal value, the “Wheaton" score” is 98-33 = 65. Again, these marker scores are summed across all markers.

Calculate this metric by

  1. Determining, for each marker, the percentage of men who do not have the particular value.
  2. Determining, for each marker, the percentage of men who do not have the modal value.
  3. Subtracting B from A to get a “net” score.[26]
  4. Summing scores for the set of tested markers.

Figure 4 shows the distributions of the resulting scores for four levels of resolution. See Appendix B for details.

Distributions “march” from left to right as more markers contribute to the scores, peaks get lower and spreads get wider; these trends also reflect increased haplotype variety with more genetic information. (The apparent bumps near the right tail of the 67-marker curve are due to collapsing of categories at the high end.)

Wheaton score distributions
Figure 4

Table 2
Wheaton Score Summary Statistics
Statistic 12
markers
25
markers
37
markers
67
markers
Average (Mean) 142 292 430 651
Standard Deviation 89.6 128.2 144.9 192.0
Minimum 0 0 39 220
Maximum 946 1,186 1,521 2,228
Median 129 283 415 636
Mode [26]   # N/A # N/A # N/A # N/A
n= 4,940 4,203 3,824 1,953

Advantages of Wheaton’s method include

However, there are disadvantages:

Wheaton partially overcame the first disadvantage following an e-mail message from Casey [28] (for R1b, 67 markers) with this categorization:

As discussed in the “Evaluation” section below, we found these interpretations to be in error. The interpretations were not applicable to the larger (and presumably more representative sample.

Evaluation:

Lacking Casey’s or Wheaton’s data, we presume they found this system of categories appropriate within their projects. However, assessment using all R1b STR results in the data set showed, more generally, a somewhat different picture.

theoretical categories vs. actual distribution
Figure 5

Using the methods outlined above we calculated “Wheaton scores”[29] for all 1,953 STR results of the eight projects in the R1b haplogroup.[30] We then compared the theoretical distribution for the actual 67-marker score distribution, as shown in Figure 5.

The actual distribution of scores did not match the Casey/Wheaton model. Few of the eight projects’ 1,953 participants had so-called “common” haplotypes and more than a third had “rare” haplotypes, an irrational result.

What was proposed as “uncommon” (500-700) for 67 markers includes -- when examining more haplotypes of greater diversity -- the median score  of 637.

The most important statistics in Table 2 above are the median scores; one-half of scores are higher and one-half lower. Thus, the median denotes the mid-point on any scale! What they proposed as “uncommon” (500-700) for 67 markers includes the median score (637) when examining more haplotypes of greater diversity. A median score should be classified as of "average" commonness or rarity.

We can use the median and differences from it to assign plain-language interpretations to the scores. An “Average” category must include the median and a percentage of scores (±~25%) immediately above and below it. Categories both more and less rare will include another ~20% each and the categories for most and least rare will contain ~5% each.[31] For a fuller discussion, see Scale-setting.

This leads us to propose the five-category interpretation in Table 3, based on the observed data. It differs significantly from that proposed by Casey, and interpreted by Wheaton

Table 3
Revised Wheaton Score Interpretation
Category Pct. 12
markers
25
markers
38
markers
67
markers
Very Common ~5% 0-0 0-99 0-249 0-349
Common ~20% 1-74 100-199 250-349 350-499
Average ~50% 75-175 200-349 350-424 500-749
Uncommon ~20% 176-249 350-499 425-649 750-899
Rare ~5% ≥250 ≥500 ≥650 ≥900

Wheaton Average per Marker (WApM)

Distributions of Wh. Avg./Mkr
Figure 6

The problem of cross-comparability across marker sets led us to consider an “average per marker” method. This metric, as displayed in Figure 6, was derived by dividing the haplotype Wheaton score by the minimum of

See Appendix B2 for details of the distributions. The effect is to bring the distribution curves into similar ranges on the X axis; the "marching" is reduced. Note that peaks are occurring at approximately the same X value (8-12) for each marker set. Also note that peaks are higher and variances ("spreads") reduced with increased marker set size.

Summary statistics are

Table 4
Wheaton Average per Marker
Statistic WApM
12
WApM
25
WApM
37
WApM
67
Average (mean) 11.85 11.69 11.62 9.78
Standard Deviation 7.46 5.27 3.93 2.89
Minimum 0.00 0.00 1.05 3.29
Maximum 78.83 81.69 42.98 34.45
Median 10.75 11.32 11.24 9.54
Mode 0.00 11.80 13.16 9.37
n= 4,940 4,203 3,824 1,953

Again, the median defines the mid-point of the scale and categories are determined from it. We can roughly interpret thus:

Table 5
WApM Score Interpretation
Category WApM
12
WApM
25
WApM
37
WApM
67
Very Common 0-0 0-4 0-6 0-6
Common 0.1-6.0 4-8 6-8 6-8
Average 6-16 8-16 8-14 8-12
Uncommon 16-24 16-22 14-18 12-20
Rare ≥24 ≥22 ≥18 ≥20

Despite the additional calculation step, this is an easier measurement to interpret. Differences between marker sets are less and they relate to differences in frequency distributions of the markers making up the sets. [32].