*Proceedings from the Seventh International Symposium on Human Identification 1996*,
Promega Corp 1997, pp 48-52

Charles H. Brenner, Ph.D. (home page)

E-mail: Discuss race

A mixed-three-race example with attractive 3d-graphs.

The ratio of profile frequencies for the same profile in different races is a likelihood ratio. By standard Bayesian reasoning it quantifies the evidence that the profile comes from one race rather than the other. Moreover there is nothing in the reasoning that depends on what "profile" means. Therefore the best strategy to compute a "profile" frequency may differ from standard casework methods. If another formula (so long as gives the frequency of something, anything) gives better predictive value it is preferable.

Suppose that a certain DNA profile *P* is calculated to occur at the rate of *f _{A}*=1/5000 for
population A and

The expression

is a likelihood ratio by definition expressing the
superiority of the hypothesis race=A over the
hypothesis race=B. This statement is true regardless of the interpretation put on the statement
"person matches *P*." For example, if "matching" is defined as having bands in the same FBI fixed
bins, then numerator and denominator correspond to the FBI definition of frequency. That will
not be a particulary useful interpretation for race-discrimination purposes however. It will
overlook any component of genetic drift that merely shuffles allelic preponderances within the
same fixed bin, and is only sensitive to gross drift that manifests as changes in the total frequency
of a bin. Hence by this definition *L* will often not be far from one.

More plausibly useful, "match" can be defined according to a ±3% window. This rule estimates
frequencies by counting a fragment from the database is counted as "matching" a fragment in *P* if
they are within 3% in size. *L* then is the ratio of frequencies as defined according what had been
dubbed a "floating bin" of size ±3%. A smaller window would give still greater discrimination
between races, but a limiting concern is the precision of measurement. After all the matching
comparisons are not made between actual sizes of fragments, but only between measurements. So
as the window becomes small, the likelihood ratio test becomes less a test of similarity between
fragments and more a test of similarity between random noise.

It is therefore desirable to define "match" in such a way as to compensate for measurement error.
The appropriate mathematical expression is implicit in Gjertson et al (1988) and several later
papers. Let *q*_{1}, . . . , *q _{N}* be the sizes in the database, be a normal distribution with mean zero
and standard deviation chosen to imitate the variation from repeat measurements of the profile
fragment

is a "Bayesian" frequency for *q* calculated by
counting each database fragment *q _{j}* on a

**Figure 1** illustrates the difference between the last
two approaches. Frequencies as calculated by the
Bayesian method are the lower, more sharply
undulating curve. For present purposes it is the
fact that it undulates more sharply that makes it
more useful.

One way to make such a prediction is by simulating a large number of cases, then summarizing them. This means, select two races or ethnic groups for which allele population frequency databases are available across several loci. Simulate a case by selecting a profile representing the first race by using the frequencies for that race. Compute the likelihood ratio of the frequencies from both races. Then construct an appropriate statistical summary.

If the statistical procedure is appropriate, then it must satisfy the following validation test. Construct a pair of databases artificially by partitioning a single population at random into two halves. The typical likelihood ratio distinguishing one half from the other must of course be unity.

At least two caveats are necessary in order to satisfy the validation experiment:

i) Include the sample profile bands in the database when computing the frequency. Of course, it is already included in the database for the race from which it is sampled. Failure to include it in the other one would create an ascertainment bias.

ii) The appropriate average of the sample likelihood ratios is the geometric, not the arithmetic mean.

Erikson et al (1991) also used these rules to compare databases, although they did not consider the racial discrimination interpretation in the present sense.

Note that point ii is equivalent to dealing with lod scores and taking an arithmetic average. Since the sum of per-locus lod scores is the lod score across several loci, it follows from the Central Limit Theorem that as the number of loci increases the distribution of likelihood ratios approaches log-normal. Applied to the validation test case, the result is, as it should be, that the likelihood ratios are log-normally distributed around unity.

The distribution between Caucasian and Black using a 3% window and the loci of Table 1 gives approximately the distribution of Figure 2. The St. Louis case turns out to be exactly typical -- a likelihood ratio of 45 is achieved half the time. The one-standard deviation tail (84% to the right) is marked at a likelihood ratio of 5.3. Assuming a 50% prior probability for each race, this means that there is an 84% chance to guess correctly with at least 84% (=5.3/6.3) confidence.

Work is in progress (based on the data discussed in Meyer et al) to evaluate the effectiveness of STRs for distinguishing racial groups. Direct comparisons are hard to make, in part because the Meyer data is for sundry populations around the world and not US groups, but broadly speaking it seems that STR's may work about as well "per unit heterozygosity." That is, one must use more STR's for the same effect in distinguishing individuals, and must also use more STR's, in roughly the same proportion, to distinguish races.

Erikson B, Svensmark O (1994) DNA polymorphism in Greenland, Int J Legal Med 106:254-257

Evett IW, Pinchin R, Buffery C (1992) An investigation of the feasibility of inferring ethnic origin from DNA profiles, JFSS 32(4):301-306

Gjertson D, Mickey R, Hopfield, Takenouchi & Terasaki P (1988) Calculation of Probability of Paternity Using DNA Sequences, Am J Hum Genet 43:860-869

Meyer E, Wiegand P, Brinkmann B (1995) Phenotype differences of STRs in 7 human populations, Int J Legal Med 107:314-322

band sizes | Caucasian vs. | |||

Black | Hispanic | |||

3% window | Bayesian | 3% window | ||

D2S44 | 2630 | 1.6 | 1.5 | 0.91 |

D1S7 | 4300
4300 |
1.1
1.1 |
1.2
1.2 |
0.99
0.99 |

D17S79 | 1520
1300 |
2
1.3 |
2.3
2.4 |
1.2
0.97 |

D4S139 | 11600
4600 |
1.8
0.8 |
1.4
1.3 |
1.4
0.75 |

D5S110 | 3610
1955 |
1.8
1.7 |
1.9
1.7 |
na |

D10S28 | 1780
1115 |
1.5
1.4 |
1.8
2.8 |
0.91
1.3 |

Likelihood ratio = |
45 |
360 |
1.33 | |

Likelihood ratio per band = | 1.41 | 1.71 | 1.03 |