Estimating the Population Frequency of a DNA Pattern

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# Estimating the Population Frequency of a DNA Pattern

DNA "exclusions" are easy to interpret: if technical artifacts can be excluded, a nonmatch is definitive proof that two samples had different origins. But DNA "inclusions" cannot be interpreted without knowledge of how often a match might be expected to occur in the general population. Because of that fundamental asymmetry, although each new DNA typing method or marker can be used for investigation and exclusion as soon as its technical basis is secure, it cannot be interpreted with regard to inclusion until the population frequencies of the patterns have been established. We discuss the issues involved in estimating the frequency of a DNA pattern, consisting of pairs of alleles at each of several loci.

Estimating Frequencies of DNA Patterns by Counting

A standard way to estimate frequency is to count occurrences in a random sample of the appropriate population and then use classical statistical formulas to place upper and lower confidence limits on the estimate. Because estimates used in forensic science should avoid placing undue weight on incriminating evidence, an upper confidence limit of the frequency should be used in court. This is especially appropriate for forensic DNA typing, because any loss of power can be offset by studying additional loci.

To estimate the frequency of a particular DNA pattern, one might count the number of occurrences of the pattern in an appropriate random population sample. If the pattern occurred in I of 100 samples, the estimated frequency would be 1%, with an upper confidence limit of 4.7%. If the pattern occurred in 0 of 100 samples, the estimated frequency would be 0%, with an upper confidence limit of 3%. (The upper bound cited is the traditional 95% confidence limit, whose use implies that the true value has only a 5% chance of exceeding the upper bound.) Such estimates produced by straightforward counting have the virtue that they do not depend on theoretical assumptions, but simply on the sample's having been randomly drawn from the appropriate population. However, such estimates do not take advantage of the full potential of the genetic approach.

Estimating Frequencies of DNA Patterns with the Multiplication Rule (Product Rule)

In contrast, population frequencies often quoted for DNA typing analyses are based not on actual counting, but on theoretical models based on the principles of population genetics. Each matching allele is assumed to provide statistically independent evidence, and the frequencies of the individual alleles are multiplied together to calculate a frequency of the complete DNA pattern. Although a databank might contain only 500 people, multiplying the frequencies of enough separate events might result in an estimated frequency of their all occurring in a given person of I in a billion. Of course, the scientific validity of the multiplication rule depends on whether the events (i.e., the matches at each allele) are actually statistically independent.

From a statistical standpoint, the situation is analogous to estimating the proportion of blond, blue-eyed, fair-skinned people in Europe by separately counting the frequencies of people with blond hair, people with blue eyes, and people with fair skin and calculating their proportions. If a population survey of Europe showed that 1 of 10 people had blond hair, 1 of 10 had blue eyes, and I of 10 had fair skin, one would be wrong to multiply these frequencies to conclude that the frequency of people with all three traits was 1 in 1,000. Those traits tend to co-occur in Nordics, so the actual frequency of the combined description is probably higher than 1 in 1,000. In other words, the multiplication rule can produce an underestimate in this case, because the traits are correlated owing to population substructure-the traits have different frequencies in different population groups. Correlations between those traits might also be due to selection or conceivably to the action of some genes on all three traits. In any case, the example illustrates that correlations within subgroups-whatever their origin-bear on the procedures for estimating frequencies.

Unlike many of the technical aspects of DNA typing that are validated by daily use in hundreds of laboratories, the extraordinary population-frequency estimates sometimes reported for DNA typing do not arise in research or medical applications that would provide useful validation of the frequency of any particular person's DNA profile. Because it is impossible or impractical to draw a large enough population to test calculated frequencies for any particular DNA profile much below 1 in 1,000, there is not a sufficient body of empirical data on which to base a claim that such frequency calculations are reliable or valid per se. The assumption of independence must be strictly scrutinized and estimation procedures appropriately adjusted if possible. (The rarity of all the genotypes represented in the databank can be demonstrated by pairwise comparisons. Thus, in a recently reported analysis of the FBI database, no exactly matching pairs of profiles were found in five-locus DNA profiles, and the closest match was a single three-locus match among 7.6 million basepair comparisons.)[13]

The multiplication rule has been routinely applied to blood-group frequencies in the forensic setting. However, that situation is substantially different: Because conventional genetic markers are only modestly polymorphic (with the exception of human leukocyte antigen, HLA, which usually cannot be typed in forensic specimens), the multilocus genotype frequencies are often about 1 in 100. Such estimates have been tested by simple empirical counting. Pairwise comparisons of allele frequencies have not revealed any correlation across loci. Hence, the multiplication rule does not appear to lead to the risk of extrapolating beyond the available data for conventional markers. In contrast, highly polymorphic DNA markers exceed the informative power of protein markers, so multiplication leads to estimates that are less than the reciprocal of the size of the databases.

Validity of Multiplication Rule and Population Substructure

The multiplication rule is based on the assumption that the population does not contain subpopulations with distinct allele frequencies-that each individual's alleles constitute statistically independent random selections from a common gene pool. Under this assumption, the procedure for calculating the population frequency of a genotype is straightforward:

• Count the frequency of alleles. For each allele in the genotype, examine a random sample of the population and count the proportion of matching alleles-that is, alleles that would be declared to match according to the rule that is used for declaring matches in a forensic context. This step requires only the selection of a sample that is truly random with reference to the genetic type; it does not appeal to any theoretical models.

It is essential that the forensic matching rule be precise and objective- otherwise it would be impossible to apply it in calculating the proportion of individuals with matching alleles in the population databank. And it is essential that the same rule be applied to count frequencies in the population databank, because this is the only way to determine the proportion of random individuals that would have been declared to match in the forensic context. (In the context of forensic applications, an estimate of the probability of a match in DNA typing has been termed conservative if on the average it is larger than the actual one, so that any weight applied to the estimate would favor the suspect. Thus, some laboratories use a more conservative rule for counting population frequencies than for forensic matches-an acceptable approach, because it overestimates allele frequency. The converse would not be acceptable.)

• Calculate the frequency of the genotype at each locus. The frequency of a homozygous genotype is calculated to be , where denotes the frequency of allele . The frequency of a heterozygous genotype is calculated to be , where and denote the frequencies of alleles a1 and a2. In both cases, the genotype frequency is calculated by simply multiplying the two allele frequencies, on the assumption that there is no statistical correlation between the allele inherited from one's father and the allele inherited from one's mother. The factor of 2 arises in the heterozygous case, because one must consider the case in which allele al was contributed by the father and allele by the mother and vice versa: each of the two cases has probability . When there is no correlation between the two parental alleles, the locus is said to be in Hardy-Weinberg equilibrium. We should note that in forensic DNA typing, a slight modification is used in the case of apparently homozygous genotypes. When one observes only a single allele in a sample, one cannot be certain that the individual is a homozygote; it is always possible that a second allele has been missed for technical reasons. To be conservative, most forensic laboratories do not calculate the probability that the sample has two copies of the allele (which is ), but rather the probability that the sample has at least one copy (which is ) leaving open the possibility of a second allele. We endorse this procedure.)

• Calculate the frequency of the complete multilocus genotype. The frequency of a complete genotype is calculated by multiplying the genotype frequencies at all the loci. As in the previous step, this calculation assumes that there is no correlation between genotypes at different loci; the absence of such correlation is called linkage equilibrium. (Some authors prefer to reserve the term linkage equilibrium for loci on the same chromosome and to use the term gametic phase equilibrium for loci on different chromosomes.) Suppose, for example, that a person has genotype . If a random sample of the appropriate population shows that the frequencies of , and are approximately 0.1, 0.2, 0.3, 0.1, and 0.2, respectively, then the population frequency of the genotype would be estimated to be [2(0.1)(0.2)][2(0.3)(0.1)][(0.2)(0.2)] = 0.000096, or about 1 in 10,417.

Again, the validity of the multiplication rule depends on the absence of population substructure, because only in this special case are the different alleles statistically uncorrelated with one another.

In a population that contains groups with characteristic allele frequencies, knowledge of one allele in a person's genotype might carry some information about the group to which the person belongs, and this in turn alters the statistical expectation for the other alleles in the genotype. For example, a person who has one allele that is common among Italians is more likely to be of Italian descent and is thus more likely to carry additional alleles that are common among Italians. The true genotype frequency is thus higher than would be predicted by applying the multiplication rule and using the average frequency in the entire population.

To illustrate the problem with a hypothetical example, suppose that a particular allele at a VNTR locus has a 1% frequency in the general population, but a 20% frequency in a specific subgroup. The frequency of homozygotes for the allele would be calculated to be 1 in 10,000 according to the allele frequency determined by sampling the general population, but would actually be 1 in 25 for the subgroup. That is a hypothetical and extreme example, but illustrates the potential effect of demography on gene frequency estimation.

Basis of Concern About Population Substructure

The key question underlying the use of the multiplication rule is whether actual populations have significant substructure for the loci used for forensic typing. This has provoked considerable debate among population geneticists: some have expressed serious concern about the possibility of significant substructure,[2,4,9,10] and others consider the likely degree of substructure not great enough to affect the calculations significantly. [1,3,6,8,11-13]

The population geneticists who urge caution make three points:

1. Population genetic studies show some substructure within racial groups for genetic variants, including protein polymorphisms, genetic diseases, and DNA polymorphisms. Thus, North American Caucasians, blacks, Hispanics, Asians, and Native Americans are not homogeneous groups. Rather, each group is an admixture of subgroups with somewhat different allele frequencies. Allele frequencies have not yet been homogenized, because people tend to mate within these groups.

2. For any particular genetic marker, the degree of subpopulation differentiation cannot be predicted, but must be determined empirically.

3. For the loci used for forensic typing, there have been too few empirical investigations of subpopulation differentiation.

In short, those population geneticists believe that the absence of substructure cannot be assumed, but must be proved empirically (see Lewontin and Hartl [10]). Other population geneticists, while recognizing the possibility or likelihood of population substructure, conclude that the evidence to date suggests that the effect on estimates of genotype frequencies are minimal (see Chakraborty and Kidd [12]). Recent empirical studies concerning VNTR loci [13,14] (Weir, personal communication, 1991) detected no deviation from independence within or across loci. Moreover, pairwise comparisons of all five-locus DNA profiles in the FBI database showed no exact matches; the closest match was a single three-locus match among 7.6 million pairwise comparisons.[13] These studies are interpreted as indicating that multiplication of gene frequencies across loci does not lead to major inaccuracies in the calculation of genotype frequency-at least not for the specific polymorphic loci examined.

Although mindful of the controversy, the committee has chosen to assume for the sake of discussion that population substructure may exist and provide a method for estimating population frequencies in a manner that adequately accounts for it. Our decision is based on several considerations:

1. It is possible to provide conservative estimates of population frequency, without giving up the inherent power of DNA typing.

2. It is appropriate to prefer somewhat conservative numbers for forensic DNA typing, especially because the statistical power lost in this way can often be recovered through typing of additional loci, where required.

3. It is important to have a general approach that is applicable to any loci used for forensic typing. Recent empirical studies pertain only to the population genetics of the VNTR loci in current use. However, we expect forensic DNA typing to undergo much change over the next decade-including the introduction of different types of DNA polymorphisms, some of which might have different properties from the standpoint of population genetics.

4. It is desirable to provide a method for calculating population frequencies that is independent of the ethnic group of the subject.

Assessing Population Substructure Requires Direct Sampling of Ethnic Groups

How can one address the possibility of population substructure? In principle, one might consider three approaches: (1) carry out population studies on a large mixed population, such as a racial group, and use statistical tests to detect the presence of substructure; (2) derive theoretical principles that place bounds on the possible degree of population substructure; and (3) directly sample different groups and compare the observed allele frequencies. The third offers the soundest foundation for assessing population substructure, both for existing loci and for many new types of polymorphisms under development.

In principle, population substructure can be studied with statistical tests to examine deviations from Hardy-Weinberg equilibrium and linkage equilibrium. Such tests are not very useful in practice, however, because their statistical power is extremely low: even large and significant differences between subgroups will produce only slight deviations from Hardy-Weinberg expectations. Thus, the absence of such deviations does not provide powerful evidence of the absence of substructure (although the presence of such deviations provides strong evidence of substructure).

The correct way to detect genetic differentiation among subgroups is to sample the subgroups directly and to compare the frequencies. The following example is extreme and has not been observed in any U.S. population, but it illustrates the difference in power. Suppose that a population consists of two groups with different allele frequencies at a diabetic locus:

If there is random mating within the groups, Hardy-Weinberg equilibrium within the groups will produce these genotype frequencies:

Suppose that Group I is 90% of the population and Group II is 10%. In the overall population, the observed genotype frequencies will be

If we were unaware of the population substructure, what would we expect under Hardy-Weinberg equilibrium? The average allele frequencies will be

which would correspond to the Hardy-Weinberg proportions of

Even though there is substantial population substructure, the proportions do not differ greatly from Hardy-Weinberg expectation. In fact, one can show that detecting the population differentiation with the Hardy-Weinberg test would require a sample of nearly 1,200, whereas detecting it by direct examination of the subgroups would require a sample of only 22. In other words, the Hardy-Weinberg test is very weak for testing substructure.

The lack of statistical power to detect population substructure makes it difficult to detect genetic differentiation in a heterogeneous population. Direct sampling of subgroups is required, rather than examining samples from a large mixed population.

Similarly, population substructure cannot be predicted with certainty from theoretical considerations. Studies of population substructure for protein polymorphisms cannot be used to draw quantitative inferences concerning population substructure for VNTRs, because loci are expected to show different degrees of population differentiation that depend on such factors as mutation rate and selective advantage. Differences between races cannot be used to provide a meaningful upper bound on the variation within races. Contrary to common belief based on difference in skin color and hair form, studies have shown that the genetic diversity between subgroups within races is greater than the genetic variation between races. [15] Broadly, the results of the studies accord with the theory of genetic drift: the average allele frequency of a large population group (e.g., a racial group) is expected to drift more slowly than the allele frequencies of the smaller subpopulations that it comprises (e.g., ethnic subgroups).

In summary, population differentiation must be assessed through direct studies of allele frequencies in ethnic groups. Relatively few such studies have been published so far, but some are under way.[16] Clearly, additional such studies are desirable.

The Ceiling Principle: Accounting for Population Substructure

We describe here a practical and sound approach for accounting for possible population substructure: the ceiling principled It is based on the following observation: The multiplication rule will yield conservative estimates, even for a substructured population, provided that the allele frequencies used in the calculation exceed the allele frequencies in any of the population subgroups. Accordingly, applying the ceiling principle involves two steps: (1) For each allele at each locus, determine a ceiling frequency that is an upper bound for the allele frequency that is independent of the ethnic background of a subject; and (2) To calculate a genotype frequency, apply the multiplication rule, using the ceiling frequencies for the allele frequencies.

How should ceiling frequencies be determined? We must balance rigor and practicality. On the one hand, it is not enough to sample broad populations defined as "races" in the U.S. census (e.g., Hispanics), because of the possibility of substructure. On the other hand, it is not feasible or reasonable to sample every conceivable subpopulation in the world to obtain a guaranteed upper bound. The committee strongly recommends the following approach: Random samples of 100 persons should be drawn from each of 15-20 populations, each representing a group relatively homogeneous genetically; the largest frequency in any of these populations or 5%, whichever is larger, should be taken as the ceiling frequency. The reason for using 5% is discussed later.

We give a simplified example to illustrate the approach. Suppose that two loci have been studied in three population samples, with the following results:

For the genotype consisting of a/b at locus 1 and c/d at locus 2, the ceiling principle would assign ceiling values of 11% for allele a, 10% for allele b, 5% for allele c, and 15% for allele d and would apply the multiplication rule to yield a genotype frequency of [2(0.11)(0.10)][2(0.05)(0.15)] = 0.00033, or about I in 3,000. Note that the frequency used for allele c is 5%, rather than 4%, to reflect the recommended lower bound of 5% on allele frequencies. Because the calculation uses an upper bound for each allele frequency, it is believed to be conservative given the available data, even if there are correlations among alleles because of population substructure and even for persons of mixed or unknown ancestry. This is more conservative, and preferable, to taking the highest frequency calculated for any of the three populations.

The ceiling principle reflects a number of important scientific and policy considerations:

• The purpose of sampling various populations is to examine whether some alleles have considerably higher frequencies in particular subgroups than in the general population-presumably because of genetic drift. It is matches at such alleles that might be accorded too much evidentiary weight, if the general population frequency were used in calculating the probability of a match.

• Determining whether an allele has especially high frequency does not require a very large sample. A collection of 100 randomly chosen people provides a sample of 200 alleles, which is quite adequate for estimating allele frequencies.

• Genetically homogeneous populations from various regions of the world should be examined to determine the extent of variation in allele frequency. Ideally, the populations should span the range of ethnic groups that are represented in the United States-e.g., English, Germans, Italians, Russians, Navahos, Puerto Ricans, Chinese, Japanese, Vietnamese, and West Africans. Some populations will be easy to sample through arrangements with blood banks in the appropriate country; other populations might be studied by sampling recent immigrants to the United States. The choice and sampling of the 15-20 populations should be supervised by the National Committee on Forensic DNA Typing (NCFDT) described in Chapter 2.

We emphasize, however, that it is not necessary to be comprehensive. The goal is not to ensure that the ethnic background of every particular defendant is represented, but rather to define the likely range of allele frequency variation.

• Because only a limited number of populations can be sampled, it is necessary to make some allowance for unexamined populations. As usual, the problem is rare alleles. Genetic drift has the greatest proportional effect on rare alleles and may cause substantial variation in their frequency. Even if one sees allele frequencies of 1% in several ethnic populations, it is not safe to conclude that the frequency might not be five-fold higher in some subgroups.

To overcome this problem, we recommend that ceiling frequencies be 5% or higher. We selected this threshold because we concluded that allele frequency estimates that were substantially lower would not provide sufficiently reliable predictors for other, unsampled subgroups. Our reasoning was based on population genetic theory and computational results, and we aimed at accounting for the effects of sampling error and for genetic drift. The latter consideration was especially important, because it scales inversely with effective population size (i.e., small populations have larger drift) and because it accumulates over generations. The use of such a ceiling frequency would correspond to a lower bound of 5% on allele frequencies. Even if one observed allele frequencies of about 1 %, one would guard against the possibility that the frequency in a subpopulation had drifted higher by using the lower bound of 5%. Thus, the lowest frequency attributable to any single locus would be 1/400 (1/20 x 1/20). In any case, it seems reasonable not to attach much greater weight to any single locus.

• The ceiling principle yields the same frequency for a genotype, regardless of the suspect's ethnic background, because the reported frequency represents a maximum for any possible ethnic heritage. Accordingly, the ethnic background of an individual suspect should be ignored in estimating the likelihood of a random match. The calculation is fair to suspects, because the estimated probabilities are likely to be conservative in their incriminating power.

Some legal commentators have pointed out that frequencies should properly be based on the population of possible perpetrators, rather than on the population to which a particular suspect belongs.[17,18] Although that argument is formally correct, practicalities often preclude use of that approach. Furthermore, the ceiling principle eliminates the need for investigating the perpetrator population, because it yields an upper bound to the frequency that would be obtained by that approach.

Some have proposed a Bayesian approach,[19-21] to the presentation of DNA evidence. However, this approach, focusing on likelihood ratios, does not avoid the kinds of population genetic problems discussed in this chapter. The committee has not tried to assess the relative merits of Bayesian and frequentist approaches, because, outside the field of paternity testing, no forensic laboratory in this country has, to our knowledge, used Bayesian methods to interpret the implications of DNA matches in criminal cases.

• Although the ceiling principle is a conservative approach, we feel that it is appropriate, because DNA typing is unique in that the forensic analyst has an essentially unlimited ability to adduce additional evidence. Whatever power is sacrificed by requiring conservative estimates can be regained by examining additional loci. (Although there could be cases in which the DNA sample is insufficient for typing additional loci with RFLPs, this limitation is likely to disappear with the eventual use of PCR.) A conservative approach imposes no fundamental limitation on the power of the technique.

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laurie.snell@chance.dartmouth.edu