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The classification of the 1600 samples described in Sect. 2.2 with the same choices of [MATH] and [MATH] (right column of Fig. is also instructive. In all three cases, all samples achieve high |
[MATH] statistics despite the fact that they are in neither [MATH] nor [MATH] , frequently with [MATH] , i.e., a nominal [MATH] -value less than [MATH] (No simulated sample genotype was identical to any true positive genotype at greater than 62% of loci, comparable to the degree of genetic identity observed in the real... |
[MATH] that is drawn on [MATH] or [MATH] , but who are not necessarily in [MATH] or [MATH] . This is unsurprising, since Eqs. quantify the degree to which [MATH] is not equidistant from [MATH] and [MATH] . Furthermore, this suggests that relatives of true positives may be misclassified (we consider this below in Sect. ... |
Classification of null samples when [MATH] and [MATH] are well-chosen. Having observed the sensitivity of the classifier to the appropriate choice of [MATH] and [MATH] , we now explore the classification of samples which are in neither [MATH] nor [MATH] |
in the case where [MATH] and [MATH] are well-chosen. Here, we randomly select 100 cases and 100 controls from CGEMS to form an out-of-pool test sample set comprising 200 individuals, and recompute the MAFs for the remaining 1045 CGEMS cases ( [MATH] ) and 1042 CGEMS controls ( [MATH] ). (Several such random subsets wer... |
For the positives samples (those in [MATH] or [MATH] ), the classifier performs fairly well, correctly classifying 2083 samples (and calling 4 as in neither [MATH] nor [MATH] ). However, of the 200 test samples which were in neither [MATH] nor [MATH] , only 62 have [MATH] , and the bulk are misclassified into the reduc... |
at the nominal [MATH] (see Table ). A plot of the [MATH] values for all samples is given in Fig. (A). A similar test, in which HapMap individuals unrelated to the CGEMS participants (90 each from CEPH and YRI groups) were classified against the same subsets of 1045 CGEMS cases [MATH] ) and 1042 CGEMS controls ( [MATH] ... |
results in a 29.5% false-positive rate amongst the 200 out-of-pool CGEMS samples, 72% false-positive rate amongst HapMap CEPHs, and 100% false-positive rate amongst HapMap YRIs. A summary of the specificity and sensitivities obtained in this test is given in Table |
The reason for the high false-positive rates in practice despite the stringent nominal false positive rate is clear from the plots Fig. (A,B): namely, it can be seen that the putative null distribution (light grey line, [MATH] , cf Eq. ) does not correspond to the observed distribution for samples for which the null hy... |
The overall shift to the right is a product of the small differences in [MATH] which accumulate as given by Eq. 17 . Because in this test we happen to know the MAFs [MATH] along with [MATH] and [MATH] for each of the CGEMS samples, we can compute [MATH] given by Eq. 17 |
as [MATH] and verify that, when divided by the average [MATH] amongst the samples, the center of the observed null distribution will be at [MATH] . Indeed, visual inspection of Fig. (A) shows that shifting each [MATH] distribution by -2 would result in [MATH] [MATH] , and null-sample distributions which lie more symmet... |
[MATH] which are not drawn on the same population as [MATH] and [MATH] may in practice have a high false positive rate. The effect of LD, derived in Sect. 3.1.2 , is also seen in these examples. In Fig. (B), we observe a narrower distribution of [MATH] for the HapMap YRI samples versus the Caucasian CGEMS participants ... |
(for which each SNP was independently sampled and hence have artificially low LD) to those of real populations. 3.2.2 Correcting for deviations from [MATH] |
Although the empirical false-positive rates obtained the the tests described above are exceedingly high, the distributions of [MATH] |
obtained in Fig. (A,B) are nonoverlapping. Hence, one might expect that if one could appropriately calibrate the thresholds of [MATH] at which classification is made, the sensitivity and specificity of the test could be considerably improved. (Note that, in practice, one does not know where the true-positive [MATH] and |
[MATH] distributions of [MATH] lie; this requires the genotypes of the [MATH] and [MATH] individuals.) Two approaches may be taken toward calibrating classification thresholds for [MATH] : an analytical approach, based on the results in Sect. 3.1 above; or an empirical approach, based on constructing a null distributio... |
Analytical approach. In order to correct for the deviations from [MATH] analytically, we need to know both the location and width of the distribution of [MATH] |
in the non-ideal circumstances under which the test is being conducted. That is, we need to know deviations from [MATH] resulting from MAF differences [MATH] and sample size differences of [MATH] and [MATH] |
(cf. Sect. 3.1.1 and Appendix), as well as the average correlation amongst SNPs [MATH] (cf. Sect. 3.1.2 , Eq. 18 ). Let us first consider the result in Eq. 17 , which shows that |
[MATH] in practice will be a function of the MAF differences [MATH] as well as the MAFs [MATH] of the population [MATH] of which [MATH] |
is a sample. If we are well-assured that [MATH] and [MATH] are large samples of the same population [MATH] and that [MATH] is also a sample of that population, an average of [MATH] and [MATH] may be used to estimate [MATH] (the [MATH] , while necessarily drawn on [MATH] , are too small a sample to be a good estimate) a... |
and Table are given in Table , in which [MATH] was estimated as [MATH] and [MATH] was computed according to Eq. 17 . A slight improvement in the performance of the method can be seen by comparing the first two columns of Table to those of Table |
However, the assumption used to compute [MATH] (i.e., that [MATH] [MATH] , and [MATH] are all i.i.d. samples of the same population [MATH] ) is one on which the accuracy of the correction is strongly dependent; consider, for instance, that the [MATH] |
obtained for the simulations in Fig. (A,B) and discussed above will produce the appropriate shift [MATH] for the 200 CGEMS samples in Fig. (A) using this method, but will not centralize the HapMap [MATH] distributions in Fig. (B) appropriately, because the [MATH] and [MATH] |
are not good estimates of the MAFs of the populations from which the HapMap samples are drawn. Applying this correction to the HapMap samples (equivalent to moving the HapMap [MATH] distribution two units to the left in Fig. (B)) results in a misclassification rate of 86% (nominal [MATH] ) and 44% (nominal [MATH] for t... |
is to be used, sound estimates of [MATH] need to be obtained. When [MATH] is not a sample of the same population as [MATH] or [MATH] , estimates of [MATH] |
are unobtainable from [MATH] [MATH] and [MATH] alone, and hence this correction relies upon the assumption that [MATH] [MATH] , and [MATH] are well-matched. |
The second influence on [MATH] , described in both Sect. 3.1.1 and the Appendix, is the effect of the sample sizes [MATH] and [MATH] . Here, corrections are readily made, provided the sample sizes of [MATH] and [MATH] |
are known. In a forensics context, where [MATH] is a sample of unknown composition, [MATH] may not be known; on the other hand, in other contexts (such as when using case and control MAFs from a GWAS), sample sizes are known and readily adjusted for. (In this test, [MATH] and the correction is negligible.) |
We also saw in Sect. 3.1.2 and Fig. (B) that the distribution of [MATH] for null samples will depend on the degree of correlation between the SNPs. To accurately derive the width of the [MATH] distribution for null samples, one would need to either select SNPs that yield vanishingly small [MATH] or know the value of [M... |
with high accuracy for the population of which [MATH] is a sample so that it can be discounted. The latter option requires knowledge beyond the MAFs of [MATH] and [MATH] and the genotype of individual [MATH] ; namely, it requires multiple genotypes from the population [MATH] from which [MATH] |
was drawn such that the average correlation [MATH] between SNPs can be computed; even with a collection of null genotypes, the computation of the average pairwise correlation for [MATH] SNPs is a computationally unfeasible task. Rather, selecting fewer SNPs in order to reduce LD is a more workable solution; the results... |
SNPs used in Fig. (A,B). 50,000 SNPs was shown in to be a reasonable lower bound to detect at nominal [MATH] one individual amongst 1000, which is the concentration of true positive individuals in this test. |
As is clear from Fig. , reducing the number of SNPs narrows the distributions considerably, yet at the same time brings them closer together such that the crisp separation previously obtained is reduced. Using this method, we see that the 200 CGEMS samples now have a distribution closer to that of the putative null [MA... |
[MATH] distributions for the [MATH] and HapMap samples. Finally, we can consider applying both the SNP reduction and the [MATH] correction applied above; the results here are given in the final two columns of Table . Because [MATH] and [MATH] are well-matched and the [MATH] correction given by Eq. 17 is slight in the c... |
Empirical approach. Another potential approach to obtaining a correct null distribution is purely empirical, namely, collecting a set of presumed-null genotypes (called [MATH] which can be assumed to be drawn from the same population as [MATH] , and determining the distribution of [MATH] for the null samples [MATH] How... |
To see this, let us once more return to Fig. In these figures, vertical bars represent the 0.05 and 0.95 quantiles of the 200 CGEMS (black), 90 HapMap CEPH (cyan) and 90 HapMap YRI (blue) |
[MATH] distributions. Let us first consider a situation in which we have [MATH] and [MATH] , along with an individual [MATH] who is one of the 200 CGEMS samples not in [MATH] |
or [MATH] , but no other genotypes. We might reasonably turn to publicly available HapMap genotypes as our group [MATH] from which we construct an empirical null distribution from which we set thresholds. The lines in Fig. (A,C) depict this case. Using thresholds obtained from the HapMap CEPH distribution (cyan lines) ... |
The converse is true as well: if we have [MATH] [MATH] , and [MATH] which are well matched—such as illustrated in Fig. (B,D), in which [MATH] [MATH] , and [MATH] all come from CGEMS data—yet [MATH] is not drawn from the same underlying population as [MATH] , the method will incorrectly classify [MATH] ; roughly a quart... |
is being used to construct a null distribution empirically—it, too, must be an i.i.d. sample of [MATH] Another empirical option is that of simulating genotypes from the [MATH] |
and [MATH] to simulate [MATH] under the alternative hypothesis, with the assumption that the null and alternative hypothesis [MATH] distributions do not strongly overlap. However, this method also requires that [MATH] and |
[MATH] are large and well-matched samples, since (as can be seen in the top- and middle-right graphs in Fig. ) poorly-matched [MATH] and |
[MATH] will not produce crisply separated distributions. Furthermore, the thresholds derived by this approach will relate not to the false-positive rate but rather to the false-negative rate, i.e., these thresholds would control the power of the test, and the specificity in practice will remain unknown. |
We have thus seen that small deviations from the assumptions that [MATH] [MATH] , and [MATH] are i.i.d. samples of the same population [MATH] |
can produce false-positive rates which greatly exceed those predicted by the null hypothesis. Even when these sources of error were adjusted for, in our tests we still observed a false positive rate that was higher than expected, such that the false positive rate was never less than 20% in practice for a nominal false-... |
and [MATH] is not obviously doable. More importantly, it is not clear that, once thresholds are chosen, the empirical specificity could be assessed without additional genotype information from subjects who are well-matched to [MATH] [MATH] and [MATH] |
3.2.3 Positive predictive value of the method. The effect of the modest specificity—even in the best of cases described above—on the posterior probability that the individual [MATH] is in [MATH] |
or [MATH] is considerable, given that the prior probability is likely to be relatively small in most applications of this method. Let us consider the positive predictive value (PPV), which quantifies the post-test probability that an individual [MATH] with a positive result (i.e., significant [MATH] ) is in [MATH] or [... |
[EQUATION] where the PPV is the posterior probability that [MATH] is in [MATH] given a prior probability of [MATH] . We can write this equivalently in terms of the positive likelihood ratio [MATH] |
[EQUATION] A plot of PPV vs. prevalence is given in Fig. . Even with the best sensitivity (99.23%) and specificity (87%) obtained in our tests—that in which [MATH] [MATH] , and [MATH] were drawn on the same underlying population [MATH] [MATH] was accurately computed, and a nominal |
[MATH] was used as a threshold (cf. Table )—the prior probability (prevalence) of [MATH] being in [MATH] needs to exceed 54% in order to achieve a 90% post-test probability that the subject is in |
[MATH] . For a PPV of 99%, the prior probability needs to exceed 72% for any specificity under 95%, assuming the observed sensitivity of 99%. We thus see the strong need for prior belief that [MATH] is in [MATH] or [MATH] |
The difficulty in assessing the (empirical) specificity of the test in absence of additional data makes the posterior probability difficult to ascertain since the false positive rate in practice is much greater than that given by the nominal false-positive rate [MATH] Eq. 21 underscores this fact; referring once more t... |
at 87% specificity and 99% sensitivity is 7.6, versus 990000 if the nominal false-positive rate [MATH] were correct. For prior probability of 1/1000, the first case yields a posterior probability of 1.1/1000, while the second yields a posterior probability of 998/1000. These differences, which are difficult to measure ... |
3.2.4 Classification of relatives We now turn to the classification of individuals who are relatives of true positives. As discussed above in Sect. 3.2.1 the results from simulations S.1–S.5 in Fig. suggest that individuals who are genetically similar, but not identical to, the subjects in pools [MATH] and [MATH] , fre... |
[MATH] = Mothers from pedigrees 1–15 and fathers from pedigrees 1–15 [MATH] = Children from pedigrees 1–15 and fathers from pedigrees 16–30 |
and then compute [MATH] for mothers and children from pedigrees 16–30 using the same SNP criteria as before. The results of these tests for both the CEPH and YRI pedigrees, given in Fig. , are as expected, with the children having a significantly higher distribution of [MATH] |
than the mothers; the [MATH] values for all the children were so large that [MATH] -values [MATH] were obtained when comparing to [MATH] By contrast, 5/15 of the YRI mothers from pedigrees 16–30 and 10/15 of the CEPH mothers from pedigrees 16–30 yielded [MATH] (with distributions roughly centered about [MATH] ). The wi... |
[MATH] for the CGEMS-based tests in Fig. ). Note, however, that without knowing the distribution of [MATH] for true positives (which necessitates knowing the genotypes of true positives) setting a threshold to distinguish between true positives and their relatives is not possible by any of the methods described above. |
In order to explore the effect of genetic similarity in a controlled, ideal situation for which [MATH] and [MATH] are known to be samples of the same underlying population and for which all SNPs are known to be independent (i.e., in the ideal situation in which the putative null distribution [MATH] should hold), we car... |
as given by the CGEMS controls; [MATH] [MATH] , and [MATH] were derived as described in Sect. 2.3 as binomial samples of [MATH] In the first of these simulations, the test samples were constrained to have a proportion [MATH] of SNPs identical to a true positive individual, with the remaining SNPs drawn on [MATH] . A pl... |
identity with a true positive individual, they universally achieve significant [MATH] , and significant values of [MATH] are found over half the time for simulated samples exceeding [MATH] identity. (It should be noted that of the real samples, no two had [MATH] fractional identity.) |
In the second set of these simulations, the test samples were drawn from a weighted mixture of MAFs: [EQUATION] i.e., the sample was drawn from MAFs [MATH] which are [MATH] percent like [MATH] and |
[MATH] like CGEMS controls (MAFs [MATH] ). By simulating 200 samples for various [MATH] computing [MATH] for each sample using the simulated [MATH] and [MATH] , and counting the number of samples that achieve significant [MATH] at [MATH] , we can see how the false positive rate varies with the percentage of [MATH] . Re... |
The misclassification of relatives follows directly from the method’s premise. Eqs. together answer whether individual’s genotype [MATH] is closer to sample [MATH] ’s MAFs [MATH] than to sample [MATH] ’s MAFs [MATH] |
than would be expected by chance , and it is unsurprising that a relative of a true member of [MATH] would appear closer to [MATH] (via Eqs. ) than to [MATH] |
Put another way, [MATH] being a member of [MATH] is sufficient but not necessary for [MATH] to be closer (via Eq. to [MATH] than to [MATH] ; it is possible for other sources of genetic variation to cause [MATH] to be closer [MATH] than to [MATH] . We can observe this by turning once again to Fig. (A,C), where the dashe... |
CGEMS cases had a distribution of [MATH] closer to the other CGEMS cases [MATH] and the not-in- [MATH] CGEMS controls had a distribution of [MATH] closer to the other CGEMS controls [MATH] , indicating that small class-specific genetic differences can yield altered values of [MATH] . The erroneous inferential leap that... |
or [MATH] is responsible for the misclassification of relatives as well as for misclassification of non-relatives in the previous examples. |
Discussion and Conclusions In this work, we have further characterized and tested the genetic distance metric initially proposed in |
. This metric, summarized here by Eqs. quantifies the distance of an individual genotype [MATH] with respect to two samples [MATH] and [MATH] using the marginal minor allele frequencies |
[MATH] and [MATH] of the two samples and the genotype [MATH] . The article proposes to use this metric to infer the presence of the individual in one of the two samples, and the authors demonstrate the utility of their classifier on known positive samples (i.e., samples which are in either [MATH] or [MATH] ) showing th... |
[MATH] . As a result, Eqs. are severely limited in their utility for discerning [MATH] ’s presence in samples [MATH] or [MATH] In this work we have shown that high [MATH] values, significant when compared against |
[MATH] , may be obtained for samples that are in neither of the pools tested under several circumstances: when pools [MATH] and [MATH] are sufficiently dissimilar such that the differences in [MATH] and [MATH] dominate, as seen in Sects. 3.1.1 and 3.2.1 as well as the Appendix; |
when [MATH] is a sample of a different population than are [MATH] and [MATH] as seen in Sect. 3.2.1 when a small amount of average LD is present such that the putative null distribution in Eq. does not hold (due to a violation of the CLT assumption of independence), as seen in Sects. 3.1.2 and 3.2.1 |
and when a sample is genetically similar, but not identical to, individuals comprising [MATH] or [MATH] (e.g., relatives of true positives), as seen in Sect. 3.2.4 |
The high false positive rates in the first two cases result from assumptions underlying the putative null distribution which are not met in practice, specifically, that the individual [MATH] along with samples |
[MATH] and [MATH] are all i.i.d. samples of the same underlying population [MATH] and that the amount of correlation between all [MATH] SNPs is vanishingly small. As we saw in Sect. 3.2.1 and 3.2.2 , these assumptions are difficult to meet; for instance, HapMap CEPH and CGEMS samples are sufficiently dissimilar that th... |
Additionally, the conclusion that high [MATH] values result from [MATH] ’s presence in [MATH] relies upon the questionable assumption that individuals in neither [MATH] nor [MATH] will be equidistant from both, resulting in false positives even when the other assumptions are met. For instance, similarly genotyped indiv... |
produce misleading classifications at a rate that is considerably greater than expected (21% vs. nominal 5% and 13% vs. nominal 0.0001% in the best cases reported in Table ). The unpredictable false positive rate in practice, resulting from the difficulty in accurately calibrating the significance of [MATH] , results i... |
[MATH] [MATH] , Eqs. correctly identify the sample of which the individual is part (Sect. 3.2.4 ). These findings have implications both in forensics (for which the method |
was proposed) and GWAS privacy (which has become a topic of considerable interest in light of ). We briefly consider each: Forensics implications. |
The stated purpose of the method—namely, to positively identify the presence of a particular individual in a mixed pool of genetic data of unknown size and composition—is difficult to achieve. In this scenario, we have [MATH] (from forensic evidence) and a suspect genotype [MATH] . To apply the method, we would need 1)... |
[MATH] which is also a sample of the underlying population [MATH] well-matched in size and composition to [MATH] 3) to obtain an estimate of the sample size of [MATH] such that sample-size effects can be appropriately discounted; and 4) to assume that the |
[MATH] -values at the selected classification thresholds are accurate. We have seen in the Results section the sensitivity to the assumption that [MATH] [MATH] and [MATH] all come from the same population, the sensitivity to the sample size of [MATH] , and the difficulties in calibrating thresholds; the high false-posi... |
GWAS privacy implications. Here the scenario of concern is that of a malefactor with the genotype of one (or many) individuals, and access to the case and control MAFs from published studies; could the malefactor use this method to discern whether one of the genotypes in his possession belongs to a GWAS subject? In thi... |
On the other hand, if the malefactor does have prior knowledge that the individual [MATH] participated in a certain GWAS but does not know |
[MATH] ’s case status, Eqs. permit the malefactor to discover with high accuracy which group [MATH] was in. Additionally, in the case of a priori knowledge, the participant’s genotype is not strictly necessary, since a relative’s DNA will yield a large [MATH] |
score that falls on the appropriate [MATH] side of null. Despite these limitations, we have found that the distance metric (Eqs. ) may still have forensic and research utility. It is clear from both our studies and the original paper |
that the sensitivity is quite high; in the (rare) case that a sample has an insignificant [MATH] , it is very likely that [MATH] is in neither [MATH] nor [MATH] . We can also see that genetically distinct groups have [MATH] distributions with little overlap (Fig. ), and so it may be worth investigating the utility of E... |
On this note, let us once more consider the quantity which Eq. measures, namely the distance of [MATH] from [MATH] relative to the distance of [MATH] from [MATH] . Referring to Fig. (right column) and Fig. (A,C), we can see that samples [MATH] which are more like those in sample [MATH] have a distribution that lies to ... |
, while failing to function as a tool to positively identify the presence of a specific individual’s DNA in a finite genetic sample, may if refined be a useful tool in the analysis of GWAS data. |
Appendix: Dependence of [MATH] on the sample size of [MATH] and [MATH] Consider [MATH] (cf. Eq. ) under the null hypothesis assumptions that [MATH] [MATH] , and [MATH] are all drawn i.i.d. from the same underlying population [MATH] with MAFs [MATH] . Writing the probability distribution of [MATH] as [MATH] [MATH] is gi... |
[EQUATION] where we exploit the fact that [MATH] [MATH] and [MATH] are independent of each other but depend on the underlying population MAFs. |
The dependence of the first (second) term in Eq. A-2 on [MATH] [MATH] ) is derived as follows. First, we note that since each [MATH] is two Bernoulli trials (two alleles) with probability [MATH] , we have the following values of [MATH] with probability [MATH] |
for each allowable value of [MATH] [EQUATION] Moreover, since each [MATH] follows a binomial distribution of size [MATH] (two alleles per person), we invoke the normal approximation to the binomial for values of [MATH] with mean [MATH] and variance [MATH] . Hence: |
[EQUATION] where we introduce [EQUATION] to simplify the notation. In consequence, the first term of Eq. A-2 can be written: [EQUATION] |
and the second term may be written analogously for [MATH] . The absolute value in Eq. A-7 is dealt with by considering the [MATH] and [MATH] cases separately, i.e., treating Eq. A-7 as the sum of integrals |
[EQUATION] Expanding the polynomials in Eq. A-8 and once more using Eq. A-6 to simplify notation, we rewrite the above as [EQUATION] |
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