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  • Matching probability isn't frequency
    type of population frequency q r r The data For purposes of analysis it s most general and best to think of the database as part of the evidence Hence the evidence E consists of Allele Q is found at the crime scene The reference database has one allele which let us suppose isn t Q The suspect is Q We re going to need Pr E H for several different hypotheses H the suspect is or is not the donor H 1 or H 0 the allele frequency is or frac12 Evidence E Pr Q likelihood L Pr E H likelihood ratio L 1 L 0 crime scene database suspect frequency q prior Pr q Pr Q H 1 suspect is donor H 0 suspect is random L 1 L 1 q Σ rL 1 L 0 L 0 q Σ rL 0 Q R Q q frac14 r Pr q q 1 q 12 64 28 128 q 2 1 q 3 64 11 128 28 11 q r frac12 16 64 8 64 Note the idea of the computation The insight of treating the database as part of the evidence leads to defining E crime scene Q database R suspect Q We need to calculate likelihood expressions such as Pr E H 1 The trick to do so is to partition E according to the set of mutually exclusive and exhaustive events q q Each of them has prior probability r so L 1 Pr E H 1 Pr q E H 1 Pr q E H 1 Pr q Pr E H 1 q Pr q Pr E H 1 q q 1 q q q 1 q q q means evaluated at q 12 64 16 64 28 128 As shown in the table a similar computation leads to L 0 11 128 and therefore to a likelihood ratio LR L 1 L 0 28 11 2 78 supporting the suspect as the donor of the crime scene allele If the database were very large and practically definitively supported q then the LR would be very nearly 4 and the smaller value we have calculated for a database of size n 1 has indeed punished the database On the other hand if somewhat less likely the larger database makes clear that q then LR 2 smaller than we have calculated Hence the result LR 28 11 represents a compromise a mathematically correct and exact compromise based on the evidence between the two frequency possibilities of and frac12 To see if in general the 1 allele database is fair we need to make a similar comparison for the other database possibility then do some sort of averaging Consider then the case that E crime scene Q database Q suspect Q Evidence E Pr Q likelihood L Pr E H likelihood ratio L 1 L 0 crime scene database suspect frequency q prior Pr q Pr Q H 1 suspect is donor H 0 suspect is random

    Original URL path: http://dna-view.com/ProbIsNotFreq.htm (2016-02-12)
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  • Brenner's Law
    tabulated 801 of them each having from 100 to 1000 chromosomes observations Singletons allelic types with a count of one are by a large margin the most popular occurring in total 930 times among the 801 databases 63 had one or more once observed allelic types or singletons On average there were 1 16 singletons per database I suggest the word popularity for the number of times something has occurred A singleton means an allelic type of count or popularity p 1 in a database If we denote by α p the number of allelic types of popularity p found in the dataset then we can say that α 1 930 is the popularity of singletons and that singletons are very popular Obviously these 930 singletons represent 930 fragments of chromosomes Doubletons types of count p 2 had a total popularity of α 2 464 among the 801 databases Since each doubleton represents two observations in total they account for 2 α 2 928 chromosomes nearly the same as the singletons And the 303 tripletons represent a similar number 3 α 3 909 of total observations Brenner s Law All of which suggests Brenner s Law the rule that The number of p α p of alleles represented by database popularity p is constant over p How well does it hold up Look at the dotted line in the image at right It s not highly accurate let s call it a rule of thumb It s moderately supported by the data shown but it is also suggested by more than the data here presented I did an earlier study based on RFLP markers they conform more closely Most importantly there is a theoretical underpinning In fact I first investigated this distribution to compare STR markers with Ewens s sampling distribution for the ideal situation of infinite alleles Brenner s Law follows from Ewens formula in the limit as the mutation rate goes to zero Of course STRs violate all of the assumptions of the infinite alleles model with 0 mutation Infinite alleles means that mutation is always to a new type STR mutations are nearly always to an already existing type STR mutation rates are about 1 350 per meiosis not zero Real populations grow and have immigration so we cannot expect accuracy But the main reason of the above that the data doesn t conform to the Law is 1 The fact of convergent mutation for STRs is an influence towards common types Point 2 compensates somewhat A very high mutation rate such as exists for Y haplotypes discourages common types The general point is that nature strongly favors rare alleles Consequences of Brenner s Law Brenner s Law is an observation about the comparative prevalence of rare and of common forensic STR allelic variants Allele frequency spectrum It is an example of a frequency spectrum the distribution of frequencies that we can expect nature to deal to us It says that for any given locus population and

    Original URL path: http://dna-view.com/BrennerLaw.htm (2016-02-12)
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  • Coherent Analysis
    There is also the finding on the suspect having the crime scene type so matching probability r 1 a n 1 a b For small a and b this is approximately equivalent to adding the suspect to the database and using approximately relative frequency estimates For large r and n the effect of conditioning on the suspect becomes unimportant Applications and comments on the β distribution Given that β p a b p a 1 1 p b 1 If a b 1 we have β p 1 1 1 uniform distribution Ewens gives f x 1 x θ 1 x θ 1 that is β x a 0 b θ for the prior probability distribution of allele frequencies under the infinite alleles model In 1999 I presented the combination of this with Dawid Mortera s formula above as a tenative solution to the matching probability for a rare haplotype Assuming the haplotype occurs r 0 times in the database the matching probability for an innocent suspect is 1 n 1 θ There are various ways to estimate θ such as one less than the reciprocal of the empirical pairwise matching rate hence θ 9000 for US Caucasians However while the idea of adopting a β prior based on Ewens result is elegant I couldn t validate the result because of the ideal assumptions so chose not to recommend it in publication for actual court use Brenner s law is Ewens distribution with θ 1 hence a 0 and b 1 I use the symbol k for D M s r hence for STR matching probabilities we get k 1 n 2 close to but not identical with the intuitive k 1 n 1 recommendation Actually for STRs it makes some kind of sense to take θ 1 5 suggesting

    Original URL path: http://dna-view.com/CoherentAnalysis.htm (2016-02-12)
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