Published in the 2003 ISFG proceedings


  1. Introduction
  2. Materials and methods
  3. Results
    1. Covert mutations
    2. Two inconsistencies
  4. Discussion
    1. Covert mutations
    2. Telling true from false
    3. What to do?


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Multiple mutations, covert mutations, and false exclusions in paternity casework

CH Brenner, Consulting in forensic mathematics, Oakland, California


The universal practice, up to now, is to make the judgement "paternity excluded" whenever there are more than some established number - such as two - of loci in which the genetic pattern, barring mutation, is inconsistent with paternity. Such a rule is founded on the implicit assumption that the probability of two mutations is vanishingly small. However the ideal procedure would of course be to evaluate the
paternity index over all loci, taking possible mutation into account. With STR's, unlike with RFLP's, a reasonably accurate mathematical model of mutation exists and hence the ideal procedure is finally possible. What happens when it is applied is somewhat surprising. Notwithstanding two or even three inconsistent loci, the posterior probability of paternity (assuming 50% prior probability) can easily be 20%. Unless the inconsistencies are particularly implausible as mutations (i.e. multiple repeat units) the posterior probability is not vanishingly small. The old rule causes bad decisions; it excludes fathers. Instead, we should compute the proper paternity index across all loci, considering the possibility of multiple mutations, and evaluate the result. The computing part is easy. The evaluation part brings a new difficulty, for it forces us to confront a question that the inaccurate policy of the past hid from view: How unlikely must paternity be in order to justify the decision "paternity excluded"?

An incidental discovery is the heretofore overlooked implication that the existence of "covert mutations" imply that most STR mutation estimates from paternity studies are wrong.

Keywords: mutation, covert mutation, stepwise, paternity index, exclusion

  1. Introduction
  2. Dealing with possible mutations in paternity casework has always been awkward. In recent years the use of STR systems have nearly supplanted RFLP's. It is time to reconsider the outmoded policies as well.
    Table 1. Paternity inference based on counting inconsistencies
    inconsistencies rate among false trios rate among true trios LR supporting paternity
    0 1/210 000 0.97 200 000
    1 1/10 000 1/35 290
    2 1/1100 1/2600 1/2.4
    3 1/190 1/320 000 1/1700
    4 1/48 1/61 000 000 1/1 300 000

    The mutation inconsistency rate for true trios at STR loci appears to average about 1/400 [AABB, Brinkmann, Kayser, unpublished data]. Assuming a 13 marker paternity test and binomial model, the expectations are shown in Table 1 [corrected August 2010 — thanks to Giuseppe Cardillo, Naples.]

    Clearly, two inconsistencies is the critical case. Prima facie it supports non-paternity by a likelihood ratio of 2.4, which is inconclusive. This paper examines more closely the case of two inconsistencies.

  3. Materials and methods
  4. Four hundred paternity trios, half true and half false trios but all with two inconsistent loci were generated by an accelerated Monte Carlo method. The simulations were allowed to continue generating Monte Carlo genotypes of trios until the desired number of trios of each type with exactly two inconsistences had been produced.

    Mutations are generated according to a modified stepwise mutation model, which assumes that most mutations are by plus or minus one repeat unit and are paternal.

    The model parameters [5, 6] are:

    Next, a likelihood ratio was computed for each of the 400 cases using the above model.

    Covert mutations

    common caseanother possibility

  5. Results
    1. Covert mutations
    2. Analyzing the results of the true-trio simulations revealed an obvious – in retrospect – phenomenon, which we might call "covert mutations" whose significance, so far as I know, has not been previously noted.

      The figures illustrates ways that a mutation may go unnoticed.

      The incidence of covert mutations might be quite large. It will vary by locus, and by preliminary calculations the rate is 18%±9%.
      Table 2. Distribution of PI's among simulated 2-inconsistency cases
      x % false trios with PI>x % true trios with PI>x
      1000 0% 4.5%
      100 0 19
      10 0.5 43
      1 3 78
      1/10 20 95.5
      1/100 47 99.5

    3. Two inconsistencies
    4. As table 1 shows, if one merely counts inconsistencies without regard to the particulars, a finding of two inconsistencies is modest evidence favoring non-paternity. Table 2 shows that taking into account the rarity of shared alleles and the plausibility as mutations of inconsistencies – i.e. computing the paternity index (PI) – somewhat distinguishes true from false trios.

  6. Discussion
    1. Covert mutations
    2. The significance of covert mutations is that since all published estimates of mutation rates are derived from paternity studies, all of the ones for autosomal loci are too low by a possibly significant amount. The rate of apparent mutations is the right number to use to calculate table 1, but for case calculations – table 2 – the covert-adjusted must be used. For example, in CSF1PO apparent μ=3/1000 but the true μ=4/1000. Failure to account for covert mutations thus inflates paternity indices, so is anti-conservative. There may also be an implication in evolutionary studies when a mutational clock is considered.

    3. Telling true from false
    4. Considering that true trios predominate over false ones in paternity laboratories, cases with two inconsistencies are false trios by a margin of only 2:1. "Paternity excluded" based on two inconsistencies is a very poor policy. Computing a likelihood ratio is the proper course. Interpreting it, though, can be a problematic when it is small.

      The 19% of true trios with PI>100 notwithstanding the two mutations, can possibly be reported (with so-called "paternity probability" > 99%) and the burden of interpretation left to the judge. A majority of the true trios will have 1/10 < PI < 100 and are obviously inconclusive. Possibly further testing will help; nothing else will.

    5. What to do?
    6. Once the untenable policy of the past – pretending in effect that PI=0 whenever some target number of inconsistencies are observed – is abandoned, one is confronted with making a policy based on interpreting the true PI. For example, if PI=1/10000 reporting "paternity excluded" may be acceptable (notwithstanding the paradox that in the symmetrically opposite case that PI=10000, no one would report "paternity certain"). But what of the less extreme cases with a smaller PI, such as 1/10 or 1/100? Science and justice collide; there is no obvious and acceptable answer. Assuming, as is customary, a 50% prior probability, the posterior probability of paternity W is 10% or 1%. It is well established social policy that the laboratory doesn't claim "definitely paternity" when W=99%, so how can it be right for the laboratory to claim "excluded from paternity" when W=1% - or 0.1% for that matter? The mathematically honest solution of reporting the small PI – whatever it is after testing resources are exhausted – to a judge who, up to now, is not prepared for such evidence, is also not very satisfactory.

      Fortunately, the situation is infrequent (see table 1). And part of the solution is clear: compute the paternity index correctly, including inconsistent loci. It that is done, it will be seen that occasionally even in some cases with three inconstencies, paternity cannot be excluded. And conversely, there are probably cases that have only one inconsistency but it is of such a nature that it alone effectively rules out paternity.


  1. AABB. Apparent Mutations Observed at the 13 CODIS STR Loci in the Course of Paternity Testing. 1999
  2. Brinkmann B, Klintschar M, Neuhuber F, Huhne J, Rolf B. Mutation rate in human microsatellites: influence of the structure and length of the tandem repeat. Am J Hum Genet. 1999 May;64(5):1473-4
  3. Kayser M, Sajantila A. Mutations at Y-STR loci: implications for paternity testing and forensic analysis. For Sci International 118(2001) 116-121
  4. unpublished data of 400 mutations
  5. May 2005 re so far as I know – I may be wrong. A while ago I saw a paper – published maybe in 1992, authors including Chakraborty – that seemed to consider the covert mutation concept with respect to VNTR loci.
    July 2005. Thanks to Jiri Drabek for the reference below. It makes theoretical predictions of the proportion of covert mutations based on a random rather than stepwise mutation model, which probably explains why my estimate of the incidence of covert mutations is a bit smaller than theirs.

    Chakraborty R, Stivers D, Zhong Y. Estimation of mutation rates from parentage exclusion data: applications to STR and VNTR loci. Mutation Research 354 (1996) 41-48

  6. Brenner CH.
  7. Brenner CH.

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