International Journal of Legal Medicine

, Volume 130, Issue 1, pp 39–57 | Cite as

Exclusion probabilities and likelihood ratios with applications to mixtures

  • Klaas-Jan Slooten
  • Thore Egeland
Original Article


The statistical evidence obtained from mixed DNA profiles can be summarised in several ways in forensic casework including the likelihood ratio (LR) and the Random Man Not Excluded (RMNE) probability. The literature has seen a discussion of the advantages and disadvantages of likelihood ratios and exclusion probabilities, and part of our aim is to bring some clarification to this debate. In a previous paper, we proved that there is a general mathematical relationship between these statistics: RMNE can be expressed as a certain average of the LR, implying that the expected value of the LR, when applied to an actual contributor to the mixture, is at least equal to the inverse of the RMNE. While the mentioned paper presented applications for kinship problems, the current paper demonstrates the relevance for mixture cases, and for this purpose, we prove some new general properties. We also demonstrate how to use the distribution of the likelihood ratio for donors of a mixture, to obtain estimates for exceedance probabilities of the LR for non-donors, of which the RMNE is a special case corresponding to L R>0. In order to derive these results, we need to view the likelihood ratio as a random variable. In this paper, we describe how such a randomization can be achieved. The RMNE is usually invoked only for mixtures without dropout. In mixtures, artefacts like dropout and drop-in are commonly encountered and we address this situation too, illustrating our results with a basic but widely implemented model, a so-called binary model. The precise definitions, modelling and interpretation of the required concepts of dropout and drop-in are not entirely obvious, and we attempt to clarify them here in a general likelihood framework for a binary model.


DNA mixtures Weight of evidence Exclusion probabilities 



The work of the second author leading to these results was financially supported from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement n 0 285487 (EUROFORGEN-NoE).


  1. 1.
    Balding D, Buckleton J (2009) Interpreting low template DNA profiles. Forensic Sci Int Genet 4(1):1–10PubMedCrossRefGoogle Scholar
  2. 2.
    Buckleton J, Curran J (2008) A discussion of the merits of random man not excluded and likelihood ratios. Forensic Sci Int Genet 2:343–348PubMedCrossRefGoogle Scholar
  3. 3.
    Buckleton J, Triggs C, Walsh S (eds.) (2005) Forensic DNA Evidence Interpretation. CRC Press, Florida, USAGoogle Scholar
  4. 4.
    Cowell R, Graversen T, Lauritzen S, Mortera J (2015) Analysis of forensic DNA mixtures with artefacts. J R Stat Soc Ser C Appl Stat 64(1):1–48CrossRefGoogle Scholar
  5. 5.
    Curran J, Gill P, Bill M (2005) Interpretation of repeat measurement DNA evidence allowing for multiple contributors and population substructure. Forensic Sci Int 148 (1):47–53PubMedCrossRefGoogle Scholar
  6. 6.
    Dørum G, Kling D, Baeza-Richer C, Magariṅos MG, Sæbø S, Desmyter S, Egeland T (2014) Models and implementation for relationship problems with dropout. Int J Leg Med 129(3):411–423CrossRefGoogle Scholar
  7. 7.
    Gill P, Gusmão L, Haned H, Mayr W, Morling N, Parson W, Prieto L, Prinz M, Schneider H, Schneider P, Weir B (2012) DNA commission of the International Society of Forensic Genetics: Recommendations on the evaluation of STR typing results that may include drop-out and/or drop-in using probabilistic methods. Forensic Sci Int Genet 6 (6):679–688PubMedPubMedCentralCrossRefGoogle Scholar
  8. 8.
    Gill P, Haned H (2013) A new methodological framework to interpret complex DNA profiles using likelihood ratios. Forensic Sci Int Genet 7:251–263PubMedCrossRefGoogle Scholar
  9. 9.
    Haned H, Slooten K, Gill P (2012) Exploratory data analysis for the interpretation of low template DNA mixtures. Forensic Sci Int Genet 6(6):762–774PubMedCrossRefGoogle Scholar
  10. 10.
    Kruijver M (2015) Efficient computations with the likelihood ratio distribution. Forensic Sci Int Genet 14:116–124PubMedCrossRefGoogle Scholar
  11. 11.
    Kruijver M, Meester R, Slooten K (2015) P-values should not be used for evaluating the strength of DNA evidence. Forensic Sci Int Genet 16:226–231PubMedCrossRefGoogle Scholar
  12. 12.
    Nothnagel M, Schmidtke J, Krawczak M (2010) Potentials and limits of pairwise kinship analysis using autosomal short tandem repeat loci. Int J Legal Med 124(3):205–215PubMedCrossRefGoogle Scholar
  13. 13.
    Slooten K, Meester R (2011) Forensic identification: the Island Problem and its generalizations. Statistica Neerlandica 65:202–237CrossRefGoogle Scholar
  14. 14.
    Slooten K, Egeland T (2014) Exclusion probabilities and likelihood ratios with applications to kinship problems. Int J Legal Med 128(3):415–425PubMedCrossRefGoogle Scholar
  15. 15.
    Slooten K, Meester R (2014) Probabilistic strategies for familial DNA searching. J R Stat Soc Ser C Appl Stat 63(3):361–384CrossRefGoogle Scholar
  16. 16.
    Steele C, Balding D (2014) Statistical evaluation of forensic DNA profile evidence. Annual Review of Statistics and Its Application 1:361–384CrossRefGoogle Scholar
  17. 17.
    Thompson E (2000) Statistical inference from genetic data on pedigrees. In: NSF-CBMS regional conference series in probability and statistics. JSTORGoogle Scholar
  18. 18.
    Westen A, Kraaijenbrink T, de Medina AR, Harteveld J, Willemse P, Zuniga S, van der Gaag K, Weiler N, Warnaar J, Kayser M, Sijen T, de Knijff P (2014) Comparing six commercial autosomal STR kits in a large Dutch population sample. Forensic Sci Int Genet 10:55–63PubMedCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Netherlands Forensic InstituteThe HagueThe Netherlands
  2. 2.Norwegian University of Life SciencesAasNorway
  3. 3.VU Department of MathematicsAmsterdamThe Netherlands

Personalised recommendations