---
title: "STR markers in forensic genetics"
author: "Mikkel Meyer Andersen, James Michael Curran and Torben Tvedebrink"
date: "`r Sys.Date()`"
output: 
  rmarkdown::html_vignette:
    fig_caption: yes
vignette: >
  %\VignetteIndexEntry{Forensic genetics}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
library(knitr)
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "", 
  fig.width = 12, 
  fig.asp = 0.62
)
options(knitr.kable.NA = '')
```

# Introduction

DNA evidence is the pre-eminent tool in the modern forensic scientists toolbox. 
It is widely accepted by the public, scientific and legal communities and it 
has been instrumental in determining both the innocence and guilt of individuals 
involved in the legal process. Despite this widespread acceptance there is unease 
regarding the statistical measures used to evaluate DNA evidence amongst some 
of members of all these communities. In particular, some people regard the random 
match probabilities associated with DNA evidence as just too small or basically 
unsupportable. In this vignette we discuss the basics of STR profiles, which
serves as a reference for the package's other vignettes:

* In the `db_vignette` ("Empirical testing of DNA match probabilities") we discuss what it means for a pair of DNA 
profiles to match or partially match, and we present how the `DNAtools` package 
allows a rational examination of the statistical properties of a DNA database.

* In the `noa_vignette` ("On the exact distribution of the numbers of alleles in DNA mixtures") we show how to calculate the distribution of the number 
of distinct alleles present in a DNA mixture constituted by an arbitrary number of
contributors.

## Matches and partial matches

Forensic genetics has its terminology which we briefly explain here. Human DNA 
consists of 23 pairs of chromosomes and those chromosomes are composed of a 
sequence of nucleotides which are labelled `A`, `G`, `C` and `T` after the bases 
adenine, guanine, cytosine and thymine that are used to form them. Modern DNA 
typing uses short tandem repeats (STRs). These are regions of DNA which are highly
variable, but are patterned in that they consist of repeats of a short sequence 
of DNA bases. The locations at which this information is collected are called loci, 
and the (length) variations in the patterns observed at each locus are called alleles. 
We have two alleles at each locus, because humans are a diploid species, meaning 
they have two copies of each chromosome. One allele comes from our mother, and the
other from our father. 


A pair of alleles at a locus is called a genotype, and 
therefore a DNA profile is actually a multi-locus genotype. Modern forensic 
laboratories genotype DNA evidence using commercial kits, called multiplexes 
which consist of 9--17 loci. The multiplex currently used in the United Kingdom 
(and until recently New Zealand and Denmark) is called [AmpFlSTR SGM Plus](https://en.wikipedia.org/wiki/Second_Generation_Multiplex_Plus), or 
SGM Plus for short, and consists of 10 loci, plus one gender specific locus, 
Amelogenin. Forensic laboratories in the United States which load profiles into 
the FBI's Combined DNA Index System (CODIS) collect a core set of thirteen loci, 
although they are not constrained to use one multiplex.

```{r, echo = FALSE}
STR_profile <- rbind(
  `**Locus:**` = c("vWA","D18","TH01","D2","D8","D3","FGA","D16","D21","D19"),
  `**Alleles:**` = c("15, 18", "14, 17",  "6, 9.3", "17, 23", "12, 15", "15, 15", "19, 23", "11, 12", "28, 28", "13, 14")
)
kable(STR_profile, caption = "A DNA profile from the SGM plus multiplex")
```

Table above shows a DNA profile from the SGM plus multiplex. There are two 
numbers at each locus representing the two alleles that make up the genotype at 
that locus. The numbers relate to the number of times the pattern or motif that 
describe the alleles at the locus are repeated. For example, this person's genotype 
at the locus TH01 is `6,9.3`. This means that on one chromosome, the motif for TH01,
`TCAT` was repeated 6 times, and on the other chromosome it was repeated 9 times, 
and then followed by `TCA`. The `.3` represents the fact that three of the four 
bases have been repeated.

# The `DNAtools` package

The aim of the `DNAtools` package is to provide statisticians and
forensic scientists with access to the specific procedures described in the
other vignettes. For example, for the database matching exercise (`db_vignette`),
early implementations by @weir2004 and then @curran2007 required custom written 
code for each new database and, in the case of @curran2007, generation of at least half a
dozen precursor files and a significant amount of memory. @tvedebrink2010; @tvedebrink2012
reduced the computational effort of @weir2004 and @curran2007 by deriving recursion formulas for 
the expectation and variance of the computed summary statistics.
`DNAtools` aims to make all of these procedures easier to use in **R**.

## Using the package `DNAtools` 

```{r, message=FALSE}
library(DNAtools)
browseVignettes(package = "DNAtools")
```

In the listed vignettes the main features of the package are described, 
which allows statisticians and forensic scientists to easily examine the properties
of a forensic DNA database. In particular, our package makes it simple
to carry out a database comparison exercise where every DNA profile in
the database is compared to every other database, and compare the
resulting numbers of observed pairs of matching and partially matching
profiles to expectation under a set of population genetic
assumptions. Similarly, evaluating the distribution of the number of distinct
alleles in high-order DNA mixtures is easily computed. 

# References

---
references: 
- id: brenner2007
  author: 
    - family: Brenner 
      given: C
  title: Arizona DNA Database Matches.
  URL: http://dna-view.com/ArizonaMatch.htm
  issued:
    year: 2007
- id: budowle_1999
  author: 
    - family: Budowle 
      given: B
    - family: Moretti 
      given: TR
  title: Genotype Profiles for Six Population Groups at the 13 CODIS Short Tandem Repeat Core Loci and Other PCR-Based Loci.
  container-title: Forensic Science Communications 1(2).
  URL: http://www2.fbi.gov/hq/lab/fsc/backissu/july1999/budowle.htm
  issued:
    year: 1999
- id: curran2010
  author:
  - family: Curran 
    given: JM 
  - family: Buckleton 
    given: JS 
  issued:
    year: 2010
  title: Re Sign mistake in allele sharing probability formulae of Curran, et al.
  container-title: "Forensic Science International: Genetics, 4(3), 215--217."
- id: curran2007
  author:
  - family: Curran 
    given: JM
  - family: Walsh 
    given: SJ
  - family: Buckleton 
    given: J 
  issued: 
    year: 2007
  title: Empirical testing of estimated DNA frequencies.
  container-title: "Forensic Science International: Genetics, 1(3-4), 267--272."
- id: Rsolnp
  author: 
  - family: Ghalanos
    given: A
  - family: Theussl
    given: S
  issued: 
    year: 2012
  title: Rsolnp General Non-linear Optimization Using Augmented Lagrange Multiplier Method. 
  container-title: R package version 1.16.
- id: kaye2009
  author:
  - family: Kaye
    given: DH
  issued: 
    year: 2009
  title: "Trawling DNA Databases for Partial Matches: What Is the FBI Afraid Of?"
  container-title: Cornell Journal of Law and Public Policy, 19(1)
- id: mueller2008
  author:
  - family: Mueller
    given: LD
  issued: 
    year: 2008
  title: Can simple population genetic models reconcile partial match frequencies observed in large forensic databases?
  container-title: Journal of Genetics, 87(2), 101--108.
- id: troyer2001
  author:
  - family: Troyer
    given: K
  - family: Gilboy T
    given: T
  - family: Koeneman
    given: B
  issued: 
    year: 2001
  title: A nine STR locus match between two apparently unrelated individuals using AmFlSTR Profiler Plus and Cofiler
  container-title: In Genetic Identity Conference Proceedings, 12th International Symposium on Human Identification.
- id: tvedebrink2010
  author:
  - family: Tvedebrink
    given: T
  issued:
    year: 2010
  title: Statistical Aspects of Forensic Genetics -- Models for Qualitative and Quantitative STR Data.
  container-title: Ph.D. thesis, Department of Mathematical Sciences, Aalborg University.
- id: tvedebrink2012
  author:
  - family: Tvedebrink
    given: T
  - family: Eriksen 
    given: PS
  - family: Curran
    given: JM
  - family: Mogensen
    given: HS
  - family: Morling 
    given: N
  issued:
    year: 2012
  title: Analysis of matches and partial-matches in a Danish DNA reference profile data set.
  container-title: "Forensic Science International: Genetics 6(3), 387-392."
- id: weir2004
  author:
  - family: Weir 
    given: BS
  title: Matching and partially-matching DNA profiles.
  container-title: Journal of Forensic Sciences, 49(5), 1009--1014.
  issued:
    year: 2004
- id: weir2007
  author: 
  - family: Weir 
    given: BS
  issued:
    year: 2007
  title: The rarity of DNA Profiles.
  container-title: The Annals of Applied Statistics, 1(2), 358--370.
- id: wiki_birthday
  author: Wikipedia 
  issued: 
    year: 2010
  title: Birthday problem.
  URL: http://en.wikipedia.org/wiki/Birthday_problem
---