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Paternity Index In parentage testing[1], paternity index (PI) is a calculated value generated for a single genetic marker or locus[2](chromosomal location or site of DNA[3] sequence of interest) and is associated with the statistical strength or weight of that locus in favor of or against parentage given the phenotypes of the tested participants and the inheritance scenario. The term “phenotype” typically refers to physical characteristics such as body plan, color, behavior, etc. in organisms[4]. However, the term “phenotype” used in the area of DNA paternity testing refers to what is observed directly in the laboratory[5]. Laboratories involved in parentage testing and other fields of human identity employ genetic testing panels that contain a battery of loci (plural for locus) each of which is selected due to extensive allelic[6] variations within and between populations. These genetic variations are not assumed to bestow physical and/or behavioral attributes to the person carrying the allelic arrangement(s) and therefore are not subject to selective pressure and follow Hardy Weinberg inheritance patterns. The product of the individual PIs is the CPI (Combined Paternity Index)[9]  which is ultimately used to calculate a the Probability of Paternity[10]  seen on paternity test reports. Minimum Probability of Paternity value requirements for state cases differ between states but the AABB[11] requires in their Standards for Relationship Testing Laboratories (currently in the 9th edition)[12] a minimum of 99.0% be reported where the tested man is ‘not excluded’ as the biological father of the child in question. U.S. Department of State requires a minimum Probability of Paternity of 99.5% for all immigration cases[13]. PI calculations utilize allele frequencies[14] generated from established population databases[15] (most commonly using STR: Short Tandem Repeats[15, 16]). The following derivation sections explore calculation methods and fundamental applications of STR allele frequencies (involving loci in commonly used multiplex[16] human identity kits) for the purposes of elucidating human identity and genetic relationships. These methods are well established, widely published and utilized by relationship and forensic testing laboratories for the purposes of paternity, human identity (such as forensic application), kinship reconstruction, siblingship, grandparentage, etc. testing worldwide[17-33]. Because allele frequencies can be either generated in-house or published, PI’s can differ between companies. This is an understood phenomenon and justifiable amongst members of the testing community.

Overview of PI Calculations
The PI is a likelihood ratio[34, 35] that is generated by comparing two probabilities where PI = X / Y: In general, X / Y can be translated as: It is X / Y times more likely to see the observed phenotypes if the tested man is the true biological father than if an untested, unrelated randomly selected man from the same racial population was the true biological father.
 * 1) Numerator[36]  (“X”)  – The probability[37] that we observe the phenotypes of the tested participants in the inheritance scenario given that the tested man is the true biological father. More simply, the probability that some event will occur given a certain set of circumstances or conditions. This calculation assumes that the individuals tested are a “true trio/duo” (which is explained two paragraphs down) or in other words, the parent(s) tested are the true biological parents.
 * 2) Denominator[36]  (“Y”) – The probability that we observe the phenotypes of the tested participants in the inheritance scenario given that a random man is the true biological father. More simply, the probability that some event will occur given a different set of circumstances or conditions. This calculation assumes that the individuals tested are a “false trio/duo” or in other words, the parent(s) tested are not the true biological parents.

There are 14 possible trio paternity combinations and 5 possible duo paternity combinations [38]. Each of these combinations has a PI formula associated with it. Because aspects of both X and Y differ between cases where the mother is available (referred to as “trios”) and where the mother is not available (referred to as “motherless” or “duos”), the following sections differentiate between how to calculate the PI’s for duos and trios.

Trio PI Calculation Derivations
X and Y can be further broken down into mutually exclusive events[39] for Trio calculations:

X: the probability that (1) a woman (biological mother) randomly selected from a population is her particular phenotype (2) a man (Alleged Father or AF[40]) randomly selected from a population is his particular phenotype and (3) a child of the biological mother (given her phenotype) and alleged father (given his phenotype) will be a particular phenotype.

Y: the probability that (1) a woman (biological mother) randomly selected from a population is her particular phenotype (2) a man randomly selected from a population and unrelated to either mother or child could contribute the obligate paternal allele and (3) a child, who is the product of the biological mother (given her phenotype) and a randomly selected man (given his phenotype), will be a particular phenotype (this point will be subsequently discussed).

Formulas, derivations, and explanations for calculating the PI for all 14 trio paternity combinations are listed below[41]. Transmission probability (t) is the likelihood that an individual will give or transmit an allele variation to a child. For instance, the probability that a biological mother giving a child an allele, P, when she herself is homozygous[42,41] for allele P (PP), the probability is 100% or a t = 1. Note that in cases where there are two copies of the P allele, standard nomenclature for paternity testing is that only a single letter “P” is used to represent homozygosity. Two copies of the allele are identified in the pedigree charts below for ease of interpretation. Also, alleles designated as “R” are non-obligate. Abbreviated designations are as follows:

'Note: All of the following combinations have been adapted from Dr. Arthur Eisenberg’s presentation, Popstats Parentage Statistcs (See Reference #41 below). '

Trio Combination 1:
Biological Mother, Child and Alleged Father are all three homozygous (contain only one allele type) for the P allele.

(*lowercase letters such as p are the allele frequencies in the given racial population of allele P)

Here is why the PI formula for this scenario is 1/p: