paternity index
each row of the table lists a possible pattern of gene sharing among child, [ mother ], and alleged father (on the left) and paternity index formulas (on the right). the paternity index formulas were derived automatically by the
is defined as the interassay standard error of measurement. it is not as important as if paternity index is evaluated by the
the paternity index under the new test for the other non - father and for the 70 fathers is computed to be 100 .
each match receives a paternity index (pi) value. the paternity index is a way of measuring the strength of a particular match based in part on the relative uniqueness of the match. in our example, the child and alleged father each show an allele value of
12
at marker d2s1338, the paternity index value is 1. 845. this is a reasonably strong match .
test the 100 cases with the new test. compute the paternity indices using the method that we are hoping to validate. convert each paternity index to w, probability of paternity, using the laboratory' s experienced prior probability of 70% .
13. combined paternity index • when multiple genetic systems are tested, a pi is calculated for each system. • the genetic systems are inherited independently, the combined paternity index (cpi) is the product of system pis. or • the combined paternity index (cpi) is determined by multiplying the individual pis for each locus tested. • this value is referred to as system pi. 9 / 27 / 2016 13saurabh bhargava
your dna paternity test results will refer to the term combined paternity index (cpi). essentially, the dna test relies on statistical probability and refers to the probability of the alleged father being the biological father of the child and not another person in a population. the test requires analyzing 20 genetic loci and the index of relationship between the loci is used to calculate the paternity index. the 21st loci is the amelogenin gene .
we begin with some identities relating paternity index to probability of exclusion (subsection v. a). then we discuss means to estimate probability of exclusion (subsection v. b). we use these results (subsection v. c) to compare the results of calculating average paternity index in different ways ,
and the problem of assigning a paternity index therefore boils down to the question of computing pr { random match }  the chance that a random allele would match (by coelectrophoresis) the present one .
the paternity index (pi) is the genetic odds in favor of an alleged father being the biological father of the child. a pi is calculated for each genetic locus, and represents the likelihood that the alleged father contributed the paternal alleles versus an untested, unrelated, random man in the population. the combined paternity index (cpi), which is calculated by multiplying the individual pis together, summarizes the overall strength of the genetic evidence .
a dna paternity test result appears all in one page. it contains a dna profile for each person in the test. the report also shows the combined parentage index (cpi) and a percentage indicating the strength of the result. all of this information produces a single answer for the question of paternity .
once this is done, the indices are put together to give a combined paternity index which is expressed as a percentage. an individual will be excluded as being the biological father by 100% whilst inclusion of paternity is 99. 99% . the inclusion rate is slightly below 100% for reasons of scientific correctness .
according to dna paternity testing accreditation guidelines, a paternity report must show a cpi of greater than 100 (and a probability of paternity greater than 99. 0 %). most courts, child support agencies, or other legal entities require an accredited report (see legal paternity testing for more information). immigration cases require a cpi value of 200 (99. 5 %) or better .
23. posterior odds in favour of paternity • posterior odds = cpi x prior odds • prior odds = p / (1 - p) • posterior odds = cpi x { p / (1 - p) } • if the prior probability of paternity is 0. 7, then the prior odds favoring paternity are 7 to 3 & if a paternity test is done & cpi is 10, 000 then - posterior odds in favor of paternity = 10000x0. 7 / 0. 3 = 23, 333 to 1 9 / 27 / 2016 23saurabh bhargava
email: comments? questions? discussion? formulas for sibship paternity and mutation go to home page of charles h. brenner
following dna paternity testing accreditation guidelines, results are reported with one of three possible statements. for example, if we tested john doe, alleged father, and james doe (child), our paternity report might say one of the following :
the error in using (ii. 3) instead of (ii. 4) is always in the direction of calculating too large an index of paternity. the error is particularly significant where large pi' s are concerned, or where the measurement is in a sparse region of f (x) but quite near a dense region .
dna paternity testing reports provide powerful and often life - changing, information. this section explains how to read and understand a standard paternity test report from dna worldwide. if you have additional questions, please contact our experienced client support team on 01373 751 131 .
10. paternity index • the paternity index (pi) compares the likelihood that a genetic marker (allele) that the alleged father (af) passed to the child to the probability that a randomly selected unrelated man of similar ethnic background could pass the allele to the child. • this is presented in the formula x / y, where x is the chance that the af could transmit the obligate allele and y is the chance that some other man of the same race could have transmitted the allele. • x is assigned the value of 1 if the af is homozygous for the allele of interest and 0. 5 if the af is heterozygous. 9 / 27 / 2016 10saurabh bhargava
once the profiles have been analysed at all locations, the product of all paternity indexes create the combined parentage index (cpi). for an alleged father to be considered the child' s father (to be
not excluded
as the biological father) the cpi value must be at least 100. a cpi value of 100 corresponds to a probability of paternity of 99. 0000% , a cpi value of 10, 000 translates to a probability of 99. 9900% and so on .
15. probability of paternity • the probability of paternity is the measure of strength of one’s belief in the hypothesis that the tested man is the father. • the correct probability must be based on all the evidences of the case. • the non - genetic evidence comes from the testimony of mother, tested man & other witnesses. • the genetic evidence comes from dna paternity tests. 9 / 27 / 2016 15saurabh bhargava
14. combined paternity index • the cpi is an odds ratio that indicates how many times more likely it is that the alleged father is the biological father than a randomly selected unrelated man of similar ethnic background. • the cpi is a measure of the strength of the genetic evidence. • the cpi is based solely on genetic evidence. • theoretically the range of cpi is 0 - ∞ (zero to infinity) – cpi = 1; no information – cpi < 1; the genetic evidence is more consistent with non - paternity than paternity – cpi > 1; the genetic evidence supports the assertion that the tested man is the father 9 / 27 / 2016 14saurabh bhargava
16. probability of paternity • the probability of paternity (w) is based upon baye’s theorem, which provides a method for determining a posterior probability based upon the genetic results of testing the mother, child and alleged father. • in order to determine the probability of paternity, an assumption must be made (before testing) as to the prior probability that the tested man is the true biological father. • the prior probability of paternity is the strength of one’s belief that the tested man is the father based only on the non - genetic evidences. 9 / 27 / 2016 16saurabh bhargava
as always, if you have any questions regarding your dna paternity test or any other dna test from dna worldwide, please call us on01373 751 131 .
for a more through explanation of paternity indexes and test results please visit the section understanding your dna test results. alterntatively visit our frequent questions section by clicking here .
18. probability of paternity • with a prior probability of 0. 5, the – probability of paternity (w) = cpi x 0. 5 _ _ _ _ { cpi x 0. 5 + (1 - 0. 5) } _ cpi _ (cpi + 1) 9 / 27 / 2016 18saurabh bhargava
why does the report say
not excluded
if the person is the father? with dna paternity testing it is possible to 100% exclude someone as the biological father. however, the only way to produce a 100% positive result, would be to test the entire genome of an individual - all of their dna. dna paternity tests use a standardised sample of dna - up to 15 dna markers - to predict the probability of paternity. these probabilities are usually very high - as high as 99. 9999% . this means that, for an alleged father who is
not excluded
, the paternity report is 99. 9999% confident that he is the biological father .
can be tested independent of delta by the binning procedure described above. this test, which validates multiplication of pi, requires experimental haplotypes, which are obtainable from disputed paternity case work .
suppose that pi' s have been computed for a series of paternity cases. in this section we discuss methods for testing whether the collection of values, as a whole, are credible .
john doe is excluded as the biological father of james doe. this means that john doe cannot be the father of james doe because the analysis shows that they do not share a paternity relationship .
a total of 252 black and white trios (mother, child, and putative father) in which the alleged father was known not to be the biological father (bf) were generated from paternity studies. the likelihood of paternity (w) and the paternity index (p) were calculated on the basis of abo, rh, mns, and hla phenotyping. the results were then reviewed to evaluate the arbitrary limits proposed by a number of states in recent paternity legislation. two hundred forty - four of 252 non - fathers (nfs) (96. 8 %) were excluded. of the 152 white nfs, 147 (96. 7 %) were excluded; 97 of the 100 blacks (97 %) were excluded. eight nfs could not be excluded by the routinely performed tests. the w value for those eight trios ranged from 1. 2 to 98. 8 percent. based on limits proposed in legislation, had the nfs actually been alleged fathers, as many as four of the eight of these men could have had the burden of proving that they were not the bfs .
john doe is not excluded as the biological father of james doe. this means that john doe is considered to be the father of james doe because the analysis shows that they share a paternity relationship .
consider a typical paternity case pattern wherein the mother and child are both heterozygous and share a single band, and an alleged father, also heterozygous, appears by coelectrophoresis to share the child' s remaining band .
suppose h, the observed rate of heterozygosity, has been determined from population data on individuals. from this a theoretical prediction can be made of the expected value for  the mean exclusion probability  for paternity casework .
therefore, if we add up all the w' s computed this way, we will be computing the total expected number of fathers. if that total is not close to 70, the paternity indices must be wrong .
while most dna paternity tests reflect this
text book
example, there are many other factors to be considered. for example, natural mutations sometimes present more complicated results. sometimes father and child share many common alleles and although there are many matches, the cpi value is not strong enough for a conclusive result. in most of these cases, including the biological mother significantly strengthens the test results. that' s why dna worldwide strongly recommends including the mother on any dna paternity test .
that the population should be in hardy - weinberg equilibrium with respect to the tested systems is an invariably stated assumption for the evaluation of blood - stain evidence in paternity and in forensics. therefore attention is beginning to be focused on the question of equilibrium with respect to dna probes .
. it can be determined experimentally, by coelectrophoresis of specimens from child and non - father. this, obviously, is the optimal method. however, estimation of mean exclusion probability by this method requires several hundred cases of non - paternity. we are aware of only one such published study (see
4. how • dna (alleles) from the mother, child, and alleged father are extracted, amplified, and identified. • a series of mathematical calculations are then used to either completely exonerate an accused man or provide an estimate of probability of his paternity (pop). 9 / 27 / 2016 4saurabh bhargava
experimental conditions for heterozygosity / homozygosity and exclusion / non - exclusion must be identical. in practice, this is easy to achieve, as data bases are often constructed from unrelated adults in disputed paternity cases, and thus heterozygosity / homozygosity is determined in the same gel (often in adjacent lanes) as exclusion / non - exclusion .
for paternity testing, the experimentally determined mean exclusion probability () is a key biostatistic, as it provides on one hand a method to evaluate the appropriateness of pi calculations and, on the other, a method to determine. once has been selected in this manner, observed heterozygosity (h) provides a test of hardy - weinberg equilibrium (wahlund' s rule) .
17. probability of paternity • to convert the genetic evidence to a probability of paternity (pop) it is necessary to use the baysian theorem. • this is a formula that tests the hypothesis that the accused is the biological father of the child. – for example, a pop of 99% reflects a 99% probability that the hypothesis is correct and a 1% probability that it is not. • the cpi is used in the bayes formula along with another variable called a prior probability (pp). – testing labs typically use a value of 0. 5 for the pp assuming this is a neutral, unbiased value. • the baysian formula is • cpi×pp { cpi×pp + (1 - pp) } 9 / 27 / 2016 17saurabh bhargava
20. prior probability • but the effect of allowing “pp” to vary is often not material because using dna tests (cpi) is usually very large. • for example, suppose that cpi = 100000. then the paternity report will probably calculate w = 100000 / 100001 and report w = 99. 999% , at 50% prior probability. 9 / 27 / 2016 20saurabh bhargava
in a dna paternity test, the analysis seeks matches of allele number values between the alleged father and child. because the child must receive one str allele from his father at all locations (or
loci
), there should be matches for each marker. (in some cases there are natural mutations, where the child has a slightly higher or lower value than the father, but these are somewhat rare. for this tutorial, we' ll assume no natural mutations exist. )
equations (v. b. 1) and (v. b. 2) do not require hardy - weinberg equilibrium. however, if the population making up the data base for computation of homozygosity / heterozygosity is heterogeneous, then equations (v. b. 1) and (v. b. 2) assume that the offspring for whom is computed or observed is heterogeneous in the same way. in practice, one must observe the same homozygosity rate for offspring in paternity matters as for the data base .
an alleged father can be excluded as the biological father with as little as one mismatch between dna profiles. when a person is excluded as the father, the profiles often reveal several, if not many, mismatches. in the following example, the 3 mismatches would exclude the alleged father (he could not be the biological father). even though there are also 3 matches, these matches are random and do not indicate a paternity relationship. almost any two people will have at least some matches, but a father - child relationship will show matches at all locations .
3. genetic markers • half of a child' s genetic material (alleles) come from the mother, while the other half is contributed by the father. • a series of genetic systems (loci) are analyzed in an attempt to ascertain the biological father of a child. • each genetic system in a person has two allele, these alleles are numerically labeled. • in paternity testing, the alleles from the child are compared to those of the
parents
to determine if it is possible for either or both parents to have contributed the particular alleles present in the child. 9 / 27 / 2016 3saurabh bhargava
a dna test profile is a set of dna markers (up to 15) used to determine paternity and identity, including the fbi codis markers used for dna evidence. geneticists developed this set of markers for consistency in identifying individuals and named them according to their location on the gene (think of
d2s1338
as gene
2
location number
1338
). each marker location (or
locus
) shows two alleles, represented by numbers on the report. in this case the alleles are simply measures of short tandem repeats (str) on the gene. one of these str alleles comes from a person' s father, one from their mother .
means there is no allele or none detected in one chromosome, such as primer dropout .
silent allele
or null allele of course includes the possibility of rflp bands that are not visible for whatever reason, and of pcr primer dropout. the only caveat is that one must somehow estimate the frequency o with which alleles are unobserved .
notes 1 *, 2 *, 3 *: the complicated formulas are not implemented. instead, these situations are approximated using the similar formulas 1, 2, 3 .
for example, if the cpi is 100, 000, it means that the alleged father is 100, 000 times more likely to be the biological father of the child than an untested, unrelated, random man in the population. in other words, the chance that he has matched the child by pure coincidence would be 1 in 100, 000. if the alleged father does not possess the paternal alleles of the child at three or more loci, then the pi at each non - matching locus is 0, the cpi is 0, and the alleged father is excluded as the biological father .
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5. mendelian segregation rules of inheritance: 1. child has two alleles for each autosomal marker (one from mother and one from biological father) 2. child will have mother’s mitochondrial dna haplotype (barring mutation) 3. child, if a son, will have father’s y - chromosome haplotype (barring mutation) c, d a, c a, b 9 / 27 / 2016 5saurabh bhargava
6. conti. . • for instance, assume that a child has a 10 and 11 allele for a particular genetic system and the child' s mother is known to possess a 10 and a 12 allele for this system. • the mother must have contributed the 10 allele and the 11 allele must be paternal. in this example, any man who does not possess an 11 allele could not be the child' s father (barring the possibility of mutation that converts one allele to another - something that is unlikely but can be taken into consideration if needed). • in the event that a man is not excluded, the likelihood that a randomly chosen man might also be able to provide the allele in question to the child can be determined by examining the allelic frequencies from a relevant population database. 9 / 27 / 2016 6saurabh bhargava
12. two possible results 1. inclusion: • tested man, alleged father, could be this child’s father 2. exclusion: • there is no way that the af could be this child’s father 9 / 27 / 2016 12saurabh bhargava
19. prior probability • if we wishes to assume a prior probability of 50% , then the (posterior) probability will be “x” [ whatever figure is given ]. • on the other hand, if we feel that some other prior probability is appropriate, then the posterior probability will be somewhat different. • note: if the report gives a w value with no mention at all of a prior probability (which is the practice in most countries), then it is flat misleading. 9 / 27 / 2016 19saurabh bhargava
21. prior probability • suppose that the defendant presents powerful testimony: the woman is unreliable and has made many false accusations in the past; he is a prominent person, a likely target; she admits that full penetration did not occur and that he failed to ejaculate. • suppose that from all of this the judge is persuaded to estimate that if testimony like this were presented in 500 cases, only about once would the man be the father and 499 he would have been falsely accused. so pp = 1 / 500. • applying the above formula, we get – w = 99. 5% , at 0. 02% prior probability. • the verdict is probably the same whether w = 99. 5% or 99. 92% . 9 / 27 / 2016 21saurabh bhargava
22. prior probability • so while the man is entitled by our system of jurisprudence to have his say in court, it is likely that nothing he can say will be significant compared to the force of the dna evidence. 9 / 27 / 2016 22saurabh bhargava
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however, if dna test analysts knew that most people would have an allele value of
12
at this location, the pi might be smaller, say. 0782. it' s true the two participants match, and that could be because one is the biological father of the child, but it could also be they match simply because most people would match with this allele value at this location .
on the other hand, what if dna scientists knew that very few people would have an allele value of
12
at this location? the match would be stronger because it' s more rare. in this case the pi value might increase; it might be 3. 452, or 12. 819 or as high as 34. 125, depending on the strength of the match. a pi value is 0. 000 indicates that there is no match between the two profiles at this location .
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this paper has not been carefully edited into html. you may, however, download a
from proceedings for the international symposium on human identification 1989, editor (none), copyright 1990 promega corporation, pp 21 - 53 .
the following question, while not essential for computation of pi, is of interest and importance for validation studies .
the simplest approach, and a common one, is to reason as follows. let
the simple method is too simple. it assumes that y is the size of the allele. in practice, y is only a measurement. the normal curve at the left is suggestive of the uncertainty of the measurement. if we make the assumption that y is normally distributed with
then as suggested by figure 2 we consider a weighted average of the shaded areas as the center of the area is wiggled back and forth, weighted according to the normal curve. thus we are led to the formula
however, the reasoning is still not complete. the normal distribution represents the probability that an allele whose true size is y would be measured somewhere else, whereas the experimental situation is the converse: y is the
refer to the figure at the right. an allele of size y is equally likely to be mis - measured as l or as r. but a measurement at y is much more likely to be a mis - measurement of r than of l, simply because, and precisely in the proportion that, r is the more common allele. by an application of bayes' theorem, then, the correct weighting function, illustrated here, is proportional to n (tx ,
in this chapter we discuss several issues pertaining to reliability of molecular weight determination .
our membranes include a ladder of standards every tenth lane or so. this amounts to three or four ladders .
we use a digitizing tablet to enter data from dna sizing membranes into a computer. this method is low - tech, but simple and reliable. the membrane is taped to the surface of the tablet, the appropriate computer program is invoked, and then the operator digitizes the dna bands one at a time using a hand - held object called a
cursor
which contains cross - hairs and buttons .
intraassay variation  that is, the difference between two different digitizations of the same membrane  averages 17 (±14) base pairs at 6. 7kb, and 9 (±7) at 3. 3kb. these differences amount to 0. 29% mw and 0. 26% mw .
as a first step to minimizing errors, a schematic image of the membrane is built on the computer display as the membrane is being digitized. this was the first of several reasons that led us to include color graphics capability as part of the computer requirements. the screen image shows colored lanes corresponding to the lanes of the membrane. as each point is digitized by virtue of the operator pressing a cursor button, a blip appears at a corresponding position on the screen. the lanes containing standards  every tenth lane or so  are in a brightly contrasting color. these lanes usually have 10 bands, as opposed to the one to four (in case of co - electrophoresis) bands in data lanes. a glance at the screen will quickly tell whether all bands are present and in the correct relative positions .
digitization is very quick  2 - 3 minutes for a 30 lane membrane .
converting digitized positions to molecular weight is a matter of interpolation. since fragment mobility will always vary from membrane to membrane, the molecular weight of the dna fragments in each data band can only be assigned by comparing the position of the band to the positions of standard markers (bands of known molecular weight) on the same membrane .
imagine for the moment that there is no variation from lane to lane. then interpolation consists of
a. choosing a curve l (m), where m is migration distance, that gives the molecular weight l of the standard bands when applied to their corresponding m .
b. evaluating l (m) for the migration distances m of the data .
a number of such curves are considered. we experimented with them, used the method of cubic splines for a while, then reevaluated and now use a hyperbola .
the method of cubic splines is a standard technique in curve fitting but has nothing particular to do with dna. it is an adequate method of interpolating, as judged by demoting one of the standard markers to a data band, and seeing if we can correctly guess it' s length by using the cubic through its neighbors. and it may be seen as a virtue that the spline passes exactly through each of the standard markers, and so
predicts
their lengths exactly .
on the other hand, when on occasion it' s necessary to extrapolate  because a data band is too close to the edge of the membrane to have 2 standards on either side  the cubic method may fail miserably, even ludicrously. it these circumstances sometimes it gives molecular weights that are negative, or 20kb too high .
according to elder & southern, such a formula is suggested by a simple model of migration. and since it somewhat mirrors reality, it is less fragile. when extrapolation is necessary the results are still poor, but they' re not awful .
passing a curve exactly through all the standard markers may be a poor idea. if some of them are out of the way, so you have to meander to get to them, perhaps they were digitized, or they migrated, inaccurately. to put it another way, rather than judging a data band by comparing it only with nearby standards, farther away standards should have some influence also, though only a modest one .
lately we have been running a standard ladder from brl as an additional control. interpreting our readings of this ladder under both methods of interpolation (
up to now we' ve pretended that the membrane is one - dimensional. in fact, rates of migration might differ from lane to lane, or the membrane might be warped (e. g. through handling). thus, if lanes 1 and 10 are standard lanes, migration distances in lane 5 should probably be interpreted by applying interpolation in the
lane
dimension as well. our algorithm is to fit a polynomial through
the standard lanes, so that if, say, 30 is also a standard lane, it too would have some influence, albeit a small one, on the interpretation of lane 5. this method is recommended by the following arguments :
suppose the following sort of deformation occurs: lie the gel flat on the table. suppose lane 30 is slid upwards while lanes 1 and 20 are held in position. to the extent that the gel is springy, lane 5 will bow downward; the lower and upper edges of the gel bend into a rough parabola .
suppose the gel is not springy. hold lanes 20 and 30 fast, and deform by sliding lane 1 up. perhaps the soft gel will absorb all the deformation at the few leftmost lanes, lane 5 not moving at all. then in interpreting lane 5 it would be a mistake to take too much account of lane 1, while ignoring lane 30 which is in fact more representative .
by visual examination some membranes appear bowed  judging both from the relative positions of the standard lanes with respect to one another, and from the way the bands slant to make a happy or sad face, it appears that the distortion is smoothly distributed across the width of the membrane .
a fundamental problem for computation of pi in single locus hypervariable dna probe systems is selection of experimental parameters. for coelectrophoresis experiments, the key parameter is delta, the discrimination power of the experimental technique. of secondary importance is the standard deviation of measurement variation, which is of primary importance for calculation of pi based only on allele measurements [ gjertson, et al am j. hum. genet .
( 1988) 860 - 869 ]. there are two separate standard deviations: interassay (between run), appropriate for accounting for measurement error in searching the data base for computation of matching frequency, and intraassay (within run), appropriate for assessing the significance of measured paternal and tested man alleles. repeat experiments will yield these parameters .
. still, it plays a role in the formula, and is pivotal if one uses gjertson' s method .
are higher than observed with the brl ladder (typically 0. 3 - 0. 5% mw) and with repeat runs on case material (typically 0. 6% mw). part of the discrepancy may be due to our standard placement of the genomic control (lane # 30), where it may be subject to edge effects .
2. are errors really normally distributed? figure 4 shows the various results of running a control subject many times. the raw data is indicated by the dotted lines. the bold dashed line is a computer - smoothed version of the same data, and it does seem similar to the normal curve included for comparison .
cannot be less than the width of a band. as the bands in a
near heterozygote
move closer together, they will begin to merge when the difference in their size (measured center - to - center) is equal to the band width. one cannot expect reliably to distinguish two such close bands. one cannot measure width of bands with great precision, but measurement of a few dozen randomly selected band widths yields a good approximation for the lower limit of, which, for our experimental conditions and ps194 and pl336, is about 1% mw .
cannot be smaller than the distance between the measured bands of the closest heterozygote. figure 6 is a scattergram showing the relative distance between band measurements for
near heterozygotes
( plotted as a function of the smaller allele), and distribution of homozygotes .
from the computer files, there are few heterozygous individuals with allelic difference < 1% mw, and many < 2% mw. these observations suggest that
note that, in practice, we do not distinguish heterozygotes with bands closer than 1% mw, although repeat runs with greater resolution occasionally will resolve a
fat band
as heterozygous (see above). thus, the limiting factor in resolution of heterozygotes appears to be delta, rather than nonexistence of heterozygotes with alleles closer than 1% mw. again, the lower limit of for our experimental conditions is about 1% mw .
from experimental data by assuming hardy - weinberg equilibrium. these methods are useful not because hardy - weinberg is a plausible assumption (which it may or may not be  see section vi), but for a slightly better reason. namely, we feel comfortable in assuming that any deviation from equilibrium will be in the direction of excess homozygosity. in that case, the true value of will be if anything lower than estimated by these methods .
the calculation is to choose pairs of alleles at random from the data base, calculating the frequency with which pairs of alleles differ by more than, and choosing so that calculated heterozygosity agrees with observed heterozygosity. this calculation assumes hardy - weinberg equilibrium. when applied to our hispanic data bases :
= 1. 4% mw for each probe, comfortably larger than the minimum estimates (1. 0% mw) obtained above. when this experiment was repeated with smaller caucasian and black data bases, anomalous results were obtained  observed heterozygosity agreed with calculated heterozygosity for caucasians when
= 2. 4% mw. observed band widths and close heterozygotes were not different among races (data not shown), so the source of the anomalous results suggested either limitations of equation v. b. 2, sampling variation, or deviation from hardy - weinberg equilibrium. most of the data used for the above were obtained from blots which are limited to deletion of alleles less than 12. 4 kb (ps194) and 8. 8 kb (pl336). less than 1% of alleles for caucasians and blacks are expected to be missed by this technique [ d. dykes, personal communication ], which is insufficient to account for the observed differences .
frequently, mean exclusion probability is determined from case material by shifting tested men by one or more positions, thus matching mother - child pairs with non - fathers. men for whom both alleles differ from the paternal allele (s) by more than are considered to be excluded. this method yields a calculated, rather than experimental, exclusion probability, as it depends on the value chosen for
. such methods have yielded illogical results - calculated larger than observed h. such results are equivalent to a claim that ones chance to win the lottery is greater with a single ticket than with two. if the limitations for equation v. b. 2 and hardy - weinberg equilibrium are satisfied, the inevitable conclusion for such anomalous results is that the chosen value for is too small. if such a is used for calculations of pi, the resulting pi' s will be too large. however, if experimental is available, agreement with calculated provides validation of the chosen value of. in the absence of several hundred exclusion cases, may be calculated from equation v. b. 2, and also calculated from case material by displacing tested men from mother child pairs, and choosing so that the two estimates of are in agreement. this method is subject to the limitations (above) for equation v. b. 2 .
through measurement, it seems worthwhile to consider what relationship should be expected theoretically. in quoting
as a percent of molecular weight we are implicitly assuming that the same percentage applies at various weights. is this a reasonable assumption ?
then if we know how l and beta behave as functions of m, it should be possible to compute how behaves .
= 0. 965 cm, and c = 64 kb - cm are typical values for the parameters, which depend on running conditions. we have found that such an equation gives an excellent fit for molecular weights from 3 - 12kb. as for the behavior of with respect to m, many assumptions are possible but we' ll consider only the possibilities
= 0. 05cm and various choices of the parameter alpha. alpha = 0 is the naive assumption that is a constant; alpha = 1 represents the pessimistic assumption that a band grows twice as thick if it migrates twice as far; and alpha = ½ corresponds to the notion that band diffusion is some sort of random walk process .
= 1. 4% mw at l = 7 kb. the result is figure 7. since at least the dotted line, corresponding to alpha = 1, is close to constant, this analysis lends plausibility to the practice of taking
some tests are given that depend on checking some formulas relating exclusion rates, expected heterozygosity rates, and simple functions of pi .
finally (subsection v. d) we present a forensic application similar in spirit to v. b .
the above relationships are independent of assumptions of hardy - weinberg equilibrium. equations (v. a. 1) and (v. a. 2) may be found in [ nijenhuis, le. pp. 103 - 114, inclusion probabilities in parentage testing. ed r. walker aabb 1983 arlington, va ]. equation (v. a. 3) may be found in [ morris, jw. p 267 - 276, of the same volume ] .
in order to make use of equations (v. a. 1), (v. a. 2), and (v. a. 3), we shall require values for the mean exclusion probability ,
calculated from h (equation v. b. 2 below) with experimental findings is excellent .
the determination that an individual is homozygous or heterozygous at a given locus may be regarded as a coelectrophoresis experiment. the maternal and paternal alleles are coelectrophoresed in the same lane and a visual determination is made as to whether or not two alleles can be distinguished. for a given population, heterozygosity (h) may be regarded as the mean exclusion probability for the hypothesis that maternal and paternal alleles are identical. this biostatistic might be useful if one is investigating the possibility that the phenotypes of a given individual arose from incest .
it is standard practice to determine exclusion / non - exclusion by coelectrophoresis in the same lane of specimens from child and tested man. if the alleles of the tested man are clearly distinguished from the paternal allele (s), an exclusion is recorded. this procedure is highly analogous to determination of homozygosity / heterozygosity - the difference is that for heterozygosity / homozygosity the maternal allele has one chance to match the paternal allele, while for exclusion / non - exclusion, there are
chances for a match between the tested man and paternal allele. it is perhaps not surprising that the mean exclusion probability (
derivations of equations (v. b. 1) and (v. b. 2) are given in appendix b. predicted values for
limitations of equations (v. b. 1) & (v. b. 2 )
it is essential that criteria for heterozygosity / homozygosity be identical to those used for exclusion / non - exclusion. in our laboratory heterozygosity depends on identifying two discrete bands in the phenotype of an individual - the same criteria (alleles of the tested man must be discretely different from the paternal allele (s) ) is used for exclusion. thus ,
fat bands
are recorded as homozygous in the data base, and as non - exclusions in case material (for purposes of analysis) although we have found in both experimental situations that retesting under conditions of increased electrophoretic resolution will sometimes permit splitting of
fat bands
into two discrete alleles .
equations (v. b. 1) and (v. b. 2) assume that silent alleles (
blanks
) occur with negligible frequency. while this appears to be the case with most single locus hypervariable dna probe systems, failure to identify alleles because of experimental conditions will cause significant distortion. if 2% of bands are missed because they are too large or too small for the conditions of electrophoresis or blotting, observed heterozygosity will be low by about four percentage points .
however pi' s are computed, relationships (v. a. 1), (v. a. 2), and (v. a. 3) must hold. optimally, a large series of case material should be used, so that and pi distribution may be determined experimentally. our case material is limited, so we have made use of simulation methods .
a representative collection of pi' s for fathers were calculated from the data bases in the following fashion. each observed allele was considered in turn to be a paternal allele, and pi
was computed, taking = 1. 4% mw and = 0. 6% mw, as :
this equation, introduced on page 1, assumes that a single paternal allele is identified and the non - excluded alleged father is heterozygous. since we are assuming that the alleged father is the biological father, pi
must be corrected for possible homozygosity of the father and for the possibility of two possible paternal alleles. we do this by the following approximation, whose derivation is given in appendix c :
[ 1 + h (1 - h) (2 - h) / 2 ] (v. c. 1 )
equation v. c. 2 can be used to predict, given the allele distribution and assuming a value for .
can be regarded as a measure of the power of a test. in this respect it is worth considering other statistics that one might consider in this role .
an arithmetic mean, or f, is not a good measure of typical performance because it is too much influenced by occasional large values .
the defect of averaging pi' s can be avoided by averaging in the domain of probabilities rather than likelihood ratios. that is, for each pi compute a probability w, find, and then backtrack to the corresponding pi. this method has the theoretical blemish of depending on a choice of prior probability, but experiment shows that the choice of prior doesn' t matter very much .
is also a sensible statistic in that likelihood ratios are meant to be multiplied. it also has the appeal of being the only candidate that naturally combines systems: the geometric mean pi for a combination of independent systems is the product of the geometric means per system .
this equation yields a
quick and dirty
estimate of typical pi for fathers from observed or calculated heterozygosity. the similarity of equations v. c. 3 and ii. 1 should be noted .
( equation v. c. 2) and calculated from observed heterozygosity (equation v. b. 2) .
our analysis yields the conclusion that for our experimental conditions typical and median pi' s should be about 3 for ps194, and about 4 - 6 for pl336. these values are substantially less than those claimed for single locus probes of equivalent polymorphism [ balazs, et al am. j. hum genet. 44 (1989) 182 - 190 ]. see endean, d. , these proceedings, for a further evaluation of these probes .
for case material consisting of roughly equal numbers of fathers and non - fathers, equation (v. a. 3) can be used. for cases in which non - exclusion is observed for a given probe, w (50% prior) for that system can be computed and averaged, yielding estimates for mean exclusion probability for comparison to observed or calculated .
for unselected case material tested in an extensive test battery, realistic prior probability can be closely estimated from exclusion rate. for each probe system, w (realistic prior) can be calculated for each case. sum of w' s (including exclusion cases) equal the expected number of fathers (realistic prior probability sample size) .
for very large collections of unselected case material with extensive testing, pi' s in each probe system may be grouped by magnitude. by definition of pi, the frequency of pi' s among fathers with a < pi < b should be more than a but less than b times greater than the frequency among non - fathers. see morris, j. w. [ transfusion 29 (1989) p281 ] for application to conventional systems .
suppose we have 100 cases and know (through thorough testing with conventional methods) that 70 of them represent fathers .
which means that 100 such men supposedly include 82 fathers, or that this one man is (probabilistically) 0. 82 fathers .
however, this method is only useful when the test is not too powerful, as in the example. suppose to the contrary that the new test shows the following results :
29 of the 30 non - fathers are also excluded by the new test .
derivation of these approximations, which assume hardy - weinberg equilibrium, can be found in appendix d, d. if coelectrophoresis is not performed, it seems to us that the method of gjertson is easily adaptable to phenotype matching, and can be modified, as suggested by gjertson, to account for
non - random
measurement error. [ see evett, i, these proceedings ].
in paternity testing , paternity index ( pi ) is a calculated value generated for a single genetic marker or locus ( chromosomal location or site of dna 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 . phenotype typically refers to physical characteristics such as body plan , color , behavior , etc. in organisms . however , the term used in the area of dna paternity testing refers to what is observed directly in the laboratory . 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 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 ) which is ultimately used to calculate the probability of paternity seen on paternity test reports . minimum probability of paternity value requirements for state cases differ between states but the aabb requires in their standards for relationship testing laboratories ( currently in the 9th edition ) 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 . pi calculations utilize allele frequencies generated from established population databases most commonly using short tandem repeats . 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 . 
in paternity testing, paternity index (pi) is a calculated value generated for a single genetic marker or locus (chromosomal location or site of dna 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. phenotype typically refers to physical characteristics such as body plan, color, behavior, etc. in organisms. however, the term used in the area of dna paternity testing refers to what is observed directly in the laboratory. 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 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) which is ultimately used to calculate the probability of paternity seen on paternity test reports. minimum probability of paternity value requirements for state cases differ between states but the aabb requires in their standards for relationship testing laboratories (currently in the 9th edition) 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. pi calculations utilize allele frequencies generated from established population databases most commonly using short tandem repeats. 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.
in paternity testing, paternity index (pi) is a calculated value generated for a single genetic marker or locus (chromosomal location or site of dna 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. phenotype typically refers to physical characteristics such as body plan, color, behavior, etc. in organisms. however, the term used in the area of dna paternity testing refers to what is observed directly in the laboratory. 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 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) which is ultimately used to calculate the probability of paternity seen on paternity test reports. minimum probability of paternity value requirements for state cases differ between states but the aabb requires in their standards for relationship testing laboratories (currently in the 9th edition) 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. pi calculations utilize allele frequencies generated from established population databases most commonly using short tandem repeats. 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.