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 .

paternity index

paternity index <EOT> 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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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 :

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 ] .