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MAmIDM-LJJ<ELIHOOD ESTIMATI<E OF THE:
PROPORTION OF NON-PATERNn'Y
by
Richard F. Potthoff and Maurice Whittinghill
Department of Statistics and Department of Zoology
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 429
May
1965
Estimating the proportion ()..) of non-paternity,
which has been done by some recent w.riters, depends
on the form of arrangement of genetic data. The
present paper provides a comprehensive discussion of
the e stima.-tion of)" via the method of maximumlikelihood.
This research was supported in part by National
Institutes of Health Research Grant 00-09358, and
also by NatianalInstitutes of Health Research
Grant GM-12868-01.
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C•
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~-LJXELIHOOD
ESTIMI\TION OF THE
PROPORTION OF NON-PATERNITY
* and Maurice
Richard F. Potthoff
Whittinghill**
Department of Statistics and Department of Zoology
University of North Carolina at Chapel Hill
The use of genetic data to estimate the proportion of non-paternity
or extrs-marital illegitimacy has been discussed in recent writings (Li,
1961, p. 35; MacCluer and Schull, 1963).
We will use the term ''proportion
of non-paternity" to mean the proportion of children ('A) for whom the putative
or supposed father is not the true (biological) father; this is essentially
the same concept that is used by Li (who speaks of "illegitimacy" rather
than "non-paternity") and by MacCluer and Schull.
As MacCluer and Schull
point out, the proportion 'A is of some interest to geneticists, since
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the results of any segregation ana:lysis or linkage detection study would be
vitiated if A were very much larger than 0 in the population being used.
It would also appear that sociologists might be interested in a measurement
technique which estimates the sociological variable A sole:ly from genetic
data; because of its objectivity, such a technique would provide a valuable
comparison with other possible techniques such as those involving questiannaires or interviews.
*Research
supported in part by National Institutes of Health Research
Grant GM-09358, and also by National Institutes of Health Research Grant
GM-12868-0l.
. **Research supported in part by National Institutes of Health Research
Grant GM-09358.
I
We suppose that a number (n) of combinations or trios--each consisting of a putative father, a mother, and a child--have been examined with
respect to sane genetic trait, and that the phenotype of each member of each
trio has been ascertained.
data.
The basic problem then is to estimate
A from such
Provided that computations are not prohibitively difficult, the
method of maximum likelihood would probably be favored as the best method of
estimation because of its many desirable properties (see, e.g., "raser, 1958,
pp. 224-228; cram6r, 1946, pp. ~98ff.).
The purpose of' this paper is to
examine how the maximum-likelihood method can be used in different situations
to estimate the proportion A of non-paternity.
In Li's brief example, he exhibits a simple and quick method (not
ma.ximum likelihood) for estimting
A in the case of a trait for which there
are two autosomal alleles with dominance.
MacCluer and Schull consider the estimation of
A for three different
cases: two autosomal alleles without dominance, two autosomal alleles with
dominance, and two sex-linked alleles with dominance.
the likelihood function Which applies after thi s grouping has already been effe'cteu; because of the predesignated grouping, the resulting estimates (which
could be considered to be maximum-likelihood estimates i,n a special ccntext)
do not utilize the entirety of the available informtion.
With our approach
in the present paper, we obtain maximum-likelihood est.imates based on the
~­
grouped data, and thereby make optimal use of the data.
~-LIKELmOOD
METHOD
Before considering specific cases, we begin with a general discussion
of the uses of the maximum-likelihood method in estimating A, the proportion
of non-paternity.
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For each case, they
designate a special grouping of the data, and then estimate A by maximizing
OOTL:mE.C1F THE
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Suppose the genetic trait we are dealing with manifests
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t
different possible phenotypes; thus, e.g., t
alleles without dominance
dominance.
1 to t.
=3
for the case of two
and t = 2 for the case of two alleles with
We rray arbitrarily identi:f'y the
t
phenotypes by the numbers from
Thus, for the case of the M-N blood system, e.g., we may assign the
numbers 1, 2, and 3 to the phenotypes M, MN, and N respectively.
Let x
ijk
denote the observed number of trios for which the putative father has phenotype i, the mother has phenotype j, and the child has phenotype k.
~ll
Thus, e. g.,
would denote the number of trios falling into the category in which the
putative father is N, the oother M, and the child M.
&11 the Xijk ' s will be
Note that the sum of
n, the total number of trios in the sample.
Next we define f ijk to be the expected frequency of trios in which
the putative father is of phenotype i, the mother is j, and the child k.
t 3 different fijk's will add :UP to 1.
will the corresponding Xijk's.
Some fijk's will autamatic&11y be 0, as
In general, each f
A and of the gene frequencies.
The
ijk will be a function of
The various fijk's for some of the more
important cases are listed in Tables 1 -
4.
We may assUJOO that the x ijk ' s follow a multinomial distribution, so
that their likelihood function is
=
L
nJ
n
xijkl
n
f
,,
i j k
x
ijk
ijk
(1)
i,j,k
The logarithm of L is equal to a constant (i. e., a term not dependent on
or the gene frequencies) plus
*=
L
~ xijklog f ijk
i,j,k
(2)
The products in (1) and the sUlll1ll8.tion in (2) are taken over all (i, j,k)
combinations except those for which f ijk is automatically O.
3
A
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To apply the method of DBXimum likel.ihood.. we IIIl.Xi.mize L (l.).. or..
equivalentl.y and more conveniently.. we my maximize L*(2).
proceed as follows.
SpecificaJ..4r.. we
Suppose there are just two allel.es.. for which the respect-
ive gene frequencies are p
and
q (= l.-p).. where p
is unl\n()w.
Considering
L*(2) as a function of A and p.. we take its partial derivatives with respect
to A and p.. and then sol.ve the system
(;a)
o L.* = 0
op
for A and p.
The resulting sol.utions.. which nay be denoted by
the ma.x:1mum.-llkel.ihood estirrates of A and p respectively.
~ and ~, are
The generalized
system of two equations in two unkIDwns.
Suppose there are three alleles, with correspcnding (unknown) gene
Then we sol.ve a system of three equations
in three unkno.wns and obtain the est:i:ma.tes
'3.. .. ~..
and
'4;
the system is
similAr to (;a-;b) except that there is a third equat:l.on invol.v1ng the partial.
derivative of L* with respect to q.
The maximum-l.1kel.ihood estirrates will be approximately normally distributed i f n
is sufficiently l.arge.
It woul.d appear that, for a given n, the
"
nornal. approximation to the distribution of A will not be as good when A is
close to 0 as it is when
when A is near
may
o.
be calculated.
as well as
A is somewhat l.arger.. because of greater skewness
The approximate variance of
'3..,
which we will cal.l
S2
A),
If, e. g., there are two alleles and we are estimating p
A.. then we first compute the el.ements of the 2 x 2 matrix
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Newton-Raphson method my be used to sol.ve the system (;a-;b), Which is a
frequencies p, q, and r (= l.-p-q).
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L f
i,j,k ijk (
L f
i j k ijk
,,
0 log f i 'k )
J
(0
aA
log f ijk\
A
/
a
using the estimated vaJ.ues
rf!-~) ,
2
(0
~
log f ijk
p
a
and
~
for
j
(4)
L
f
~ log f ijk) 2
i"j"k ijk\-
ap
A and p respectively.
To obtain
l
we calculate the element in the upper left-hand corner of I.... and
divide it by n.
Ass'U!Ji>tions
One matter which we have so far ignored is that of the ass'U!Ji>tions
upon which the above method rests. For our assumptions, we follow much the same
which were
ones/used (either explicitly or implicitly) by MacCluer and Schull and by Li.
They are as follows:
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1.
We assume that the phenotypes of all of the 3n individua.ls con-
tained in the sample have been correctly ascertained, and that no errors
in diagnosis have been made.
2.
For every trio we assume that the "mother" is indeed the true
mther of the child.
In other 'WOrds" we assume that aJ.l mothers are correctly
identified and that the frequency of non-maternity is
o.
Although questions
of non-maternity might be of interest in certain situations, the present paper
IIBkes no attempt to consider such problems.
3.
We assume that the Hardy-Weinberg equilibrium conditions hold.
Furthermore, with respect to the mother, putative father, and true father
(if different from the putative father) pertaining to any child, we assume
that, for the trait under examination" the genotypes of these three (or two)
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individuals are determined ran.dom1.y and DDltually independently.
This set of
assumptions concerning Hardy-Weinberg equil.ibrium and random choice of
partners wou.ld probably not be satisfied, e. g., in any popul.ation comprised
of different sociol.ogical. groups whose respective gene frequencies are not
all equal.
4. We assume that the n trios in the sampl.e are sel.ected independentlz
of each other (and randomJ.y).
'!'bis means, e.g., that the investigator shouJ.d
avoid a sampl.ing scheme which deliberately sel.ects mre than one chUd frau.
a fam:tly.
Perhaps the mat severe adverse effect of systematical.ly using
more than one chil.d from a family woul.d be with respect to s2A) rather than
~
i tsel.f: it woul.d appear that the est imator S2 A) coul.d seriously under-,
!J.
estimate the true variance of 7\ if the n trios in the sam;pl.e represent far
fewer than n different famil:Les.
dependent sel.ection of the n trios.
(and, in fact, exhibits an
MacCl.uer and Schul.1. evidently assume inAJ.though Li makes no such assumption
exampl.e where the sampl.ing is by families), neither
does he make any attempt to estimate the variance of
!J.
7\.
Further uses of the ma:x::iJm.un-llkellhood technique
Provided again that the four assumptions Which we just discussed are
satisfied, our estimation of
A can be effected via the ma.x:i.mum-l.ikel.ihood
technique under circumstances other than those for which the basic description
of the method was given above.
l..
We now indicate several. such situations:
Instead of being unknown, the gene frequencies for the population
rray sanet:t.mes be rather precisely known as a resul.t of previous
studies.
In such circumstances, what we do is to substitute these known val.ues of
the gene frequencies into equation
(:~a)
and then sol.ve this singl.e
equation
1\
for
A.
The sol.ution, w'hich we again call
6
A, is our estimate of
A.
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Its approx:i.mte variance s2(1) is (lin) times the reciprocal of the element
-
in the upper left-hand corner of I (4), where this element is evaluated by
using the known values for the gene frequencies and using the estimate
for A.
1\
A
It is apparent that the application of the maximum-likelihood
method becomes somewhat simpler if the gene frequencies are known, because
now we need to solve only a single equation (3a) rather than a system of
two or mre equations such as the system (3a-3b).
2.
Even when the gene frequencies are really unknown, one might still
like to use a scheme sim:Uar to the one just described in order to reduce
the computational burden.
One might be inclined, e. g., to estimate the gene
frequencies on the basis of the phenotypes of the mothers and putative fathers,
and then substitute these estimates into (3a) and solve the resulting equation
1\
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for A in order to get an estimate A.
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exact maximum-likelihood estimate, but would appear to provide an amply close
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Although it would be favored with
definite computational advantages, such a
approximation if
3.
The
n
3n
1\
A obviously would not be the
is moderately large.
individuals in the sample might sometines be examined
with respect to more than just one genetic trait.
We would expect that the
extra information provided by an additional trait would lead to an estimate
of A with reduced variance.
If we can assume independence among the two or
lOOre traits with respect to the distribution of their genotypes in the population, then the f
ijk
formulas are easily obtained.
The maximum-likelihood
method can thus stUl be applied, but becomes more complicated than before.
However, these complications will be minimized if all the gene frequencies
are known (refer to point #1 just above) or if the scheme just described in
point
#2
is used, since then only a single lengthy equation in
7
A
will have
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to be solved.
An example below will show how to obtain
t
and the f
.-I
ijk
formulas when there is JOOre than one trait.
4.
Instead of the individual x ijk f s, we may sametines only have
available a tally of the trios with respect to a coarser and m:>re oondensed
classification, and it nay not be possible to recover the Xi k's if the
.
original experimental data is no longer accessible.
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Thus, e. g., the trios
having an M putative father, N JOOther, and MN child might have been combined
with the trios baving an N putative father, M mother, and MN child, so that
'the sum (~2 + ~12) is available but the individual vaJ..ues xl32 and ~12
are not. In such situations the maximum-likelihood metJ:x>d can still be
applied if we work with the expected frequencies which pertain to the coarser
classification.
Thus, e.g., the expected frequencies for the category just
desribed would be (f
+f
), which we would find (via Table 1 below) to
l32
312
be equal to ~q2(2_}"). By IDB.Ximizing the likelihood function which pertains
to the coarser classification, we obtain an estimate of
A which is the maxi-
IlD.lIll-likelihood estimate based on the limited available data, but which obviously
is not the same as the maximuJll-likelihood estimate Which would ha'v-e resulted
from the complete data consisting of the Xijk's •
5. Even when all the x ijk ' s are available, one could still deliberately
group the data in some fashion and then obtain an estimate of A by maximizing
the likelihood function which pertains to the resulting coarser classification.
Such a procedure would not utilize all the available data and would thus
generally produce an estimator with greater variance; a second difficulty would
be that one would have to make a decision as to how to do the grouping.
However,
there might be offsetting advantages (SUCh as reduced computational requirements,
in particular).
The deliberate grouping of the categories of trios could be
effected in many different ways, one of which is utilized by MacCluer and
Schull.
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APPLICATIONS TO DDTERENT CASES
Now that we have completed our general discussion of the estimation
of
A via the method of maxinmm likelihood, we may turn to the rore specific
matter of applications.
The first thing that will be needed in any appli-
cation will be the formulas for the fijk's.
Tables 1-4 present these
formu1.a.s for the following four cases respectively:
two autosomal alleles
without dominance (as exemplified by the M-N blood system); two autosomal
alleles (denoted by C and c) with dominance; two sex-linked alleles (denoted
by G and g) with dominance; and the A-B-O blood system when four phenotypes
(A,B,AB, and 0) are distinguished.
For the first three tables there are two
gene frequencies, p and q (=l-p), which are the respective
frequencies of M
and N in Table 1, of C and c in Table 2,and of G and g in Table .3; in Table 4,
the respective gene frequencies for A,B, and 0 are p,q, and r (=l-p-q).
the top of each table are given the identifications for the
t
At
possible
phenotypes (note that t = .3 for Table 1, t = 2 for Tables 2 and .3, and t = 4
4).
for Table
For each of the t 3 possible categories of trios (some of which
are null), the body of each table lists the formulas for f
Table
4,
ijk
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Except in
each f ijk formula is first presented as the product of three
elements, which are respectively the expected frequency of the putative
father's phenotype, the expected frequency of the lWther's phenotype, and
the expected frequency of the child's phenotype given the phenotypes of the
putative father and the lWtber.
Thus, e. g., Table 2 tells us that the
expected frequency for the category of trios having a c putative father, C
lWther, and C child is
f
2ll
= (q2)(p2+2pq) {ell (q+l) ](l-A)+[(pq+l)/ (q+l)]A )
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cf is the expected frequency of a c putative father, (p2-+2pq) is the
eJq>ected frequency
of a C DX>ther, 1/ (q+l) is the probability of a C child
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if the putative father (Who is c) is the true father, and (pq+l)/ (q+l) is
the eJq>ected frequency of a C child if the putative father is not the true
father.
The formula for f
(Printer:
211
simplifies to pcf(l-+pq~).
start inserting Tables 1-4 about here.)
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The f ijk formulas in Table 3 pertain to the situation where the
only examined trios are those in which the child is a girl.
In the case of
a trait determined by sex-linked alleles, data on trios with male children
will provide no information about", since a son receives a Y chromosome
from his father no matter who his father is.
Thus better estimation of "
will be achieved if the available resources are expended only on the examination of trios with female children, in which event Table 3 will be fully
applicable.
In case trios "lith male children have also been examined,
however, then the situation may be a little less simple; in some circumstances
we might simply discard these trios and still utilize Table 3, but in other
circumstances IOOre complicated techniques might be caJJ.ed for (in order to
take advantage, e. g., of the information about gene frequencies which these
trios provide).
Following MacCluer and Schull, we use D to denote the expected frequency
of trios Which constitute detectable instances of non-paternity (i.e., instances
in which it is genetically impossible for the putative father to be the true
father).
Then the quantity D/A is the fraction of non-paternity which is
detectable.
At the top of each of the four tables we indicate the applicable
fornruJ.a. for D/A , because this formula can be utilized to advantage to obtain
a preliminary value for the estimate of A •
The D/A fornruJ.a.s of the first
three tables were already presented by MacCluer and Schull.
Tables 1-4 can be utilized in the situation where the individuals in
the sample have been ex.a.mi.ned with respect to IOOre than one trait.
eJa3JlIPle should serve to make this clear.
An
Suppose we are dealing with two
traits, one involving two autosomal alleles (M and N) without dominance and
the other involving two autosomal alleles (C and c) with dominance.
11
In this
I
situation, there are 3 x 2 = 6 possible phenotype combinations, and so we
consider t to be equal to 6.
of each of the t
3
We need a formula for the expected frequency
= 216 categories of trios.
As an example, we consider just
one of these categories, the one for which the lUtative father of the trio has
phenotypes N and C, the mther is M and c, and the child is MN and C.
Then
the expected frequency for this category is
(~(p2+2PQ»
(~~)([l/ (Q+l) ](l-A) + qPA )
,
where p and q are the respective gene frequencies far M and N, and P and Q (in
place of p and q) denote the respective gene frequencies for C and c.
The
above formula can be obtained very simply if we just look at 1'312 in Table 1
and 1'121 (first formula) in Table 2:
q2(p2 + 2PQ) is the product of the
first element of 1'312 by the first element of 1'121'
~~ is the product of
the two second elem:!nts, l/(Q+l) is the product of the two coefficients of
(l-A), and
qP
is the product of the two coefficients of A •
The last column of Table 1 displays some unpublished data of Gershowitz
concerning n = 265 trios, cOD:!Prised wholly of Detroit Negroes, which were
examined with respect to the M-N blood system.
The distribution of these
265 observed trios into the various categories is exhibited in the table.
At the end of the paper, we will use this data to work out a numerical
~le.
Incidentally, the 265 trios were obtained independently, and re-
present 265 separate families and 795 (=3 x 265) distinct individuals;
hence, our assumption
#4
is certainly tenable.
Gershawitz's 265 trios in Table 1 include the 243 trios which were pre~
viously r~ported by MacCluer and Schull (p. 200) in connection with their numerical example.
The
ma.ximum~likelihood
equations
The system of maximum-likelihood equations is the system (3a-3b), or
sometimes there may be just one equation or three or mre equations
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rather than two.
Although nothing IOOre than elementary differential calculus
is involved in the obtaining of the equations (3a-3b), it may nevertheless
serve to clarify the maximum-likelihood estimation procedure if we present
the explicit equations for two of the IOOst common cases.
The two cases we will
look at will be the ones covered by Tables 1 and 2.
For a trait involving two autosomal alleles without dominance, we
use Table 1 and find that the system of maximum-likelihood equations (3a-3b)
is
~ (p-i)
p}..+i(l-}..)
~(q-i)
:+ q}..+'Rl-}..)
~q
x4P
p}..+(l- A) -
Xn
q"'l.:+(l_"'l.):+
~
1'\
1\
1\ = 0
(5a)
'Where we define
~
= ~1l:+~~32
(6a)
'
(6b)
~ • ~12+~3~33 '
~
= xlll~l+x132
,
x4
= ~12~23:+~33
'
(6c)
(6d)
Xn = ~:+~3~33+X:;1l+X:;21+~32
(6e)
'
XM = 4xlll+4~12+3 xl2l+3~:+3~3
:+ 2~2+2~3:+3~+3~12+2~21
+ 2~22+2~3+ ~32+ ~33+3~1l
+ 2~12+2~21:+ X;22+ ~23:+ ~32
13
'
(6f)
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and
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~ = XlJ2+ ~+ ~-+e~3~2
+ 3JU3+ ~ll+ ~J2~1~22
+ 2~3 +3~32 +3X233-+eX,ll-+eX,J2
+ 3X,21+3X,22+3X,23+4X,32 +4~33
(6g)
•
Equations (5a-5b) are solved simultaneously for 'A and p.
It may be of
interest to note that ;.r(6f) is the number of M genes
all putative
aJOOIlg
fathers and all mothers, plus the small number of children 'sM genes which
could not possibly have been received either from the mother or the putative
father; ~(6g) has a similar interpretation with respect to N genes.
Note
also that" (6e) is the number of trios which represent detectable instances
of non-paternity.
In the event that the gene :frequencies p and q are known, we sub-
stitute these known values into equation (5a).
Then we discard equation (5b)
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aJ.together and just solve the single equation (5a) for 'A •
For the case of a trait involving two autosomal alleles with dominance,
we use Table 2 and find that the maximum-likelihood equaticns (3a-3b) are
:x:w.q3
...-;;;~-
2q+l-q3 'A
+
(JU2~)q
-
~lcf + ~llPq
-=.~-
l_q2 'A
1 + q'A
1 + pq'A
(X2J2+~)p x..
--...=--..;;;;.'-'=-- + ~
1 - p'A
=0
'A
and
:x:w. (3q 'A-2)
2
-
2q+l..q3'A
(JU2~)'A
1 + q'A
+
Xe_
p
2XJ.2lq'A
~(1-2p)'A
(~12~22)'A
+ --.;;;=- +
- --...;=~=;.....1 - q2 'A
1 + pq'A
1 - P'A
Xc
- = 0,
q
where we define
Xe
=2XUl +2~J2 +~l +~ +~ +~J2 +X22l
14
(7a)
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and
Xc
= 2~
+ 2~1 + 3~ + 2XZu + 3~]2 + 4~ + 4~22
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We solve equations (7a) and (7b) simultaneously for" and p; or, if p
knovln" we solve (7a) for " •
Numerical emmp1e
By using the data of Gershowitz which is exhibited in the last column
of Table 1, we now present a numerical example to illustrate the estimation
of" by the maximum-likelihood method.
obtain the seven numbers (6a-6g).
XJ. = 1.3
~
+ 20 + 15
= 536,
= 48.
To start our calculations, we
The first one (6a), e.g., is
We also find
X2 = 49, X:; = 47,
= 40,
x4
~
= 10,
and ~ = 534.
Our next step is to substitute these seven numbers
then solve this equation system (5a-5b) for" and p.
into (5a-5b), and
To obtain the solution,
we may employ the generalized Newton-Raphson JlEthod, which is an iterative
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procedure.
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values for" and p which can be used in the initial iteration, and which
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In order to utilize this iterative method, we first need preliminary
(preferably) will tend to be fairly close to the solution values
We might, e.g., use
xJ(~ +
XN)
of p for the initial iteration.
could (e.g. ) proceed as follows.
that
D/" = pq(l-pq).
.1875.
If
p
= 536/(536 + 534)
= .5009
~ and~.
as our value
To obtain a preliminary value for '"
we
Looking at the top of Table 1, we find
is .5009, the numerical value of
D/"
is then
As our value of" for the initial iteration, we can use
(xn!n)/.l875 = (10/265)/.1875 = .2013.
The generalized Newton-Raphson method is a standard technique which
is adequately dis'cussed in textbooks on numerical analysis, and also
15
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MacC1uer and Schull (pp. 200-201) have exhibited a particular example of
its use.
For these reasons, it is perhaps best to save space and omit a.ll of
the detailed computations associated with the Newton-Raphson procedure (except
for the material in the preceding paragraph, which demonstrates one way of
caJ.culating the initial vaJ.ues).
We present simply the finaJ. result of the
computations, which is that
I:!J.
~ = .5012
A = .2062,
constitute the solution of (5a-5b), and hence constitute the ma.x:innunlikelihood estimates of A and p.
To obtain
need the matrix
S2~),
-I(4).
the estimated variance of the estimate '\ we first
The calculation of
I
involves numerical evaJ.ua.tion of
the fijk's of Table 1 and of the first derivatives of their logarithms,
using .2062 for A and. 5012 for p.
Again we omit the cumbersome details and
present just the result, which is
1.01384
!
=
- .00152 ]
16.2362
[ -.00152
Now we find
=
=
1
265 x
16.2362
(1.01384)(16.2362) - (-.00152)2
1
16.2362 _ OO'~~
265 x 1~.4609 -. ~ ,....
•
In case we are also interested in the estimated variance of ~ , we write
.When n is large enough that the normal approximation to the
~
distribution of A is reasonably accurate, the formula
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(8)
provides a 95% confidence interval for A.
For the present example, (8)
becomes
•
However, the normal approrlmation probably is still not as accurate as we
might like it to be even in this situation with an n of 265.
AJ.though a.l1 the above results were obtained with a desk calculator,
the use of a computer might have reduced the labor.
maximum-likelihood estimate of
would a..l.Joost be a necessity.
~a.ns
Which requires
For obtaining the
A in more complicated situations, a computer
It is always possible to estimate A by a
virt~
no calculation, such as by the procedure which
we utilized earlier which consisted simply of dividing (xJn) by an approximate value of (D/A).
The variance of a simple estinRtor such as this may
not be too much la.rger than the variance of the ma.ximum-likelihood estimator
in some situations (among Which our example appears to be included), but may
be markedly la.rger in other situations.
indicated earlier, 243
As i'repf the 265 trios of Table 1 above are included in the tabulation
which MacCluer and Schull made of Gershowitz' s data and which they utilized
for their numerical example (pp. 200-201).
Professor Gershowitz has informed
us that the estimation procedure of MacCluer and Schull has been re-run using
the data as it is in Table 1 (rather than the for~r tabulation), and he
haS
kindly sent us a copy of these re-calcula.tions.
The results of the
estimation procedure of Ma.cClu,er and Schull, correct to four significant
figures, are as follows:
g2 ~) = .ooo4611.
~ = .2105, ~ = •5192,
g2
(h) = '.003907,
The confidence interval for A of the form (8) thus turns
17
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out to be • 0880 ~ 7\ ~ .3330.
Thus both
S2~)
and
S2~)
are larger with the MacCluer-Schull pro-
cedure than with our naximum.-likelihood procedure.
This is not surprising
in view of the asymptotic efficiency properties of naximum-likelihood esti/}.
However, the improvement in aR(7\) which the maximum-likelihood pro-
mates.
cedure produces over the MacCluer-Schull procedure is not especially large
in this particular eJaUllPle.
If p had not been so close to
l,
though, it
would appear that the contrast would have been somewhat greater, because
more informtion about 7\ would have been extractable from the categories
with an MN putative father and thus a larger contribution to the element
in the upper left-hand corner of NI (4) would have resulted.
SUMMA.RY
This paper attempts to clarify the use of the maximum-likelihood
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method for estimating the proportion (7\) of non-paternity or extra-marital
illegitimacy.
The phenatn>es of the putative father, the IOOther, &Ild the
child in each of a number of trios constitute the basis for classifying each
trio into just one of many categories.
Each category is uniform as to the
combination of the three phenotypes; i.e., dissimi]ar categories are not
merged.
The expected frequency for each category will generally depend on
7\, and 7\ is estimated by utilizing the expected frequency and the observed
number of trios for each category.
The paper concludes with a numerical
example.
18
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TAmE 1.
FORMULAS FOR THE f ijk I s FOR THE CASE OF ~ AUTOSOMAL ALIELES
WUHOUT nOmNANCE (AS EXEMPLD'mn BY THE M-N BLOOD SYS'lEM), AND
DATA (x
ijk
' s) FOR A :NUMERICAL EXAMPLE
Gene frequencies: p = frequency of M, q = fre quency of N.
Phenotype identifications: 1 refers to M, 2 refers to MN,
3 refers to N •
ForIlDl1a. for n/A: n/A = pq(l-pq)
Phenotype of
1>utatJ.vE
father lOOther Child
M
M
M
f
M
M
MN
f
M
MN
M
f
M
MN
MN
f
M
MN
N
f
M
N
MN
M
N
N
MN
M
M
MN
M
MN
MN
MN
M
!>iN
MN
MN
f 222
MN
MN
N
f
MN
N
MN
f
MN
N
N
f
N
M
M
N
M
MN
N
MN
M
N
MN
lIN
N
MN
N
N
N
MN
N
N
N
Any
M
N
Any
N
M
Totals
lll
112
121
122
123
f
132
f
133
f
211
f
212
f 221
223
232
Expected frequency
for category
= (p2)(p2)(I_A) + pAl
Data of
Gershowitz
J<J.ll = 14
J<J.12 = 5
= (p2) (p2) (qA)
= (p2)(2pq)(¥I_A) + ~A)
J<J.21
~
x
123
J<J.32
= (p2)(2pq)(~l
= (p2)(2pq)(~A)
= (p2)(q2) ((I-A) + pAl
= 20
J<J.33 = 1
~11 = 13
~12 = 9
= (p2)(q2) (qA)
= (2pq){p2) (¥1-A) + pA)
= (2pq)(p2)(¥I_A) + <].A)
= (2pq) (2pq) (t(l-A)
= (2pq)(2pq){~1
+ ~ A)
~1 = 20
~ = 41
= (2pq) (2pq) (t(l-A) + ~A)
~3 = 21
= (2pq){q2) (¥I-A) + pAl
~32
~33
= (2pq)(q2) (¥1-A) + qA)
233
= (q2){if) (PAl
f
311
f
= (q2)(p2) «I-A) + qA)
3J2
f
= (q2)(2pq) (~A)
321
f
= (q2)(2pq)(~l
322
f
= (q2)(2pq) (-MI-A) + ~A)
323
f
= (q2)(q2) (PAl
332
f
= (q2)(q2) «I-A) + qAl
333
fill = f 213 = f 313 = 0
f
= f
= f 331 = 0
131
231
I:
i
= 13
=16
= 0
I:
j
~ f iJk
-
= 191
~11
= 21
~12 = 14
~21 = 1
~22 = 14
~23 = 16
X,32 = 1
~33 = 10
=1
n
._ ...
19
= 15
= 265
'-'-"--
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TA.BIE 2.
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FORMUIAS FOR THE f ijk I s FOR THE CASE OF ':M> AUTOSOMAL ALLEIES
wr.rH I><>MmANCE
Gene frequencies: p = frequency of C, q = frequency of c.
Phenotype identifications: 1 refers to C, 2 refers to c.
Formu.1.a for D/"A: D/"A = pq4 •
Phenotype of
~tative
Expected frequency
for category
father M:>ther Child
C
C
C , flU
= (p2+2pq){p2+2pq) {[(2q+l)~q-l:l)2J(1-'A)+[(pq+l)/(q+l)Th
= p2(2q+l_q3 'A)
C
C
c
f
l12
= (p2+2pq) (p2+2pq) ([q2/(q+l)2)(1-"A)+[q2/(q+l)]"A)
= p2qa(1+q'A)
C
c
C
f
C
c
c
f
c
C
C
f
l2l
122
211
= (p2+2pq) (q2) ([1/ (q+l) ](1-'A)+P"A)=pq2(1-q2"A)
= (p2+2pq)(q2)([q/(q+l)](1-"A)+q"A) = pq3(1+q"A)
= (q2)(p2+2pq)([1/(q+l)](1-"A)+[(pq+l)/(q+l)]"A)
= pqa(l+pq"A)
c
C
c
f
212
= (q2)(p2+2pq)([q/(q+l)](1_"A)+[q2/(q+l)]"A)
= pq3(1_p'A)
c
c
C
f22l.
= (q2)(qa) (p"A) = pq4 "A
c
c
c
f
= (qa)(q2) ((l-'A) + q"A) = q4(1_p'A)
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T.ABIE 3.
FORMULAS FOR THE f
IS FOR THE CASE OF TWO SEX-LINKED ALLELES
ijk
WITH DOmNANCE, WHEN THE CHILD IN EACH TRIO IS A GIRL
Gene frequencies: p = frequency of G, q = frequency of g.
Phenotype identifica.tions: 1 refers to G, 2 refers to g.
Formula for D/"A: D/"A = q2(1_q2).
Phenotype of
Expected frequency
for ca.tegory
tputative
father Mother Daughter
G
G
G
f
G
G
g
f
G
g
G
f
G
g
g
f
g
G
G
f
lll
l12
121
122
211
= (p)(p2+2pq) ((1-"A)+[(pq+l),(q+1)]"A}=p2(q+l-q2"A)
= (p)(p2+2pq) {[q2/(q+l)]"A}
= p2 q2 "A
= (p)(q2) {(l-A)-f1>A} = pcr(l-q"A)
= (p)(q2) {q"A} = pq3 "A
= (q)(p2+2pq) {[l/(q+l)](l-"A)+[(pq+l)/(q+l)]"A}
= pq(l-f1>q"A)
g
G
g
f
212
= (q)(p2+2pq) {[q/ (q+l) ](l-A)+ [q2/(q+l) ]"A}
= pq2(1_p"A)
g
g
G
f
g
g
g
f
221
222
= (q)(q2) (PA) = pq3A
= (q)(q2) {(l-"A)+qA) = q3(1_PA)
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TABLE 4.
FORMULAS FOR THE fijk's FOR THE CASE OF THE A-B-O BLOOD SYSTEM
WHEN FOUR PHENC7.rn?ES ARE DISTmGUISHED
Gene frequencies: p frequency of A, q = frequency of B,
r = frequency of O.
Phenotype identifications: 1 refers to A, 2 refers to B,
3 refers to AB, 4 refers to o.
FormulB for D{A: D/7\ = pq(p-+2r)(p+r)2+pq(q+2r)(q+r)2+4pqr2 -+2pqr3 +r 4 (p+q).
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..----------.,..--------------------------Phenotype of
Expected frequency
of category
Putative
father M:7ther ChUd
= F(p+r){p+3r){1-7\)+p2(p-+2r)(p2+-3pr-..r2)7\
A
A
A
f
A
A
B
f
A
A
A
A
o
f
A
B
A
f
A
B
B
f 122 = pqr(q+r)(1-7\)+pq(p+2r)(q2+3qrif)7\
A
B
AB
f
A
B
o
f
A
AB
A
f
A
AB
B
f
A
AB
AB
f
A
o
A
f
A
o
B
f
A
o
o
f
B
A
A
f
B
A
B
f
B
A
AB
f
B
A
o
f
B
B
A
f
B
B
B
f
f
lll
ll2
ll3
U4
l21
123
124
131
132
= p2qr(p+2r)7\
= p2q (p+2r)(p+r)7\
= p 2r2(1_7\)+p2r 2(p+2r)7\
= pqr(p+r)(1-7\)+p2qr (p+2r)7\
= pq(p+r){q+r)(1-7\)+p2q(p-+2r)(q+r)
= pqr2(l-7\)+pqr2 (p+2r)7\
= p2q(p+2r)(1-7\)+p2q(p+2r)(p+r)7\
= Fqr(1-7\)+p2q(p+2r)(q+r)7\
133
= Fq(p+r)(l-7\)~q(p+2r)(p+q)7\
141
= pr2(p+r)(1-7\)+p2r2(p-+2r)7\
l42
l44
211
212
213
214
221
222
7\
= pqr2 (p-+2r)7\
= pr3 (l-7\)+pr3 (p+2r)7\
= pqr(p+r)(1-7\)+pq(q-+2r){p2+3pr+r2 )7\
= pqr(q+r){1-7\)+pq2r (q+2r)7\
=pq(p+r)(q+r)(l-7\)+p~(q+2r)(p+r)7\
= pqr2(l-7\)+pqr2(q-+2r)7\
= pq2r(q+2r)7\
= q2(q+r)(q+3r)(1-7\)+q2(q+2r)(~+3qr~)7\
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TABLE 4 (continued)
f
B
B
AB
B
B
0
B
AB
A
B
AB
B
f
B
AB
AB
f
B
0
A
f
B
0
B
f
B
0
0
f
AB
A
A
f
AB
A
B
f
AB
A
AB
f
AB
A
0
f311~
AB
B
A
f
AB
B
B
f
AB
B
AB
f
AB
B
0
f
AB
AB
A
f
AB
AB
B
f
AB
AB
AB
f
AB
0
A
f
AB
0
B
f
AB
0
0
f
0
A
A
f
0
A
B
f
0
A
AB
0
A
0
223
f 4
22
f
231
232
233
241
242
244
311
312
313
321
322
323
324
331
332
333
341
342
344
411
412
= pq2 (q+2r ) ( q+r ) A
= q2r2(l_A)+q2r 2(q+2r)A
= pq2r(1-A)+pcf(q+'.2r){p+r)A
= pq2(q+2r)(l-A)+pq2(q+2r)(q+r)A
= pq2(q+r ){1- A)+pq2 (q+2r )(p+q) A
= pqr2(q+2r)A
= qr2(q+r){1-A)+q2r2(q+2r)A
= qr3 (1-A)+qr3 (q+2r)A
= p2q(p+'.2r){1-A)+2;p2q(p2+3pr+r2)A
= p2qr(l_A)+2p2q2rA
= p2q(p+r ){1_A)+2p2q2(p+r)A
= 2ifqr2A
= pq2r(1-A)+ep2cfrA
= pq2(q+er){1-A)+2pq2(q2+3qr+r2)A
= pq2(q+r )(1_A)+2p2q2(q+r)A
= 2pc!r2A
= p2 cf(1_A)+ep2q2(p+r)A
= p2 q2(1_A)+2p2q2(q+r)A
= 2p2cf(1-A)+2p2q2(p+q)A
= pqr2(l-A)+2p2qr2 A
= pqr2(l-A)+2pcfr2 A
= 2pqr3 A
= pr2(p+r){1-A)+pr2(p2+3pr+r2)A
= pqr3 A
f 413 = pqr2 (p+r ) A
f
4 = pr3 (1_A)+pr4 A
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4 (continued)
0
B
A
f42l. = pqr3 1\
0
B
B
f
0
B
AB
f
0
B
0
f 424 = qr3 (1-1\)+qr"1\
0
A'B
A
f
0
AB
B
f
0
AB
A'B
f
0
0
A
f 441 = pr"1\
0
0
B
f 41J2 = qr"1\
0
0
0
f
Any
AB
0
f1.34 = f 234 = f 334
Any
0
AB
422
423
431
432
433
= qr2(q+r Hl-1\)+qr2 (q2+3qr+r2 )1\
= pqr2(q+r)1\
= pqr2(1-1\)-+pqr2 (p+T)1\
= pqr2(1-1\)-+pqr2 (q+r)1\
= pqr2(p+q)1\
5
444 = r"(1-1\)+r 1\
f 143
= f 243
= f
343
= f 434 = 0
= f 443 = 0
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ACKNOWLEDGMENT
The authors are indebted to Professor Henry Gershowitz of the
University of Michigan, for his generous efforts in providing us with the
detailed break-down of his data in the form in which it appears in Table 1
above, and for furnishing us with certain re-calculations based on the data.
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REFERENCES
CRAMfR,
H. 1946.
MathematicaJ. Methods of Statistics.
Princeton, New Jersey:
Princeton University Press.
FRASER, D.A.S. 1958.
Statistics--An Introduction.
GERSHOWITZ, H.
Unpublished data.
LI,C. C. 1961.
Human Genetics.
New York: McGraw-H:U1.
MacCWER, J. W., and SCHULL, W. J. 1963.
of non-paternity.
New York: Wiley.
On
the estimation of the frequency
.A1rsr. J. Hum. Genet. 15: 191-202.
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