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A likelihood approach to HLA serology has been developed in which the aim is not to define a recognition set for a serum but to describe the serum&amp;amp;#39;s ability to react with each and every antigen in the test cells, this ability being

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European Journal
of
lmmunogenetics
zyxwv
1
992),
zyxw
9,
3
1-322
A
LIKELIHOOD
zyxw
PPROACH
TO
HLA
SEROLOGY
J.
F.
CLAYTON,*
.
LONJOU,*
.
BOURRET‘t
A. CAMBON-THOMSEN,*
E.
OHAYON,*
.
HORS’
zyxw
E.
D.
ALBERT^
*CNRS-CRPG, and ‘CERT-ONERA, Toulouse, and ‘France Transplant, Paris, France, and ‘lrnrnunogenetics Laboratory, Munich, Germany Received
9
March
1992;
revised
19
June
1992;
accepted
22
June
1992)
SUMMARY
A likelihood approach to
HLA
serology has been developed in which the aim
is
not to define a recognition set for a serum but to describe the serum’s ability to react with each and every antigen in the test cells, this ability being quantified in terms of the probability
of
a positive reaction.
For
a given set of probabilities, one for each antigen, it
is
possible to derive the probability
of
the observed set
of
reactions the likelihood of the set
of
probabilities). The maximum possible value of the likelihood for any possible combination of the probability set can then be sought, but this requires a maximization
of
likelihood with respect
to 60-100
independent parameters. Theoreti- cal considerations of the shape of the likelihood surface prove that, in this particular case, this is a feasible proposition. This approach allows the recognition
of
three groups
of
antigens: those for which there is considerable evidence of a specificity, those for which there is either no specificity or a very weak specificity, and those for which there is insufficient evidence on which
to
base a conclusion. The existence
of
a specificity can be tested using a
log
likelihood ratio as a statistic, but the usual assumption
of
a
zyx
z
distribution
of
this statistic cannot automatically be made in this situation. Therefore, the distribution is estimated by simulation. A serologist using this approach would receive considerably more information as to the serum’s reaction patterns and valid statistics for the existence, or not, of a specificity. INTRODUCTION The basic experimental result of HLA serology
is
the reaction pattern between a number of cells and
a
particular serum Bodmer, 1986,1987). However, except in the most straightforward
cases,
it is not at
all
easy
to
establish which antigens on the cells’ surfaces cause a positive reaction with the serum and consequent cell death.
To
aid in this, serologists have invoked the concept
of
the recognition set
of
a serum, which is an ordered list of antigens with which the serum appears to be
Correspondence:
Dr
John Clayton, CNRS-CRPG,
Hopital
Purpan,
Ave
de Grande Bretagne, 31300 Toulouse, France. 31
312
zyxwvu
zyxwvu
.
Clayton
et
al.
able to react Albert
et
zyxwvut
l.,
1984;
Fergusson
et
al.,
1987).
However, this concept requires a further definition which has not yet been standardized: what statistic is to be used in the ordering
of
these antigens and what value of it
is
to be used to terminate the list? A number
of
statistics have been proposed and used both in laboratory serum typing, international workshops and in computer programs: the most prominent
of
these being the
z
2
statistic and correlation coefficient both based
on
representing the data as two by two contingency tables) and the Q-score, which incorporates the ability of serologists to recognize varying levels of reaction Sierp
zyxwvu
Albert,
1984).
zyx
major drawback to this approach is that the definition of a positive recognition set might be wrongly taken to imply that unnamed antigens are unrecognized by the serum, there being no simple method
of
assessing the quantity of evidence in favour
of
an antigen’s positivity
or
negativity. In order to avoid this complication, a scheme
of
serological typing has been developed, based on likelihood principles, which are widely used in genetics. This scheme starts with the assumption that there
is
no such thing
zyxwv
s
a serum’s recognition set. Instead, the serum is seen as having a finite probability of a positive reaction with each of the antigens in the set of test cells, this probability being different for different antigens. The aim is to estimate, for each antigen, the value
of
this probability. The theory involved in this approach and an implementation of it in a computer program are presented here.
Overview
of
the theory
The basic parameter
of
this approach
is
the probability
of
a positive reaction
4.J
between a serum and a particular antigen x, or its converse the probability
of
a negative reaction
Ox).
It is considerably easier to formulate the algebra and calculus
of
this approach using the negative probability, but, in accordance with the wishes of serologists, results are expressed in terms
of
positive reactions. For a cell which has antigens x, y, and z on its surface, the probability
of
a negative reaction will be:
P-
=
0x
8
BZ
assuming that the reactions between the serum and antigens occur independently. Similarly the probability of a positive reaction will be:
P+
=
zyxwv
0,
ey
0*
The assumption of independent reactions seems reasonable in that there is no suggestion that the antigenic properties change accordance with the other antigens present on the cell, for example. This assumption does not preclude cross reactions. Further,
if
a serum reacts with two or more antigens on a cell, the probability of cell death would be increased over that which would be the case
if
only one of these antigens was present. However, although it is intuitively reasonable, the assumption remains unproven. Thus for a given set of values
0,
it is possible to calculate the probability
of
finding exactly the set of observed data. This figure, the probability of the data for given
0,
is
also termed the likelihood
of
the assumed values of
0.
In general, there is a set of values for
0
which has the maximum possible likelihood
(Lmax).
hat is, all possible changes in the values
of
the
B’s
either singly
or
in combinations, would produce a smaller value
of
the likelihood. This set
of
values provides the maximum likelihood estimates
of
the true values
of
8.
For details on the properties
of
such estimates see Edwards
(1984).
A
likelihood approach
zyxw
o
HLA
serology
zy
13
z
theoretical study of the shape of this likelihood surface has been undertaken and is presented in more detail in the appendix. The principle conclusions
of
this study are: 1) that all stationary points on the likelihood surface are maxima,
2)
hat there are no saddle points,
(3)
that there
is
a single maximum, and
4)
that the unique maximum can be formally demonstrated by calculation of the likelihoods at 2n+l points, where n is the number
of
antigens present. Hence, the procedure is a practical proposition which can be accomplished in a reasonable time. Under the
Null
hypothesis
of
no specificity to any antigen, each cell has exactly the same probability
of
a positive,
or
negative, reaction. Pnull= n n
zyxwvuts
m1-I where
n
is the number
of
positively reacting cells, and m the number
of
negatively reacting cells. Therefore, the likelihood for the null hypothesis is
Lnull
=
p,,,,ll”
l-pnull)m. The binomial coefficient, which is common to both L,,, and Lnull, has been ignored since it is irrelevant to their ratio. The Lnull and L,,, can be used to generate a measure of the evidence for the specificity of the serum. According to large sample theory, the statistic: is asymptotically
x2
distributed, with degrees
of
freedom equal to the number
of
antigens in the data set.
In
practice, this will only rarely apply to data sets such as are used in serological analysis since many alleles will be represented in very few cells. It is predictable that the distribution
of
this statistic will be dependent on the repartition
of
the antigens in the test cells and therefore, must be assessed for those cases where a relatively low value has been achieved. This
is
discussed in more detail below. However, as
a
first approximation this parameter can be thought of as a
z
2
est for the existence of a specificity of the serum. Therefore, for each antigen, the maximum likelihood estimate
of
the
0
is
provided. However, there is also the need for an evaluation of the reasonable dispersion
of
this estimate. For this a ‘support interval’ is provided, which is defined as that interval for the
6
outwith which it is not possible to achieve a likelihood greater than L,,,/e2 by varying the values of all the other
z
’s
see the appendix for further details). Again, given the uneven sampling
of
antigens in the test cells, it is not possible to be sure of the level of significance
of
such a support limit. RESULTS Typical results for sero-typing are shown in Tables 1,2, and
3,
the sera and the test cells having been taken from the study ‘Provinces Francaises’ Ohayon
&
Cambon-Thomsen, 1986). The first line
of
each listing shows the general parameters
for
the serum: the
Pnull,
he natural logarithm
of
the likelihood of the null hypothesis, the logarithm of the maximum likelihood achievable under the hypothesis
of
allele specificity and twice the log likelihood difference called the
Xlld(l)).
These parameters have already been defined. There then follows the list
of
all
alleles which have a non-zero estimate
of
0.
For each such, there are displayed the maximum likelihood estimate
of
0
and its support limit as previously defined. There is also given a further statistic which quantifies the support for the existence of a specificity against the allele and which has been used to order the list
of
alleles. This statistic
(Xlld(2))
is defined as follows:
314
zyxwvu
F.
zyxwv
layton
et
al.
zyx
ABLE
.
The result
of
analysing
data
for
serology between Workshop Serum 026 and 200 test cells
allele
zyxwvutsrq
r
zyxwvutsrqp
XII,dz
B39 0.100
0.00
0.38 {2.201} A23 0.888 0.59 0.99 {29.867} Bw58 0.334 0.00 0.85 {1.924} A29 0.062 0.00 0.25 10.916)
zy
SI
alleles
0-0.9 Bw53 Bw56 DRwlO 0-0.7 B49
(M.65
A25 Aw33 DR9
0-0.5
Bw50 DRw8 (w.45 DRwl2 0-0.4 A26 B37 Bw41
04 35
B38 Bw47 Bw61 Cwl
0.3
845 Bw52 0-0.25 A30 B27 Bw5S 0.2 B13 B14 0-0.15
zyxwvutsr
w.
0-0.05
A28 A31 A32 B18 Bw62 Cw2 All A24 B35 B44 B51 Bw57 Bw60 Cw3 Cw4 CwS DR4 DRwll DRwl4 A1 A2 A3 BI B8 Cw6 Cw7 DRl DR2 DR3 DR7 See
zyxwvutsr
ext
for
explanation
of
symbols.
SI
=
support
interval.
where
L,,,
is the maximum likelihood, and
Lo
is the maximum likelihood achievable under the assumption that there is no specificity against the allele,
i.e.
0
=
1.
Asymptotically, this statistic is
x
distributed with one degree
of
freedom. However, this cannot be assumed to be the case for many
of
the alleles in a study. The significance
of
this statistic
is
discussed later. After this, the alleles for which there is a zero estimate
of
0
are listed. These are organized on the basis
of
the upper value
of
the support limits. It is not possible to declare that there is no specificity against such antigens, but, for those antigens low in the list, if such specificity exists, it is weak. Table
1
hows the results
for
Workshop Serum number 026, when tested against a sample
of
200 cells, representative of the French Caucasian population. The global
Xlld
61.559)
is significant for the rate of positivity
(p,,11)
and this particular
cell
set, as is discussed below.
A
specificity against A23 is established, and any specificity against Al, A2,
or
A3
or
any other of the antigens at the foot of the table must be weak.
In
the central part
of
the table, each of the antigens Bw53, Bw56, DRwlO, A25, Aw33 and
DR9
have a very wide support limit because
of
the paucity of the number of cells expressing these antigens. None of these antigens is present on more than 3 cells. By contrast,
B49
was present on five cells and it would have been expected that a narrower value
of
the support limit could have been found. The support limit actually found
0
to 0.7) is double that
of
Cwl,
also
present on
5
cells. This
is
explained by the strong linkage disequilibrium between A23 and
B49,
both antigens being present
on
four cells. This disequilibrium has acted to flatten the likelihood surface, thus increasing the support intervals
of
both antigens.
A
likelihood approach
to
HLA
serology
zy
15
Tables
2
and
3
show the results for Workshop Serum
199,
when tested against
200
and
800
cells respectively. B7 heads both lists despite the fact that the maximum likelihood estimate
of
its
z
is smaller than those of Bw60, B13 and Bw61. This is because its
Xlldr
hich reflects the quantity of evidence favouring a specificity,
is
the largest. It is important to note that the estimates
of
the
8 s
and their support limits are independent of the method
of
ordering. The order of the antigens
is
unchanged between the two lists
for
all
antigens with an
Xlld
reater than
5
with the
200
cell set. However, the
Xlld s
re much larger with
800
cells, and the support intervals much narrower. With the exception
of
B7, the estimates of
8 s
are little different. There is a much longer list of alleles at the foot of the table with the
800
cell set.
Statistical considerations
The use
of
the log likelihood difference is central to the approach presented here and, for those antigens present on the test cells with high frequency, there is little problem in its interpretation. However, for the others some caution is required. This caution applies equally to the parameter used to test the
null
hypothesis
of
no specificity for any antigen and those used to test for specificity to a particular antigen. Therefore, assessment of the distributions
of
these statistics is required: that is, it
is
necessary to be able to estimate the probability
of
obtaining any particular value
of
the statistic
or
higher if the null hypothesis was to be true. This distribution is a function of the number
of
cells in the test sample, the number
of
alleles present, the patterns
of
association of the alleles, and the overall rate of positive reaction for the serum under the null hypothesis pnull). Hence, there can be no globally determined level of significance and the distributions must he estimated for any serum typing for which there
is
doubt.
TABLE
. The result of analysing data
for
serology between
Workshop
Serum 199 and 200 test cells
serum
199
allele
B7 Bw60 B 13 Bw61 Bw47 Bw41 A29 DRw
10
Bw52
zyxwvut
SI
G0.9
zyxwvutsr
M.65
C0.5
0.45 s0.35 (M.3 0-0.25 0-0.2 (M.15
zyxwvutsr
-0.1
0.05
J
0.888
0.934
1.000
1.000 0.684
0.500
0.126 1 000 0.202
SI
0.77 0.96 0.73 0.99
0.80
1.00 0.67
1
zyxwv
00
0.17 0.98 0.09 0.90 0.00 0.36 0.00
1
oo
0.00
0.64
zyx
Xlld 2.J
(96.014} {43.493) (42.414)
{
15.093} (8,920) (5.745) (1.454) {0.130)
{
0,029)
alleles
A25 Bw53 Bw56 DRw8 Aw33 Bw50 Bw58 DR9 DRwl2 B37 Cwl A28 B27 Bw55 A30 A31 A32 838 B45 B49 A23 A26 B39
Cw2
DRwl4 All B14 B18 A3 A24 B44 Bw57 DR7 B8 B35 B51
Bw62
Cw3 Cw4
Cw5
Cw6 Cw7 DRl DR2 DR4 DRwl1 A1 A2 DR3

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