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
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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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