# Preliminary analysis of a KIR haplotype estimation algorithm: a simulation study

P. Gourraud, Katia Gagne, J. Bignon, A. Cambon-Thomsen, D. Middleton

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P. Gourraud, Katia Gagne, J. Bignon, A. Cambon-Thomsen, D. Middleton. Preliminary analysis of a KIR haplotype estimation algorithm: a simulation study. Tissue Antigens, 2007, 69 (s1), pp.96-100. 10.1111/j.1399-0039.2006.762_4.x. inserm-05029284

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# Preliminary analysis of a KIR haplotype estimation algorithm: a simulation study

P. A. Gourraud¹, K. Gagne², J. D. Bignon², A. Cambon-Thomsen¹ & D. Middleton³,⁴

1 Inserm U558, Department of Epidemiology, Faculty of Medicine, France

2 Department of Research on NK cell Alloreactivity, EE0503, Etablissement Français du Sang, Nantes, France

3 Histocompatibility and Immunogenetics Laboratory, Northern Ireland, Belfast, Northern Ireland, UK

4 University of Ulster, Coleraine, Northern Ireland

**Correspondence**

Dr Pierre-Antoine Gourraud

Service d'epidemiologie

CHU Toulouse

Unité INSERM 558 - Faculté de medicine-UPS

37 allée Jules Guesde

F-31073 Toulouse

France

Tel: 33 (0)5 61145959

Fax: 33 (0)562264240

e-mail: gourraud@cict.fr

doi: 10.1111/j.1399-0039.2006.762_4.x

**Abstract**

The analysis of Killer cell immunoglobulin-like receptors (KIRs) in terms of haplotypes have only been done through genotyping numerous and selected families. Consequently and schematically, KIR haplotypes have been roughly described by two groups (A and B) based on their gene contents. No further KIR adapted methods have been applied to the estimation of haplotype frequencies using unrelated data. We propose here a maximum likelihood (ML) estimation of KIR haplotype frequencies. ML estimation was developed as an extension of those successfully applied to human leukocyte antigen (HLA) data including the handling of missing values and HLA nomenclature. It has been implemented using an adapted Expectation Masimisation algorithm. KIR types on 11 loci in more than 40 Irish families were used to validate the method in a simulation study. Estimated haplotype frequencies are compared to the phase known. Various allele or gene frequency estimation methods were also compared. We demonstrated the interest and reliability of the haplotype method and underline the effect of the sample size on the quality of the estimation. The ML haplotype method also provides by collapsing more accurate estimation of allele or gene frequencies in population. Such an algorithm opens new perspectives in the analysis of KIR genotypes. Large sample size studies are required using phase-known data and/or simulations. It would allow a genotype-based approach to explore the KIR gene haplotype diversity. The haplotype frequencies may be used to compare populations.

Thirty years ago in the field of human leukocyte antigen (HLA) population genetics, applications of maximum likelihood (ML) estimation of haplotypes were successfully applied (1-3). To overcome the lack of phase information provided by the (HLA) typing techniques, likelihood-based calculations have been formalised by Dempster and further developed in the general framework of Expectation Maximisation algorithm (EM) (4). The EM algorithm is primarily set to handle the missing phase information. It was shown that it could be adapted to handle at the same time complete and incomplete genotypes (missing phase information and missing values within a genotype) (3, 4). Many of the properties of the EM algorithm for ML estimation method have already been discussed (5-8). Two recent reviews covering the single nucleotide polymorphism area of

application focused on haplotype estimation both theoretically and practically (8, 9).

The analysis of killer cell immunoglobulin-like receptor (KIR) haplotypes has only been achieved through the genotyping of families. Consequently and schematically, KIR haplotypes have been described by two groups (A and B) based on their gene content (10). A method to estimate KIR haplotypes will contribute to a better understanding of KIR polymorphisms. It will be of interest to determine significant differences in KIR haplotype frequencies, in order to compare frequencies across populations, or, in order to determine if such differences might account for variation in disease susceptibility. Knowing the haplotype polymorphism, even if estimated, is more informative than relying on phase unknown information only. It provides
a more powerful approach as was foreseen by the initial developers of estimation methods for the HLA polymorphism haplotype frequencies (1).

Because KIR genotyping not only fails to provide phase information but also remains ambiguous as regards to the heterozygous status of the individuals, we have further extended the ML and the EM algorithms in order to handle KIR types. No methods adapted for KIR have been so far applied to the estimation of haplotype frequencies using unrelated data. In this paper, we propose to extend the general principle of haplotype ML estimation to the estimation KIR haplotype frequencies to unrelated individual genotypes.

Five KIR loci were typed at allelic level, 2DL3, 2DL4, 2DS4, 3DL1/3DS1 and 3DL2 and the following six loci were typed at the gene level (presence or absence), 2DL2, 2DL5, 2DS1, 2DS2, 2DS3 and 2DS5, using methods

previously described (11). Typing at the allele level determined whether there were one or two copies of the gene present in most of the cases. The 3DL3 locus was present in all individuals but was not typed at the allele level and not included. We did not differentiate 2DL5 into 2DL5A and 2DL5B, neither did we type for the pseudo-genes 2DP1 and 3DP1. The presence/absence of 2DL1 was known for all family members but was of no help to elucidate the haplotypes and was omitted. Several assumptions were made: if 2DL3 was positive on a haplotype 2DL2 was considered as negative and vice versa. Within a family if only one allele of a gene was found in an individual the individual was considered homozygous for that allele. Data from 40 Irish families were used to validate the method in a simulation study. All parents are unrelated. All individuals tested in families gave an informed consent.

<table><caption>**Table 1** Medians and standard deviations (SD) of first 25 haplotype frequencies over 10 simulations of samples of 2000 individuals</caption><thead><tr><th>KIR haplotype loci</th><th>Reference frequencies derived from families (%)</th><th>Median of estimations. Exact counts on phase-known simulated samples (%)</th><th>SD (%)</th><th>Median of estimations by EM algorithm on simulated KIR typed samples (%)</th><th>SD (%)</th></tr></thead><tbody><tr><td>Haplotype N°1</td><td><strong>9.210</strong></td><td>9.263</td><td>0.47</td><td>9.241</td><td>0.48</td></tr><tr><td>Haplotype N°2</td><td><strong>5.921</strong></td><td>5.613</td><td>0.59</td><td>5.441</td><td>0.56</td></tr><tr><td>Haplotype N°3</td><td><strong>5.263</strong></td><td>4.988</td><td>0.35</td><td>4.808</td><td>0.40</td></tr><tr><td>Haplotype N°4</td><td><strong>4.605</strong></td><td>4.450</td><td>0.40</td><td>4.294</td><td>0.47</td></tr><tr><td>Haplotype N°5</td><td><strong>3.947</strong></td><td>3.988</td><td>0.28</td><td>3.778</td><td>0.38</td></tr><tr><td>Haplotype N°6</td><td><strong>3.947</strong></td><td>4.000</td><td>0.32</td><td>3.944</td><td>0.44</td></tr><tr><td>Haplotype N°7</td><td><strong>3.947</strong></td><td>3.938</td><td>0.26</td><td>5.320</td><td>0.42</td></tr><tr><td>Haplotype N°8</td><td><strong>3.289</strong></td><td>3.375</td><td>0.21</td><td>3.279</td><td>0.22</td></tr><tr><td>Haplotype N°9</td><td><strong>3.289</strong></td><td>3.263</td><td>0.25</td><td>3.222</td><td>0.26</td></tr><tr><td>Haplotype N°10</td><td><strong>2.632</strong></td><td>2.738</td><td>0.19</td><td>2.528</td><td>0.18</td></tr><tr><td>Haplotype N°11</td><td><strong>2.632</strong></td><td>2.588</td><td>0.16</td><td>2.424</td><td>0.21</td></tr><tr><td>Haplotype N°12</td><td><strong>2.632</strong></td><td>2.600</td><td>0.20</td><td>2.405</td><td>0.29</td></tr><tr><td>Haplotype N°13</td><td><strong>2.632</strong></td><td>2.625</td><td>0.21</td><td>2.696</td><td>0.25</td></tr><tr><td>Haplotype N°14</td><td><strong>1.974</strong></td><td>2.000</td><td>0.15</td><td>1.984</td><td>0.18</td></tr><tr><td>Haplotype N°15</td><td><strong>1.974</strong></td><td>1.975</td><td>0.12</td><td>1.881</td><td>0.12</td></tr><tr><td>Haplotype N°16</td><td><strong>1.974</strong></td><td>1.813</td><td>0.15</td><td>1.875</td><td>0.12</td></tr><tr><td>Haplotype N°17</td><td><strong>1.974</strong></td><td>1.975</td><td>0.23</td><td>1.975</td><td>0.23</td></tr><tr><td>Haplotype N°18</td><td><strong>1.974</strong></td><td>1.963</td><td>0.28</td><td>1.909</td><td>0.24</td></tr><tr><td>Haplotype N°19</td><td><strong>1.974</strong></td><td>2.038</td><td>0.19</td><td>2.032</td><td>0.21</td></tr><tr><td>Haplotype N°20</td><td><strong>1.316</strong></td><td>1.275</td><td>0.12</td><td>1.095</td><td>0.22</td></tr><tr><td>Haplotype N°21</td><td><strong>1.316</strong></td><td>1.375</td><td>0.14</td><td>1.302</td><td>0.20</td></tr><tr><td>Haplotype N°22</td><td><strong>1.316</strong></td><td>1.313</td><td>0.22</td><td>1.239</td><td>0.25</td></tr><tr><td>Haplotype N°23</td><td><strong>1.316</strong></td><td>1.350</td><td>0.16</td><td>1.266</td><td>0.16</td></tr><tr><td>Haplotype N°24</td><td><strong>1.316</strong></td><td>1.175</td><td>0.16</td><td>1.154</td><td>0.19</td></tr><tr><td>Haplotype N°25</td><td><strong>1.316</strong></td><td>1.313</td><td>0.19</td><td>1.217</td><td>0.18</td></tr></tbody></table>

The frequencies used in the simulations are indicated in **Bold** in the 'Reference Frequencies derived from families' column. For each of the 25 haplotypes, the median of the estimates obtained by simple counting of the observed haplotypes is indicated in the simulated sample (phase-known simulated samples allow 'Exact counts on phase-known simulated samples') in the 'Median of the estimates, exact counts of phase-known simulated samples' column. Next column gives the corresponding SD. In the two last columns ['Median of Estimations by Expectation Maximisation (EM) algorithm on simulated killer cell immunoglobulin-like receptor (KIR) typed samples' and 'SD'] are presented the median of the estimates obtained on the simulated KIR typed samples (phase unknown and using KIR typing like data) and the corresponding SD. These are not counts on observed data but estimated by computers.
<figure><img src="image_2.png" /><figcaption>**Figure 1** Overview of the use of the pedigree data in the simulation method.</figcaption></figure>

The pedigrees were used to identify the set of haplotypes that segregate within the 40 families. Eight KIR haplotypes were not used due to incomplete typing. Thus, 152 parental haplotypes were further used as the founder set of haplotypes to simulate population of unrelated individuals. The exact counts of each of the founder haplotypes are given in Table 1 (left side column entitled: 'Reference Frequencies derived from families'). These are actual observed frequencies derived from the family segregation data. In the simulations, each individual resulted from the random matching of two haplotypes that were sampled from the founder set of haplotypes at fixed probabilities. The simulated samples are thus phase known for the KIR types. They were computed in the algorithm as experimental result (unphased and results with ambiguities regarding the homozygosity status). Finally, the estimated haplotype frequencies were compared with initial frequencies of the founder set of haplotypes. Figure 1 summarises the method principle and its main steps.

ML estimation was developed as an extension of those successfully applied to HLA data including the handling of missing values and HLA nomenclature. It has been implemented using an adapted Expectation Maximisation (EM) algorithm previously described in details (4). The major steps were the following: 1 – the Expectation step (E-step) and 2 – the Maximisation step (M-step). The presence of the KIR typing ambiguities invited to underline the main adaptations in the counting procedure of the M-step.

1 – The E-step generally computes the likelihood of the sample using haplotype estimations of the previous iteration or the initial values at first step (here chosen at random, no multiple starting conditions were used).

2 – In the M-step, haplotype frequency estimation is based on a gene-counting procedure (12): in the count for each genotype, the presence of a haplotype is weighted by the probability of its phase. We extended this procedure to

KIR genotypes experimental data, below with an example using notation for three loci, indexed by *i*, *j* and *k*. This notation generalises to any number of loci by extending the number of indices.

$$h_{ijk}^{(t+1)} = \frac{1}{2N} \times \sum_{n=1}^{N} \left[ P_{1}^{(t)}(h_{ijk}/n) + P_{2}^{(t)}(h_{ijk}/n) \right]$$

where $h_{ijk}$ is the estimation of haplotype *i-j-k* at iteration *t* + 1, *N* is the number of genotypes observed, $P_1(h_{ijk}/n)$ and $P_2(h_{ijk}/n)$ are the probabilities to observe the haplotype *i-j-k* as first and second possible haplotypes given the observed genotype *n* (complete or not). $P_1(h_{ijk}/n)$ and $P_2(h_{ijk}/n)$ could be calculated as a function of haplotype estimations at iteration (*t*):

$$P_1(h_{ijk}/n) = \frac{\sum_{h_c^{(t)}/(h_{ijk};n)} (2 \times h_{ijk}^{(t)} \times h_c^{(t)} - \delta_{h_{ijk};h_c} \times h_{ijk}^{(t)} \times h_c^{(t)})}{\sum_{(h_1,h_2)/n} (2 \times h_1^{(t)} \times h_2^{(t)} - \delta_{h_1;h_2} \times h_1^{(t)} \times h_2^{(t)})}$$

where $P_1(h_{ijk}/n)$ is the probability to observe the *i-j-k* haplotype for a given genotype *n*, $h_c^{(t)}$ is the complementary haplotype or possible haplotypes to observe the *i-j-k* haplotype in genotype *n*, $h_1^{(t)}$ and $h_2^{(t)}$ are the pseudo haplotype frequency estimations at iteration (*t*) for genotype *n*; $(h_1, h_2)$ notation refers to all possible pairs of haplotypes which may result in the observation of the genotype *n*, whereas $h_{ijk}^{(t)}$ is the *i-j-k* haplotype frequency estimated at iteration (*t*); $\delta_{h_1;h_2}$ is the Kronecker delta defined by:

$$\delta_{h_1;h_2} = \begin{cases} 0 & \text{for } h_1 \neq h_2 \\ 1 & \text{for } h_1 \neq h_2 \end{cases} \quad h_1 = h_2 \Leftrightarrow \text{homozygous genotype.}$$

Such M-step was performed for all $h_{ijk}$, i.e. all haplotype estimations at each iteration. The software also allows
<table><caption>**Table 2** Comparison of the median number of haplotypes between estimation by the algorithm and reference set</caption><thead><tr><th>Sample size</th><th>Estimated</th><th>SE</th><th>Added</th><th>SE</th><th>Shared</th><th>SE</th><th>IH</th><th>SE</th></tr></thead><tbody><tr><td>100</td><td>110</td><td>39.6</td><td>73</td><td>40.3</td><td>36</td><td>1.8</td><td>0.433</td><td>0.09</td></tr><tr><td>250</td><td>104.5</td><td>16.8</td><td>53.5</td><td>16.2</td><td>50.5</td><td>2.8</td><td>0.628</td><td>0.07</td></tr><tr><td>500</td><td>90</td><td>15.1</td><td>34.5</td><td>15.9</td><td>58.5</td><td>2.0</td><td>0.743</td><td>0.09</td></tr><tr><td>1000</td><td>108.5</td><td>33.6</td><td>48.5</td><td>33.6</td><td>60</td><td>0.0</td><td>0.709</td><td>0.11</td></tr><tr><td>1500</td><td>106</td><td>13.9</td><td>46</td><td>13.8</td><td>60</td><td>0.3</td><td>0.719</td><td>0.07</td></tr><tr><td>2000</td><td>87</td><td>18.0</td><td>27</td><td>18.0</td><td>60</td><td>0.0</td><td>0.812</td><td>0.09</td></tr></tbody></table>

The comparisons were performed over 10 simulations for each of the six different sample sizes. The first column indicates the size of the simulated samples, the column 'Estimated' indicates the median of the number of haplotypes estimated by the algorithm and in the third column the corresponding standard error (SE). The fourth column 'Added' indicates the median number of added haplotypes which were not present in the reference set. The fifth column gives the corresponding SE. The sixth column 'Shared' indicates the median number of shared haplotypes between those provided by the Expectation Maximisation algorithm estimation and the reference set. The seventh column gives the corresponding SE. The 'IH' column provides the median of the haplotype identification index as described in (ref) and the corresponding SE.

constraining the likelihood space using a user-defined set of KIR haplotypes. Asymptotic properties of the EM algorithm were not modified (4).

No multiple starting conditions were used in routine, the iterations were stopped using a direct measure of the whole stability of the estimations between two consecutive iterations: the sum of absolute differences between each of the *t*...(t + 1) haplotype estimations. The computations were stopped when it reached 10E-8.

In order to compare the haplotype estimation obtained on the simulated data with the founder set of frequencies used in the simulation, we measured the difference between haplotype estimations using five classical indexes: the haplotype identification index (IH), the sum of square errors and mean square error, the sum of absolute errors and the mean absolute error and the Normalised Similarity index. Details can be found in Gourraud et al. (3).

In order to evaluate the minimal sample size required to provide reliable estimations, we investigated the effect of the sample size on the exactness of the method: we simulated 10 samples of 100, 150, 200, 500, 1000, 1500, 2000 individuals.

The comparison indicators were calculated for each of the simulation batches.

The software is developed in C++ as a newer version of the Loginserm_Estihaplo (3). All computations were achieved on the bioinformatics platform of Genopole Toulouse Midi Pyrenees (40 2-GHz 64-Bits AMD bi-processors nodes run under Linux RedHat 9.0 with 2 Go of RAM). Computational details are available from the authors upon request.

Table 1 presents numerical examples of: the 25 most common haplotype frequencies obtained over 10 simulations of samples of 2000 individuals. The table shows in its right columns numerical examples of the estimated frequencies (cf. column entitled: 'Median of Estimations by EM algorithm on simulated KIR typed samples'). The estimated frequencies using the EM algorithm are numerically very close to the reference frequencies. Global indices of the numerical comparisons of estimated and references frequencies can be found for the same sample size (2000) in the bottom line of Table 3.

Table 2 presents an overview of the performance of the algorithm in terms of KIR haplotype identification

<table><caption>**Table 3** Indexes measuring the differences between the haplotype frequencies estimated and the reference haplotype frequencies</caption><thead><tr><th>Sample size</th><th>Median of the sums of absolute errors</th><th>SE</th><th>Median of the means absolute errors</th><th>SE</th><th>Median of the sums of squared errors</th><th>SE</th><th>Median of the means of the squared errors</th><th>SE</th><th>Median of the Ifn index</th><th>SE</th></tr></thead><tbody><tr><td>100</td><td>3.21E-01</td><td>3.29E-02</td><td>9.17E-03</td><td>9.69E-04</td><td>5.65E-03</td><td>1.70E-03</td><td>1.57E-04</td><td>5.02E-05</td><td>0.492</td><td>2.54E-02</td></tr><tr><td>250</td><td>2.51E-01</td><td>2.75E-02</td><td>5.14E-03</td><td>5.34E-04</td><td>2.51E-03</td><td>5.34E-04</td><td>5.03E-05</td><td>1.02E-05</td><td>0.716</td><td>3.93E-02</td></tr><tr><td>500</td><td>2.03E-01</td><td>2.28E-02</td><td>3.47E-03</td><td>3.75E-04</td><td>1.24E-03</td><td>3.36E-04</td><td>2.13E-05</td><td>5.81E-06</td><td>0.859</td><td>2.78E-02</td></tr><tr><td>1000</td><td>1.54E-01</td><td>1.71E-02</td><td>2.57E-03</td><td>2.86E-04</td><td>9.08E-04</td><td>1.97E-04</td><td>1.52E-05</td><td>3.28E-06</td><td>0.908</td><td>8.43E-03</td></tr><tr><td>1500</td><td>1.27E-01</td><td>1.44E-02</td><td>2.11E-03</td><td>2.38E-04</td><td>6.09E-04</td><td>1.42E-04</td><td>1.02E-05</td><td>2.37E-06</td><td>0.919</td><td>7.68E-03</td></tr><tr><td>2000</td><td>1.15E-01</td><td>1.23E-02</td><td>1.92E-03</td><td>2.05E-04</td><td>5.77E-04</td><td>1.35E-04</td><td>9.61E-06</td><td>2.23E-06</td><td>0.927</td><td>6.04E-03</td></tr></tbody></table>

Table 3 data were computed on the shared haplotypes between the Expectation Maximisation algorithm estimations and the reference set used for the simulation. They correspond to haplotypes considered in the column 'shared' of Table 2. Medians of the different indicators are given in the first line of the table. Standard errors (SE) are also given for all the indexes. The Normalised Similarity Index (Ifn) ranges from 0 to 1. It reaches 1 when haplotype frequencies are completely similar between the estimation and the reference set.
according to different sample sizes. It is worth noticing
that for the smallest sample size, the algorithm cannot
exclude the existence of many haplotypes that are of low
frequency. According to IH index, the identification of
haplotypes performs better with increasing sample size.
Estimation based on larger sample size have two major
characteristics: (i) it provides better estimations of the
haplotype frequencies found in the reference set and (ii) it
better excludes many of the possible haplotypes.

Table 3 shows various indices which measure the
difference between the estimated frequencies and the
reference set haplotype frequencies for different sample
sizes. As suggested by the numerical examples of Table 1, it
shows that the estimations of frequency are generally close
to the reference expected value and are the closer the larger
the sample size. As for the haplotype identification, sample
size influences the results in a non-linear way. Taken
together these findings suggest that KIR haplotype estima-
tion can be used for sample size of 500 unrelated individuals
with acceptable performances.

These tables provide evidence that the proposed
estimation method can achieve valuable KIR haplotype
estimations using unrelated samples, thus opening the
perspective of non family based KIR haplotype studies.
Such an algorithm opens new perspectives in the analysis
of KIR genotypes. It would allow a genotype-based
approach to explore the KIR genes haplotype diversity.
The haplotype frequencies may be used in order to
compare populations. To our knowledge, this is the first
successful attempt of setting population based haplotype
estimation in KIRs.

It would be particularly interesting to test whether the
algorithm is still efficient in cases with missing typing data.
The impact of other data imperfections such as departure
from Hardy-Weinberg equilibrium should be studied too.
Classical EM algorithm has been shown to be robust to
these problems but KIR data challenge the method in ways
which have not been evaluated and which may affect its
general properties. Obviously, increasing the sample size
can minimise these foreseen problems. Interestingly, the
simulation study can be a good tool to decide on the sample
size relevant for a study. Our preliminary results need to be
confirmed by larger studies involving more families and
non-typed loci as well as confirmatory experimental work
on allele typing.

Acknowledgments

The statistical work was supported by the Amalyse du
Resistre de Dormeuns Volontaires Potentiels de cellules
souches hematopoietiques en vue d'Optimisation pon bio
Informatique, Statistique et Economie: prototype de
modèle theonique d'Optimisation des performances
(ARDOISE) project of Etablissement français des greffes.

**Conflict of Interest Statement**

All authors have declared no conflicts of interests.

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