 Methodology article
 Open Access
 Published:
Haplotype block partitioning as a tool for dimensionality reduction in SNP association studies
BMC Genomicsvolume 9, Article number: 405 (2008)
Abstract
Background
Identification of diseaserelated genes in association studies is challenged by the large number of SNPs typed. To address the dilution of power caused by high dimensionality, and to generate results that are biologically interpretable, it is critical to take into consideration spatial correlation of SNPs along the genome. With the goal of identifying true genetic associations, partitioning the genome according to spatial correlation can be a powerful and meaningful way to address this dimensionality problem.
Results
We developed and validated an MCMC Algorithm To Identify blocks of Linkage DisEquilibrium (MATILDE) for clustering contiguous SNPs, and a statistical testing framework to detect association using partitions as units of analysis. We compared its ability to detect true SNP associations to that of the most commonly used algorithm for block partitioning, as implemented in the Haploview and HapBlock software. Simulations were based on artificially assigning phenotypes to individuals with SNPs corresponding to region 14q11 of the HapMap database. When block partitioning is performed using MATILDE, the ability to correctly identify a disease SNP is higher, especially for small effects, than it is with the alternatives considered.
Advantages can be both in terms of true positive findings and limiting the number of false discoveries. Finer partitions provided by LDbased methods or by markerbymarker analysis are efficient only for detecting big effects, or in presence of large sample sizes. The probabilistic approach we propose offers several additional advantages, including: a) adapting the estimation of blocks to the population, technology, and sample size of the study; b) probabilistic assessment of uncertainty about block boundaries and about whether any two SNPs are in the same block; c) user selection of the probability threshold for assigning SNPs to the same block.
Conclusion
We demonstrate that, in realistic scenarios, our adaptive, studyspecific block partitioning approach is as or more efficient than currently available LDbased approaches in guiding the search for disease loci.
Background
After emerging as one of the main sources of subjectspecific variation in the human genome, Single Nucleotide Polymorphisms (SNPs) are now routinely used to investigate the role of genetics in a wide spectrum of diseases [1]. The number of known SNPs is continuously growing and it is presently approaching twelve million http://www.ncbi.nlm.nih.gov/projects/SNP/. Technological progress is now enabling the genotyping of up to one million SNPs at a time, a number also expected to increase rapidly. This provides scientists with a considerable amount of information for the study of genedisease associations [2]. The ability to identify associations by statistical analyses of SNP data is challenged by such high dimensionality. Strategies to organize SNP information for discovery of disease susceptibility loci have been proposed [3]. Some of these methods are especially useful when dealing with binary covariates [4], while others require exceptional computer power [5].
By studying the distribution of Linkage Disequilibrium (LD) across the genome, several authors observed that LD is related to the distance between markers [6–10]. The relationship between intermarker distance and LD does not follow a regular pattern and is related to the particular location in the human genome [11]. From these observations, it has been suggested that genetic information could be clustered into smaller sets of genomic regions [12–15] possibly separated by recombination hot spots [16]. Although the exact genetic basis for the existence of these regions is still controversial, empirically, the statistical dependence of neighboring SNPs was shown to be high. The evidence that SNPs cluster more than by chance alone suggests that treating SNPs as independent entities in association studies could be inefficient, and prone to missing true loci if multiple testing adjustments are applied. Recently, haplotype block partitioning was successfully used to accommodate the multiple testing concern while detecting genetic association in prostate cancer [17]. However, block partitioning methods differ substantially in their results [18, 19]. Most comparisons between blocking methods have focused on their similarity in boundary calling or SNP membership, rather than on their ability to detect true associations.
In the present article we develop and validate a new methodology for DNA block partitioning, with a focus on improving power for association studies. Partitioning is viewed pragmatically as a genetically motivated approach to address the challenge of dimensionality. Our goal is to improve power in multiple testing and to make association testing units that are biologically meaningful. We consider each block as a single entity, by estimating a within block haplotype, thus reducing a sequence of S consecutive SNPs into K consecutive haplotype blocks. For inference on blocks we propose a probabilistic approach based on the LD map: the key idea is that pairwise LD statistics can arise from one of two separate probability distribution functions, one being the LD distribution, the other the independence distribution. This is, of course, a simplification because real LD is not binary, but this assumption has been the essence of the haplotype blocking concept. From this standpoint, blocking is similar to a classification problem and can be handled using an optimal Bayes classifier. The result is a vector of probability scores for each candidate block border SNP.
To implement this plan, we developed an MCMC Algorithm To Identify blocks of Linkage DisEquilibrium (MATILDE) and a framework for using MATILDE partitions in genetic association analysis. Our implementation presents several advantages over existing approaches, including: a) the estimation of the distribution of chance LD is specific to the population, the technology and the sample size of the study considered; b) the uncertainty about block boundaries and about whether any two SNPs are in the same block is assessed probabilistically, and c) the option for users to tune the probability threshold for assigning SNPs to the same block.
From the perspective of association studies, a block partitioning algorithm is more appealing than other ones if it provides the researcher higher chance to detect a SNP truly associated to the study trait. With detection of association in mind, we compared MATILDE and the most commonly used methods for haplotype block partitioning with respect to their ability to capture truly associated SNPs, rather than on boundary or membership agreement as in previous comparisons [18–20].
Results
In our analysis we considered a representative data set from the HapMap project [21] (release 200509 phase II 6 chr). We considered the first 500 non redundant SNPs in region 14q11, with minor allele frequency (MAF) greater than 0.05 and Hardy Weinberg Equilibrium (HWE) at α = 0.01. For simplicity, we focused on unrelated individuals from a homogeneous population, by choosing the 45 Japanese, who represent the largest group of unrelated individuals within HapMap. On this data set we first carried out descriptive comparisons of block partitioning approaches, and then we performed controlled simulated experiments to assess the ability of our method to identify disease loci.
Block partitioning of HapMap data
To illustrate how MATILDE captures LDblock information, we compared it to commonly used methods for block partitioning. Among the many methods available, we chose the limited haplotype diversity method by Patil et al. [13] and extended by Zhang et al. [22], as implemented in the HapBlock software [23] (HapBlock), and the three LDbased methods implemented in the Haploview software [24]: the Gabriel et al. approach [15] (DprimeCI), the Solid Spine of LD (SSD), and the four gamete test [25] (4Gamete). The computational speed of MATILDE was comparable to that of the HapBlock algorithm, with both being significantly slower than the rest. As expected, we observed pronounced differences in the LD map, depending on the LD statistic (Figs. 1A and 1B, upper triangles). When LD was estimated with D', many contiguous SNPs were clustering in blocks, but strong LD was also observed between very distant SNPs, in a pattern characterized by noisy stripes. This trend is clearer when zooming in on the region from the 400^{th}to the 500^{th}SNP (Fig. 1A1). This made identification of block partitions more difficult. A cleaner picture was given by r^{2} (Figs. 1B and 1B1), which identified a few big blocks, interspersed by a number of smaller ones, and areas with no blocks.
After 100,000 iterations of MATILDE, the posterior distribution of LD blocks resulted in the partition represented by the triangles on the lower right of the four panels of Fig. 1. The representation is based on a threshold of 0.5 on the marginal probability that each location is a block boundary: MATILDE isolated plausible LD blocks when based on r^{2}, while the noise in D' results in a less appealing partition. The number of estimated blocks was 114 with r^{2} and 215 with D', including singletons. On the same data, HapBlock estimated 53 bigger blocks, DprimeCI 284 (217 of which were singletons). Intermediate values were observed when 4Gamete and SSD were used.
When increasing the sample size from 45 to 1000, using a resampling approach, the number of blocks estimated by DprimeCI decreases slightly from 209 to 191 (CV = 4.4%). 4Gamete and SSD were stable (CV < 2.0%), while HapBlock (CV = 3.0%) was intermediate. MATILDE with r^{2} and a 0.5 probability cutoff for block boundaries had a CV of 3.7%. The relatively high variation of DprimeCI and MATILDE reflects their ability to take advantage of a more favorable signaltonoise ratio to provide a more refined block partition.
A different trend was observed when MATILDE was applied to D'. With increasing sample size the number of blocks quickly degenerates to 1. This effect can be explained by a pronounced clustering of D' values to the maximum, which amplifies noise patterns at distant loci. This "ceiling effect" was also reported in a study comparing population recombination rates [26]. The ceiling effect is sensitive to noise, especially when the sample size is small or the allele frequency is extreme, in which case many observed high disequilibrium pairs would only be due to missing allelic combinations at one locus. Using r^{2} results in a much reduced sensitivity to this problem [27, 28]. For this reason, we only used r^{2} in the simulation studies.
An overview of the partitions obtained with each method is given in Fig. 2, for a sample size of 1000. By modulating the probability cutoff, MATILDE can generate a fine partition, as do LDbased methods, or a coarse one, as HapBlock (see Additional files 1, 2, 3, and 4 for additional sample sizes). MATILDE proved stable over varying cutoff, with little variation in the break points occurring for cutoffs between 0.1 and 0.9. In most instances, MATILDE estimated fewer singleSNP blocks than DprimeCI and 4Gamete, but a greater number of smaller blocks than HapBlock. Moderate to good agreement of break points was observed between DprimeCI, SSD, and 4Gamete: κ between DprimeCI and 4Gamete ranged between 0.67 and 0.76, depending on sample size; κ's between SSD and DprimeCI were 0.52–0.60; while they were 0.48–0.53 between SSD and 4Gamete. DprimeCI, SSD, and 4Gamete were not in agreement with HapBlock (κ < 0.10 under all conditions). Generally, MATILDE was in an intermediate position between the LDbased approaches and HapBlock. κ between MATILDE and HapBlock was low but not null, often taking values greater than 0.10. When comparing MATILDE to the three LDbased approaches, we observed that κ was nearly the same, usually ranging between 0.20 and 0.50. The highest agreement was observed between MATILDE and SSD. In general, as the probability cutoff increased, the agreement between MATILDE and DprimeCI, 4Gamete, and SSD decreased. When HapBlock was considered, the agreement with MATILDE was higher for central probability cutoffs (see Additional file 5 for an extensive overview). When a break point was concomitantly recognized by the common methods, it was typically detected by MATILDE as well.
Comparison of performance in association studies
In our simulation studies, described in detail in the Methods section, we generated artificial casecontrol studies with a single disease SNP, using two genotypephenotype association models (dominant or recessive) and a range of odds ratios and sample sizes. We applied this approach in turn to all SNPs in the chosen region. This strategy preserves the observed LD in the HapMap sample. After estimating within block haplotypes, we used the likelihood ratio statistics (LRS) applied to the marginal distribution of haplotypes for each block, i.e., we performed a haplotypebased comparison rather than a diplotypebased comparison, such that each individual contributes two haplotypes, rather than one diplotype to the statistic. SNPs not in a block were considered a block of size one and in this situation, the LRS was an allelic SNP test. The sensitivity and specificity for detecting the causal SNP are reported in Fig. 3. For each method, block, and simulated dataset, we declare a positive if the pvalue, after multiple testing adjustment with the BenjaminiHochberg method [29], is smaller than .05. MATILDE can be used at different cutoffs for the probability that a SNP is a boundary point between blocks. Varying this threshold generates the receiver operating characteristic (ROC) curve shown. The other methods produce a single sensitivity/(1specificity) pair. DprimeCI, 4Gamete and SSD had high specificity for all OR's, but very low sensitivity. At the other extreme, sensitivity was generally high for HapBlock, but this method had a poor specificity thus giving a high number of false positives. MATILDE was performing generally at equal or better sensitivity/specificity tradeoffs than the existing methods, and had the additional advantage that it could be tuned to have a higher sensitivity than the LDbased approaches. When compared to HapBlock, for pvalue thresholds that achieve the same sensitivity level, MATILDE had about 10% greater specificity, and for the same specificity, nearly half the probability of missing a true effect – a practically important difference especially in screening studies. A better performance of MATILDE over other methods was observed for all sample sizes considered, as shown in the Additional files 6, 7, 8, and 9.
In addition to blocking approaches, we performed two types of singleSNP association analysis: allelebased, indicated with an 'x' in the graph, and genotypebased, indicated with an '+' in the graph. These are described in more detail in the Methods section. While the genotypebased analysis is more appropriate and more common in practice, the allelebased singleSNP analysis is reported because it is more directly comparable with the blocking methods, as it does not use phase information. Any gains seen in comparing the "x" with the blocking algorithms can be attributed to blocking. The sensitivity of the allelebased singleSNP analysis is zero in all scenarios, though some positive calls would be made at a higher false discovery rate (FDR) of 0.1. In practice, even in SNPbySNP studies, SNPs in close proximity with the one with the lowest pvalue may be examined closely, as SNPs close to the causal SNP may have low pvalues as a result of linkage disequilibrium. To capture this practice, we relaxed our definition of a "correct call" in our sensitivity/specificity calculations by considering as true positives all loci who were within a given distance from the causal SNP, and satisfied the FDR threshold. We examined SNP windows of 1, 2, 3, and 4 SNPs on each side. In all cases, results were similar to those reported in Fig. 3, and the gain in sensitivity was very modest.
Fig. 4 summarizes results obtained using two additional comparison criteria that better highlight important properties of the blocking approaches. Criterion R represents the ratio of the rank of the block including the causal SNP, and the total number of blocks. On the left sides of the four panels, we reported the distribution of R at ORs ranging from 1.2 to 1.8. The better methods are those with distributions of R closer to 1. Boxplots represent variability over simulated datasets. For small effects, that is OR = 1.2, the median R's for DprimeCI, SSD and 4Gamete were comparable, and all are higher than for HapBlock. The median for MATILDE at several cutoffs was the highest, by a sizeable margin, even when compared to the single SNP analysis. This is because, for small effects, there are often several SNPs that are ranked better than the causal one in the single locus analysis. At increasing OR's the performance of DprimeCI and 4Gamete improved and for values bigger than 1.4, they were on average slightly better than MATILDE. For effects ≥ 1.4, the analysis at single locus outperformed the other methods (see Additional files 10, 11, 12, and 13 for additional sample sizes).
Criterion B is the count of SNPs belonging to blocks ranked as high or higher than the block including the correct SNP (Fig. 4, right sides). Lower values of B are preferable. For small OR's, the blocking methods performed comparably, with the exception of MATILDE at cutoffs ≤ 0.1, which had a better performance. At higher OR's (see Additional files 10, 11, 12, and 13) the methods with the highest number of singleSNP blocks (DprimeCI, 4Gamete and MATILDE with cutoffs ≤ 0.1) had a significantly better performance than SSD, HapBlock and MATILDE with bigger cutoffs. As expected, the single SNP analysis performed better than blocking methods by this criterion. Consistently, lowcutoff MATILDE provided the best performance in both R and B.
Discussion
Overall, our experimental results suggest that probabilistic modeling of LD patterns is a useful approach to summarize a high dimensional collection of SNPs into a smaller set of haplotype blocks when searching for diseaserelated loci. Our methodology, implemented in the MATILDE program, adapts to the available data, provides an assessment of uncertainty, and can be used flexibly as a dimension reduction tool compared to the alternatives available so far. In our HapMapbased simulation experiments, MATILDE showed the best ability to rank loci when looking for small effect sizes. This is a critical strength, since most SNP association studies involve small effect sizes. An important, empirical example, in this sense, was recently illustrated in the field of prostate cancer [30]. DprimeCI and 4Gamete perform well in ranking, though at the cost of a large number of singleSNP blocks, which makes these methods less efficient when using multiple comparisons corrections. MATILDE also provides significant gains in sensitivity when a low specificity is appropriate – as in SNP screening studies – and is comparable to the other methods considered in the high specificity range.
For a broad range of sample sizes and effect sizes, traditional singleSNP analyses fail to find the causal locus. These analyses only become effective when the sample size is greater than 1000 and the effect size is at least 1.8 – a rare case in genomics. Otherwise, grouping SNPs into blocks with any method is a better solution. This conclusion reinforces the suggestion of Zhang et al. [31] that haplotypebased analysis can be much more powerful than single locus analysis. Their study was based on HapBlock. In our simulations MATILDE shows better performance than HapBlock, so the case for blocking is now stronger.
While our simulations consider a large number of scenarios (over 850,000) and are closely mimicking real data, there remain some limitations. First, because of the computational burden, it would have been prohibitive to consider the joint distribution at two chromosomes after blocking. Thus our comparisons are based on the simpler unphased haplotype estimation, whereby each subject contributes two separate haplotypes, and association is assessed by comparing the distribution of cases' haplotypes to that of the controls. This approach is still the most prevalent in applications, but may negatively affect the performance of all blocking methods, and may favor the single marker analysis for big effects, especially with regard to the R and B performance criteria. To explore the potential gains in efficiency that can be expected when using the phase information, we carried out a genotypebased singleSNP analysis. This is indicated by a '+' in Fig. 3 and should be compared only to the 'x' symbol, which represents the results of the allelebased singleSNP analysis. We also reported both analyses in Fig. 4. For R and B the results are similar, while a difference is observed at an OR of 1.8 in Fig. 3. The latter, however is partly the result of a sensitivity to the choice of the significance level, and is not as pronounced when a stricter level of .01 is required.
As a second limitation, we focused our comparison on the most commonly used block partitioning methods. Minimum Description Length (MDL) methods [32–34], including the MDBlock implementation [32], have also been shown to reliably locate boundaries between blocks at regions of rapid LD decay, and produce block partitions of intermediate size between those of LDbased approaches and those given by limited haplotype diversity methods. Additionally, future work could consider the comparison between blocking and using tagging SNPs. Two useful approaches, HaploBlockFinder [35] and htSNPer [36], produce both haplotype or LD blocks, and tagging SNPs. As they yield block partitions similar to those of methods already covered by Haploview and HapBlock, they were not considered here, but would be natural choices if tagging SNPs were studied. Lastly, the iHAP (integrated haplotype analysis pipeline) [37] integrates several block partitioning and tagging SNPs methods with web resources for gene finding. It was explicitly defined to mine the HapMap dataset by means of the HapBlock software and it has not the aim to process user's genotype data.
Our results include a descriptive analysis of the agreement among blocking approaches. Our goal is to provide further intuition about the reasons behind the performance of different blocking methods in identifying disease SNPs, rather than fully characterizing their behavior from a population genetics viewpoint. In our study, block partitioning is an intermediate step towards identifying genotypephenotype associations, which is ultimately assessed through statistical models. This bypasses the need for a gold standard for haplotype blocks, and also brings the evaluation closer to practical study goals. To account for the potential instability of estimated blocks when small sample sizes are taken [20], we also examined large sample sizes.
While several measurements of agreement between blocks are available in the literature, we chose the simple κ statistic on the between block break points. Alternatively, the SB_{2} statistic [38] would have been useful when comparing two populations one of which is considered less diverse than the other one, that is, in the case where block boundaries could vary among populations. In our case, however, we were considering a homogeneous sample of subjects from the same geographical location and testing different methods over the same small chromosomal segment. Our results were consistent with those of Schwartz et al. [18] who defined an agreement statistic based on the number of shared boundaries. The block partition given by MATILDE was more similar to the LDbased methods than to HapBlock. Since the MATILDE block estimation is based on the LD map, this finding was not surprising. Other authors [19] compared the LDbased method of Gabriel et al. [15] and the limited haplotype diversity method of Patil et al. [13], in the Zhang et al. formulation [22]: they found that block partitions given by the two methods were different, strongly dependent on minor allele frequencies, and sensitive to changes in the algorithms' parameters. We confirm the previous observation [18, 19] that the method from Gabriel et al. [15] generates a higher number of smaller blocks than that of Patil et al. [13].
Block partitioning criteria can potentially perform at different levels of sensitivity and specificity in different populations. Spatial correlation in the genome can be influenced by a variety of factors, including demographic history and recombination hotspots [39]. Depending on how these factors contribute to the block structure in a population, different partitioning criteria may differ in their ability to identify associations. An assessment of how different methods could perform in populations with different demographic evolution is an interesting question for further research. MATILDE, however, differs from biologically based methods such as the four gamete rule [25] as it was designed pragmatically, without any reference to biological theories about the origin of blocks. We can speculate that MATILDE may be more powerful than methods based on biological hypotheses in situations where there is noise in the LD pattern, as is the case of outbred populations. In isolated populations, where population growth followed a bottleneck event, haplotype heterogeneity is much smaller and individuals share longer chromosomal regions. When this situation is also accompanied by a reduced number of external individuals, one may expect less noise in the LD pattern, and most of the block partitioning methods should give more similar results.
Our method allows users to specify a pairwise measure of LD. This choice matters: in our analysis MATILDE's performances varied depending on whether r^{2} or D' was used. Both measures have a clear genetic interpretation. The expected value of r^{2} is a direct function of the population recombination rate, and r^{2} is the standard χ^{2} test statistic divided by the number of chromosomes. Thus, it is a natural candidate for testing the disequilibrium between loci [9, 40, 41]. Strengths and limitations of D' have already been described [28]. LD can be assessed by many other statistics. An extensive list is provided by Devlin and Risch [42]. An example is Levin's population attributable risk [43]. Statistics that show a robust behavior in case control studies are the difference in proportion d suggested by Nei and Li [44], the odds ratio, and the Yule's Q [45]; d and Q are bounded between 0 and 1 and between 1 and 1, respectively. More recently, entropy was suggested as a measure of LD for multiallelic loci [46], and the volume measures of LD proved to be robust in case of small samples [47]. In addition, potential candidates are the Morton's rho [48], which models LD by a linear mixture of SNPs under nonLD and in perfect LD, and the Delta statistic [49] which is less noisy than r^{2} and D', and is robust to allele frequency.
The ability to adapt to SNP density is an advantage of using a Bayes classifier like MATILDE. Marker density affects the LD distribution [50], though this is not an issue when clustering is used only as a dimension reduction step. In our formulation, block partitioning is related to the specific set of SNPs typed. This is different from estimating blocks on the basis of recombination hotspots [51], which aims at uncovering an underlying genetic structure.
While our implementation was successful as a proof of principle, additional work remains necessary before the full potential of dimensionreduction by blocking can be realized. For example, computational obstacles remain before the current implementation of MATILDE can be used efficiently on studies of the size of current genomewide association analyses. We plan to address these computational issues in future versions of the program.
Finally, we hope that the idea of using probabilistic blocking for dimension reduction of DNA information can in the future become the foundation for a comprehensive analysis, including haplotype reconstruction, missing data imputation, and modeling of the genotypephenotype relationship. It has been shown that the best method for haplotype reconstruction when the phase is unknown is also probabilistic and based on MCMC [52, 53]. The issue of integrating block partitioning and haplotype reconstruction was already undertaken by some authors [54, 55]. Additionally, a potentially important extension available within an integrated approach is the ability to construct blocks that optimally capture association signal, a feature which is not presently implemented in our approach.
Conclusion
We demonstrated that at low signaltonoise ratio, blocking SNP's via a classification approach can lead to significant increases in efficiency in identifying disease related loci. For this task, we provided a flexible methodology and software.
Methods
A probabilistic formulation of LD maps
LD is the nonrandom association between alleles at different loci [56]. Let us now consider a sequence of S SNPs, ordered by chromosomal location. The set of all the S(S  1)/2 pairwise LD statistics is Θ = {θ_{ ij }, i = 1, ..., S  1; j = 2, ..., S}. Note that θ can be any measure of LD among those varying in [0, 1] [42].
Denote by Θ_{1} the subset of θ s estimated from SNPs in true LD, and by Θ_{0} the subset of θ s estimated from SNPs which are not in LD. Since no other intermediate option is allowed between the LD and the absence of LD status, then Θ_{1} ∪ Θ_{0} ≡ Θ. Under the assumption that two SNPs are in LD only if they belong to the same haplotype block, the partition of Θ can be uniquely identified by a binary vector γ = [γ_{1}, γ_{2}, ..., γ_{i1}, γ_{ i }, γ_{i+1}, ..., γ_{ S }, γ_{S+1}]' where γ_{ i }= 1 means that the a border of a haplotype block falls between SNPs (i  1) and i; γ_{ i }= 0 means that SNP (i  1) and SNP i belong to the same block. By definition, γ_{1} = 1 and γ_{S+1}= 1. SNPs not belonging to any block are classified as blocks by themselves, with borders γ_{i1}= γ_{ i }= 1. In the following, γ will be referred to as block border vector.
Empirical evidence and theoretical studies [57–59] showed that the distribution of the θ s, f(θ), is generally skewed to the right, often with a mass close to 1. The magnitude of this mass depends on the LD statistic used and on the study sample size. This property of θ, made us to assume that f(θ) is composed of two underlying distributions, so that
the θ s being drawn from f_{1} when the SNPs are in LD, from f_{0} otherwise.
For given f_{0}, f_{1}, and γ, and assuming conditional independence of the θ s, the likelihood is
The assumption of conditional independence does not correspond closely to how the data are generated, and is made pragmatically, to simplify an otherwise nearly intractable problem. We consider it unlikely that this assumption will significantly affect the accuracy of the classification, although it may affect the uncertainty assessment. Alternatively, one can model the joint distribution of haplotypes directly and address blocking, for example, as a model selection problem [60]. This approach is more realistic but not yet scalable to the number of SNPs generated by current technology.
Because of the onetoone correspondence between γ and {Θ_{0}, Θ_{1}}, the (1) can be written as
f_{0} can be estimated nonparametrically by randomly permuting the genotypes between subjects. LD is estimated from the genotype distribution via an EM algorithm [61], and the empirical ${\widehat{f}}_{0}$ is finally estimated with a kernel smoothing method [62]. As permutation affects LD estimation, these steps were repeated several times and the final estimate ${\widehat{f}}_{0}$ of the density was the average of each of the densities, evaluated on a grid of 1000 percentile points.
Let's assume θ ∈ Θ_{1} follows a Beta distribution, θ_{ ij } Θ_{1}, α, β ~ Beta(α, β), such that
with α > 0, β > 0, and assume that β > α to ensure that the mode of this distribution is greater than 0.5. Substitutions and simple algebra allows to write the loglikelihood l_{ θ }(γ, α, β ${\widehat{f}}_{0}$, f_{1}) as
with the unknown parameters being α, β and γ.
The (4) was explored by means of a MetropolisHastings algorithm, using uniform priors on all unknowns, within the constraints described above (see the Appendix for a detailed description of the algorithm). With respect to the block border vector, starting values can be chosen using a threshold criteria [10] in order to have a block border where θ_{i, i+1} θ_{i+1, i+2}> τ, with τ that can be defined by the user on the basis of the particular LD statistic being used.
The algorithm was tested using the absolute value of the Lewontin's Dprime, D', and the square of the correlation coefficient for 2 × 2 tables, r^{2} [56, 63, 42]. Indeed, the described approach applies to any measure of LD between two loci.
Several tests demonstrated that, when a sufficient number of iteration is performed, starting values do not influence the results. The posterior distribution for (γ, α, β) was estimated after eliminating the first half of the Markov Chain, as burnin. The chain can be used to estimate the vector of the S + 1 probabilities of each point being a block border. Partitions at varying probability cutoff can be derived from these estimates.
Software and blocking algorithm definitions
MATILDE was written in the R language and requires the package genetics [64]. The software is available at http://astor.som.jhmi.edu/~gp/software/matilde/. Haploview 3.2 [24] was used for: (i) LD map estimation, (ii) genotype data cleaning, (iii) block partitioning, and (iv) for estimating the within block haplotype distributions. The blocking methods implemented in Haploview that were used in our analysis were the following: DprimeCI is the method proposed by Gabriel et al. [15] and based on the D' statistic; SSD is the Solid Spine of LD method (for a detailed description see the support documentation of the software at http://www.broad.mit.edu/mpg/haploview/); 4Gamete is the "Four Gamete Rule" by Wang et al. [25], which assumes that a recombination took place when all the four possible twomarker haplotypes between couples of contiguous SNPs occur. HapBlock v3.0 [23] was used for the limited haplotype diversity approach suggested by Patil et al. (Hapblock) [13]. While the original method was based on a greedy algorithm which did not ensure an optimal solution to the problem of block partitioning, the program is based on the dynamic programming algorithm for haplotype partitioning introduced by Zhang and colleagues [22] which guarantees to find an unique optimum. Hapblock provides the possibility to use one of three definitions of haplotype blocks: we selected the "common haplotype" option. Under this definition, "a set of consecutive SNPs with size one or more forms a block if the number of common haplotypes account for at least a percent of all the observed haplotypes (see the manual available at the software's homepage http://wwwhto.usc.edu/msms/HapBlock/ for more details). For the parameters α and β we used the recommended values of 5% and 80%, respectively. R 2.6.0 [65] was used to perform the whole analysis and to interface Haploview and Hapblock.
Descriptive analysis of block partitioning approaches
To facilitate the comparison between different methods, we defined all the single SNPs outside blocks to be blocks by themselves. This is required because with DprimeCI, 4Gamete, and SSD the SNP blocking may not be exhaustive of all the SNPs in the series. To explore the sensitivity to sample size we obtained samples of 200, 400, 600, 800, and 1000 then by drawing, with replacement, the 45 original subjects, leaving their SNP profiles unchanged, to preserve the LD structure. Empirical block structures of the study chromosomal region were obtained from all methods for each sample size. A much finer inspection was run for MATILDE to assess the performance under different cutoff levels (0.01, 0.02, ..., 0.1, ..., 0.9, 0.95, ...). The variability of the number of estimated blocks was assessed via the coefficient of variation (CV). The agreement between methods was assessed through the κ statistics [66] on the number of shared break points.
Comparison of performance in association studies
In our simulations, we generated casecontrol studies each including a single disease SNP. For each SNP in the sequence, we created several artificial casecontrol studies each with a 1:1 ratio of cases to controls. For both dominant and recessive genotypephenotype association models, subjects were classified into risk allele carriers and non carriers; then subjects were assigned to cases or to controls in a random way, satisfying the constraint of a prespecified Odds Ratio (OR), that is the proportion of risk allele carriers in cases and controls was fixed in advance. ORs used are 1.2, 1.4, 1.6, 1.8, and 2.0. This was repeated for five choices of sample sizes. In this way, we covered a wide spectrum of scenarios, while preserving the empirically observed LD.
Block partition and haplotype distribution were estimated on the pooled samples. In this way we could reuse the partitions estimated in the previous section. Within block haplotype distributions were estimated using the EM algorithm [61], separately for cases and controls. Subject's chromosomes were considered to be independent so that each subject carried two haplotypes. Because blocks were determined without consideration for case status, they are not optimized statistically to maximize the block associations.
Our analysis proceeds as follows: given a haplotype block, we estimated the haplotype. Each subject contributes two phased haplotypes, one for each chromosome. Using the Likelihood Ratio Statistics (LRS) we compared the distribution of haplotypes in cases and in controls. Specifically, within the k^{th}block, the LRS ${G}_{k}^{2}=2{\displaystyle {\sum}_{i=1}^{2}{\displaystyle {\sum}_{j=1}^{{m}_{k}}{n}_{ijk}log}\left(\frac{{n}_{ijk}}{{\nu}_{ijk}}\right)}$ was used to test the hypothesis of independence of the haplotype distribution in cases (i = 1) and in controls (i = 2), with m_{ k }being the number of observed haplotypes in the k^{th}block, n_{ ijk }the observed frequency of the haplotype j in the group i, ν_{ ijk }the expected frequency of the haplotype j in group i under independence. For large sample sizes, ${G}_{k}^{2}~{\chi}_{{m}_{k}1}^{2}$[67]. Because the choice of the best method, on the basis of genotypephenotype association, depends on the study goals, the efficiency of the block partition algorithms was ranked under different criteria. First, for each block partitioning method, G^{2} and the relative pvalue were estimated; then, the pvalues were sorted in descending order: p = {p_{(1)}, ..., p_{(k)}, ..., p_{(K1)}, p_{(K)}}, with K being the number of blocks. In the following we define k* as the index of the block containing the SNP that is truly associated with the disease. For singleSNP analyses, we examined two strategies. The first one is to consider each locus as a block of length one, and apply the procedure above. For example if we have 3 subjects with genotypes, 'AA', 'Aa' and 'Aa', respectively, then the marginal allelic distribution is 'A' with frequency 4 and 'a' with frequency 2. This type of distribution will be compared across cases and controls using the LRS. We refer to this as allelebased singleSNP analysis. The second one is a genotypebased singleSNP analysis, where the marginal allelic distribution is replaced by the genotype distribution, that is: AA with frequency 1, Aa with frequency 2. The reason for considering the allelebased analysis is to allow a fair comparison with other blocking approaches, where a genotypebased analysis would have been too onerous to implement. Sensitivity/specificity comparisons are based on mimicking the association testing situation. For each method, block, and simulated dataset, we declare a positive if the pvalue, after multiple testing adjustment with the BenjaminiHochberg method [29], is smaller than .05. In more detail, let T_{k, j}be an indicator variable for the k^{th}block at the j^{th}simulation: T_{k, j}= 1 when the null hypothesis is rejected, 0 otherwise. Let J be the number of simulations and ${k}_{j}^{\ast}$ the indicator for the right block at the j^{th}simulation, then $Se={\displaystyle {\sum}_{j=1}^{J}{T}_{{k}_{j}^{\ast},j}/J}$ is the Sensitivity, that is the probability of deciding that the block k* contains the right SNP, when this is true. The Specificity is the probability of deciding that a block does not contain the right SNP when it actually does not contain the SNP. Thus $Sp=\frac{1}{J}{\displaystyle {\sum}_{j=1}^{J}S{p}_{j}}$, where $S{p}_{j}=\frac{1}{{K}_{j}1}{\displaystyle {\sum}_{k=1,k\ne {k}_{j}^{\ast}}^{{K}_{j}}(1{T}_{k,j})}$ is the specificity at the j^{th}simulation. To assess the behavior of MATILDE at different probability cutoffs (that is the probability to classify a specific location as a block border), a Receiver Operating Characteristic (ROC) curve fitted by means of a local polynomial regression (loess), each point of the curve being the sensitivity/(1specificity) combination for one specific probability cutoff. At this scope we used the function loess.smooth implemented in the R package stats.
RELATIVE POSITION OF THE CORRECT BLOCK
From the standpoint of evaluating the quality of the dimension reduction methodology, it is useful to reward approaches that give a high ratio R = (k*  1)/K, with R ∈ [0, (K  1)/K]. This statistic is a way to reward the method which is faster in finding the area where the right SNP is, irrespective of the dimension of blocks.
RELATIVE POSITION OF THE CORRECT SNP
When the dimension of blocks matters, it could be more interesting to count the number of SNPs classified as good as, or better than, the right SNP, that is $B={\displaystyle {\sum}_{k={k}^{\ast}}^{K}\#{B}_{k}}$, where #B_{ k }is the number of SNPs in the k^{th}block. B is the number of SNPs that should be screened before discovering the true SNP, thus the smaller the B the better the method.
Appendix
Description of the MATILDE's core algorithm.
The MetropolisHastings algorithm
Here the t^{th}iteration of the MetropolisHastings algorithm used to explore the (4) is described. The parameters (γ_{t1}, α_{t1}, β_{t1}) were updated in three steps as follows:
1^{st}step
i) sample γ_{ t }as described below;
given γ_{ t }, split Θ into ${\Theta}_{0}^{\ast}$ and ${\Theta}_{1}^{\ast}$
compute l_{ t }(γ_{ t }, α_{t1}, β_{t1} ${\widehat{f}}_{0}$, f_{1}); r = exp {l_{ t } l_{t1}};
ii) sample u ~ U(0, 1);
if u <min(r, 1) then {Θ_{0}, Θ_{1}} ← {${A}_{0}^{\ast}$, ${A}_{1}^{\ast}$};
else γ_{ t }← γ_{t1}and l_{ t }← l_{t1};
2^{nd}step
i) sample β_{ t }~ U(β_{t1} 1, β_{t1}+ 1);
compute l*(γ_{ t }, α_{t1}, β_{ t } γ_{t1}, f_{1}); r = exp {l*  l_{ t }};
ii) sample u ~ U(0, 1);
if u <min(r, 1) then l_{ t }← l*;
else β_{ t }← β_{t1};
3^{rd}step

i)
sample a* ~ U(α_{t1} 1, α_{t1}+ 1);
α_{ t }← max(β_{ t }, α*);
compute l*(γ_{ t }, α_{ t }, β_{ t } γ_{t1}, f_{1}); r = exp {l*  l_{ t }};
ii) sample u ~ U(0, 1);
if u <min(r, 1) then l_{ t }← l*;
else α_{ t }← α_{t1}.
Sampling the block border vector
At each iteration, t, the key point is the proposal of the new block border vector, which is sampled as follows: first, let's decide either to move a boundary (i) or to change the number of blocks. Option (i) corresponds to changing the size of two neighboring blocks, option (ii) corresponds to joining or splitting two neighboring blocks. The choice is done by sampling from a Bernoulli(p), with p defined by the user on the basis of sample size and number of SNPs.
Under the option (i), one of the existing boundaries, γ_{t,1}··γ_{t, S+1}, is sampled with equal probability; then the border is moved one step to the left or to the right at random: of the two blocks sharing the boundary, one will increase its size of one SNP, while the other will be shortened by one. When this move is chosen, the total number of blocks does not change.
Under the option (ii), one of two actions is sampled with equal probability: I) split one block: one block is sampled at random and one point inside the block is also chosen at random and turned into a border, generating two contiguous and smaller blocks; II) modify a random value of γ_{ t }: one point, γ_{t, i}, between γ_{t,2}and γ_{t, S}, is randomly chosen; if γ_{t, i}= 0 then γ_{t, i}← 1 (this means to join two contiguous blocks into a bigger one), else γ_{t, i}← 0 (this is equivalent to splitting one block into two smaller ones).
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Acknowledgements
IR was supported by NIH grant CA 074841, DMF was supported by the grant R01AG020688 from NIA, and GP was supported by the NSF grant DMS034211.
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The scientific motivation and statistical models were defined by CP and GP with the support of IR for the computational expertise and MDF for epidemiological expertise. Simulations were performed by CP. The article was written mainly by CP with substantial contribution by all the other authors. The project was supervised by GP.
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Keywords
 Linkage Disequilibrium
 Haplotype Block
 Likelihood Ratio Statistics
 Linkage Disequilibrium Block
 Block Partitioning