To identify the optimum htSNP set for a given ϕ_T, the full enumeration strategy is a straightforward approach, but is computationally very intensive (Discussed by Dr. David Clayton, please see Electronic Database Information Section [1]). The greedy methods [Carlson et al., 2004; Stram et al., 2003] dramatically lighten the computational burden by reducing the number of SNP sets to be considered, which can also accommodate various user-specified ϕ levels. Similar to SMET, these approaches organize the search space into tree structures, but only grow the branch with the largest ϕ at each level of the tree. Therefore, these greedy methods traverse a much smaller subspace than the entire space of the full tree. Although the greedy algorithms can run faster than SMET due to its heuristic nature (note than the SMET algorithm traverses a sparse version of the full tree), their drawbacks include (1) there is no mathematical guarantee that the greedy algorithm can identify the optimal htSNP sets; and (2) when there are multiple optimum htSNP sets, the greedy algorithm may not identify them all. Recently, principle component analysis (PCA) has been applied in assessing multivariate SNP correlations to identify htSNPs [Horne and Camp 2004; Lin and Altman 2004].

However, the PCA algorithm does not directly take haplotypes into account and cannot guarantee finding the optimal sets. In contrast, SMET traverses all non-singular tree nodes (i.e. SNP sets) with the potential to be optimal htSNP set(s), and mathematically guarantees the global optimum. Moreover, SMET can identify multiple optimal htSNP sets if they exist. Recently, Ding et al. [2005] developed a computer program named htSNPer1.0 with a graphical user interface for characterizing the haplotype block structure and for selecting htSNPs. Similar to SMET, htSNPer1.0 first estimates haplotypes within each haplotype block, and based on the estimated haplotypes, htSNPs are selected according to three htSNPs performance criteria (α-percent coverage [Patil et al., 2001], explained proportion of Clayton's haplotype diversity (please see Electronic Database Information Section [2]), and weighted-average haplotype r² [Weale et al., 2003]) and four haplotype block definitions [chromosome coverage [Weale et al., 2003], average pairwise LD |D’| [Reich et al., 2001], estimated pairwise LD confidence limits [Gabriel et al., 2002] with minor modifications by Wall and Prichard [2003], and no historical recombination [Wang et al., 2002]). The htSNPer1.0 software package takes advantages of a novel tree-based heuristic algorithm called the Generalized Branch-and-Bound (GBB) algorithm to search the minimal htSNPs set. However, the GBB algorithm of htSNPer 1.0 does not use the exactly same ϕ criterion for htSNP selection as we used in SMET.

The reduction of computational steps by SMET over the full enumeration strategy becomes more substantial as the search tree depth becomes greater. This is because there is an exponential growth of the number of nodes (and hence the search space) for the exhaustive enumeration method to traverse when tree depth increases linearly. Generally speaking, the tree depth correlates with the haplotype block size (measured by the number of SNPs within the block). Thus, the advantage of SMET algorithm over the full enumeration method was more evident for htSNP selection for large haplotype blocks such as the two largest blocks of the UK Graves’ disease dataset. In comparison with the full enumeration method (e.g. htSNP program, please see Electronic Database Information Section [3]), the factor of computational step reduction by SMET becomes as large as 18 for the largest block in the UK Graves’ disease dataset - Block 8 with 26 SNPs and an average D’ of 0.777.

In our study, D’, a measure of LD between pairs of sites [Lewontin 1964], was chosen for defining haplotype blocks because it is directly related to the goal of detecting historical recombination, which is pivotal to the block concept, and because it can be applied directly to unphased diploid data. Two widely-cited previously performed studies - Reich et al. [2001] and Gabriel et al. [2002], used similar D’-based block definitions as that of ours. In particular, our haplotype block definition is based on minor modifications of Gabriel et al. [2002], which appears to perform reasonably well by controlling the random noise inherent in D’. In contrast, the r² measure of LD is typically used for tagging SNP selection rather than for block partitioning [Ahmadi et al., 2005; Goldstein et al., 2003].

As shown in Figure 4, ϕ measured by H sometimes cannot clearly distinguish different htSNP sets because of its narrow range of values. Comparatively speaking, the Shannon entropy (E) is a more sensitive measure of haplotype diversity coverage than H. Besides H and E, another metric of haplotype diversity, , was used in the tagSNP program [Stram et al., 2003]. Zhang et al. [2004] surveyed the choices of diversity measurements (e.g., E or ), and found they led remarkably consistent results. The tagSNP algorithm employs the “greedy stepwise inclusion method”, instead of an exhaustive search. Therefore, the tagSNP inherits the two drawbacks discussed in last paragraph. As acknowledged by Stram et al. [2003], this stepwise procedure does not fully explore the entire search space, and thus does not guarantee the finding of the globally “best pick” of htSNPs [Stram et al., 2003]. Figure 4 also shows that, when the htSNPs ratio (the size of the htSNP set/the size of the lossless htSNP set) increases, ϕ rises rapidly at the early stage, and then enters a plateau phase before reaching the maximum value. In the plateau phase, we gain little extra information and statistical power (Figure 8) by adding more tagging SNPs. Thus, appropriate haplotype diversity coverage is a key factor to be considered in designing cost-effective association studies.

Figure 6 presents the results of the assessment of the impact of the sample size for the developing panel, N_d, as an important parameter to be taken into consideration for htSNP selection. There are several reasons to choose an adequately large N_d. First, a small N_d leads to an instability in the performance of various haplotype phasing algorithms [Niu et al., 2002]. Second, when N_d is small, certain common haplotypes in the large-scale cohort may appear only once or may not even be sampled in the developing panel. Such “unlucky” haplotypes thus may be inadvertently removed before the htSNP selection step. Our simulation study suggested a N_d at least 50 (i.e. 100 chromosomes) in order to ensure a sufficient coverage of common haplotypes in the large-scale sample.

As shown in Figure 7, we found that selecting htSNPs based on a predefined, genuinely LD-based block structure could be more efficient than selecting htSNPs based on an arbitrarily-defined, pseudo-block structure. The gain in efficiency becomes more noticeable for high-LD blocks. The findings of this simulation are reasonable because the haplotype diversity for each pseudo-block is higher than that of its corresponding genuinely LD-based haplotype block, and the total number of htSNPs needed to tag each pseudo-block would be greater than the total number of htSNPs needed to tag each corresponding genuinely LD-based block. We regard the SMET algorithm to be more appropriately applied within LD-based blocks after the genomic segment of interest (regardless of gene-wide, region-wide, or chromosome-wide scales) was partitioned into LD-based blocks. The rationales are: (1) LD-based blocks typically represent haplotype blocks of common ancestry [Halldorsson et al., 2004], which are transmitted from generation to generation with little evidence of historical recombination. Thus, the LD-based “haplotype block model” has important implications for association mapping, because it implies that, by identifying htSNPs for each LD-based block, it is possible to predict the likely configurations of alleles at an unobserved SNP site within the same block. Indeed, this is one of the motivations of the International HapMap Project, which aims to produce a genomewide haplotype map that can be used for genomewide haplotype block detection and to streamline association mapping. (2) For an increasing number of linked SNPs, the accuracy of haplotype phasing results of various statistical algorithms of haplotype inference is decreasing [Niu et al., 2002], because the number of possible phases grows exponentially with an increasing number of SNPs [Niu, 2004]. This is especially the case when N_d (i.e. the size of the developing panel) is as small as 25 or 50, because the phasing accuracy also decreases pronouncedly with a decreasing sample size [Niu et al., 2002]. To overcome this obstacle, pairwise LD-based haplotype block partitioning was performed as a first step, and then SMET algorithm was applied just within each LD-based block such that the haplotype reconstruction is only necessary within each LD-based block of limited haplotype diversity, and the phasing results would be reasonably accurate for each individual block. However, if we consider each region or each entire chromosome as a single block (i.e. we totally disregard the underlying block structure) for selecting htSNPs using SMET, we have to directly reconstruct the haplotype phase for a large region or even a whole chromosome. The accuracy of haplotype reconstruction for such large blocks would then be computationally unstable which can lead to untenable results. Recently, Ahmadi et al. [2005] developed a “block-free” tagging SNP selection strategy based on a multiple-marker criterion – haplotype r², which selects tagging SNPs independently of any underlying haplotype block structure [Goldstein et al., 2003]. Their strategy required the minimum estimated r² between the tagged and tagging SNP set to be at least 0.85. Halldorsson et al. [2004] similarly developed a “block-free” tagging SNP selection strategy based on finding neighborhoods of a target SNP, which uses a metric that is in spirit similar to haplotype r². Although ignoring heuristically defined block boundaries, “long-range” LD can be captured by “block-free” tagging SNP selection strategies, such approaches also have a limitation: to achieve a high r² value, the frequencies of the SNPs must be matched that could be difficult to achieve in a “SNP-by-SNP” manner or even between the haplotypes defined by the tagging SNPs and the other SNPs, thus requiring an excessively large number of htSNPs [Goldstein et al., 2003].

We used the program HUDSON [Schierup and Hein, 2000a; Schierup and Hein, 2000b], which is based on the coalescent model, to simulate data to assess the impact of genetic diversity (haplotypic diversity) on htSNP selection efficiency. We found that the Pop II, the population with a longer evolution history and thus a greater genetic diversity, required 1.51 times more computational steps than Pop I. Thus, haplotypic diversity will determine the proportion of non-singular nodes in the dataset, i.e., a highly diverse population (e.g. the African sample) will have fewer redundant SNPs, and therefore the number of nodes traversed by the SMET algorithm will approach the number in the full enumeration approach.

What is the cost-effectiveness, in term of statistical power in association study, between htSNP sets of ϕ=80 or 90% and the lossless sets? We found that htSNP sets with a ϕ of 80% or 90% can achieve 87−95% of the statistical power attained by the lossless htSNP sets. More importantly, htSNP sets with ϕ = 80 or 90% incurred relatively insubstantial biases in relative risk ratio estimation (Figure 8). Because the size of the htSNP set is much smaller at ϕ = 80 or 90% than that of a lossless set (Figure 2), scientists can save considerable genotyping cost without much reduction of the statistical power or the relative risk ratio in detecting a haplotype-based association. It should be noted that the disease model and the association test we chose in our paper are both simple and straightforward, which solely serve the purpose of demonstrating the application of htSNPs selected based on the SMET algorithm. Thus, these were not necessarily the most realistic ones among all possible disease models and all possible types of association tests, and further extensive evaluations of the performances of the htSNPs under a spectrum of disease models using a variety of different association tests are clearly needed (such extensive evaluations would go beyond the scope of our paper).

Practically speaking, whether or not haplotype tests are more powerful than single-marker tests remains a highly debatable topic [Akey et al., 2002; Clark, 2004; Roeder et al., 2005; Schaid, 2004]. Using simulated case-control data sets, Nielsen et al. [2004] demonstrated that either approach has its merits: when moderate to high levels of multilocus LD exist, haplotype tests tend to be more powerful, which could be by a very large degree. Single-marker tests tend to prevail when pairwise LD is high (note that the power of the single-marker tests can be seen to rely on pairwise LD of the observed marker with the functional site). If single-marker tests are to be used, Nielsen et al. [2004] pointed out that a multiple-testing adjustment should be applied, which would reduce the power of single-marker tests. Based on standard chi-square statistics, the simulation studies of Akey et al. [2001] showed that the power of haplotype tests is influenced by critical population genetic parameters, such as genetic distances between the observed markers and the causative mutation, maker allele frequency, age of the causal mutation. Given a single founder mutation and the absence of phenocopy, haplotype tests are more powerful and more robust than single-marker tests in case-control studies. An example of the superiority of haplotype tests over single-marker tests is a study of the adenine phosphoribosyltransferase (APRT) deficiency [Kuno et al., 2004]. In single-SNP analyses, even at SNP loci close to the mutation site (APRT*J), no significant results were found; however, the use of haplotypes based on the haplotype block data gave sufficient significance, and thus, haplotype tests based on the haplotype-block structure is more powerful than single-marker analyses for the detection of disease-related loci. Because of the lack of consensus as to which (haplotype vs. single-marker) tests are more powerful in case-control studies, the trade-off between these two types of tests need to be weighed carefully on a case-by-case basis.