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Nucleic Acids Research 2007 35(17):e113; doi:10.1093/nar/gkm621
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Nucleic Acids Research, 2007, Vol. 35, No. 17 e113
© 2007 The Author(s)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/2.0/uk/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Methods Online

Efficacy assessment of SNP sets for genome-wide disease association studies

Andreas Wollstein1,2, Alexander Herrmann2,3, Michael Wittig2, Michael Nothnagel4, Andre Franke2, Peter Nürnberg1, Stefan Schreiber2, Michael Krawczak4 and Jochen Hampe2,3,*

1Cologne Center for Genomics, Cologne, 2Institute of Clinical Molecular Biology, Christian-Albrechts University, 3Ist Department of Medicine and 4Institute of Medical Informatics and Statistics, Christian-Albrechts University, University Hospital Schleswig-Holstein Campus Kiel, Kiel, Germany

*To whom correspondence should be addressed. Tel: +49 431 597 1246; Fax: +49 431 597 1842; Email: J.Hampe{at}1med.uni-kiel.de

Received May 31, 2007. Revised July 30, 2007. Accepted July 30, 2007.


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
The power of a genome-wide disease association study depends critically upon the properties of the marker set used, particularly the number and physical spacing of markers, and the level of inter-marker association due to linkage disequilibrium. Extending our previously devised theoretical framework for the entropy-based selection of genetic markers, we have developed a local measure of the efficacy of a marker set, relative to including a maximally polymorphic single nucleotide polymorphism (SNP) at the map position of interest. Using this quantitative criterion, we evaluated five currently available SNP sets, namely Affymetrix 100K and 500K, and Illumina 100K, 300K and 550K in the CEU, YRI and JPT + CHB HapMap populations. At 50% relative efficacy, the commercial marker sets cover between 19 and 68% of the human genome, depending upon the population under study. An optimal technology-independent 500K marker set constructed from HapMap for Caucasians, in contrast, would achieve 73% coverage at the same relative efficacy.


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
Genome-wide association studies with large sets of single nucleotide polymorphisms (SNP) (1) are a new option for mapping the genetic variants underlying complex human diseases. However, the power and cost-effectiveness of such studies depends critically upon the properties of the SNP sets used. Consequently, the choice between one of the commercially available marker panels and the construction of a new set is of strong practical significance. No objective criteria other than descriptive measures (e.g. marker number) have so far been used to compare the utility of genome-wide marker sets. More importantly, any sensible assessment of a marker panel requires that recent discoveries about the biology of meiotic recombination are appropriately taken into account (2–4). For example, it has been shown (2) that the ‘geodesy’ of the human genetic map is fairly homogenous above the centi-Morgan level, but that the correlation between physical and genetic distance is weak at a finer scale, due to rapidly evolving recombination hotspots. Consequently, SNP selection strategies that are based upon the assumption of static linkage disequilibrium (LD) blocks, or that merely employ pairwise LD, may result in sub-optimal marker sets.

The utility of a marker set for disease association analysis is determined by a number of factors, including marker number, informativity and spacing, in addition to the local level of LD. In practice, genotyping technologies may pose serious restrictions upon the usability of an individual SNP, irrespective of whether its inclusion might be desirable or not. If such limitations can be ignored, however, then the utility of a marker set should ideally be evaluated by a criterion that:

  1. allows the assessment of the coverage of a genomic region in a single quantity,
  2. is computationally practicable,
  3. is applicable to the limited genotype information typically available for large marker sets and
  4. draws upon a theoretical framework that allows meaningful interpretation of the numerical results.
Shannon entropy (5) is a well-established mathematical concept for assessing the utility of genetic markers. We have recently devised an entropy-based SNP selection approach (6) that can in principle be adapted to a genome-wide setting. Furthermore, the methodology facilitates estimation of the relative, region-specific efficacy of a given marker set by {tau}, a quantity that approximates to the relative sample size required to map a causative variant at a given map position, compared to including a maximally polymorphic SNP at the same position (see Methods section). We calculated {tau} across the genome using publicly available genotype data for HapMap (Phase 2, built 35) (7) and for the five commercial marker sets of Affymetrix (8) (100K and 500K) and Illumina (9) (100K, 300K and 550K). The results were compared to an ‘ideal’ SNP set constructed from HapMap via entropy-based marker selection.


   METHODS
TOP
 ABSTRACT
 INTRODUCTION
 METHODS
RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
Estimation of inverse swept radii
Parameter {varepsilon}, which denotes the inverse of the swept radius, was used as a local measure of LD strength (10,11) and was estimated from HapMap genotype data on the basis of all markers with a minor allele frequency ≥10%. To this end, pairwise haplotype frequencies were estimated from the genotype data using an EM algorithm. Then, the pairwise allelic association was quantified as

Formula 1 (1)
where P is the haplotype frequency matrix (pij)i,j=1...2, Q = p11 + p12 and R = p11 + p21 (10), and where det|P| denotes the determinant of P. Marker-specific {varepsilon} values were estimated by a log-linear regression analysis of {rho} and the physical distance to all other markers Xi in a 500 kb window surrounding the marker Y of interest (12), i.e. by fitting model log({rho}) = –{varepsilon}·|xi – y| to marker locations xi and y.

Here and in the following, we assumed that the population of interest was characterized by monophyletic inheritance and by a lack of association between unlinked loci, a simplification of the original model of LD decay that was justified by empirical observations made for autosomal markers in Europe and the US (11).

At inter-marker positions z1 < z < z2, {varepsilon}(z) was estimated by linear interpolation, i.e.

Formula 2 (2)

Entropy-based SNP selection
We have previously devised a method for assessing the utility of marker sets for disease association studies (6), based upon Shannon entropy (5). In brief, for a locus X with k alleles of frequency pi (i = 1...k), entropy H(X) is defined as

Formula 3 (3)

For the purposes of disease association analysis, a genomic region is assumed to be covered by markers X1, ... , Xn at map positions x1 < ... < xn. Then, the problem of SNP selection reduces to deciding, on the basis of existing genotype or haplotype data, which single marker out of some additional markers Y1, ... , Ym to include in order to maximize the mapping utility of the extended panel. Without loss of generality, it can be assumed that this choice is confined to maximizing the utility of the marker set in a given interval, centred at map position z. A utility score {kappa}(Y:X, z) is then constructed that reflects the benefit, with respect to mapping a disease gene at position z, of adding Y to a single marker X,

Formula 4 (4)

Here, H(Y|X) = H(X,Y) – H(X) denotes the conditional entropy of Y given X. The quantity in formula (4) can be calculated directly from pairwise haplotype frequencies, known swept radii and known marker locations. The best marker to include into the existing marker panel X1, ... , Xn is then chosen according to

Formula 5 (5)

Application to genome-wide marker panels
Application of the above-mentioned framework to large-scale genome-wide data sets poses additional computational problems since the comprehensive evaluation of all pairwise haplotype frequencies, as required by formulas (4) and (5), is no longer feasible. Thus, {kappa}(Y:X, z) was replaced by

Formula 6 (6)
when the distance between Y and z exceeded 3/{varepsilon}(z) (11). In this way, the number of pairwise haplotype frequency estimations was limited and the computing time scaled linearly (instead of quadratically) with marker number. Formula (6) was also used for selecting the first few markers on a given chromosome, successively breaking the chromosome down into shorter intervals by applying formula (6) to the corresponding interval centers. Marker selection according to formula (4) commenced for an interval when it was shorter than three times the internal median swept radius.

Evaluation of genome-wide marker sets
Following Hampe et al. (6), we define criterion {tau}(z) for the local evaluation of a marker set around map position z as

Formula 7 (7)
where qX(z) is the minor allele frequency of that marker, X(z), that maximizes the right-hand side of formula (7) [note that {tau}(z) is similar, but not equivalent, to 1 – {kappa}min(z) as defined in the original paper (6)]. Since

Formula 8 (8)
equals the predicted allelic association (11) between X and a maximally informative biallelic marker Z at map position z, it follows that

Formula 9 (9)
On the other hand, the number n of individuals required to detect association {rho} between X and Z at significance level {alpha} and with power 1 – ß is approximately equal to

Formula 10 (10)
where z1–{alpha}/2 and zß are the respective quantiles of the Gaussian distribution (for a detailed derivation of formula (10), see Appendix). For any two marker sets A and B, let {tau}A(z) and {tau}B(z) be the {tau} values obtained with respect to the same location z. Then,

Formula 11 (11)
which implies that {tau}(z) is a good approximation of the relative efficacy of a marker set, measured by the inverse of the sample size required to map a maximally informative SNP at position z.

Computer implementation
The methodology described above has been implemented into a suite of JAVA programs interacting with a MySQL relational database for the storage of genotypes and intermediate results. Since the HapMap data set was the most exhaustive one, calculation of swept radii was based upon these markers and genotypes. The software is available as a web service under http://www.ikmb.uni-kiel.de/snpselection/.

SNP data sources and genotyping
Caucasian genotype data for HapMap (Phase II, built 35), Affymetrix 100K and 500K were retrieved from the respective web sites (www.hapmap.org, www.affymetrix.com). The marker identities of the Illumina 100K, 300K and 500K sets were retrieved from the Illumina website (www.illumina.com); the corresponding genotypes were taken from HapMap or from the Illumina website.


   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS
DISCUSSION
 APPENDIX
 REFERENCES
 
Quantity {tau} measures the relative efficacy of a given marker set to map a causal variant at a specified map position, compared to including a maximally polymorphic SNP at the very same position (see Methods section). Therefore, {tau} = 1 corresponds to full local efficacy of a marker panel whereas {tau} = 0 indicates that no information can be extracted locally. For the purpose of comparing different marker sets, {tau} was calculated here at 10 kb intervals along the human genome (NCBI build 34), except for annotated gaps, heterochromatic, telomeric and centromeric regions. Y chromosomal SNPs were also excluded. Variation of the interval size between 5 and 10 kb for chromosomes 3 and 19 did not yield notably different results (data not shown). It may be argued that, in many instances, only markers located in gene-coding regions are of practical interest for genome-wide disease association studies. In order to take this issue into account, ‘coding’ regions were defined here as all sequences containing one of the ‘RefSeq’ genes of the Golden Path (http://genome.ucsc.edu), including exons, introns and 10 kb of flanking sequence. Marker sets were evaluated on the basis of publicly available genotype data (Table 1). Our analyses included CEPH samples from Northern and Western Europe (CEU), from Yoruba in Nigeria (YRI) and from Japanese and Han Chinese people (JPT + CHB).


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  		Table 1. Sources of marker and genotype data

 
Swept radii 1/{varepsilon} were estimated for different genomic regions on the basis of the available HapMap genotype data. As is exemplified by chromosomes 12 and 19 in the CEU population (Figures 1A and 2A), the distribution of 1/{varepsilon} was found to vary considerably along chromosomes and therefore resembled recently published recombination plots in this respect (2). The median 1/{varepsilon} of ~500 kb corresponds closely to previous estimates (11). A graphical representation of all swept radii and {tau} values obtained in the present study is available at http://www.ikmb.uni-kiel.de/snpselection. In the following, our results will be exemplified by a more detailed consideration of chromosomes 12 and 19, which are typical in terms of their size and gene density.


Figure 1
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  		Figure 1. Distribution of local LD and SNP set efficacy on chromosome 19 in the CEU population. Panel A: swept radius 1/{varepsilon} as estimated around each marker from the HapMap genotype data (median 1/{varepsilon}: 191 kb, interquartile range: 166–230 kb). Panel B: relative efficacy {tau} of the HapMap set, calculated at 10 kb intervals, excluding gaps, centromers, telomers and heterochromatin. Panel C: as Panel B, but for the Affymetrix 100K set. Note: physical positions (in Megabases, Mb) are given according to NCBI build 35.

 

Figure 2
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  		Figure 2. Distribution of local LD and SNP set efficacy on chromosome 12 in the CEU population. Panel A: swept radius 1/{varepsilon} as estimated around each marker from the HapMap genotype data (median 1/{varepsilon}: 181 kb, interquartile range: 159–214 kb). Panels B and C: see legend to Figure 1.

 
When all 180 613 HapMap SNPs on chromosome 12 were included in the analysis, {tau} values larger than 0.5 were obtained for most of the chromosome (Figure 1C). By contrast, the 5253 chromosome 12 markers of the Affymetrix 100K set left many intervals with {tau} close to 0, indicating low efficacy (Figure 1B). Similar results were obtained for chromosome 19 (Figure 2). Figures 3 and 4 provide an overview of the distribution of {tau} along the coding’ regions and the full genomic sequences of the two chromosomes. When all HapMap SNPs were included, the median {tau} values obtained were 0.70 (interquartile range: 0.56–0.82) for chromosome 12 and 0.66 (interquartile range: 0.52–0.78) for chromosome 19. By contrast, the best commercial marker sets yielded a median {tau} of 0.59 (interquartile range: 0.45–0.73) for chromosome 12, and of 0.56 (interquartile range: 0.41–0.70) for chromosome 19 in the case of Illumina 550K, and of 0.52 (interquartile range: 0.36–0.67) for chromosome 12 and of 0.41 (interquartile range: 0.26–0.58) for chromosome 19 with the Affymetrix 500K set.


Figure 3
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  		Figure 3. Relative efficacy of SNP sets on chromosome 12 in the CEU population. For each marker set, the blue histogram depicts the distribution of relative efficacy {tau} in the full genomic sequence and the coding regions, respectively (for definition, see main text). Frequencies have been normalized such that the modal frequency equals unity. The distribution of {tau} as obtained for a similarly sized, hypothetical marker set, constructed from HapMap by entropy-based marker selection, is included for each marker set (open histograms).

 

Figure 4
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  		Figure 4. Relative efficacy of SNP sets on chromosome 19 in the CEU population. For details, see legend to Figure 3.

 
A comparison of the two commercially available 100K sets revealed the impact of both, the genotyping technology and the selection strategy upon the mapping efficacy. If only the coding sequence was considered on chromosome 12, the median {tau} for Affymetrix 100K was 0.21 (interquartile range: 0.08–0.41), as compared to 0.44 (interquartile range: 0.27–0.61) for Illumina 100K (Figure 4). The Illumina 100K set, designed primarily for a good coverage of sequences containing annotated transcripts, provides essentially the same efficacy for the coding sequence on this gene-rich chromosome as the Affymetrix 500K set (median {tau}: 0.41, interquartile range: 0.26–0.58). Similar, albeit less pronounced results were obtained for chromosome 12 (Figure 3). A genome-wide overview of the efficacy of all SNP sets is given in Table 2 and, on a chromosome-wise basis, in Figure 5.


Figure 5
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  		Figure 5. Chromosome-specific estimates of relative SNP set efficacy in full genomic (Panel A) and coding (Panel B) sequences. Chromosome-wide median {tau} values and interquartile ranges obtained for the CEU population are plotted in chromosomal order.

 

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  		Table 2. Median estimated efficacy and coverage of the human genome (excluding the Y chromosome) in different populations, provided by different marker sets.

 
Let Cx denote the local coverage of a chromosome or chromosomal region at relative efficacy x, achieved by a particular marker set (i.e. Cx equals the proportion of a given genomic region for which {tau} ≥ x). For the coding regions of chromosome 12, for example, C0.5 = 0.16 for the Affymetrix 100K set and C0.5 = 0.48 for Affymetrix 500K (Figure 3). This means that the two sets cover 16 and 48% of the gene containing sequence, respectively, at 50% or higher relative efficacy. At 80% relative efficacy, the respective figures decrease to 2 and 8%, respectively. A genome-wide overview of the coverage of the different marker sets at 50 and 80% efficacy is given in Table 2 and, on a chromosome-wise basis, in Figures 6 and 7.


Figure 6
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  		Figure 6. SNP set coverage of full genomic (Panel A) and coding (Panel B) sequences at 50% relative efficacy. The chromosome-wide coverage C0.5 is plotted in chromosomal order. HYP 500K: hypothetical, optimal marker set constructed from HapMap so as to include the same number of SNPs per chromosome as the Affymetrix 500K set.

 

Figure 7
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  		Figure 7. SNP set coverage of full genomic (Panel A) and coding (Panel B) sequences at 80% relative efficacy. The chromosome-wide coverage C0.8 is plotted in chromosomal order. HYP 500K: hypothetical, optimal marker set constructed from HapMap so as to include the same number of SNPs per chromosome as the Affymetrix 500K set.

 
The HapMap markers provide the ‘gold standard’ for the currently achievable coverage of the human genome with informative SNPs. If a fully flexible genotyping technology were available, optimal SNP sets could thus be constructed from HapMap using, for example, entropy-based marker selection. As exemplified for chromosomes 12 (Figure 3) and 19 (Figure 4), such customized panels would significantly improve the coverage provided by a given number of markers. With 5253 SNPs on chromosome 12, which corresponds to the size of the respective Affymetrix 100K set, HapMap would yield C0.5 = 0.81, i.e. a more than four times higher coverage than the commercial product. Replacing the Affymetrix 500K set by a similarly sized HapMap set would increase C0.5 from 0.48 to 0.81 whereas C0.8 would increase from 0.07 to 0.21.

More detailed information about the present study can be found on our web server at http://www.ikmb.uni-kiel.de/snpselection. The same site also provides routines for the customized selection of optimal SNP sets from HapMap build 19, using the available Caucasian, Asian and Yoruba genotype data.


   DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS
 DISCUSSION
APPENDIX
 REFERENCES
 
Justification of an entropy-based SNP selection framework
Currently available technologies do not allow full re-sequencing of the human genome in samples that are appropriately sized for mapping complex disease genes. Instead, the success of genome-wide association studies depends heavily upon the presence of sufficient LD between the causal variant(s) and at least one marker in the study panel. Whilst the level of inter-marker LD may indeed be fully known, however, LD is inherently unknown in relation to the causal variant itself, and therefore has to be extrapolated. This implies that the markers of an ideal study panel should be selected in such a way as to maximize the information extracted about any possible location of a disease variant in the genome.

Under a model of spatially homogenous LD, with constant recombination and mutation rates and a common evolutionary history shared by all chromosomal regions, disease association markers would ideally be spread evenly along the genome. However, the systematic evaluation of both LD and local recombination rates has revealed an inherent non-uniformity of these characteristics (2,13,14). Thus, recombination rates differ between chromosomal segments and between populations, which implies that even closely linked genomic regions may be of substantially different ancestry in individuals from one and the same population (15). Consequently, the relationship between LD and physical distance is complex, and combinations of unevenly spaced SNPs may prove more informative than equally spaced markers, depending upon the genomic region of interest.(16)

Previous studies have suggested the existence of ‘haplotype blocks’, i.e. clearly identifiable chromosomal segments that are characterized by a reduced rate of recombination, low haplotype diversity and a high level of internal LD (2–4). In addition, haplotype-tagging SNPs (htSNPs) have been proposed to be capable of identifying haplotypes for substantially larger marker sets from within these blocks (17–19). The practical relevance of this block concept arises from the expectation that htSNPs extract sufficient information from an LD block with respect to co-ancestry while, at the same time, reducing genotyping costs (2–4). A number of computational methods for the construction of htSNP sets have been developed (16,19,20) but for these techniques to be efficient, detailed knowledge of the extended haplotype frequency distribution in the population of interest is required. Moreover, the size and location of haplotype blocks depend critically upon the SNP density and the method of marker selection (13,14,21–24). Therefore, haplotype tagging appears feasible only when large samples and appropriate family structures are available for the necessary (deterministic or probabilistic) haplotype assignments, the reliability of which decreases with the number and complexity of the haplotypes present (25–27).

The idealized picture of static LD blocks, separated by hot spots of recombination, (28) has recently been challenged by new insights into the biology of meiotic recombination (2–4). The correlation between physical and genetic distance is weak below the centi-Morgan level so that the inference of marker genotypes from htSNP haplotypes is far from being reliable (24). Moreover, block-like structures may even occur merely because of genetic drift (29). It thus appears as if the tacit assumption underlying the use of the haplotype block concept for disease association mapping, namely that all genetic variation in a block follows the same hierarchical pattern, is often not fulfilled. As a consequence, the usefulness of htSNPs for such studies has generally been questioned (30–32).

SNP selection based upon pairwise LD alone has been suggested to avoid the conceptual and computational problems of extended haplotype (or ‘block’) approaches. The use of some SNPs as proxies for other SNPs that are in high LD with the former (2–4), measured by r2, reduces the redundancy of a SNP set. Thresholds for r2 of at least 0.8 are generally regarded as sufficient to provide good marker coverage for association studies (21,33–38). The rationale underlying the pairwise approach is the expectation that high inter-marker LD translates into high LD between some of the markers and potentially causative variants, an assumption that is however unlikely to hold true in general (2–4). Selection of SNPs based upon pairwise LD alone is therefore likely to perform well only with a particularly high and uniform SNP density (6). Irrespective of the approach taken, the inherently unknown LD between markers and unknown causal variants has to be extrapolated in one way or another from both physical distance and the local strength of LD. However, marker selection based upon pairwise LD alone does not take distance or individual marker informativity into account. As a consequence, simple pairwise ‘haplotype tagging’ potentially leads to inhomogeneous marker spacing with less than maximum efficacy.

Here, we have adapted a recently proposed method for selecting maximally informative marker sets for association studies (6) to a genome-wide comparison of marker sets. The original approach combines the information content, physical spacing and pairwise LD of individual markers with information on the local LD structure, extracted from available data in the form of swept radii (10,11). All of these determinants are included in a single, position-specific utility measure that corresponds to the distance-weighted haplotype entropy of the marker set, approximated however by a pairwise score of the same form (see Methods section). The approach is therefore not affected by the computational and conceptual problems of block-based methods and, at the same time, takes physical distance and local LD structure into account when extrapolating LD between markers and causal variants from pairwise inter-marker LD. An extension of the approach has led to the development of a quantitative criterion ({tau}) that approximates the efficacy of a given marker set to map a disease-causing variant at a position of interest. It should be emphasized that the interpretation of {tau} as a measure of efficacy is only valid in relative terms, i.e. by comparison to the inclusion of a maximally polymorphic SNP at the site of the causal variant. In general, since the properties of the underlying disease model are unknown, no marker-based quantity can on its own provide information about the absolute power of a marker set to map genetic variants underlying a given phenotype.

Quality of currently available marker sets
Owing to recent successes (39) and its theoretical appeal (1), significant funds have been allocated to the concept of genome-wide association analysis in the context of various phenotypes. Researchers are however facing the practical problem of choosing the ‘right’ genotyping technology. In many countries, universal control genotype pools are in the process of being established, and these pools will pre-determine the choice of technology for future studies. Of the currently available marker sets, the Affymetrix 500K (C0.5 = 0.68, C0.8 = 0.19) and Illumina 550K (C0.5 = 0.79, C0.8 = 0.29) products provide the best genomic coverage in Caucasians. The Illumina 550K marker set provides a higher coverage than the 500K Affymetrix set, probably because of the higher flexibility of the Illumina genotyping technology. Pronounced differences between full genomic and ‘coding’ region coverage were only observed for the 100K sets, probably because of the relatively small marker numbers. The good ‘coding’ region coverage provided by the Illumina 100K set highlights the fact that this panel was primarily designed for gene-based association mapping. It should be emphasized, however, that all of the above conclusions were based upon the assumption that all markers were callable, and that practical factors such as genotyping quality, departure from Hardy–Weinberg equilibrium and DNA requirements could be neglected. Furthermore, interesting differences became apparent in terms of in different ethnic groups. Whilst their relative efficacy was approximately the same in the Caucasian and African populations, SNP coverage was notably poorer for all products for the East Asian populations.

The analytical method used here to compare the utility of different marker sets provides a means to weight the costs and benefits of closing gaps in a given marker set. Additional genotyping costs incurred by a flexible (and thus more expensive) genotyping method can be contrasted directly with the relative efficacy gained from using additional, customized SNPs. If genotyping costs would be negligible, the complete current HapMap set would provide 90% coverage of the genome with at least 50% relative efficacy, and 47% coverage with at least 80% relative efficacy. These figures represent the gold standard with which all other marker panels have to be compared. Interestingly, when our entropy-based SNP selection approach was used to construct an optimum SNP set, the size of the Affymetrix 500K product from HapMap, this technology-independent, hypothetical set would nearly double the coverage at 80% relative efficacy.

In summary, we have devised a methodology that helps researchers make rational choices between different marker sets for genome-wide disease association studies and to assess the trade-off between genotyping costs and gain in power when expanding existing marker sets. Furthermore, use of the {tau} criterion facilitates judging the position-specific ‘completeness’ of a genome-wide association study and may thus help to improve the practicability of complex disease gene mapping.


   APPENDIX
TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
REFERENCES
 
In general, the sample size n required to detect the difference between proportions {pi}1 and {pi}2 by means of a {chi}2 test can be approximated by

Formula 12 (A.1)
where {pi} = ({pi}1 + {pi}2)/2, {alpha} and 1 – ß are the significance level and power of the applied test, respectively, and z1–{alpha}/2 and z1–ß denote the corresponding quantiles of the Gaussian distribution (40). If q is the minor allele frequency of marker X, and if the two alleles of marker Z are equally frequent, then the corresponding haplotype frequency matrix equals

Formula 13 (A.2)
with

Formula 14 (A.3)
Furthermore, since Q = p11 + p12 = 0.5{pi}1 + 0.5(1 – {pi}1) = 0.5 and R = p11 + p12 = 0.5{pi}1 + 0.5{pi}2, it follows that

Formula 15 (A.4)
Solving Equations (A.3) and (A.4) for {pi}1 and {pi}2 yields {pi}1 = 1 – q(1 – {rho}) and {pi}2 = 1 – q(1 + {rho}), so that {pi} = 1 – q and {pi}1 – {pi}2 = 2q{rho}. Replacing {pi}1, {pi}2 and {pi} by these expressions in formula (A.1) yields

Formula 16 (A.5)
for sufficiently small {rho}. This proves formula (10) of the main text.


   ACKNOWLEDGEMENTS
 
This study was supported by the German Federal Ministry of Education and Research as part of the National Genome Research Network (01GS02105, 0313437A) and the MediGrid project (01AK803G), and by the German Research Council (Ha 3091/1-2). We are most grateful to Ulf Leser, Humboldt-University, Berlin, for helpful discussions and to Uwe Mordhorst and Marcus Will, Christian-Albrechts-University, Kiel, for computing support. Funding to pay the Open Access publication charges for this article was provided by the Medical Faculty of the University of Kiel.

Conflict of interest statement. None declared.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 

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  4. Jeffreys AJ, Neumann R, Panayi M, Myers S, Donnelly P. Human recombination hot spots hidden in regions of strong marker association. Nat. Genet. (2005) 37:601–606.[CrossRef][Web of Science][Medline]

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