# Genome-wide association tests by using block information in family data

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## Abstract

By applying an association test to analyze the data sets from Genetic Analysis Workshop 15 Problem 3, we compare power using different haplotype-block information. The results from using both of the two different coding schemes show that the test using tight blocks with limited haplotype diversity within each block is more powerful than that using evenly spaced blocks, and the latter is more powerful than that using single-marker blocks. By using carefully chosen haplotype blocks, the power of association tests may be enhanced.

### Keywords

Code Scheme Haplotype Block Genetic Analysis Workshop Block Information Linkage Disequilibrium Measure## Background

Genome-wide association is a promising approach to mapping complex disease genes. Currently, either single-marker tests or haplotype-based tests are used to test association for genome-wide association studies. There is evidence that the approaches based on haplotypes are more powerful than the single-marker approaches [1]. For genome-wide association studies, a haplotype approach usually uses a sliding-window method to test one short chromosome region at a time [2]. Recent studies have suggested that linkage disequilibrium (LD) in the human genome can be partitioned into blocks with limited haplotype diversity within each block [3]. If we conduct haplotype-based tests in each haplotype block, we may gain power due to the small number of haplotypes in one haplotype block because there would be a smaller number of degrees of freedom. Furthermore, with hundreds of thousands of single-nucleotide polymorphisms (SNPs) tested for association, the *p*-values need to be adjusted for controlling type I error rates. When we test association in each block, the number of haplotype-based tests is smaller than that of single-marker tests and the correlation between haplotype-based tests is small. Thus, multiple testing would require less correction.

In this article, based on two coding schemes, we extend the general score test statistic proposed by Schaid [4] for case-parents from one child, to include multiple children. We use this extended method to test the association between a disease locus and one haplotype block. Then, by analyzing data sets in Genetic Analysis Workshop 15 (GAW15) Problem 3, we compare the power of the single-marker test and that of the haplotype-based test considering each haplotype block at a time. We also compare the power of haplotype-based tests by using different methods to find haplotype blocks. The results show that the haplotype-based approach is more powerful than the single-marker approach. When we use the haplotype-based test to test one block at a time, the haplotype diversity within the carefully chosen blocks is limited, which results in obtaining higher power than by using evenly spaced blocks.

## Methods

Consider a sample of *n* nuclear families. Suppose that there are *M* genotyped markers across the genome or in a candidate region for each sampled individual, also, that all children in the nuclear families are affected. Schaid et al. [1] proposed a general score test for association of a multi-allelic genetic marker using case-parents design. We first extend this method to include multiple diseased children in one family and deal with multi-marker haplotypes. Because each family has two diseased children in GAW15 Problem 3 data, at this point, we just consider the case with two affected children. It is straightforward to extend the approach to a general situation with more than two affected children in each family.

### General score tests for multiple children

*D*

_{1}and

*D*

_{2}to represent the first and the second affected children, respectively. Let

*g*

_{c1},

*g*

_{c2},

*g*

_{ m }, and

*g*

_{ f }denote the genotypes of the first child, the second child, mother, and father, respectively. The probability of genotype of the diseased child, given the genotypes of the parents is

*G*is the set of the four possible genotypes the parents can produce. Choosing a baseline genotype, let

*r*(

*g*) be the relative risk of genotype

*g*to the baseline genotype. Following Schaid et al. [1], we use log-linear model to model the relative risk, that is,

*r*(

*g*) = exp(

*X*

^{ T }

*β*), with

*X*representing the numerical coding of the genotype

*g*(see Coding section). Then, the conditional likelihood of one family is given by

If there are *n* families, denote the corresponding numerical coding of *g*_{c1 }and *g*_{c2 }in the *i*^{th} family as *X*_{i1 }and *X*_{i2}, respectively. The likelihood function can be shown as $L={\displaystyle \prod _{i=1}^{N}\frac{\mathrm{exp}({X}_{i1}^{\text{'}}\beta +{X}_{i2}^{\text{'}}\beta )}{({\displaystyle {\sum}_{{g}_{i}^{*}\in {G}_{i}}\mathrm{exp}({X}_{i}^{*\text{'}}\beta ){)}^{2}}}}$, where ${X}_{i}^{\ast}$ is the coding vector associated with a genotype ${g}_{i}^{\ast}$ and *G*_{ i }is the set of the four possible genotypes the parents of *i*^{th} family can produce. Following the general form of Rao's score test, the score test statistic *S* = *UV*^{-1}*U*' has a *χ*^{2} distribution *S* = *UV*^{-1}*U** ~ ${\chi}_{r}^{2}$, where the degrees of freedom *r* is the rank of matrix *V*, which is the information matrix of likelihood function *L* with element ${V}_{ij}=-E\left[\frac{{\partial}^{2}\mathrm{ln}L}{\partial {\beta}_{i}\partial {\beta}_{j}}{|}_{\beta =0}\right]$, and *U* = ∂ln*L*/∂*β*|_{β=0}. The quantities *U* and *V* can be expressed as $U={\displaystyle \sum ({X}_{i1}+{X}_{i2}-2{\overline{X}}_{i}^{*})}$, $V={\displaystyle \sum _{i=1}^{N}2{V}_{i}}$, with ${\overline{X}}_{i}^{*}=\frac{1}{4}{\displaystyle \sum _{j=1}^{4}{X}_{ij}^{*}}$, ${V}_{i}=\frac{1}{4}\left[{\displaystyle \sum _{j=1}^{4}{X}_{ij}^{*}{X}_{ij}^{*\text{'}}}\right]-{\overline{X}}_{i}^{*}{\overline{X}}_{i}^{*\text{'}}$, where ${X}_{ij}^{\ast}$, *j* = 1, 2, 3, 4 are the numerical coding corresponding to the four possible genotypes that the parents of the *i*^{th} family can produce.

### Coding

Suppose for one haplotype block there are *m* distinct haplotypes, denoted by *h*_{1},...,*h*_{ m }. For each person, the genotype in this block, denoted as *g*, can be a combination of any two haplotypes selected from *h*_{1},...,*h*_{ m }. Under the assumption that the phase information of the genotype is known, we use two different ways to code the genotypes.

The first coding scheme is defined as follows. Let *X* denote a *m*-dimensional indicator vector, *X* = (*x*_{1},...,*x*_{ m }). The *j*^{th} element *x*_{ j }, is the number of haplotype *h*_{ j }in the genotype *g*, so *x*_{ j }can only take three possible values – 0, 1, or 2 – when *g* has 0, 1, or 2 haplotypes *h*_{ j }, respectively. We also consider the second coding in which we test whether a specific haplotype *h*_{ L }is associated with the disease. In this case, *X* is a scalar value, taking 0, 1, or 2 when *g* has 0, 1, or 2 haplotypes *h*_{ L }, respectively. Using this coding, if there are *m* distinct haplotypes in one block, we will have *m* tests for this block. Let *p*_{1},...,*p*_{ m }denote the *p*-values of the *m* tests. In order to have an overall test between the haplotype block and the disease, we test the null hypothesis *H*_{0}, where at least one haplotype is associated with the disease. The *p*-value of testing *H*_{0} is given by *p* = min{*p*_{1},...,*p*_{ m }) × *m*. Thus, using either of the two coding schemes, we have a *p*-value corresponding to each haplotype block (or a single marker).

### Select significant SNPs by controlling false-discovery rate (FDR)

*B*haplotype blocks. Let

*P*

_{ i }denote the

*p*-value of the test of association between the

*i*

^{th}haplotype block and the disease by using the score test statistic discussed above. Denote the ordered

*p*-values by

*P*

_{(1)},...,

*P*

_{(B)}. A block is considered to be associated with the trait if its

*p*-value is less than a threshold

*δ*

_{ B }. The threshold

*δ*

_{ B }is determined by controlling the FDR at level

*α*[5]. The threshold

*δ*

_{ B }can be calculated by

We choose those blocks with associated *p*-values satisfying *p* ≤ *δ*_{ B }as the blocks that have a significant association with the disease.

### Haplotype blocks

*A*

_{1}and

*A*

_{2}and marker B with alleles

*B*

_{1}and

*B*

_{2}. Let

*p*

_{11}denote the population frequency of haplotype

*A*

_{1}

*B*

_{1}, and ${p}_{{A}_{i}}$, ${p}_{{B}_{i}}$ denote the population frequency of allele

*A*

_{ i }and

*B*

_{ i }(

*i*= 1, 2), respectively. One of the LD measures (

*r*

^{2}), which is proportional to the statistical power of association tests, is defined by

The approach in Zhu et al. [6] to find tight blocks is roughly the same as finding blocks in which all markers have a pair-wise *r* value > *r*_{0}. For the purpose of the power comparison, we choose *r*_{0} = 0.2 for our analysis.

We also use the program HaploBlockFinder V0.7 [7] to find the tight blocks. The power calculations resulting from each of these two approaches are very similar. Thus, we only report the results based on tight blocks found by the approach in Zhu et al. [6].

## Results

### GAW15 data analysis

We use our proposed screening procedures to analyze the dense SNP data of chromosome 6 of the GAW15 Problem 3 simulated rheumatoid arthritis (RA) data. The data contains 100 replications total. Each includes 1500 nuclear families with two disease children and 2000 unrelated controls. In this analysis, we used only family data. For each individual, there are 17820 SNPs on chromosome 6, and the phase information for the genotype is known. From the data provided, we know that there are three disease loci – Locus DR, Locus C, and Locus D – on chromosome 6. Locus DR affects the risk of RA. Locus C increases RA risk only in woman. These two loci are in the same position. The typed SNP 3437 on chromosome 6 is in the same position where Loci DR and C are located. The rare allele of Locus D increases RA risk by five-fold. SNP 3917 is the nearest SNP to Locus D. The genetic distance between Locus D and SNP 3917 is 0.00171 cM, and the physical distance is 1565 bp. We use SNPs 3437 and 3917 as disease-associated SNPs to study the behavior of the score test by using different haplotype information.

The distribution of haplotype blocks using LD measure of Zhu et al. [6]

No. blocks | No. markers in each block |
---|---|

1 | 1331 |

2–5 | 2554 |

6–10 | 641 |

11–15 | 120 |

16–20 | 47 |

<20 | 20 |

Total | 4713 |

The evenly spaced blocks may depend on which SNP is considered the "first" SNP. There are three possible frames of three-SNP blocks. We report the results from all three frames. Finally, we compare the two ways of partitioning with the one that does not use block information, that is, we set each marker as one block, which results in 17820 blocks in total.

### The validity of the test and power comparison

*p*-values of the test. For each block scenario, the total number of tests we performed is

*N*= 100 × {number of blocks}. The estimated type I error for nominal level 0.05 is given by {number of tests with

*p*-value < 0.05}/

*N*. From Table 2, we see that the type I error rates are very consistent with the nominal level, which indicates that the score test is valid regardless of which kind of haplotype block we use. For evenly spaced blocks, we only report the results from the frame that starts from SNP1. For the other two frames, the results are similar.

Type I error rates of the tests at nominal level 0.05^{a}

SNP ID | Single-marker blocks | Evenly spaced blocks | Tight blocks |
---|---|---|---|

The first coding | |||

<2000 | 0.051 | 0.063 | 0.052 |

>4000 | 0.044 | 0.047 | 0.039 |

The second coding | |||

<2000 | 0.042 | 0.078 | 0.050 |

>4000 | 0.036 | 0.060 | 0.050 |

The powers of the score tests using different kinds of haplotype blocks^{a}

Evenly spaced blocks | |||||
---|---|---|---|---|---|

SNP | Single-marker blocks | SNP1 | SNP2 | SNP3 | Tight blocks |

The first coding | |||||

3437 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

2917 | 0.27 | 0.37 | 0.28 | 0.24 | 0.43 |

The second coding | |||||

3437 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

2917 | 0.27 | 0.47 | 0.44 | 0.37 | 0.51 |

It is worth noting that for evenly spaced blocks, the results depend on which SNP is considered to be the first SNP. When SNP ID1 is considered the first SNP in the partition, SNP 3917 falls into the middle of a block, which shows the most powerful result among the three evenly spaced block formations. The power of this partition is smaller than that of the tight block partition, but is not statistically significant at level 0.05. When ID2 or ID3 are considered as the first SNP in the partition, SNP 3917 is not located in the middle of a block. They both have significantly less power than the tight block partition at level 0.05.

## Conclusion

In this paper, we first extend the score test of Schaid [4] from dealing with one affected child to the case of dealing with multiple affected children in each nuclear family. Applying this test to the dense SNP data in GAW15 Problem 3, we compared the power of the test by using different haplotype block information. The conclusion we reach is that the test using tight block with limited haplotype diversity within each block is more powerful than that using evenly spaced blocks, and the latter is more powerful than that using single-marker blocks. The reason may be that, when using tight blocks, there is limited diversity within each block, and thus the degrees of the freedom of the test is small, which may in turn increase the power of the test.

One thing we need to mention is that for the multi-marker blocks (tight block and evenly spaced block) we assume that the phase information is known. Further investigation is needed to evaluate the performance of the test using multi-marker blocks when the phase information is unknown.

## Notes

### Acknowledgements

Research supported by National Institute of Health (NIH) grants R03 AG024491, R01 GM069940, R03 HG003613, and R01 HG003054.

This article has been published as part of *BMC Proceedings* Volume 1 Supplement 1, 2007: Genetic Analysis Workshop 15: Gene Expression Analysis and Approaches to Detecting Multiple Functional Loci. The full contents of the supplement are available online at http://www.biomedcentral.com/1753-6561/1?issue=S1.

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