Similarity-based multi-marker association tests for continuous traits

Summary

Testing multiple markers simultaneously not only can capture the linkage disequilibrium patterns but also can decrease the number of tests and thus alleviate the multiple-testing penalty. If a gene is associated with a phenotype, subjects with similar genotypes in this gene should also have similar phenotypes. Based on this concept, we have developed a general framework that is applicable to continuous traits. Two similarity-based tests (namely, SIMc and SIMp tests) were derived as special cases of the general framework. In our simulation study, we compared the power of the two tests with that of the single-marker analysis, a standard haplotype regression, and a popular and powerful kernel machine regression. Our SIMc test outperforms other tests when the average r-square (a measure of linkage disequilibrium) between the causal variant and the surrounding markers is larger than 0.3 or when the causal allele is common (say, frequency = 0.3). Our SIMp test outperforms other tests when the causal variant was introduced at common haplotypes (the maximum frequency of risk haplotypes > 0.4). We also applied our two tests to an adiposity data set to show their utility.

1. Introduction

Multi-marker association testing has become an important approach for investigating putative loci affecting common human disorders (de Bakker et al., 2005). Testing multiple genetic markers simultaneously not only can capture the joint effects of these adjacent markers while considering the linkage disequilibrium (LD) patterns between them but also can decrease the number of tests. For detecting multi-marker associations, in addition to the traditional regression-based approach, testing associations by using the similarities constructed by DNA sequences among pairs of subjects is another approach. If a gene is associated with a phenotype, subjects with similar genotypes in this gene should have similar phenotypes as well. A traditional regression analysis usually captures the main effects of genetic variants without considering higher-order interactions among them. However, similarity-based methods can capture some complex higher-order interactions by using genomic similarity metrics (Han & Pan, 2010).

A number of similarity methods have been proposed for detecting multi-marker associations. For binary traits, Allen and Satten proposed a general framework that can be reduced to a p test or a cross test, corresponding to two different choices of a parameter vector in their test statistic (Allen & Satten, 2009). Their p test compares within-case similarities with within-control similarities. It is based on the same concept as that in some previous works (Schaid et al., 2005, Tzeng et al., 2003). Their cross test contrasts within-group similarities (including within-case similarities and within-control similarities) with between-group similarities and is based on the same concept as that in other works (Lin & Lee, 2010, Nolte et al., 2007, Sha et al., 2007). Allen and Satten (2009) have established a general framework for similarity-based approaches. However, their method can only deal with binary traits. To analyze continuous traits such as body-mass index (BMI) and triglyceride, we here extend Allen and Satten's (2009) framework. Based on our framework for continuous traits, we propose two tests corresponding to Allen and Satten's (2009) p test and cross test (we call them SIMp and SIMc, respectively).

We then compare the performance of the two proposed tests with that of the single-marker analysis, a standard haplotype regression (Schaid et al., 2002), and a popular and powerful kernel machine regression (referred to as KMR) (Kwee et al., 2008, Wu et al., 2010), through systematic simulations. In addition, we apply our tests to a data set on human adiposity study. Our work provides a useful general framework of similarity-based approaches for continuous traits.

2. Methods

Suppose that the chromosomal region under study contains L single nucleotide polymorphisms (SNPs). A measure of genomic similarity can be constructed with genotypes or haplotypes formed by these L SNPs.

2.1 Genotype-based similarity measure

Let S_ij be the similarity between the i^th and the j^th subjects for this multi-marker set. Summing the similarities over the L loci, we have $S_{\mathit{ij}}={\sum }_{l=1}^{L}s(g_i^l,g_j^l)$, where $s(g_i^l,g_j^l)$ is the similarity of the i^th and j^th subjects at the l^th locus, and $g_i^l$ and $g_j^l$ are the genotypes of the i^th and j^th subjects at the l^th locus, respectively. A commonly used measure of genotype similarity is the allele-match measure. This measure counts the number of alleles that match between a pair of subjects, i.e., $s(g_i^l=a_{i1}^l{a}_{i2}^l,g_j^l=a_{j1}^l{a}_{j2}^l)={\sum }_{m=1}^2{\sum }_{n=1}^{2}I\left[a_{\mathit{im}}^l=a_{\mathit{jn}}^l\right]$, where $a_{i1}^l$ and $a_{i2}^l$ are the two alleles at the l^th locus for the i^th subject, and I[·] is the indicator function with a value of 1 or 0 according to whether the argument is true or false. Note that the similarity metric can be standardized to a range of from 0 to 1. We ignore this standardization part because it makes no essential change to the test results. This allele-match measure has been widely used in genetic linkage (Weeks & Lange, 1988) and association studies (Schaid et al., 2005, Tzeng et al., 2003, Tzeng et al., 2009), and it is equivalent to the counting measure of haplotypes (see below).

2.2 Haplotype-based similarity measure

Genomic similarities can also be quantified with haplotype data. L SNPs can form at most k ≡ 2^L “haplotype categories” (i.e., unique haplotypes in the sample, two haplotypes are classified into a same category if all observed alleles on the two haplotypes are the same), denoted as H = {h¹,h²,…, h^k}. Suppose that the haplotype phases can be directly observed, we denote h_i1h_i2 and h_j1h_j2 as the diplotypes (a diplotype is a pair of haplotypes a subject possesses) of the i^th and j^th subjects, respectively. Let S_ij be the similarity between the i^th and the j^th subjects, and $S_{\mathit{ij}}=s(g_i=h_{i1}{h}_{i2},g_j=h_{j1}{h}_{j2})={\sum }_{m=1}^2{\sum }_{n=1}^{2}s(h_{\mathit{im}},h_{\mathit{jn}})$, where g_i and g_j are the multi-marker genotypes of the i^th and j^th subjects at the L SNPs, respectively. However, in most situations, haplotype phases cannot be directly observed and need to be inferred from genotypes. Let h_iu1h_iu2 be the u^th possible diplotype of the i^th subject. P(h_iu1h_iu2∣g_i) is the posterior probability that the i^th subject has the u^th possible diplotype, given the unphased multi-marker genotypes as g_i. Under the assumption of the Hardy-Weinberg equilibrium (HWE) (Excoffier & Slatkin, 1995, Hawley & Kidd, 1995, Long et al., 1995, Schaid et al., 2002), the posterior probability P(h_iu1h_iu2∣g_i) can be inferred by statistical methods such as the expectation-maximization (EM) algorithm (Dempster et al., 1977). The expected haplotype similarity over the posterior distribution of haplotype pairs, given the observed unphased genotypes, is

$$S_{\mathit{ij}}={\sum }_u{\sum }_{v}P(h_{\mathit{iu}1}{h}_{\mathit{iu}2}\mid g_i)P(h_{\mathit{jv}1}{h}_{\mathit{jv}2}\mid g_j)s(g_i=h_{\mathit{iu}1}{h}_{\mathit{iu}2},g_j=h_{\mathit{jv}1}{h}_{\mathit{jv}2})={\sum }_u{\sum }_{v}P(h_{\mathit{iu}1}{h}_{\mathit{iu}2}\mid g_i)P(h_{\mathit{jv}1}{h}_{\mathit{jv}2}\mid g_j){\sum }_{m=1}^2{\sum }_{n=1}^{2}s(h_{\mathit{ium}},h_{\mathit{jvn}}).$$

If the counting measure of haplotypes (Tzeng et al., 2003) is used, i.e., $s(h_{\mathit{im}},h_{\mathit{jn}})={\sum }_{l=1}^{L}s(a_{\mathit{im}}^l,a_{\mathit{jn}}^l)={\sum }_{l=1}^{L}I\left[a_{\mathit{im}}^l=a_{\mathit{jn}}^l\right]$, where $a_{\mathit{im}}^l$ is the allele on haplotype h_im at the l^th locus and I[·] is the indicator function, then

$$S_{\mathit{ij}}={\sum }_u{\sum }_{v}P(h_{\mathit{iu}1}{h}_{\mathit{iu}2}\mid g_i)P(h_{\mathit{jv}1}{h}_{\mathit{jv}2}\mid g_j){\sum }_{m=1}^2{\sum }_{n=1}^{2}s(h_{\mathit{ium}},h_{\mathit{jvn}})={\sum }_{l=1}^L{\sum }_u{\sum }_{v}P(h_{\mathit{iu}1}{h}_{\mathit{iu}2}\mid g_i)P(h_{\mathit{jv}1}{h}_{\mathit{jv}2}\mid g_j){\sum }_{m=1}^2{\sum }_{n=1}^{2}I\left[a_{\mathit{ium}}^l=a_{\mathit{jvn}}^l\right]={\sum }_{l=1}^L{\sum }_{m=1}^2{\sum }_{n=1}^{2}I\left[a_{\mathit{im}}^l=a_{\mathit{jn}}^l\right].$$

This equation holds because all possible diplotypes of a subject should be consistent with his/her multi-marker genotypes. Therefore, the similarity metric based on the counting measure of haplotypes is equivalent to that based on the allele-match measure of genotypes (Section 2.1).

2.3 Test statistics

With certain haplotype phases

If the haplotype phases can be observed directly, the haplotype-frequency vector of the i^th subject is

$$\mathbf{\Gamma }\left(h_i\right)=\frac{1}{2}\left(\begin{array}{c}I\left[h_{i1}={\mathrm{h}}^1\right]+I\left[h_{i2}={\mathrm{h}}^1\right]\\ \begin{array}{c}I\left[h_{i1}={\mathrm{h}}^2\right]+I\left[h_{i2}={\mathrm{h}}^2\right]\\ \begin{array}{c}⋮\\ I\left[h_{i1}={\mathrm{h}}^{\mathrm{k}}\right]+I\left[h_{i2}={\mathrm{h}}^{\mathrm{k}}\right]\end{array}\end{array}\end{array}\right),$$

where I[·] is the indicator function. In Sections 2.1 and 2.2, we have defined the similarities of pairs of subjects. The average of similarities between any two subjects is $\stackrel{{\sum }_{i<j}{S}_{\mathit{ij}}}{}/\underset{\left(\begin{array}{c}n\\ 2\end{array}\right)}{}={\widehat{\mathit{p}}}^{\mathit{T}}\mathbb{S}\widehat{\mathit{p}}+O\left(\frac{1}{n}\right)$ (Schaid et al., 2005, Tzeng et al., 2003), where n is the number of subjects, p̂ is the vector of the average haplotype frequency of the n subjects, and is a k×k matrix whose (υ, ν) element is the similarity between the υ ^th and ν ^th categories of haplotypes. Using similarities of pairs of ‘haplotype categories’ (i.e., unique haplotypes in the sample) instead of similarities of pairs of subjects in the following test statistic, we can reduce the dimension of the similarity matrix from n×n to k×k and decrease the computational burden (because the number of haplotype categories observed in a sample is usually smaller than the number of subjects under study). If we let $a_{{\mathrm{h}}^{\mathrm{m}}}^l$ be the allele on h^m at the l^th locus and I[·] be the indicator function, then the counting-measure similarity between the υ ^th and ν ^th categories of haplotypes is $s({\mathrm{h}}^{\mathit{\upsilon }},{\mathrm{h}}^{\mathit{\nu }})={\sum }_{l=1}^{L}I\left[a_{{\mathrm{h}}^{\mathit{\upsilon }}}^l=a_{{\mathrm{h}}^{\mathit{\nu }}}^l\right]$.

Let Y be the vector of continuous traits of n subjects, let C be an n×(m + 1) matrix with the i^th row of ${\mathit{c}}_{\mathit{i}}^T=\left[1c_{i,1}{c}_{i,2}\dots c_{i,m}\right]$ coding 1 (for the intercept term) and m covariates (age, gender, ethnicity, etc.) of the i^th subject, and let x be an n-length vector with the i^th element x_i = γ^T· ·Γ(h_i) coding the genetic information (regarding the region under investigation) of the i ^th subject. For x_i = γ^T· ·Γ(h_i), Γ(h_i) is the haplotype-frequency vector of the i^th subject, γ is a specified vector accounting for the aggregate haplotype information of all the n subjects, and is a k×k matrix whose (υ, ν) element is the similarity between the υ ^th and ν ^th categories of haplotypes. The scalar x_i = γ^T· ·Γ(h_i) is a quantity comparing the i ^th subject's haplotypes against haplotypes of all the other subjects. We consider the model:

$$E\left(\mathit{Y}\right)=\mathit{C}\mathit{\alpha }+\mathit{x}\mathit{\beta },$$ (1)

where α is the (m+1)-element vector of covariate effects including the intercept term, and β is the regression coefficient of the genetic information (regarding the region) represented by x. Note that the regression model of Equation (1) is different from the standard haplotype regression model, in which trait values are regressed on haplotypes of subjects, i.e., E(Y) = Cη+ℑψ, where ℑ is an n×k matrix with the i^th row of Γ(h_i)^T (the transpose of the haplotype-frequency vector of the i^th subject), η is the (m+1)-element vector of covariate effects including the intercept term and ψ is the k -element vector of regression coefficients for the k categories of haplotypes in the region. We can see that the standard haplotype regression tests the association between phenotypes and haplotypes, while the regression model of Equation (1) tests the association between phenotypes and quantities of contrasting individual haplotypes with haplotypes of all the other individuals (eventually, this creates ‘similarity’).

Let ${\widehat{\mathit{\mu }}}_i={\mathit{c}}_{\mathit{i}}^T{\left({\mathit{C}}^{\mathit{T}}\mathit{C}\right)}^{-1}\left({\mathit{C}}^{\mathit{T}}\mathit{Y}\right)$ be the predicted mean of y_i under the null hypothesis of no association between the gene variants (in the region under investigation) and the traits, i.e., H₀ : β = 0 in Equation (1). Based on the model in Equation (1) and under the assumption of gene-covariate independence, the score statistic is

$$U={\mathit{\gamma }}^{\mathbf{T}}\mathbb{S}\sum _{i=1}^n\left(y_i-{\widehat{\mathit{\mu }}}_i\right)\mathbf{\Gamma }\left(h_i\right)$$ (2)

(see Appendix I for derivation). In Equation (2), (y_i−μ̂_i)'s are the residuals obtained from regressing the traits on the covariates. We consider two choices for γ. One is $\mathit{\gamma }=\frac{1}{n}{\sum }_{i=1}^n\mathbf{\Gamma }\left(h_i\right)\equiv \widehat{\mathit{p}}$, where p̂ is the vector of the average haplotype frequency of all the n subjects. We call the resulting test the ‘SIMp’ test, which extends Allen and Satten's p test (Allen & Satten, 2009) to deal with continuous traits. Another choice for γ is $\mathit{\gamma }={\sum }_{i=1}^n\left(y_i-{\widehat{\mathit{\mu }}}_i\right)\mathbf{\Gamma }\left(h_i\right)$, and the resulting test is called the ‘SIMc’ test, which extends Allen and Satten's cross test to be applicable to continuous traits. In the following, we consider the asymptotic properties of the two tests respectively.

(1) The SIMp test: The test statistic

$$T_{\mathit{\text{SIMp}}}=\frac{{\left[{\hat{\mathit{p}}}^{\mathbf{T}}\mathbb{S}\sum _{i=1}^n\left(y_i-{\hat{\mathit{\mu }}}_i\right)\mathbf{\Gamma }\left(h_i\right)\right]}^2}{{\hat{\mathit{p}}}^{\mathbf{T}}\mathbb{S}\hat{\Omega }\mathbb{S}\hat{\mathit{p}}}$$ (3)

has an asymptotic chi-square distribution with one degree of freedom (because it is the square of a standard normal variable), where Ω is the variance-covariance matrix of ${\sum }_{i=1}^n\left(y_i-{\widehat{\mathit{\mu }}}_i\right)\mathbf{\Gamma }\left(h_i\right)$. The matrix Ω is estimated by

$$\widehat{\Omega }={\sum }_{1=1}^n\left[\left(y_i-{\widehat{\mathit{\mu }}}_i\right)\mathbf{\Gamma }\left(h_i\right)\right]{\left[\left(y_i-{\widehat{\mathit{\mu }}}_i\right)\mathbf{\Gamma }\left(h_i\right)\right]}^{\mathbf{T}}-\frac{1}{n}\left[{\sum }_{1=1}^n\left(y_i-{\widehat{\mathit{\mu }}}_i\right)\mathbf{\Gamma }\left(h_i\right)\right]{\left[{\sum }_{1=1}^n(y_i-{\widehat{\mathit{\mu }}}_i)\mathbf{\Gamma }\left(h_i\right)\right]}^{\mathbf{T}}.$$

(2) The SIMc test: The test statistic

$$T_{\mathit{\text{SIMc}}}={\left[{\sum }_{i-1}^n\left(y_i-{\widehat{\mathit{\mu }}}_i\right)\mathbf{\Gamma }\left(h_i\right)\right]}^{\mathbf{T}}\mathbb{S}{\sum }_{i=1}^n\left(y_i-{\widehat{\mathit{\mu }}}_i\right)\mathbf{\Gamma }\left(h_i\right)$$ (4)

is asymptotically distributed as ${\sum }_{i=1}^{\varpi }{\mathit{\lambda }}_i{\mathit{\chi }}_{1,i}^2$, where ${\mathit{\chi }}_{1,i}^2$'s are independent chi-square variables with one degree of freedom, an λ₁ ≥ λ₂ ≥ … ≥ λ_ϖ are the ordered eigenvalues of the matrix Ω̂ (by the theory of quadratic forms of normal variables (Scheffe, 1959)). The distribution of T_SIMc can be approximated by the three-moment approximation method (Allen & Satten, 2007, Imhof, 1961, Tzeng et al., 2009, Allen & Satten, 2009). The P value of the observed SIMc test statistic is given by

$$P\left({\chi }_b^2>\left(T_{\mathit{\text{SIMc}}}-c_1\right)\times \sqrt{\frac{b}{c_2}}+b\right),$$

where $c_j={\sum }_{i=1}^{\varpi }{\lambda }_i^j,b=\stackrel{c_2^3}{}/\underset{c_3^2}{}$, and ${\mathit{\chi }}_b^2$ is the chi-square distribution with b degrees of freedom.

To perform the SIMp and the SIMc tests, the trait values Y are first regressed on the covariates (C) alone, and the subject-specific predicted means are obtained by ${\widehat{\mathit{\mu }}}_i={\mathit{c}}_{\mathit{i}}^T{\left({\mathit{C}}^{\mathit{T}}\mathit{C}\right)}^{-1}\left({\mathit{C}}^{\mathit{T}}\mathit{Y}\right)$, where i = 1, 2, …, n. Then the test statistics and the P values can be computed through the formulas of T_SIMp and T_SIMc. An R code to perform these two tests is available upon request.

With uncertain haplotype phases

If the haplotype phases are uncertain, the haplotype-frequency vector of the i^th subject needs to be inferred by his/her multi-marker genotypes:

$$\begin{array}{l}\mathbf{\Gamma }\left(h_i\mid g_i\right)=\frac{1}{2}\left(\begin{array}{c}{\sum }_u\left[P\left(h_{\mathit{iu}1}={\mathrm{h}}^1\mid g_i\right)+P\left(h_{\mathit{iu}2}={\mathrm{h}}^1\mid g_i\right)\right]\\ \begin{array}{c}{\sum }_u\left[P\left(h_{\mathit{iu}1}={\mathrm{h}}^2\mid g_i\right)+P\left(h_{\mathit{iu}2}={\mathrm{h}}^2\mid g_i\right)\right]\\ \begin{array}{c}⋮\\ {\sum }_u\left[P\left(h_{\mathit{iu}1}={\mathrm{h}}^{\mathrm{k}}\mid g_i\right)+P\left(h_{\mathit{iu}2}=h^{\mathrm{k}}\mid g_i\right)\right]\end{array}\end{array}\end{array}\right),\end{array}$$

under the assumption of the Hardy-Weinberg equilibrium (HWE). The HWE is a typical assumption when inferring haplotypes from multi-marker genotypes (Excoffier & Slatkin, 1995, Hawley & Kidd, 1995, Long et al., 1995, Liu et al., 2008, Liu et al., 2006), but how to make this assumption is an issue for case-control studies (e.g., assuming the HWE in a pooled sample or only in a control sample) (Allen & Satten, 2008). However, the issues related to phase ambiguity are not crucial here because we use a phase-independent similarity measure (i.e., the counting measure of haplotypes, or equivalently, the allele-match measure of genotypes). Although the test statistic (Equation (2)) is presented with a haplotype-frequency vector, using the inferred haplotypes in the downstream test statistic will be equivalent to using the observed multi-marker genotypes in the test statistic (In Section 2.2, we have shown that the expected haplotype similarity over the posterior distribution of haplotype pairs is equivalent to the similarity based on the allele-match measure of multi-marker genotypes.) We will elaborate on the pros and cons of using phase-independent and phase-dependent similarity measures in our discussion.

3. Simulation Study

With simulations, we evaluate the performances of our methods. We first generated 200 data sets, each containing 10,000 chromosomes of 1 Mb regions with the Cosi program (Schaffner et al., 2005). The chromosomes were generated in consistency with the HapMap CEU (CEPH people from Utah, U.S.A.) samples. We considered five levels of causal allele frequency: q = 0.01, 0.05, 0.1, 0.3, and 0.5. For each data set, we chose SNPs with MAFs within the region of [0.01 ± 0.01/100], [0.05 ± 0.05/100], [0.1± 0.1/100], [0.3 ± 0.3/100], or [0.5 ± 0.5/100] as the causal SNPs, respectively. In each data set, we randomly chose 120 from the 10,000 chromosomes to mimic the Phase II HapMap CEU data, and we randomly paired them to form 60 subjects. Based on the LD patterns of the 60 subjects, we selected tag SNPs according to the conventional cut-off r² = 0.8 and MAF > 0.05, with the H-clust method (Rinaldo et al., 2005, Roeder et al., 2005). These tag SNPs were served as markers used in our simulations. To generate chromosomes of one individual, we randomly selected two chromosomes from the remaining 9,880 (= 10,000 − 120) chromosomes. Following Wu et al. (2011), we created two covariates when generating trait values. Trait values were generated by Y = 0.5C₁ + 0.5C₂ + g + e, where C₁ was a continuous covariate generated from a standard normal distribution, C₂ was a dichotomous covariate taking values 0 and 1 with a probability of 0.5, g was the genetic effect (2, 1, or 0, depending on the genotype of the causal SNP), and e was the random error. The random error, e, was assumed to have a normal distribution with a mean of zero and a variance of V_e, where V_e was determined so that the heritability was 2% under the alternative hypothesis. After generating the trait values, the genotypes of the causal SNP were removed from our data sets. We randomly selected a ten-SNP region to encompass the causal SNP. We then evaluated the performances of the proposed SIMp and SIMc tests. We compared our similarity tests with the standard haplotype regression (referred to as HAP) (Schaid et al., 2002), the single-marker analysis with P values adjusted by the Sidak correction (Sidak., 1967) (referred to as Single-marker), the single-marker analysis with P values adjusted by 5,000 permutations (referred to as Single-marker-perm), and a popular kernel machine regression (referred to as KMR) (Kwee et al., 2008, Wu et al., 2010).

When analyzing the simulated data, we first fitted a model under the null hypothesis of no association between the gene variants (in the region under investigation) and the traits. That is, we first regressed the trait values on the covariates C₁ and C₂ to obtain the predicted means of trait values under the null hypothesis (μ̂_i's). Then the residuals (y_i−μ̂_i)'s were used in the test statistics of SIMp and SIMc, in the single-marker analysis, and in the package ‘haplo.stats’ (Schaid et al., 2002).

The single-marker analysis was performed with the ‘--assoc’ command implemented by PLINK-1.07 (Purcell et al., 2007). In our simulations, each analysis region was formed by ten SNPs. We adjusted the minimum P value (P_min) of the ten SNPs with the Sidak correction (Sidak., 1967), P_min,adjusted =1−(1−P_min)^M_eff, where M_eff was the effective number of independent tests and was estimated by the Keffective program (Moskvina & Schmidt, 2008) (referred to as Single-marker). This single-marker analysis is used to be compared with some popular multi-marker association tests (Tzeng et al., 2009, Wu et al., 2010, Tzeng et al., 2011), and is commonly used in real data analyses (Harold et al., 2009, Kirov et al., 2009, Jones et al., 2010). Therefore, we also include it into our comparisons.

We also performed the single-marker analysis with P values adjusted by permutations (referred to as Single-marker-perm). For each replication, we permuted the phenotypes 5,000 times and obtained a minimum P value of the ten SNPs from each of the 5,000 permuted data sets (so there were 5,000 minimum P values). We then estimated the significance by comparing the minimum P value (of the ten SNPs) from the observed data set with the 5,000 minimum P values from the permuted data sets. This analysis was performed by using PLINK-1.07 (Purcell et al., 2007) with the command ‘-assoc --mperm 5000’.

The HAP was performed by the package ‘haplo.stats’ (Schaid et al., 2002). The haplotypes with frequencies less than a threshold α₀ were pooled into a single baseline group by the default of the package ‘haplo.stats’. In our simulations and the following data analysis, the threshold α₀ was set at 0.01.

In addition, we compared our tests with a popular multi-marker association test, the kernel machine regression (referred to as KMR) (Kwee et al., 2008, Wu et al., 2010). We performed this test with the R code of KMR (http://genetics.emory.edu/labs/epstein/software/lskm/index.html). The trait value Y was treated as phenotype and the two covariates C₁ and C₂ were treated as environmental covariates when the KMR package was performed. The similarity measure used in the KMR package is the MAF-weighted identical-by-state (IBS) sharing (see Equation (5) of (Kwee et al., 2008)). This measure downweights the similarities contributed by common SNPs and upweights the similarities contributed by rare SNPs. The implication is that subjects sharing rare alleles may have more similar genomes than do subjects sharing common alleles (Lin & Lee, 2010, Wessel & Schork, 2006).

4. Simulation Results

Type-I Error Rates

We created scenarios under the null hypothesis of no association by fixing the genetic effect g at zero. For each of the 200 data sets, we performed 1,000 replications. In each replication, the sample size was set at 1,000 subjects. Figure 1 presents the type-I error rates averaged over all of the 200,000 replications. The results show that the type-I error rates of five tests (SIMp, SIMc, HAP, Single-marker, and Single-marker-perm) are under control. KMR is slightly anticonservative (will be discussed later).

Figure 1. Type-I error rates.

The x-axis is nominal significance level, and the y-axis is type-I error rate. The gray line represents y = x.

Statistical Power for Low-Frequency Variant Detection

For each of the 200 data sets, we performed 1,000 replications given each q (we have five levels of causal allele frequency, q = 0.01, 0.05, 0.1, 0.3, or 0.5), and so the total number of replications was 200,000 for each q. The heritability (H²) was set at 2%, and the sample size was set at 1,000. Figure 2 presents the power performances of the six tests (SIMp, SIMc, HAP, Single-marker, Single-marker-perm, and KMR) for low-frequency variant detection (q = 0.01 or 0.05). We stratified the results based on the average R² between the causal SNP and the surrounding markers (the average R² was obtained by averaging the ten R²'s of the causal SNP and each of the ten surrounding markers, where R was the correlation coefficient between two SNPs and was obtained by using the package ‘genetics’ (Warnes et al., 2011)). We considered three ranges of the average R²: smaller than 0.2, between 0.2 and 0.3, and larger than 0.3. For q = 0.01, the average R²'s of all the 200,000 replications were smaller than 0.2 (the left panel of Figure 2). For q = 0.05, the average R²'s of 198,896 replications were smaller than 0.2 (the middle panel of Figure 2) and the remaining 1,104 replications were between 0.2 and 0.3 (the right panel of Figure 2). Overall, HAP is the most powerful test among the six tests, when the frequency of the causal variant is smaller than or equal to 0.05.

Figure 2. Power curves of the six tests for low-frequency variant detection.

The x-axis is significance level, and the y-axis is power. The left panel shows the results when the causal allele frequency q is set at 0.01. The middle and the right panels present the results when the causal allele frequency q is 0.05, where the average R² (between the causal SNP and the surrounding markers) is smaller than 0.2, and between 0.2 and 0.3, respectively. In the bottom of each panel, we list the number of replications meeting the conditions of the causal allele frequency and the average R².

The test statistics of similarity-based methods are averages of genomic similarities weighted by some forms of covariate-adjusted phenotypes (see Equations (3) and (4)), depending on different choice of γ. When the causal variant is uncommon, the similarity-based method is an inferior choice because few subjects have this causal variant and most subjects are similar by having no causal variants. However, basically, HAP let each haplotype category (common or uncommon, as long as the frequency is larger than the pooling threshold α₀) account for one degree of freedom in the test statistic (see Equation (3) of (Schaid et al., 2002)). Therefore, the association contributed by uncommon haplotypes is more likely to be preserved by HAP, compared with the similarity-based tests.

Statistical Power for Common Variant Detection

Figure 3 presents the power performances of the six tests, according to various combinations of causal allele frequency q and the average R² (between the causal SNP and the surrounding markers). With a causal allele frequency of 0.1, HAP is the most powerful test when the average R² is smaller than 0.2; KMR and Single-marker-perm are the two most powerful tests when the average R² is between 0.2 and 0.3; SIMc and KMR are the most powerful tests when the average R² is larger than 0.3. With a causal allele frequency of 0.3 or 0.5, SIMc and KMR are the most powerful tests. Note that KMR is generally more powerful than SIMc when the causal allele frequency q <= 0.1. However SIMc is more powerful than KMR when the causal allele frequency q >= 0.3. This result is reasonable because the similarity measure used in the R code of KMR is the MAF-weighted IBS sharing that downweights the similarities contributed by common SNPs and upweights the similarities contributed by rare SNPs (Kwee et al., 2008). This similarity measure favors the detection of uncommon variants, but suffers from power loss when the causal variant is common.

Figure 3. Power curves of the six tests for common variant detection.

The x-axis is significance level, and the y-axis is power. The top, middle, and bottom rows are the results when the causal allele frequency q is set at 0.1, 0.3, and 0.5, respectively. The left, middle, and right columns present the results based on the replications with average R² (between the causal SNP and the surrounding markers) smaller than 0.2, between 0.2 and 0.3, and larger than 0.3, respectively. In the bottom of each panel, we list the number of replications meeting the conditions of the causal allele frequency and the average R².

For single-marker analyses, adjusting for multiple testing with permutations (Single-marker-perm) is much more powerful than adjusting with the Sidak correction (Single-marker) (Sidak., 1967). Overall, Single-marker-perm has a good power performance, although SIMc, KMR, or HAP can outperform it in almost all combinations of causal allele frequency and average R². From Figures 2 and 3, SIMp is generally not powerful. However, SIMp has a good power performance when the causal SNP was introduced at common haplotypes. Figure 4 presents the results (only for q = 0.5) stratified by the Maximum Frequency of Risk Haplotypes (MFRH) (where a risk haplotype is defined if the causal variant was introduced at that haplotype). We can see that SIMp outperforms other tests when the MFRH is larger than 0.4.

Figure 4. Power curves of the six tests when q = 0.5, stratified by the Maximum Frequency of Risk Haplotypes (MFRH).

The x-axis is significance level, and the y-axis is power. The left and the right panels present the results based on the replications with MFRH smaller than 0.4 and larger than 0.4, respectively. In the bottom of each panel, we list the number of replications meeting the condition of MFRH.

SIMp is a generalization of the p test (Tzeng et al., 2003, Allen & Satten, 2009) that compares within-case similarities with within-control similarities. The p test is more powerful than the cross test when most cases share a common haplotype and controls have many different categories of haplotypes. Likewise, SIMp is more powerful than SIMc (a generalization of the cross test) when most subjects with undesirable traits (say, large BMIs that lead to complex disorders) share a common haplotype and subjects with normal traits have many different categories of haplotypes. When the MFRH is large, many subjects with undesirable traits share the risk haplotype with the highest frequency, and SIMp can become more powerful than SIMc. Note that even for phase-independent similarity metrics, as long as no severe errors in the haplotype inference, the MFRH is still a good index of the genomic similarity (in the region under study) between any two subjects with undesirable traits.

5. Application to Adiposity Data

We then applied our method to a data set on human adiposity (Chung et al., 2009). The data set contains 1,982 unrelated European Americans living in the New York City metropolitan area. We investigated the association of 23 SNPs in the carboxypeptidase E (CPE) gene (located on chromosome 4q32.3) with BMI. To replicate the previous single-marker analysis results (Chung et al., 2009), we first followed their analysis. BMI was log-transformed and then was adjusted for the covariates sex, age, age², and their respective interactions. Associations between BMI and the joint additive and dominance effects of each of the 23 SNPs were tested using the ordinary-least-squares (OLS) regression method.

To perform multi-marker analyses, we first need to specify the number of SNPs used in an analysis unit. Partitioning the gene into segments according to the LD patterns is a commonly used practice (Gabriel et al., 2002, Twells et al., 2003, Zhang et al., 2002). Therefore, we first used the Haploview (Barrett et al., 2005) and learned that the 22 SNPs with MAFs larger than 0.01 constitute three haplotype blocks, as shown in Figure 5. Then we applied the six tests (SIMp, SIMc, HAP, Single-marker, Single-marker-perm, and KMR) to each of the three haplotype blocks. Table 1 presents the P values based on the six tests, for the three blocks, respectively. Given a significance level of 0.05, only SIMp suggests that the CPE gene, especially from 166,668,633 bp to 166,727,266 bp (Block 1), may potentially be interesting to investigators (0.05 > 0.012 × 3 blocks/tests, if the Bonferroni correction is used to adjust for the three blocks/tests in the CPE gene). In Table 2, frequencies and P values of haplotype categories (with frequencies >= 0.01) in Block 1 are listed. We see that the second significant haplotype (Haplotype M) has a high frequency of 0.37. Although the P value of the score statistic of Haplotype M, 0.075, is not smaller than the commonly used significance level of 0.05, aggregating the information of these individual haplotypes in a SIMp test yields a significant result (P value = 0.012, from Table 1). Based on our simulation results, SIMp has a good power performance when the causal variant was introduced at a common haplotype. The situation of Block 1 may correspond to this scenario, and therefore SIMp yields the most significant result among all the six tests we evaluated. Note that studies have shown that a single point mutation in the CPE gene is sufficient to produce an animal with multiple disorders including obesity (Fricker et al., 1996, Naggert et al., 1995). Moreover, an independent study (Jeffrey et al., 2008) also pointed out the association of the CPE gene with human obesity and diabetes. The role of the CPE gene in human adiposity may deserve further investigation.

Figure 5. The haplotype structure of the carboxypeptidase E (CPE) gene.

The number in each cell (the quantity between each pair of SNPs) is 100 × D′, a measure of linkage disequilibrium.

Table 1. Real data analysis: P values for the three haplotype blocks.

	Block 1	Block 2	Block 3
SIMp	0.012	0.745	0.857
SIMc	0.049	0.262	0.964
HAP	0.571	0.490	0.844
Single-marker1	0.178	0.517	0.800
Single-marker-perm2	0.139	0.454	0.712
KMR	0.092	0.382	0.883

¹Single-marker: The P values have adjusted with the Sidak correction (Sidak., 1967), where the effective numbers of tests were estimated as 7.92, 5.80, and 2.99 for the three blocks, respectively. The effective numbers of tests were estimated by the Keffective program (Moskvina & Schmidt, 2008).

²Single-marker-perm: The P values for the three blocks have adjusted with 50,000 permutations.

Table 2. Real data analysis: Frequencies and P values of haplotype categories in Block 1.

Haplotype	loc-1	loc-2	loc-3	loc-4	loc-5	loc-6	loc-7	loc-8	loc-9	loc-10	loc-11	loc-12	Hap-Freq1	p-val2
A	2	1	2	1	1	2	2	1	2	1	2	2	0.01678	0.04491
B	2	1	2	2	2	1	1	2	1	1	2	2	0.01093	0.20489
C	2	1	2	1	1	2	2	1	2	2	2	2	0.24596	0.22458
D	1	1	1	1	1	1	1	2	1	1	1	1	0.10447	0.34698
E	1	2	2	2	2	1	2	1	2	2	2	2	0.02043	0.64974
F	1	2	2	2	1	2	2	1	2	2	2	2	0.02225	0.69037
G	1	1	1	1	1	1	1	2	1	1	2	2	0.01599	0.85261
H	1	2	2	1	1	1	1	2	1	1	1	1	0.01621	0.87198
I	1	1	2	1	1	1	1	1	1	1	1	2	0.02046	0.92624
J	1	1	1	1	1	2	1	2	1	1	2	2	0.01229	0.92330
K	2	1	2	1	1	2	1	2	1	1	2	2	0.01630	0.65679
L	1	2	2	2	2	1	1	2	2	2	2	2	0.05067	0.24820
M	1	2	2	2	2	1	1	2	1	1	2	2	0.37006	0.07503

This table came from the result of the function ‘haplo.score’ in the R package ‘haplo.stats’ (Schaid et al., 2002), where all the 1,982 unrelated European Americans were used for analyses. The haplotype categories with frequencies less than 0.01 were not listed here.

¹Hap-Freq: The frequencies of haplotypes.

²p-val: The P values of the score statistics of individual haplotypes, based on the chi-square distribution with one degree of freedom.

6. Discussion

Many previous similarity-based methods can only deal with binary traits (Allen & Satten, 2009, Lin & Lee, 2010, Lin & Schaid, 2009, Nolte et al., 2007, Schaid et al., 2005, Tzeng et al., 2003, Yuan et al., 2006, Sha et al., 2007). In this work, we developed a general framework for similarity-based methods that can handle continuous traits. To evaluate the robustness of our tests to the underlying distributions, we deliberately set the error term to be very far away from the normal distribution, and we performed more simulations. We find that our tests are very robust to the assumption of normality (this assumption is made in Appendix I).

Using multiple genetic markers in association analyses has been shown to be superior to the single-marker analysis in some situations (Schaid, 2004b, Schaid, 2004a). The method proposed here is applicable to chromosomal regions or genes, as well as genome-wide search for disease susceptibility loci. An issue is how to choose an ‘appropriate’ size of a multi-marker set. Multi-marker analyses with larger window sizes may allow for measuring sharing over longer genomic sequences and lead to more power gains, though this also increases the computational burdens (Allen & Satten, 2009). How to choose the size of a multi-marker set is still an open question (Schaid, 2004b). Analyses can be chosen based on sliding windows or haplotype blocks (Guo et al., 2009). However, similar to single-marker analysis, a harsh multiple-testing penalty will be encountered by using the sliding-window strategy. Partitioning the chromosomal region into segments according to the LD patterns is a commonly used practice (Gabriel et al., 2002, Twells et al., 2003, Zhang et al., 2002), as we have shown in the analysis of the adiposity data.

The computational burden of performing the SIMp and SIMc tests is reasonable, although it increases with the number of ‘haplotype categories’ (i.e., unique haplotypes in the sample). For data sets simulated based on the LD patterns of the HapMap CEU samples, the burden of analyzing 20 SNPs in a multi-marker test is still reasonable. It takes 1.5 hours to run 1,000 replications (each with 1,000 subjects), given an Intel Xeon workstation with 3.0 GHz CPU and 2.0 GB of memory. When using 20 SNPs to form a multi-marker set for simulations, the relative power performances of the four tests were very similar to our Figures 2 and 3. However, each method was slightly more powerful than that by using ten SNPs in a multi-marker set. This is because larger multi-marker sets allow for measuring sharing over longer genomic sequences and lead to more power gains.

The genomic distance-based regression (GDBR) (Wessel & Schork, 2006) and the gene-trait similarity regression (GTSR) (Tzeng et al., 2009, Tzeng et al., 2011) can also deal with continuous traits. These two approaches, our SIMc test, the sum-of-squared-score (SSU) test (Pan, 2009), the variance-component score test (Tzeng & Zhang, 2007), and the KMR method (Kwee et al., 2008, Wu et al., 2010) are closely related to one another. Motivated differently, these test statistics end up with very similar forms. We show the theoretical connection between these methods in Appendix II. Note that we observed a slight inflation of type-I error rates for KMR (Figure 1). The KMR uses a Satterthwaite procedure to approximate the mixture chi-square distribution, while SIMc and other related tests use the three-moment approximation (Allen & Satten, 2007, Imhof, 1961, Tzeng et al., 2009, Allen & Satten, 2009). This may be a possible reason for the slight inflation of type-I error rates of KMR.

More than just SIMc, we would like to emphasize that we have proposed a general framework of similarity-based tests for continuous traits. By employing different γ in Equation (2), our test statistic results in various tests. SIMc (and thus GDBR and GTSR) is a special case of our general framework. The SIMp test is another special case (which was shown to identify the signal of the CPE gene that was pointed out to be associated with obesity by several independent studies (Jeffrey et al., 2008, Fricker et al., 1996, Naggert et al., 1995)). In addition, we have performed systematic simulations to evaluate the performance of several popular methods for detecting causal variants, common or uncommon. Through these simulations, we further clarify the merits and limitations of similarity-based methods.

Our proposed framework results in various test given different choice of γ, and no one test performs well universally. There is no uniformly most-powerful test in the situation of multi-marker association testing (Cox & Hinkley, 1974, Han & Pan, 2010). These different tests capture various aspects of information from the data. One test can outperform other tests in certain scenarios, as we have shown in Figures 2-4. Note that the three features (1) causal allele frequency, (2) the average R² between the causal SNP and the surrounding markers, and (3) the maximum frequency of risk haplotypes (MFRH) are unknown in real-data analysis (because we do not know where the causal variant is). Therefore, we cannot choose the most powerful test in advance. Our aim is to provide a general framework harboring various tests that are complementary to one another. A research direction is to come up with a third test based on our general framework, by specifying a different γ. More interestingly, developing a strategy that can determine an optimal γ based on data, as illustrated by the test selection of a previous study (Pan et al., 2010).

The similarity measure used in this work is a phase-independent metric (i.e., the counting measure of haplotypes, or equivalently, the allele-match measure of genotypes), although we infer haplotype phases with the EM algorithm to facilitate the computation of the test statistic (Equation (2)). In Section 2.2, we have shown that the expected haplotype similarity over the posterior distribution of haplotype pairs is equivalent to the similarity based on the allele-match measure of multi-marker genotypes. There are other similarity measures, including phase-independent and phase-dependent metrics (Durrant et al., 2004, Lin & Schaid, 2009, Tzeng et al., 2003, Van der Meulen & te Meerman, 1997, Wessel & Schork, 2006). These similarity measures can also be plugged into our framework (Equation (2)). The merit of a phase-independent metric is its robustness to genotyping errors and the HWE assumption, although it might be less powerful than a phase-dependent metric that can capture the identical-by-descent sharing more precisely (Tzeng et al., 2009). Although a phase-dependent metric (e.g., a matching measure of haplotypes (Tzeng et al., 2003)) may be more powerful, the mis-specification of haplotypes given multi-marker genotypes can cause adverse results such as inflated type-I error rates (Allen & Satten, 2008).

Finally, low-frequency (0.01 ≤ MAF < 0.05) and rare (MAF < 0.01) variants are thought to be responsible for a non-negligible portion of heritability that cannot be explained by common variants (Manolio et al., 2009, Zeggini, 2011). Recently, Wu et al. (2011) proposed a sequence kernel association test (SKAT) to analyze rare variants. SKAT is closely related to KMR (Kwee et al., 2008, Wu et al., 2010) and our SIMc test. However, there are two fundamental differences between Wu et al.'s (2011) work and our work. First, SKAT was developed for sequencing data, and most variants were uncommon in their simulations (Wu et al., 2011). Our work is mainly for association analyses using tag SNPs as markers, and the MAF of each marker was larger than 5% (see our Simulation Study). Second, Wu et al. (2011) simulated multiple uncommon causal variants for each simulated region, while we simulated only one uncommon or common causal variant in this work. In addition, Tzeng et al. (2011) proposed a GTSR for uncommon and common variants. The GTSR is also closely related to the SIMc test here. Tzeng et al. (2011) found a weighting scheme on GTSR that can improve the power for detecting rare variants and have comparable power for detecting common variants (referred to as ‘SIM1’ by Tzeng et al. (2011)). However, the standard haplotype regression (HAP) (Schaid et al., 2002) is still more powerful than that ‘SIM1’ test, when the causal variant is rare and when the average R² between the causal variant and the surrounding markers is low (this is also a feature of rare causal variants) (Tzeng et al., 2011). Unifying the standard haplotype regression (Schaid et al., 2002) with the similarity-based tests may further increase the power to detect rare causal variants. This topic may deserve further research.

Appendix

Appendix I: Derivation of the score statistic, given phase-certain data

Consider the model in Equation (1) : E(Y) = Cα+xβ, where the i^th element of the n-length vector x is x_i = γ^T · · Γ(h_i), β is the regression coefficient of the genetic information (in the region) represented by x, C is an n×(m+1) matrix with the i^th row ${\mathit{c}}_{\mathit{i}}^T=\left[1c_{i,1}{c}_{i,2}\dots c_{i,m}\right]$ coding 1 (for the intercept term) and the m covariates of the i^th subject, and α is the (m+1) -element vector of covariate effects including the intercept term. Assume that the error term ${\varepsilon }_i=y_i-{\mathit{c}}_{\mathit{i}}^T\mathit{\alpha }-\mathit{\beta }{x}_i$ is normally distributed with a mean of zero and a variance of σ², where i = 1, …, n. The likelihood is

$$L=\prod _{i=1}^n\frac{1}{\sqrt{2\mathit{\pi }{\mathit{\sigma }}^2}}exp\{-\frac{{\left[y_i-{\mathit{c}}_{\mathit{i}}^T\mathit{\alpha }-\mathit{\beta }{x}_i\right]}^2}{2{\mathit{\sigma }}^2}\}.$$

The log-likelihood is

$$l\propto \sum _{i=1}^n{\left[y_i-{\mathit{c}}_{\mathit{i}}^T\mathit{\alpha }-\mathit{\beta }{x}_i\right]}^2.$$

The score function for β is

$$\frac{\partial l}{\partial \mathit{\beta }}=-2\sum _{i=1}^n{x}_i\left[y_i-{\mathit{c}}_{\mathit{i}}^T\mathit{\alpha }-\mathit{\beta }{x}_i\right].$$

The score function for α is

$$\frac{\partial l}{\partial \mathit{\alpha }}=-2\sum _{i=1}^n{\mathit{c}}_{\mathit{i}}\left[y_i-{\mathit{c}}_{\mathit{i}}^T\mathit{\alpha }-\mathit{\beta }{x}_i\right].$$

When β = 0, solving α from the equation

$$0\equiv \frac{\partial l}{\partial \mathit{\alpha }}=-2\sum _{i=1}^n{\mathit{c}}_{\mathit{i}}\left[y_i-{\mathit{c}}_{\mathit{i}}^T\mathit{\alpha }-\mathit{\beta }{x}_i\right]=-2\sum _{i=1}^n{\mathit{c}}_{\mathit{i}}\left[y_i-{\mathit{c}}_{\mathit{i}}^T\mathit{\alpha }\right],$$

and we obtain $\widehat{\mathit{\alpha }}={\left(\sum _{i=1}^n{\mathit{c}}_{\mathit{i}}{\mathit{c}}_{\mathit{i}}^T\right)}^{-1}\left(\sum _{i=1}^n{\mathit{c}}_{\mathit{i}}{y}_i\right)={\left({\mathit{C}}^{\mathit{T}}\mathit{C}\right)}^{-1}\left({\mathit{C}}^{\mathit{T}}\mathit{Y}\right).$ Then we have

$$\frac{\partial l}{\partial \mathit{\beta }}=-2\sum _{i=1}^n{x}_i\left[y_i-{\mathit{c}}_{\mathit{i}}^T\mathit{\alpha }-\mathit{\beta }{x}_i\right]=-2\sum _{i=1}^n{x}_i\left[y_i-{\mathit{c}}_{\mathit{i}}^T{\left({\mathit{C}}^{\mathit{T}}\mathit{C}\right)}^{-1}\left({\mathit{C}}^{\mathit{T}}\mathit{Y}\right)-\mathit{\beta }{x}_i\right],$$

and

$${\frac{\partial l}{\partial \mathit{\beta }}\mid }_{\mathit{\beta }=0}=-2\sum _{i=1}^n{x}_i\left[y_i-{\mathit{c}}_{\mathit{i}}^T{\left({\mathit{C}}^{\mathit{T}}\mathit{C}\right)}^{-1}\left({\mathit{C}}^{\mathit{T}}\mathit{Y}\right)\right].$$

We thus have the score statistic,

$$U=\sum _{i=1}^n{x}_i\left[y_i-{\mathit{c}}_{\mathit{i}}^T{\left({\mathit{C}}^{\mathit{T}}\mathit{C}\right)}^{-1}\left({\mathit{C}}^{\mathit{T}}\mathit{Y}\right)\right].$$

Because x_i = γ^T· ·Γ(h_i), the score statistic is

$$U=\sum _{i=1}^n{\mathit{\gamma }}^{\mathit{T}}\mathbb{S}\mathbf{\Gamma }\left(h_i\right)\left[y_i-{\mathit{c}}_{\mathit{i}}^T{\left({\mathit{C}}^{\mathit{T}}\mathit{C}\right)}^{-1}\left({\mathit{C}}^{\mathit{T}}\mathit{Y}\right)\right]={\mathit{\gamma }}^{\mathbf{T}}\mathbb{S}\sum _{i=1}^n\left[y_i-{\mathit{c}}_{\mathit{i}}^T{\left({\mathit{C}}^{\mathit{T}}\mathit{C}\right)}^{-1}\left({\mathit{C}}^{\mathit{T}}\mathit{Y}\right)\right]\mathbf{\Gamma }\left(h_i\right).$$

Let ${\widehat{\mathit{\mu }}}_i={\mathit{c}}_{\mathit{i}}^T{\left({\mathit{C}}^{\mathit{T}}\mathit{C}\right)}^{-1}\left({\mathit{C}}^{\mathit{T}}\mathit{Y}\right)$ be the predicted mean of y_i under the null hypothesis of no association between the gene variants (in the region under study) and the traits, i.e., H₀ : β = 0.

The score statistic is

$$U={\mathit{\gamma }}^{\mathbf{T}}\mathbb{S}\sum _{i=1}^n\left[y_i-{\widehat{\mathit{\mu }}}_i\right]\mathbf{\Gamma }\left(h_i\right).$$

Appendix II: Relationship between GDBR, GTSR, SSU, KMR, and our SIMc test

(I) GDBR

The relationship between the pseudo-F statistic in the GDBR and the SSU test in logistic regression has been elucidated by Han and Pan (2010). This relationship also holds for continuous traits. Let G be the Gower matrix (Gower, 1966) harboring the information of genomic similarities among subjects. If G is an outer product matrix, we can find an n×p matrix Z such that G = ZZ^T. If G is not an outer product matrix, we can still approximate G ≈ ZZ^T by using the principal coordinates analysis (Gower, 1966). Suppose we have an n×p matrix Z such that G = ZZ^T. Then the GDBR can be reformulated as a multivariate linear model Z = y_cB+ε, where y_c is the n×1 vector of covariate-adjusted trait values, B is a 1×p vector of unknown regression coefficients, and ε is an n×p matrix of random errors (Han & Pan, 2010) (in contrast to the standard regression approaches, the GDBR treats the genetic part as the response variable and treats the trait value as the explanatory variable). Therefore, we have the projection matrix $\mathit{H}={\mathit{y}}_{\mathit{c}}{\left({\mathit{y}}_{\mathit{c}}^{\mathbf{T}}{\mathit{y}}_{\mathit{c}}\right)}^{-1}{\mathit{y}}_{\mathit{c}}^{\mathbf{T}}$ and the fitted matrix Ẑ = HZ. The association between y_c and Z can be assessed with a simple linear regression: y_c=Zβ + ξ, where β is a p×1 vector of unknown regression coefficients and ξ is an n ×1 vector of random errors. With permutations, the kernel of the pseudo-F statistic in the GDBR is

$$\mathit{tr}\left({\widehat{\mathit{Z}}}^{\mathbf{T}}\widehat{\mathit{Z}}\right)=\mathit{tr}\left({\mathit{Z}}^{\mathbf{T}}{\mathit{H}}^{\mathbf{T}}\mathit{HZ}\right)=\mathit{tr}\left({\mathit{Z}}^{\mathbf{T}}{\mathit{y}}_{\mathit{c}}{\left({\mathit{y}}_{\mathit{c}}^{\mathbf{T}}{\mathit{y}}_{\mathit{c}}\right)}^{-1}{\mathit{y}}_{\mathit{c}}^{\mathbf{T}}\mathit{Z}\right)={\left({\mathit{y}}_{\mathit{c}}^{\mathbf{T}}{\mathit{y}}_{\mathit{c}}\right)}^{-1}{\mathit{y}}_{\mathit{c}}^{\mathbf{T}}{\mathit{ZZ}}^{\mathbf{T}}{\mathit{y}}_{\mathit{c}}.$$

The scalar ${\left({\mathit{y}}_{\mathit{c}}^{\mathbf{T}}{\mathit{y}}_{\mathit{c}}\right)}^{-1}$ does not vary with permutations. Therefore, the pseudo-F statistic in the GDBR is proportional to ${\mathit{y}}_{\mathit{c}}^{\mathbf{T}}{\mathit{ZZ}}^{\mathbf{T}}{\mathit{y}}_{\mathit{c}}$, which is equivalent to the SSU test (Pan, 2009).

(II) GTSR

The kernel of the test statistic of GTSR is ${\mathit{y}}_{\mathit{c}}^{\mathbf{T}}{\mathit{S}}_{\mathit{0}}{\mathit{y}}_{\mathit{c}}$ (Equation (4) of Tzeng et al. (2009), where S₀ is an n×n matrix with diagonal elements equal to 0 (not 1) and off-diagonal elements equal to S_ij (the similarity between the i^th and the j^th subjects). If we specify G = ZZ^T = S₀ in the GDBR, the GDBR will be equivalent to the GTSR.

(III) Our SIMc test

The test statistic is

$$T_{\mathit{\text{SIMc}}}={\left[\sum _{i=1}^n\left(y_i-{\widehat{\mathit{\mu }}}_i\right)\mathbf{\Gamma }\left(h_i\right)\right]}^{\mathbf{T}}\mathbb{S}\sum _{i=1}^n\left(y_i-{\widehat{\mathit{\mu }}}_i\right)\mathbf{\Gamma }\left(h_i\right),$$

where is the k×k matrix whose (υ, ν) element is the similarity between the υ ^th and ν ^th categories of haplotypes; y_i and Γ(h_i) are the trait value and haplotype-frequency vector of the i ^th subject, respectively; μ̂_i is the predicted mean of y_i under the null hypothesis of no association between the gene variants (in the region under investigation) and the traits. Let S be an n×n matrix with diagonal elements equal to 1 and off-diagonal elements equal to S_ij (the similarity between the i^th and the j^th subjects). The test statistic of SIMc can be re-expressed as

$$T_{\mathit{\text{SIMc}}}={\left[\sum _{i=1}^n\left(y_i-{\widehat{\mathit{\mu }}}_i\right)\mathbf{\Gamma }\left(h_i\right)\right]}^{\mathbf{T}}\mathbb{S}\sum _{i=1}^n\left(y_i-{\widehat{\mathit{\mu }}}_i\right)\mathbf{\Gamma }\left(h_i\right)=\sum _{i=1}^n\sum _{j=1}^n\left[\left(y_i-{\widehat{\mathit{\mu }}}_i\right)\left(y_j-{\widehat{\mathit{\mu }}}_j\right)\mathbf{\Gamma }{\left(h_i\right)}^{\mathbf{T}}\mathbb{S}\mathbf{\Gamma }\left(h_j\right)\right]=\sum _{i=1}^n\sum _{j=1}^n\left[\left(y_i-{\widehat{\mathit{\mu }}}_i\right)\left(y_j-{\widehat{\mathit{\mu }}}_j\right)S_{\mathit{ij}}\right]={\mathit{y}}_{\mathit{c}}^{\mathbf{T}}{\mathit{Sy}}_{\mathit{c}}.$$

If we specify G = ZZ^T = S in the GDBR, the GDBR will be equivalent to our SIMc test. There is a subtle difference between the GTSR (with S₀ in the test statistic) and the SIMc test (with S in the test statistic). To be more specific, the SIMc test is closer to the variance-component score test (VC-score) (Tzeng & Zhang, 2007, Tzeng et al., 2009), which also includes S (instead of S₀) in the test statistic. However, the amount of information contributed by the comparison of two haplotypes within a subject is relatively small compared with the between-subjects comparisons (Tzeng et al., 2009). Therefore, the performances of the SIMc test, VC-score, and GTSR are very similar.

In addition, from $T_{\mathit{\text{SIMc}}}={\left[\sum _{i=1}^n\left(y_i-{\widehat{\mathit{\mu }}}_i\right)\mathbf{\Gamma }\left(h_i\right)\right]}^{\mathbf{T}}\mathbb{S}\sum _{i=1}^n\left(y_i-{\widehat{\mathit{\mu }}}_i\right)\mathbf{\Gamma }\left(h_i\right)={\mathit{y}}_{\mathit{c}}^{\mathbf{T}}{\mathit{Sy}}_{\mathit{c}}$, we can see that our SIMc test has a similar form as KMR (see Equation (9) of (Kwee et al., 2008)). A major difference between these two tests is that the similarity measure (or ‘kernel’ called by the KMR paper) used in the R code of the KMR method is the MAF-weighted IBS sharing (see Equation (5) of (Kwee et al., 2008)). This measure downweights the similarities contributed by common SNPs and upweights the similarities contributed by rare SNPs. Consequently, KMR is more powerful than SIMc when the causal variant is uncommon (MAF <= 0.1, based on our simulation results), while SIMc is more powerful than KMR when the causal variant is common, as we can see from our simulation results.

In summary, the GDBR, GTSR, VC-score, SSU, KMR, and our SIMc test are all closely related. They detect the same aspect of information with a difference caused by various choices of similarity measures.