Were Genome-Wide Linkage Studies a Waste of Time? Exploiting Candidate Regions Within Genome-Wide Association Studies

Abstract

A central issue in genome-wide association (GWA) studies is assessing statistical significance while adjusting for multiple hypothesis testing. An equally important question is the statistical efficiency of the GWA design as compared to the traditional sequential approach in which genome-wide linkage analysis is followed by region-wise association mapping. Nevertheless, GWA is becoming more popular due in part to cost efficiency: commercially available 1M chips are nearly as inexpensive as a custom-designed 10K chip. It is becoming apparent, however, that most of the on-going GWA studies with 2,000~5,000 samples are in fact underpowered. As a means to improve power, we emphasize the importance of utilizing prior information such as results of previous linkage studies via a stratified false discovery rate (FDR) control. The essence of the stratified FDR control is to prioritize the genome and maintain power to interrogate candidate regions within the GWA study. These candidate regions can be defined as, but are by no means limited to, linkage-peak regions. Furthermore, we theoretically unify the stratified FDR approach and the weighted p-value method, and we show that stratified FDR can be formulated as a robust version of weighted FDR. Finally, we demonstrate the utility of the methods in two GWA datasets: Type 2 Diabetes (FUSION) and an on-going study of long-term diabetic complications (DCCT/EDIC). The methods are implemented as a user-friendly software package, SFDR. The same stratification framework can be readily applied to other type of studies, for example, using GWA results to improve the power of sequencing data analyses.

INTRODUCTION

Even though current genome-wide association (GWA) studies use sample sizes considerably larger than past genetic studies, it is evident that the power of a GWA study as large as 2000 cases and controls can remain low [e.g. The Wellcome Trust Case Control Consortium (WTCCC), 2007; Manolio et al., 2008; Rodriguez-Murillo and Greenberg, 2008; Tenesa et al., 2008]. Besides sample size, low power may result from small genetic effects, as anticipated for susceptibility loci associated with complex traits/diseases (e.g. odds ratios of 1.1~1.5) [Ioannidis et al., 2006; WTCCC, 2007; Manolio et al., 2008]. Most of all, simultaneous investigation of several hundred thousands of single nucleotide polymorphisms (SNPs) requires very strict criteria to properly adjust for multiple hypothesis testing [e.g. WTCCC, 2007; Dudbridge and Gusnanto, 2008].

To improve the power of GWA, several studies have adopted meta-analysis approaches to combine data from multiple sources [e.g. Scott et al., 2007; Tenesa et al., 2008; Zeggini et al., 2008]. Alternatively, a number of studies have investigated the benefits of utilizing available prior information. The information considered in these studies includes biological knowledge of SNPs and genes [Thomson et al., 2004; Sun et al., 2006; Chen and Witte, 2007; Li et al., 2008], features of different genotyping technologies and platforms [Greenwood et al., 2007], results of previous studies of related measures and populations [Pe'er et al., 2006; Roeder et al., 2006], or the union of different types of prior information [Chen and Witte, 2007]. The proposed methods often adopt the false discovery rate (FDR) methodology, because FDR control allows for a more sensible balance between type I error rate and power for large-scale hypothesis testing compared to the traditional control of family-wise error rate [Benjamini and Hochberg, 1995; Storey, 2002]. However, the relative merits of different methods remain unknown.

In this article, we unify and compare the stratified FDR (SFDR) [Sun et al., 2006] and the weighted FDR (WFDR) [Genovese et al., 2006] control methods; both of which have been shown to improve the power of GWA studies and can be easily implemented. The SFDR method has been evaluated using either multiple phenotypes or the minor allele frequency as the prior information [Sun et al., 2006], as well as using the genotyping technology [Greenwood et al., 2007] or candidate gene information [Huang et al., 2007; Li et al., 2008], while WFDR has used genome-wide linkage (GWL) results [Roeder et al., 2006]. Here, we focus on GWL data as the available prior information. Other prior types are possible, but GWL results are commonly available for many traits/diseases of interest from previous family studies. Most importantly, this allows us to illustrate the practical value of taking advantage of complementary linkage and association information and combining all available data generated from these studies. We show, via theoretical arguments, simulation studies and data applications, that the agnostic approach to the genome implied by the GWA study design might not be the optimal one. In the era of GWA it is both feasible and beneficial to prioritize the genome using methods such as SFDR and WFDR to exploit other available information.

We also demonstrate the trade-off between power and robustness by theoretical, simulation, and application studies. Robustness is desirable to safeguard against potentially uninformative or even misleading prior information due to, e.g. low power of GWL, ascertainment and phenotypic differences between GWL and GWA, or population heterogeneity. We find that WFDR loses power when the prior information consists of random noise and the power loss can be considerable if the prior is in fact misleading. In contrast, SFDR is robust to both uninformative and misleading priors. When the prior is indeed informative, both methods improve power, with marginally better performance for WFDR.

METHODS

FDR, SFDR AND WFDR CONTROL

Let P_i be the p-value of an association test for SNP i, i=1,…,m. FDR control can be achieved by converting the p-values to the corresponding q-values [Storey, 2002] (q-value calculation provided in the Appendix). SNPs with q-values less than the FDR threshold value (e.g. γ= 0.05) are declared significant. The expected proportion of false positives among all the positives is then controlled at the γ level.

Let Z_i be the linkage score of SNP i obtained from a previous GWL study using either allele-sharing or parametric approaches. For the SFDR method, m SNPs are divided into K disjoint strata based on the prior linkage information. Without loss of generality, consider K = 2 and assign each SNP i to stratum 1 (the high priority group) or stratum 2 (the low priority group) according to whether the linkage score Z_i exceeds a threshold C (e.g. C=1.64, equivalent to a one-sided linkage p-value of 0.05, for Z, a normally distributed NPL score). FDR control is then applied separately in each stratum at the same γ level [Sun et al., 2006].

In contrast, WFDR calculates a weighting factor W_i for each SNP i with weights subject to two constraints: W_i ≥ 0 and W̄ = Σ_i W_i/m = 1. The weight W_i is proportional to the linkage signal Z_i for SNP i (e.g. W_i = exp(B·Z_i)/v, v=Σ_i exp(B·Z_i/m)and B=1 [Roeder et al., 2006], and the FDR procedure is applied to the set of weight-adjusted p-values, P_i/W_i, i=1,…,m. This continuous weighting scheme can be converted to a binary one (e.g. a fraction ε of SNPs with weight W₁=B/(Bε+1-ε) and the remaining SNPs with weight W₀=1/(Bε+1-ε)) [Roeder et al., 2006], but we focus on the continuous SNP-specific weight recommended by Roeder et al. [2006] for quantitative prior GWL data.

UNIFICATION OF SFDR AND WFDR

The SFDR and WFDR approaches model the prior distinctively. WFDR uses the SNP-specific linkage score to adjust the original association p-value at each SNP, while SFDR dichotomizes the linkage results and assigns each SNP to one of the two strata according to the chosen threshold. To better understand the differences as well as the similarities between the methods, we show that SFDR can be formulated as a robust version of WFDR, and that SFDR control of K strata of SNPs, each at the same FDR level of γ, is equivalent to WFDR control applied to P_i/W_i*. However, the new weighting factor W_i* is no longer determined in a single-point fashion (i.e. SNP-specific) nor externally (i.e. solely based on the prior). Instead, all tests in the same stratum k have the same weight, $W_i^{\ast }\equiv W^{\left(k\right)}$ for SNP i in stratum k. We show in the Appendix that $W^{\left(k\right)}={\alpha }^{\left(k\right)}∕{\Sigma }_{i=1}^K(m^{\left(i\right)}\cdot {\alpha }^{\left(i\right)}∕m)$, where ${\Sigma }_{i=1}^K{m}^{\left(i\right)}=m$ and m⁽¹⁾, …m^(K) are the stratum sizes (i.e. the number of tests), and , α⁽¹⁾,…,α^(K) are the critical values (for the unadjusted association p-values) that control FDR at the γ level for the K strata. In stratum k, ${\alpha }^{\left(k\right)}=\{(1-{\pi }_0^{\left(k\right)})\cdot \gamma \cdot (1-\stackrel{‒}{\beta \left({\alpha }^{\left(k\right)}\right)})\}∕\left\{{\pi }_0^{\left(k\right)}(1-\gamma )\right\}$, where ${\pi }_0^{\left(k\right)}=m_0^{\left(k\right)}∕m^{\left(k\right)}$ is the proportion of the null SNPs/noise, and $1-\stackrel{‒}{\beta \left({\alpha }^{\left(k\right)}\right)}$ is the average power to detect the truly associated SNPs/signals each at the α^(k) level. Note that this stratum-specific weight is implicitly determined by the characteristics of a family of tests and is driven by two FDR-relevant parameters, i.e. ${\pi }_0^{\left(k\right)}$ and $1-\stackrel{‒}{\beta \left({\alpha }^{\left(k\right)}\right)}$. Because these two parameters are features of the current data, the corresponding weights depend on both the prior information (i.e. the linkage results used to define the strata) and the current data (i.e. the association results used to evaluate the patterns of the observed p-values in a stratum).

In practice, to calculate the stratum-specific weight for SFDR, one must estimate ${\alpha }^{\left(k\right)}=\{(1-{\pi }_0^{\left(k\right)})\cdot \gamma \cdot (1-\stackrel{‒}{\beta \left({\alpha }^{\left(k\right)}\right)})\}∕\left\{{\pi }_0^{\left(k\right)}(1-\gamma )\right\}$ for each stratum k. We estimate π₀ using the well known estimator, ${\widehat{\pi }}_0^{\left(k\right)}=\{\text{number of tests with p-values}>\lambda \}∕\{(1-\lambda )\cdot m^{\left(k\right)}\}$ and λ=0.5. And we estimate $1-\stackrel{‒}{\beta \left({\alpha }^{\left(k\right)}\right)}$ by $\{R^{\left(k\right)}\cdot (1-\gamma )\}∕\{m^{\left(k\right)}\cdot (1-{\widehat{\pi }}_0^{\left(k\right)})\}$, where R^(k) is the number of rejections/positives, and the numerator is the expected number of true positives and the denominator is the estimated total number of alternatives in stratum k. Thus, ${\widehat{\alpha }}^{\left(k\right)}=\{R^{\left(k\right)}\cdot \gamma \}∕\{m^{\left(k\right)}\cdot {\widehat{\pi }}_0^{\left(k\right)}\}$ and ${\widehat{W}}^{\left(k\right)}={\widehat{\alpha }}^{\left(k\right)}∕\widehat{\stackrel{-}{\alpha }}$, where $\widehat{\stackrel{-}{\alpha }}={\Sigma }_{i=1}^K{m}^{\left(i\right)}\cdot {\widehat{\alpha }}^{\left(i\right)}∕m,\forall k$.

SIMULATIONS

SIMULATION STUDY DESIGN

We first generated data using the same simulation models as in Roeder et al. [2006]. GWA association p-values were simulated for 500,000 independent SNPs under a normal model, i.e. p-values of null SNPs were derived from N(0,1) following the Unif(0,1) distribution, and p-values of causal SNPs from N(μ_a, 1) where μ_a varied between 4.0 and 6.0 quantifying different levels of power to detect the association, and μ_a is a function of both the GWA study sample size and the genetic effect size of a causal variant. The number of truly associated SNPs, m₁, was 10 and these were randomly placed on 10 different chromosomes for each simulation replicate. Nonparametric GWL scan signals (NPL scores) were simulated under a Gaussian autoregressive moving average model (ARMA (2,1) with parameters ar₁ = 1.51, ar₂=−0.51, and ma₁=0.22) [Bacanu, 2005]. This model was shown to adequately capture the features of actual linkage traces without the computationally intensive simulation of the original genotype data via gene dropping methods (details in Appendix A of Roeder et al. [2006]). For the informative GWL, the average linkage signal strength at the same causal SNPs as in the GWA simulation was set to a mean value μ_l =1, 2.5 or 3.5, and the average linkage signal strength at a non-causal locus was determined from the correlation value (ρ) generated from the ARMA model as ρ·μ_l. For the uninformative GWL, the average linkage signal strength was set to μ_l =0. Then 100 GWL scans were generated for each of 10 GWA studies, specifying C=1.64 for SFDR and B=1 for WFDR. FDR was controlled at γ=0.05. Power was measured by 1-NDR (1- non-discovery rate) defined as the ratio of the number of true positives divided by m₁ [Craiu and Sun, 2008], averaged over 100×10=1000 replicates for each simulation scenario. For SFDR, the same γ=0.05 was used for both strata, and power was calculated as the overall power summarizing both strata.

We then investigated additional simulation models. We considered a different threshold value for SFDR, C=1.96, and allowed for more than 10 truly associated SNPs in the genome, setting m₁=100. The 100 causal SNPs were randomly placed in 100 independent chromosomal regions, and the rest of the GWA and GWL data simulations were as described above. We also considered a different type of uninformative linkage data in which m₁ randomly selected SNPs (possibly differing from the ones chosen in the GWA simulation) were given positive linkage signals (μ_l =1, 2.5 or 3.5) representing misleading GWL results. This consideration was mainly motivated by potential inaccurate prior information due to population heterogeneity, for example, or phenotypic differences between GWL and GWA. In addition, we generated data with different ARMA parameter values (ar₁ = 1.45, ar₂=−0.51, and ma₁=0.22) allowing for a faster decay of the linkage scores away from a linkage peak location.

POWER AND ROBUSTNESS OF SFDR AND WFDR

Roeder et al. [2006] concluded via simulations that WFDR improves the power of GWA when the linkage scan is informative but decreases power otherwise. Following the same set of models as described above, we replicated their WFDR results and compared them to those of SFDR (Figure 1).

Figure 1. Power of FDR (black solid), SFDR (blue dot-dashed) and WFDR (red dotted) under different linkage and association conditions.

GWA association p-values were simulated for 500,000 independent SNPs among which 10 are true causal ones with association signal strength varying from μ_a=4.0 to 6.0. FDR was controlled at 0.05, and power = (number of true positives/10) averaged over 1,000 simulated replicates. Figures (a)-(c) show power of the methods when the prior linkage scans are informative with mean NPL scores at the 10 causal loci set at μ_l=1, 2.5 and 3.5 respectively and Figure (d) shows uninformative prior μ_l=0. C=1.64 for SFDR and B=1 for WFDR.

Figures 1 (a)-(c) confirm that power increases when the prior GWL scan is informative (the average Z is μ_l=1, 2.5 or 3.5 at a causal SNP and ρ·μ_l at a null SNP) and the increase is, as expected, proportional to the informativeness of the linkage signal. Remarkably, under the informative condition, the performance of the two methods is rather similar despite their distinctive use of the prior information. However, when the prior is not informative (the average Z equals 0 for all SNPs), differences between the methods are observed. Unlike WFDR, SFDR is robust to misspecification of the prior and retains power when the linkage results are random (Figure 1(d)).

Furthermore, when we consider the scenario that the prior is misleading, we observe that SFDR remains robust while WFDR can lose power substantially (Figure 2). Similar results are obtained under the additional simulation models considered (Figures S1-S6). The choice of the SFDR threshold value, C=1.64 or 1.96, can noticeably affect the performance in some cases. As expected, a larger C value is preferred when the prior is highly informative (e.g. μ_l =3.5 in Figure S1 (c)) because it increases the proportion of truly associated SNPs in the high-priority stratum. This is consistent with the observation made by Roeder et al. [2006] that a higher B value can further increase the power of WFDR when the prior provides more precise information about the locations of the associated variants.

Figure 2. Power of FDR (black solid), SFDR (C=1.64 blue dot-dashed) and WFDR (red dotted) when 10 causal loci generated independently for linkage and association data.

Figures (a)-(c) show power of the methods when the prior linkage scans provide misleading information at 10 independently chosen loci with μ_l=1, 2.5 and 3.5 respectively. For other details see Figure 1.

EMPIRICAL DISTRIBUTION OF THE WEIGHTING FACTORS FOR SFDR AND WFDR

When linkage scans are uninformative, the SNP-specific weight calculated in WFDR (W~exp(Z)) for a causal SNP can deviate substantially from 1 (Figure 3), and 69% of the simulated replicates have weight below 1 (Table S3). In contrast, the corresponding weighting factors inferred from the SFDR approach center around 1 with considerably less variation (Figure 3 and Table S3).

Figure 3. Empirical distributions of weights obtained from SFDR and WFDR at causal loci when linkage scans are uninformative.

100 datasets were simulated, each with 100 causal SNPs with association signal strength μ_a=4.0 and linkage μ_l=0. The distributions of weights at all causal loci in all simulated datasets (10,000 observations in total) are shown.

Figure 4 illustrates the distribution of weights over a chromosomal region including 3 causal loci from one dataset simulated with uninformative linkage data. For SFDR the weights are approximately 1 across the region, whereas for WFDR the weights fluctuate considerably depending on the observed NPL scores at the loci. As a result, some of the null SNPs are up-weighted (i.e. 12% of the replicates have an WFDR weight above 2 for a null SNP) and the second and third causal SNPs are down-weighted (i.e. 42% of the replicates have an WFDR weight below 0.5 for a causal SNP) leading to inefficiency and power loss (see Figure S7 for an example of informative linkage data).

Figure 4. An example of weights obtained from SFDR and WFDR when the linkage scan is uninformative.

The linkage results in the form of NPL scores (top), corresponding weights (middle) and association p-values (bottom, dots for the original association p-values and triangles for the weight adjusted association p-values) are shown for (a) SFDR and (b) WFDR in a chromosomal region from one simulated dataset with mean association signal strength μ_a=4.0 and linkage μ_l=0. Stratum specific weights for SFDR were calculated using C=1.64 and FDR controlled at 0.05, and B=1 was used for the WFDR weight calculation. Arrows indicate the location of the causal SNPs.

APPLICATIONS

To demonstrate practical relevance, we applied SFDR and WFDR to two GWA studies including the Finland-United States Investigation of NIDDM Genetics (FUSION) study of Type 2 Diabetes (T2D) [Scott et al., 2007] and the on-going Diabetes Control and Complications Trial/Epidemiology of Diabetes Interventions and Complications study (DCCT/EDIC) of diabetes-related retinal and renal phenotypes [Al-Kateb et al., 2008].

FUSION T2D GWA Study

For the FUSION Stage 1 GWA study by Scott et al. [2007], the previous FUSION GWL scan by Silander et al. [2004] served as the prior information. The FUSION Stage 1 GWA study included 1,161 T2D cases and 1,174 normal glucose tolerant controls. All samples were genotyped on the Illumina HumanHap300 BeadChip (v1.1). The 10 T2D SNPs reported in Table 1 of Scott et al. [2007] were confirmed by a powerful meta-analysis (14,586 cases and 17,968 controls). The p-values of the FUSION Stage 1 association analyses of 306,284 autosomal SNPs were computed by logistic regression with additive SNP coding. We downloaded them from NCBI dbGaP (phs000100.v1.p1). Among the 10 T2D SNPs listed in Table 1 of Scott et al. [2007], 5 are imputed SNPs (rs4402960, rs7754840, rs10811661, rs1801282, rs5219) and were not available from dbGaP. Therefore, we added the Stage 1 association p-values and the base-pair positions of these 5 SNPs obtained from Table 1 of Scott et al. [2007] to the dbGaP FUSION Stage 1 GWA data.

Table 1.

Comparison of the FDR ranks of the ten confirmed T2D SNPs reported in FUSION GWA study (FUSION stage 1 p-values) with the ranks using the SFDR and WFDR methods incorporating both real and simulated prior GWL scans

		No Prior Information		Fusion Linkage		Simulated Linkage
SNP	Chr.	P-value	Rank		Rank	Null	μl=1	μl=2	μl=3
rs7903146	10	1.25E-05	4	SFDR	1	13	9	1	1
				WFDR	4	7	3	2	1
rs9300039	11	5.99E-05	19	SFDR	19	25	21	6	6
				WFDR	21	36	16	9	7
rs4402960	3	1.20E-04	50	SFDR	50	57	53	12	12
				WFDR	50	77	35	17	12
rs13266634	8	0.001	418	SFDR	439	420	424	390	93
				WFDR	297	602	273	123	71
rs1801282	3	0.0011	530	SFDR	497	416	416	387	89
				WFDR	471	619	289	138	72
rs5219	11	0.0022	902	SFDR	902	909	902	902	902
				WFDR	1048	1169	566	266	160
rs7754840	6	0.021	6740	SFDR	6572	6843	6870	6634	6442
				WFDR	7252	11388	6375	4012	2840
rs1111875	10	0.039	12593	SFDR	12196	12843	12570	12081	11361
				WFDR	14374	20600	9008	4407	2263
rs10811661	9	0.082	26338	SFDR	24947	26480	26484	26799	30880
				WFDR	33467	43820	18201	9180	4608
rs8050136	16	0.578	178670	SFDR	275498	178662	178114	182293	237108
				WFDR	113281	183640	111687	57309	30430

306,289 GWA autosomal SNPs were investigated. The Fusion linkage LOD scores from Silander et al. [2004] were used as the real prior GWL scan, and C=0.5 and B=1 were used for SFDR and WFDR respectively. The simulated linkage data mimic the allele-sharing NPL scores, C=1.64 and B=1 were used for SFDR and WFDR respectively, and the median ranks across 1,000 simulated linkage scans were reported.

The FUSION GWL analysis [Silander et al., 2004] included 1,709 affected subjects in 737 families (FUSION 1 and 2 samples), with samples genotyped at 408 microsatellite markers for FUSION 1 and 392 for FUSION 2 (34 markers in common). The combined linkage LOD scores were based on the likelihood-based allele-sharing method [Kong and Cox, 1997], and the genome-wide marker-specific numerical results were kindly provided by Dr. Boehnke and his colleagues. The maximum LOD score was 2.98 on chromosome 11, and LOD scores at the 10 T2D loci ranged from 0.0 to 0.73 with an average of 0.27.

The corresponding linkage score for each of the 306,289 SNPs (306,284 dbGaP SNPs + 5 imputed SNPs) was interpolated from the flanking linkage markers. (See Supporting Information for details of the interpolation method.) WFDR was performed with B=1, and SFDR was performed with C=0.5. A LOD score value of 0.5 is approximately 1.5 on the NPL score scale and was chosen because of the relatively small sample size of the GWL study.

The 10 confirmed T2D loci had FUSION Stage 1 association p-values ranging from 1.25×10⁻⁵ to 0.578 and ranks from 4 to 178,670 (Table 1), but none met significant q-value criteria (Table S1). Using the GWL scan data [Silander et al., 2004], SFDR ranked the 4^th SNP as number one and WFDR gave the 418^th SNP an improved rank of 297 (Table 1). The lack of substantial overall improvement may be due to low power of either the FUSION GWL scan, or the original GWA study or both.

To uncouple the two contributing factors, we also simulated GWL scans with various levels of informativeness at these 10 loci. The method of generating simulated GWL NPL scores was the same as in the simulation studies above, with mean linkage signal values varying from μ_l=0 (uninformative linkage) to 1, 2 or 3 (informative linkage) at each of the 10 T2D loci, using ARMA (2,1) with parameters ar₁ = 1.51, ar₂=−0.51, and ma₁=0.22. We simulated 1,000 linkage datasets for each μ_l considered, and we applied SFDR (C=1.64 for the simulated NPL scores) and WFDR (B=1) to the FUSION stage 1 GWA p-values incorporating the linkage results from each of the simulated GWL scans. The ranks reported in Table 1 are the median ranks of the 10 T2D SNPs over the 1,000 simulated GWL scans. Ranks were obtained according to the SFDR and WFDR q-values and the original association p-values were used to break any ties among the q-values. The q-values reported in the Table S1 are the mean values averaged over the 1,000 replicates.

When the simulated prior GWL scan is truly informative, both SFDR and WFDR increase the chance of finding the top 5 associated SNPs, with comparable performance. However, none of the remaining 5 SNPs achieve top 100-ranking even when the prior is highly informative, due to their extreme low power in the GWA study. Notably, the performance of WFDR in improving the rank of a true SNP is superior to SFDR when confirmed loci from the original GWA study have large association p-values, e.g. rs8050136 with stage 1 p-value=0.578. However, when the prior is truly uninformative (column 7 of Table 1), WFDR tends to give the 10 T2D SNPs considerably worse ranks compared to those obtained without the prior, indicating power loss, while SFDR largely does not change the ranks reflecting its robustness to prior misspecification.

DCCT/EDIC RETINOPATHY GWA STUDY

Samples from the on-going DCCT/EDIC GWA study of long-term diabetic complications which includes 1,304 white probands were genotyped by the Illumina 1M BeadArray assay. SNPs that passed quality control criteria (MAF >1%, HWE > 10⁻⁶, and not significantly associated with gender) were analyzed for association with time-to-event phenotypes for several retinal and renal outcomes. We report on the severe retinopathy phenotype defined as severe non-proliferative diabetic retinopathy or scatter laser treatment. The time to event was modelled by the Cox proportional hazards regression for each SNP with an additive genotype coding. We analysed 841,347 autosomal SNPs for the SFDR and WFDR applications.

Prior GWL scan data were provided by a linkage study of diabetic retinopathy in the Pima Indians (607 siblings from 211 families) [Looker et al., 2007]. Genome-wide LOD scores were obtained with the variance component method [Amos et al., 1996], using 516 autosomal microsatellite markers, and the phenotype was the grade score of the worse eye retinal photograph adjusted for age and sex. The genome-wide marker-specific numerical results were kindly provided by Dr. Hanson and his colleagues. The maximum multipoint LOD score was 2.59 on chromosome 1. The corresponding linkage LOD score for each of the 841,347 GWA SNPs was interpolated from the flanking GWL markers in the same fashion as the FUSION study. SFDR was performed using a LOD score threshold value C=0.5 and WFDR with B=1.

Table 2 lists the 12 SNP with SFDR q-values ≤ 0.1; these SNPs are all from stratum 1. (See Table S2 for a list of SNPs that are among the top 20 by any of the FDR, SFDR and WFDR methods.) The original FDR q-values without linkage stratification are considerably larger and none of the WFDR q-values is smaller than 0.1. SFDR and WFDR give similarly improved rankings to the SNPs on chromosome 1, but the remaining SNPs from chromosomes 15 and 18 are given higher priority by SFDR than WFDR. A replication study of these SNPs is in progress.

Table 2.

Comparison of ranks and q-values of the top 12 SNPS (based on SFDR) of the DCCT/EDIC GWA study of diabetic retinopathy, using the SFDR and WFDR methods incorporating the GWL results of Looker et al. [2007]

				Rank			q-value
SNP	Chr.	Position (bp)	p-value	FDR	SFDR	WFDR	FDR	SFDR	WFDR
rs918837	18	49666395	4.67E-06	4	1	7	0.64	0.07	0.31
rs2744772	1	25935243	7.30E-06	9	2	1	0.66	0.07	0.31
rs2744768	1	25929896	8.71E-06	11	3	2	0.66	0.07	0.31
rs4777949	15	92110772	1.14E-05	13	4	20	0.69	0.07	0.32
rs4238481	15	92110385	1.18E-05	14	5	21	0.69	0.07	0.32
rs2065970	1	25924110	1.25E-05	15	6	4	0.69	0.07	0.31
rs807263	1	25970937	1.59E-05	17	7	5	0.76	0.07	0.31
rs2786873	1	25913549	1.67E-05	18	8	6	0.76	0.07	0.31
rs12439760	15	92123817	1.73E-05	19	9	23	0.76	0.07	0.42
rs2246979	1	25937873	2.87E-05	25	10	10	0.81	0.09	0.31
rs4777700	15	92118937	2.92E-05	27	11	27	0.81	0.09	0.60
rs2744761	1	25913101	3.31E-05	31	12	11	0.81	0.10	0.31

841,346 GWA autosomal SNPs were investigated. The LOD scores from Looker et al. [2007] were used as the prior GWL scan, and C=0.5 and B=1 were used for SFDR and WFDR respectively. The SNPs reported here are the 12 SNPs with SFDR q-values ≤ 0.1.

DISCUSSION

In this study, we theoretically unify two distinctive approaches, stratified and weighted FDR control, proposed to improve power of genome-wide association studies via incorporating available prior information. We show that the SFDR approach can be formulated as a robust version of the WFDR method, and the corresponding stratum-specific weight is implicitly determined by two FDR-relevant features for a family of tests: the proportion of true associations and the average power to detect them. The WFDR method is equivalent to using a different significance threshold for each test, while SFDR adjusts the threshold for a family/stratum of tests.

Specifying a common within-stratum weighting factor, as in SFDR, provides robustness. When the prior is not informative, SNPs are effectively randomly assigned to different strata leading to similar characteristics across strata, i.e. ${\pi }_0^{\left(k\right)}$ and $1-\stackrel{‒}{\beta \left({\alpha }^{\left(k\right)}\right)}$. As a result, the α^(k) that satisfies the equation, ${\alpha }^{\left(k\right)}=\{(1-{\pi }_0^{\left(k\right)})\cdot \gamma \cdot (1-\stackrel{‒}{\beta \left({\alpha }^{\left(k\right)}\right)})\}∕\left\{{\pi }_0^{\left(k\right)}(1-\gamma )\right\}$, would be the same for all strata subject to sampling variation. Therefore α^(k) = α and ᾱ = α, and the stratum-specific weight is data-driven to be W^(k) = α^(k) / ᾱ = 1 for all K strata. In contrast, the observed prior linkage NPL/LOD score Z from a single dataset is not necessarily zero when the prior is not informative (although the expected linkage score Z equals zero at any given locus, the NPL/LOD scores from a GWL scan simulated under the null of no linkage fluctuate around zero across the genome given finite sample size). Thus, the SNP-specific weight calculated in WFDR, W ~ exp(Z), can deviate substantially from 1 leading to inefficiency and power loss.

When the prior provides misleading information (e.g. a linkage peak is expected to appear at a locus many cM away from the true underlying location due to for example phenotypic or population differences between GWL and GWA data), WFDR gives high weight to a null SNP but not to a truly associated SNP. In contrast, in SFDR, if the high prior stratum fails to contain the majority of truly associated variants due to the misleading nature of the linkage results, those true positives will be captured by the other stratum which now may have a higher proportion of true associations than the overall value, leading to a higher weight for the low prior stratum. This highlights a robustness feature of SFDR that is absent from WFDR in which the externally assigned weights are independent of the relationship between the prior information and the current data.

In their work, Roeder et al. [2006] also considered binary weights, i.e. a fraction ε of SNPs with weights W₁=B/(Bε+1−ε) and the remaining SNPs with weights W₀=1/(Bε+1−ε), where B=2, 6 or 50. However, they concluded that the continuous weighting scheme is more suitable for GWL prior data because the linkage statistics are correlated between markers. Therefore, the performance of WFDR with binary weights remains to be determined for genome-wide genetic data. Nevertheless, it is not difficult to see that the binary weights proposed by Roeder et al. [2006] do not provide robustness to WFDR, because W₁>1 (as long as B >1) regardless of the prior information. The simulation studies confirm our theoretical argument and show that only SFDR is robust to both uninformative and misleading prior information.

It is also important to note that there are cases when WFDR is superior to SFDR, as demonstrated in the application to the FUSION dataset (Table 1). When causal loci have large association p-values in the original GWA study (e.g. p-value=0.578 for rs8050136 in FUSION stage 1), the fact that such SNPs belong to the high-priority stratum may not change their statistical evidence significantly, because the stratum typically also contains many null SNPs some of which are expected to have small p-values by chance. (Realistic prior never unambiguously separates all the nulls from the alternatives. Otherwise, the prior alone has 100% testing power.) The stratum-specific weight could be greater than 1 but it will be applied to all SNPs in that stratum, and SFDR does not change the original rank order of significance within a stratum. Thus, a test/SNP with p-value around 0.5 would remain mid-ranked in a stratum based on SFDR. In contrast, WFDR could decrease the p-value of a single SNP if the corresponding SNP-specific weight is high. This observation indicates some of the limitations of our simulation studies. Compared to the FUSION stage 1 GWA study where 6 of the 10 confirmed T2D loci had p-values >10⁻³, the power of the simulated GWA studies was considerably higher: the median p-value of a causal SNP with μ_a=4 is of the order of 10⁻⁵ (10⁻⁹ for μ_a=6). Another possible limitation is the independence assumption made for the simulated GWA SNPs, although correlation was modeled for the linkage simulation component.

In general, the WFDR method is less applicable than SFDR due to the need for SNP-specific prior information. For example, in the case of using GWL results as the prior in the WFDR setting, the precise linkage result (e.g. LOD or NPL score) must be obtained or estimated for each SNP to up- or down-weight the corresponding association p-value. In contrast, to apply the SFDR method, one only needs to know if the linkage result of a SNP is above or below a threshold value. Other types of prior information, such as knowledge about SNPs, genes, phenotypes or platforms could be also used to stratify and prioritize the SNPs but are not so straightforward to apply using WFDR with continuous weights. Although Roeder et al. [2006] argued in favor of continuous weights for GWL prior data because of the correlation between linked markers, a binary weighting scheme of WFDR could be considered for those situations. As discussed above, however, the binary option does not provide robustness to WFDR.

In some cases, the readily available prior information may be so limited that even the SFDR method is not directly applicable. For example, the most commonly reported results for GWL studies are the peak LOD scores, the chromosomal location of such peaks and possibly the locations of the markers flanking the 1-LOD-unit down support intervals. In an affected-sib-pair design in which only the peak NPL/LOD scores are available, one could consider estimating the size of the intervals for stratification using the equation, $d_{t,\tau }\approx 25\times (\ln \left(E\left[\sqrt{{\mathit{LOD}}_{\tau }}\right]\right)-\ln \left(E\left[\sqrt{{\mathit{LOD}}_t}\right]\right))$, where d is the distance in cM between locus τ and locus t, and LOD_τ and LOD_t are the LOD scores at a peak and a nearby locus t. (See Appendix for details of the derivation.) We applied this method to the FUSION study for which the published numerical GWL results are available for peak regions only (Table 4 of Silander et al. [2004]), and we compared the estimated results to the original GWL data provided by Dr. Boehnke and his colleagues. Figure S8 shows the original GW linkage trace (black solid) and the estimated one (red dashed). Among the 306,289 GWA SNPs, 276,392 have both an observed and estimated LOD < 0.5, and 17,413 have both LOD ≥ 0.5. However, 6,526 had an observed LOD ≥ 0.5 but an estimated LOD < 0.5, and 5,958 had an observed LOD < 0.5 but and estimated LOD ≥ 0.5. Table S4 contrasts the SFDR application results using the observed linkage results with those from the estimated intervals. Although the proposed method provides rough estimation, further improvement is needed and acquisition of the original linkage results remains beneficial.

Ideally, maximal power improvement could be achieved by choosing optimal SFDR threshold(s) and WFDR weighting function(s). However, how to accomplish this is still an open question. For the weighted approach, a two-stage design has been suggested to obtain optimum group-specific weights in the context of family-wise error rate control [Roeder et al., 2007]. A related question is how to optimally combine a set of available priors, e.g. GWL results from various linkage studies of the same phenotype, results from both GWL and candidate gene studies, and information such as the heterozygosity of the linkage markers, in addition to the minor allele frequency and genotyping quality score of the GWA SNPs.

GWA studies have produced an avalanche of data, and the results of previous genome-wide linkage scans seem to be mostly relegated to the waste bin. Although few GWL studies produced significant results at the strict genome-wide level, the chance of a region with modest linkage evidence harboring susceptibility loci is more than random. Moreover, although the GWA study design allows us to interrogate common variation across the whole genome, its agnostic nature means that by design only a very small proportion of the SNPs are expected to be truly associated among all SNPs genotyped and tested (e.g. the estimated proportion based on the GWA p-values is <3% for FUSION stage 1 and <2% for DCCT/EDIC). This signal sparsity feature is detrimental from the multiple hypothesis testing point of view, and as a result it is extremely difficult to separate the wheat from the chaff unless the genetic effect of the causal variant and/or the sample size of the study are very large. The SFDR and WFDR methods provide simple ways to combine all available data, and SFDR in particular allows us to prioritize specific parts of the genome within the analysis of GWA data, while maintaining robustness to both uninformative and misleading prior information. The applications to the FUSION and DCCT/EDIC data show that previous GWL studies were not a waste of time and can provide information that improves the power of current GWA studies. However, our work also demonstrates that both GWL and GWA studies must be reasonably powered in the first place, so that the combined analysis achieves a meaningful gain in efficiency. The same SFDR and WFDR framework can be easily applied to other types of studies. For example, results of current GWL and GWA studies could be utilized as the available prior information to improve the power of statistical analyses of the impending sequencing data.

SOFTWARE

A user-friendly software package, SFDR, tailored for genome-wide association and linkage data is available at http://www.utstat.toronto.edu/sun/. SFDR implements the false discovery rate, stratified false discovery rate and weighted false discovery rate control methods and computes the corresponding ranks and q-values.

APPENDIX

DERIVATION OF STRATUM-SPECIFIC WEIGHTING FACTOR FOR SFDR

Controlling the FDR at level γ is equivalent to calling all tests with q-values ≤γ significant. The estimates of the q-values can be obtained by the use of the following recursive formula

$${\widehat{q}}_{\left(i\right)}=\min \left\{\frac{{\widehat{\pi }}_0{\mathit{mP}}_{\left(i\right)}}{i},{\widehat{q}}_{(i+1)}\right\},$$

where P₍₁₎ ≤…≤ P_(m) is the ordered sequence of the m available p-values, q̂_(m) = π̂₀P_(m), and π̂₀ is an estimate of π₀ = m₀/m , the proportion of null hypotheses among the total tested.

The weighted p-value method [Genovese et al., 2006] applies the FDR control procedure to a set of weight adjusted p-values, $P_i^W=P_i∕W_i$ where W_i is the weighting factor for test i with two constraints: W_i ≥ 0 and W̄ = Σ_iW_i/m = 1. To achieve FDR control at the nominal γ level, one could re-estimate the q-values corresponding to $P_i^W$ and reject all hypothesis tests with q-values ≤γ. Alternatively, one could declare a test significant if its weighted but otherwise unadjusted p-value, $P_i^W\le \alpha$, where α satisfies the following equation [Sun et al., 2006]:

$$\alpha =\frac{(1-{\pi }_0)\cdot \gamma \cdot (1-\stackrel{‒}{\beta \left(\alpha \right)})}{{\pi }_0(1-\gamma )}.$$

Here $1-\stackrel{‒}{\beta \left(\alpha \right)}=1-{\Sigma }_{i=m_0+1}^m{\beta }_i\left(\alpha \right)∕m_1$, the average (single-hypothesis) power to detect the m₁ alternatives, each at the α level.

In the context of SFDR, the set/family of p-values are assigned to K disjoint strata/groups depending on the prior Z for the corresponding hypotheses. The FDR procedure is then applied separately to p-values in each stratum at the same γ level. The expression of α^(k) for the k^th stratum is identical to the above equation with the addition of superscript ^(k) for π₀ and $\stackrel{‒}{\beta \left(\alpha \right)}$. That is, rejecting tests/SNPs with P_i ≤ α^(k) controls FDR at the γ level in stratum k, where P_i is the unadjusted p-value of SNP i that was assigned to stratum k, and

$${\alpha }^{\left(k\right)}=\frac{(1-{\pi }_0^{\left(k\right)})\cdot \gamma \cdot (1-\stackrel{‒}{\beta \left({\alpha }^{\left(k\right)}\right)})}{{\pi }_0^{\left(k\right)}(1-\gamma )}.$$

To make a connection between SFDR and WFDR, note that with the following re-parameterization, α^(k) = W^(k) ᾱ, P_i is equivalent to P_i / W^(k) ≤ ᾱ, a weighted p-value approach. Since α^(k) = W^(k) ᾱ must hold for all k=1,…,K with the constraint $\left({\displaystyle {\Sigma }_k}{m}^{\left(k\right)}{W}^{\left(k\right)}\right)∕m=1$, one can solve the set of equations and obtain

$$W^{\left(k\right)}={\alpha }^{\left(k\right)}∕\stackrel{-}{\alpha },\stackrel{-}{\alpha }={\displaystyle \underset{k}{\Sigma }}\frac{m^{\left(k\right)}}{m}{\alpha }^{\left(k\right)},\forall k.$$

Therefore, the original SFDR method can be formulated as WFDR control by re-scaling the p-values with the above weight, i.e.

$$P_i^{W^{\ast }}=P_i∕W_i^{\ast },$$

where $W_i^{\ast }\equiv W^{\left(k\right)}$ if test i belongs to stratum k.

ESTIMATION OF STRATA FOR SFDR USING LOD SCORES AT PEAK REGIONS ONLY

Under a set of reasonable assumptions (e.g. the affected sib-pair, ASP, design), we show that

$$d{}_{t,\tau }\approx 25\times (\ln \left(E\left[\sqrt{{\mathit{LOD}}_{\tau }}\right]\right)-\ln \left(E\left[\sqrt{{\mathit{LOD}}_t}\right]\right)),$$

where d is the distance in cM between locus τ and locus t, and $E\left[\sqrt{{\mathit{LOD}}_{\tau }}\right]$ is the expected square root of the LOD score at a peak (assumed to be the disease locus) and LOD_t is the LOD score at locus t. The derivation of the above equation requires 3 steps.

Step 1 specifies the relationship between IBD sharing at two different loci. Under the assumption of random mating, linkage equilibrium, no more than one susceptibility gene in the region of interest and generalized single ascertainment, Liang et al. [2001] derived a simple representation relating the expected IBD sharing at any position, t, to the sharing at the disease locus, τ , and the distance between t and τ:

$$E[{\mathit{IBD}}_t\mid \mathit{ASP}]-1=f({\theta }_{t,\tau },\mathit{ASP})(E[{\mathit{IBD}}_{\tau }\mid \mathit{ASP}]-1),$$

where f(θ_t,τ,ASP) is a function that depends on the recombination fraction between t and τ and the relationship type of the two individuals. Assuming the no-interference model (Haldane's map function) and the ASP design,

$$\begin{array}{c}\hfill f({\theta }_{t,\tau },\mathit{ASP})=2({\theta }^2+{(1-\theta )}^2)-1\hfill \\ \hfill =2({\left(\frac{1-\exp (-.02d{}_{t,\tau })}{2}\right)}^2+{\left(1-\frac{1-\exp (-.02d{}_{t,\tau })}{2}\right)}^2)-1=\exp (-0.04d{}_{t,\tau }),\hfill \end{array}$$

where d_t,τ is the genetic distance between t and τ in cM. Thus,

$$E[{\mathit{IBD}}_t\mid \mathit{ASP}]-1=\exp (-0.04d{}_{t,\tau })(E[{\mathit{IBD}}_{\tau }\mid \mathit{ASP}]-1).$$

.

Note that the coefficient will be different for designs other than the ASP design, but the derivation requires no assumptions about the mode of inheritance, except that the region contains no more than one susceptibility gene. (See Biernacka et al. [2005] for two linked genes.) Since genetic distance in cM is additive, assuming t1 < t2 < τ, it is not difficult to show that

$$\begin{array}{c}\hfill E[{\mathit{IBD}}_{t1}\mid \mathit{ASP}]-1=\exp (-0.04d{}_{t1,\tau })(E[{\mathit{IBD}}_{\tau }\mid \mathit{ASP}]-1)\hfill \\ \hfill =\exp (-0.04(d{}_{t1,t2}+d{}_{t2,\tau }))(E[{\mathit{IBD}}_{\tau }\mid \mathit{ASP}]-1)\hfill \\ \hfill =\exp (-0.04d{}_{t1,t2})\exp (-0.04d{}_{t2,\tau })(E[{\mathit{IBD}}_{\tau }\mid \mathit{ASP}]-1)\hfill \\ \hfill =\exp (-0.04d{}_{t1,t2})(E[{\mathit{IBD}}_{t2}\mid \mathit{ASP}]-1).\hfill \end{array}$$

However, similar results cannot be obtained for t1 < τ < t2.

Step 2 specifies the relationship between the expected IBD sharing and the expected NPL score. Assuming that the GWL analysis was conducted using n ASPs, the NPL score at a locus t can be calculated as

$${\mathit{NPL}}_t={\displaystyle \underset{i}{\Sigma }}\frac{S_t^i-1}{\sqrt{1∕2}}∕\sqrt{n},$$

where $S_t^i$ is the observed or inferred IBD sharing statistic for the i^th ASP at locus t. The expected NPL score is

$$E\left[{\mathit{NPL}}_t\right]={\displaystyle \underset{i}{\Sigma }}\frac{E\left[S_t^i\right]-1}{\sqrt{1∕2}}∕\sqrt{n}={\displaystyle \underset{i}{\Sigma }}\frac{E[{\mathit{IBD}}_t\mid \mathit{ASP}]-1}{\sqrt{1∕2}}∕\sqrt{n}=\sqrt{2n}(E[{\mathit{IBD}}_t\mid \mathit{ASP}]-1).$$

Step 3 specifies the relationship between the NPL score and the LOD score. Assuming the LOD score was calculated based on the likelihood ratio test with 1 d.f., the NPL is approximately normally distributed, and the score test and likelihood ratio test yield similar results. Then

$$D={\log }_{10}(L_1∕L_0)=\frac{1}{2\ln 10}\cdot 2\ln (L_1∕L_0)~\frac{1}{2\ln 10}{\chi }_1^2\text{and}{\mathit{NPL}}^2~{\chi }_1^2.$$

Therefore, LOD ~ NPL² /(2ln10).

Putting everything together, we obtain

$$\begin{array}{c}\hfill E\left[\sqrt{{\mathit{LOD}}_t}\right]\approx \frac{E\left[{\mathit{NPL}}_t\right]}{\sqrt{2\ln 10}}=\frac{\sqrt{2n}(E[{\mathit{IBD}}_t\mid \mathit{ASP}]-1)}{\sqrt{2\ln 10}}\hfill \\ \hfill =\frac{\sqrt{2n}\exp (-0.04d{}_{t,\tau })(E[{\mathit{IBD}}_{\tau }\mid \mathit{ASP}]-1)}{\sqrt{2\ln 10}}=\frac{\exp (-0.04d{}_{t,\tau })E\left[{\mathit{NPL}}_{\tau }\right]}{\sqrt{2\ln 10}}\hfill \\ \hfill \approx \exp (-0.04d{}_{t,\tau })E\left[\sqrt{{\mathit{LOD}}_{\tau }}\right],\text{and}\hfill \\ \hfill d{}_{t,\tau }\approx 25\times (\ln \left(E\left[\sqrt{{\mathit{LOD}}_{\tau }}\right]\right)-\ln \left(E\left[\sqrt{{\mathit{LOD}}_t}\right]\right)).\hfill \end{array}$$

Based on this result, LOD_τ = 3 and LOD_t = 2 leads to d ≈ 5cM, and LOD_τ = 3 and LOD_t = 0.5 leads to d ≈ 22cM . Note that the derivation provides the expected values. This leads to a “sharper” LOD curve. Therefore d_t,τ typically underestimates the actual distances observed in GWL studies with finite sample size. In addition, the derivation does not take into account the edge effect (i.e. when the peak is at or close to the telomeres or centromere of a chromosome).