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Defining the power limits of genome-wide association scan meta-analyses

Kay Chapman

Wellcome Trust Centre for Human Genetics, Roosevelt Drive, University of Oxford, Oxford, United Kingdom

Botnar Research Centre, Institute of Musculoskeletal Sciences, Nuffield Department of Orthopaedics, Rheumatology and Musculoskeletal Sciences, Windmill Road, Oxford, United Kingdom

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Teresa Ferreira,

Teresa Ferreira

Wellcome Trust Centre for Human Genetics, Roosevelt Drive, University of Oxford, Oxford, United Kingdom

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Andrew Morris,

Andrew Morris

Wellcome Trust Centre for Human Genetics, Roosevelt Drive, University of Oxford, Oxford, United Kingdom

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Jennifer Asimit,

Jennifer Asimit

The Wellcome Trust Sanger Institute, Wellcome Trust Genome Campus, Hinxton, Cambridge, United Kingdom

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Eleftheria Zeggini,

Corresponding Author

Eleftheria Zeggini

[email protected]

The Wellcome Trust Sanger Institute, Wellcome Trust Genome Campus, Hinxton, Cambridge, United Kingdom

Wellcome Trust Sanger Institute, The Morgan Building, Wellcome Trust Genome Campus, Hinxton, Cambridge CB10 1HH, UK===Search for more papers by this author

Kay Chapman,

Kay Chapman

Wellcome Trust Centre for Human Genetics, Roosevelt Drive, University of Oxford, Oxford, United Kingdom

Botnar Research Centre, Institute of Musculoskeletal Sciences, Nuffield Department of Orthopaedics, Rheumatology and Musculoskeletal Sciences, Windmill Road, Oxford, United Kingdom

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Teresa Ferreira,

Teresa Ferreira

Wellcome Trust Centre for Human Genetics, Roosevelt Drive, University of Oxford, Oxford, United Kingdom

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Andrew Morris,

Andrew Morris

Wellcome Trust Centre for Human Genetics, Roosevelt Drive, University of Oxford, Oxford, United Kingdom

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Jennifer Asimit,

Jennifer Asimit

The Wellcome Trust Sanger Institute, Wellcome Trust Genome Campus, Hinxton, Cambridge, United Kingdom

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Eleftheria Zeggini,

Corresponding Author

Eleftheria Zeggini

[email protected]

The Wellcome Trust Sanger Institute, Wellcome Trust Genome Campus, Hinxton, Cambridge, United Kingdom

Wellcome Trust Sanger Institute, The Morgan Building, Wellcome Trust Genome Campus, Hinxton, Cambridge CB10 1HH, UK===Search for more papers by this author

First published: 15 September 2011

https://doi.org/10.1002/gepi.20627

Citations: 17

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Abstract

Large-scale meta-analyses of genome-wide association scans (GWAS) have been successful in discovering common risk variants with modest and small effects. The detection of lower frequency signals will undoubtedly require concerted efforts of at least similar scale. We investigate the sample size-dictated power limits of GWAS meta-analyses, in the presence and absence of modest levels of heterogeneity and across a range of different allelic architectures. We find that data combination through large-scale collaboration is vital in the quest for complex trait susceptibility loci, but that effect size heterogeneity across meta-analyzed studies drawn from similar populations does not appear to have a profound effect on sample size requirements. Genet. Epidemiol. 2011. © 2011 Wiley Periodicals, Inc. 35:781-789, 2011

INTRODUCTION

The advent of genome-wide association scans (GWAS) has undoubtedly changed the field of complex trait genetics dramatically over the last few years. GWAS became feasible following an alignment of possibilities, including large-scale sample size availability, an improved understanding of human genome sequence variation, advances in high-accuracy, high-throughput genotyping technologies, and a deeper appreciation of analytical considerations for the interpretation of data. The first wave of GWAS led to the successful identification of multiple common variants and, in general, reaped the low-hanging fruit, i.e. variants closely tagging causal alleles with modest to large effect sizes [WTCCC, 2007]. However, individual studies are limited in power by finite sample sizes. Wide collaborative networks culminated in consortium formation with the purpose of meta-analyzing at the genome-wide scale, increasing sample size without the need for de novo genotyping. By synthesizing summary association statistics across multiple GWAS, power is increased and novel discoveries have been made, for example in type 2 diabetes, colorectal cancer, coronary artery disease, and fat distribution [Voight et al., 2010; Houlston et al., 2010; Preuss et al., 2010; Heid et al., 2010].

The vast majority of GWAS meta-analyses to date have focused on populations of similar ancestry (primarily European), circumventing the complications of locus heterogeneity that can have profound effects on power when combining data across different genetic architectures. Statistical heterogeneity is a challenge not only for meta-analyses across diverse populations but also across similar-ancestry data sets when effect sizes are dissimilar. Allelic heterogeneity, which is conceivably the case for loci containing multiple low frequency/rare variants, represents an additional challenge.

The field is poised to continue with ever-increasing large-scale GWAS meta-analyses in parallel to entering the era of next generation association studies involving whole-exome or whole-genome sequencing. In this study, we investigate when GWAS meta-analyses may reach a point of diminishing returns in terms of power to detect disease-associated variants, in the presence and absence of modest levels of heterogeneity and across a range of different allelic architectures.

Replication of findings before declaring success, i.e. before claiming the detection of established disease associations, has become the sine qua non in GWAS. The larger the discovery sample, the smaller the effect sizes that a GWAS meta-analysis can detect. This factor, in combination with winner's curse (i.e. the fact that the originally identified effect size is likely to be overestimated in comparison to its true value), can translate into a requirement for replication data sets that supersede the discovery set in sample size. For current meta-analysis efforts that involve several tens of thousands of samples (even surpassing 100,000, for example for quantitative anthropometric traits such as height), identifying appropriately sized replication data sets can be extremely challenging. In this study, we also explore the combined power of the discovery (Stage 1) and replication (Stage 2) sets to detect associations at genome-wide significance, with a focus on sample size requirements.

METHODS

We obtained estimates for the sample sizes required for a range of genetic effect sizes and different risk allele frequencies, keeping power constant at 80%. We investigated different scenarios involving a combination of Stage 1 (discovery) and Stage 2 (replication) samples, to mimic realistic GWAS meta-analysis studies and their followup. We considered a sample of cases and an equal number of controls. This 1:1 ratio was kept constant throughout all simulations for Stages 1 and 2 and when considering the effects of heterogeneity. The disease trait was set to have a population risk of 0.05. We assumed that we were measuring the effect of the causal variant itself and examined power under the additive model in the log-odds ratio of the risk allele.

STAGE 1 (DISCOVERY)

Using the software package Quanto version 1.2.4 [Gauderman, 2002], we estimated the sample sizes required for 12 different ORs ranging from 1.05 to 3.0 (with increments of 0.05) and 15 different risk allele frequencies ranging from 0.001 to 0.5 (with initial increments of 0.001, followed by 0.01 then 0.1). We applied three different thresholds of significance: P<1 × 10⁻⁴, P<1 × 10⁻⁵, and P<5 × 10⁻⁸ (genome-wide significant; Fig. 1).

Details are in the caption following the image — **Figure 1**
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Overview of study design. Stage 1 sample sizes were estimated using the following design parameters: population risk of 0.05, additive inheritance model, case-control ratio of 1:1, 80% power. Replication sample sizes were established using the following design parameters: 80% power to detect 100 signals from a GWAS of 2 million markers, α = 5 × 10⁻⁸.

STAGE 2 (REPLICATION)

Using the software package CaTS [Skol et al., 2006], we assumed that 100 signals (0.005%) from 2 million markers of a GWAS meta-analysis (for example, using HapMap data to impute genotypes at untyped variants) were selected for replication in Stage 2. This number of markers is representative of the first followup step for most GWAS meta-analysis efforts. The sample size necessary to replicate Stage 1 signals at a final significance threshold of 5 × 10⁻⁸ in a combined meta-analysis was calculated for each of the scenarios for each of the three Stage 1 significance thresholds. A combined P-value surpassing genome-wide significance is typically taken as evidence for robust association at disease-associated loci. Sample sizes could not be estimated beyond a threshold of 1,000,000. A total of 540 scenarios were examined.

HETEROGENEITY

We selected three of the above scenarios for further investigation through simulations incorporating heterogeneity. These were:

(1)
Allele freq = 0.30, OR = 1.25 (common SNP, moderate effect size)
(2)
Allele freq = 0.05, OR = 1.25 (low-frequency SNP, moderate effect size)
(3)
Allele freq = 0.01, OR = 3.0 (low-frequency SNP, large effect size)

Evidence for heterogeneity of genetic effects is commonly investigated using two statistics: Cochran's Q statistic of homogeneity and the I² measure [Higgins et al., 2003]. We used I² to quantify the effect of heterogeneity. The I² measure has a very intuitive interpretation and allows assessing statistical significance and the extent of heterogeneity simultaneously. Different extents of heterogeneity were considered for each of the three scenarios. We assigned adjectives of none, low, moderate, and high heterogeneity to I² values less than 15%, between 15 and 35%, between 35 and 70%, and greater than 70%, respectively (calculated as the mean across all simulations carried out for the scenario considered) [Higgins et al., 2003].

For our simulations, we assumed a meta-analysis of 10 studies with equal sample size and with a 1:1 case-control ratio. Disease prevalence was fixed at 0.05 (as above). For a given scenario, the allele frequency was fixed, while the ORs varied across studies to produce the selected, combined meta-analysis OR. The amount of variability depends on the level of heterogeneity considered in the scenario under analysis.

In each scenario, 10,000 replicates of data were simulated. In each replicate, and for each of the 10 studies, the genotypes of the SNP were simulated independently based on the allele frequency considered and assuming Hardy-Weinberg equilibrium. Conditional upon the genotype data, the case-control status was simulated according to the additive model. The association analysis was then performed in a logistic regression modeling framework. Based on this, fixed-effects meta-analysis was carried out, including a test of heterogeneity.

Based on these simulations, power to detect association at a significance level of 5 × 10⁻⁸ was estimated, with mean I² determined to evaluate heterogeneity. These simulations were repeated for different sample sizes in order to verify which would reach a detection power of 80%.

RESULTS

Estimated sample sizes for Stages 1 and 2 are given for each of Stage 1 significance thresholds of 1 × 10⁻⁴ (Table I), 1 × 10⁻⁵ (Table II), and 5 × 10⁻⁸ (Table III) in the absence of heterogeneity. As expected, the results demonstrate a relationship between sample size, risk allele frequency (RAF), and the size of the genetic effect (Figs. 2, 3, and 4). As the genetic variant becomes rarer and the odds ratio becomes smaller, the sample size required to detect the signal becomes increasingly large.

Table I. Sample size for Stages 1 and 2 for a range of RAFs and ORs

		RAF
OR		0.001	0.002	0.003	0.004	0.005	0.01	0.02	0.03	0.04	0.05	0.1	0.2	0.3	0.4	0.5
1.05	Stage 1	>1,000,000	>1,000,000	>1,000,000	>1,000,000	>1,000,000	930,100	470,000	316,700	240,100	194,200	102,700	58,000	44,400	39,000	37,600
	Stage 2						760,991	384,545	259,118	196,445	158,891	84,027	47,455	36,327	31,909	30,764
1.1	Stage 1	>1,000,000	>1,000,000	790,000	593,100	475,000	238,800	120,700	8,141,400	61,700	50,000	26,500	15,000	11,600	10,200	9,900
	Stage 2			646,364	485,264	388,636	195,382	98,755	66,600	50,482	40,909	21,682	12,273	9,491	8,345	8,100
1.15	Stage 1	>1,000,000	539,800	360,300	270,500	216,600	108,900	55,100	37,200	28,200	22,800	12,100	6,900	5,300	4,700	4,600
	Stage 2		441,655	294,791	221,318	177,218	89,100	45,082	30,436	23,073	18,655	9,900	5,645	4,336	3,845	3,764
1.2	Stage 1	621,900	311,300	207,800	156,000	125,000	62,900	31,800	21,500	16,300	13,200	7,000	4,000	3,100	2,800	2,700
	Stage 2	508,827	254,700	170,018	127,636	102,273	51,464	26,018	17,591	13,336	10,800	5,727	3,273	2,536	2,291	2,209
1.25	Stage 1	407,800	204,100	136,300	102,300	82,000	41,200	20,900	14,100	10,700	8,700	4,600	2,700	2,100	1,850	1,800
	Stage 2	333,655	166,991	111,518	83,700	67,091	33,709	17,100	11,536	8,755	7,118	3,764	1,800	1,400	1,514	1,473
1.3	Stage 1	290,000	145,200	96,900	72,800	58,300	29,300	14,900	10,000	7,600	6,200	3,300	1,900	1,500	1,350	1,300
	Stage 2	237,273	118,800	79,282	59,564	47,700	23,973	9,933	8,182	6,218	4,133	2,700	1,555	1,227	900	1,064
1.35	Stage 1	218,000	109,200	72,900	54,700	43,800	22,100	11,200	7,600	5,700	4,700	2,500	1,400	1,100	1,000	1,000
	Stage 2	178,364	89,345	48,600	36,467	29,200	14,733	7,467	5,067	4,664	3,133	1,667	1,400	1,100	1,000	818
1.4	Stage 1	170,700	85,500	57,100	42,900	34,300	17,300	8,800	5,900	4,500	3,700	2,000	1,100	900	800	800
	Stage 2	139,664	69,955	38,067	28,600	22,867	11,533	5,867	3,933	3,000	2,467	1,333	1,100	600	655	655
1.5	Stage 1	114,200	57,200	38,200	28,700	23,000	11,600	5,900	4,000	3,000	2,700	1,300	800	600	550	550
	Stage 2	93,436	38,133	25,467	19,133	15,333	7,733	3,933	2,667	2,000	1,667	1,064	431	491	450	450
2	Stage 1	34,700	17,400	11,600	8,700	7,000	3,500	1,800	1,200	900	800	400	250	200	200	200
	Stage 2	23,133	11,600	7,733	5,800	4,667	2,333	1,200	800	736	431	327	167	133	108	133
2.5	Stage 1	18,200	9,100	6,100	4,600	3,700	1,900	950	650	500	400	250	150	100	100	100
	Stage 2	12,133	6,067	3,285	2,477	1,992	1,023	512	350	269	267	83	64	300	122	300
3	Stage 1	11,800	5,900	4,000	3,000	2,400	1,200	600	400	350	300	150	100	100	100	100
	Stage 2	6,354	3,177	2,154	1,615	1,292	646	400	267	117	100	81	43	18	18	25

Sizes are calculated for detection of 80% power at a significance level of 1 × 10⁻⁴ for Stage 1 and a combined Stages 1 and 2 significance level of 5 × 10⁻⁸. Estimates are based on assumptions of 1:1 case:control ratios and a population risk of 0.05. RAF, risk allele frequency.

Table II. Sample size for Stages 1 and 2 for a range of RAFs and ORs

		RAF
OR		0.001	0.002	0.003	0.004	0.005	0.01	0.02	0.03	0.04	0.05	0.1	0.2	0.3	0.4	0.5
1.05	Stage 1	>1,000,000	>1,000,000	>1,000,000	>1,000,000	>1,000,000	>1,000,000	580,400	391,100	296,500	239,800	126,900	71,700	54,900	48,200	46,500
	Stage 2						>1,000,000	193,467	130,367	98,833	79,933	42,300	23,900	18,300	16,067	15,500
1.1	Stage 1	>1,000,000	>1,000,000	975,600	732,500	586,600	294,900	149,100	100,500	76,200	61,700	32,700	18,600	14,300	12,600	12,200
	Stage 2			325,200	244,167	195,533	98,300	49,700	33,500	25,400	20,567	10,900	6,200	4,767	4,200	4,067
1.15	Stage 1	>1,000,000	666,600	444,900	334,000	267,500	134,500	68,000	45,900	34,800	28,200	15,000	8,500	6,600	5,800	5,700
	Stage 2		222,200	148,300	111,333	89,167	44,833	22,667	15,300	11,600	9,400	5,000	2,833	2,200	1,933	1,800
1.2	Stage 1	768,000	384,400	256,600	192,700	154,300	77,600	39,300	26,500	20,100	16,300	8,700	5,000	3,900	3,400	3,350
	Stage 2	192,000	96,100	64,150	48,175	38,575	25,867	9,825	8,833	6,700	4,075	2,175	1,250	975	1,133	1,117
1.25	Stage 1	503,600	252,100	168,300	126,400	101,200	50,900	25,800	17,400	13,200	10,700	5,700	3,300	2,600	2,300	2,250
	Stage 2	125,900	63,025	42,075	31,600	25,300	12,725	6,450	4,350	3,300	2,675	1,900	825	650	575	750
1.3	Stage 1	358,100	179,300	119,700	89,900	72,000	36,200	18,300	12,400	9,400	7,600	4,100	2,400	1,800	1,650	1,600
	Stage 2	89,525	44,825	29,925	22,475	18,000	9,050	4,575	3,100	2,350	1,900	1,025	600	600	550	533
1.35	Stage 1	269,300	134,800	90,000	67,600	54,100	27,200	13,800	9,300	7,100	5,800	3,100	1,800	1,400	1,250	1,250
	Stage 2	67,325	33,700	22,500	16,900	13,525	6,800	3,450	2,325	1,775	1,450	775	450	350	417	313
1.4	Stage 1	210,900	105,600	70,500	52,900	42,400	21,300	10,800	7,300	5,600	4,500	2,400	1,400	1,100	1,000	1,000
	Stage 2	52,725	26,400	17,625	13,225	10,600	5,325	2,700	1,825	1,400	1,125	600	350	367	250	250
1.5	Stage 1	141,000	70,600	47,100	35,400	28,400	14,300	7,200	4,900	3,700	3,000	1,600	1,000	750	700	700
	Stage 2	35,250	17,650	11,775	8,850	7,100	3,575	1,800	1,225	925	750	533	250	188	175	175
2	Stage 1	42,800	21,400	14,300	10,800	8,600	4,400	2,200	1,500	1,150	950	500	300	250	250	250
	Stage 2	10,700	5,350	3,575	2,700	2,150	1,100	550	375	288	238	125	75	63	44	63
2.5	Stage 1	22,400	11,200	7,500	5,700	4,500	2,300	1,200	800	600	500	300	200	150	150	150
	Stage 2	5,600	2,800	1,875	1,006	1,125	575	212	141	150	88	33	2	26	17	26
3	Stage 1	14,600	7,300	4,900	3,700	3,000	1,500	800	550	400	300	200	100	100	100	100
	Stage 2	2,576	1,288	865	653	529	265	89	61	71	100	11	43	18	18	25

Sizes are calculated for detection of 80% power at a significance level of 1 × 10⁻⁵ for Stage 1 and a combined Stages 1 and 2 significance level of 5 × 10⁻⁸. Estimates are based on assumptions of 1:1 case:control ratios and a population risk of 0.05. RAF, risk allele frequency.

Table III. Sample size requirements for a range of RAFs and ORs

OR	0.001	0.002	0.003	0.004	0.005	0.01	0.02	0.03	0.04	0.05	0.1	0.2	0.3	0.4	0.5
	RAF
1.05	>1,000,000	>1,000,000	>1,000,000	>1,000,000	>1,000,000	>1,000,000	831,100	560,000	424,600	343,400	181,700	102,600	78,500	69,000	66,600
1.1	>1,000,000	>1,000,000	>1,000,000	>1,000,000	>1,000,000	422,300	213,500	143,900	109,200	88,300	46,800	26,600	20,400	18,000	17,450
1.15	>1,000,000	954,500	637,100	478,300	383,100	192,700	97,400	65,700	49,900	40,400	21,500	12,200	9,400	8,400	8,150
1.2	>1,000,000	550,500	367,400	275,900	221,000	111,200	56,200	38,000	28,800	23,300	12,400	7,100	5,500	4,900	4,800
1.25	721,100	361,000	241,000	180,900	144,900	72,900	36,900	24,900	18,900	15,300	8,200	4,700	3,700	3,300	3,200
1.3	512,800	256,700	171,400	128,700	103,100	51,900	26,300	17,700	13,500	10,900	5,800	3,400	2,600	2,400	2,300
1.35	385,600	193,000	128,900	96,800	77,500	39,000	19,800	13,400	10,200	8,200	4,400	2,600	2,000	1,800	1,800
1.4	301,900	151,200	100,900	75,800	60,700	30,600	15,500	10,500	8,000	6,500	3,500	2,000	1,600	1,400	1,400
1.5	201,900	101,100	67,500	50,700	40,600	20,500	10,400	7,000	5,300	4,300	2,300	1,400	1,100	1,000	1,000
2	61,300	30,700	20,500	15,400	12,400	6,200	3,200	2,200	1,650	1,350	750	450	400	350	350
2.5	32,100	16,100	10,800	8,100	6,500	3,300	1,700	1,150	900	700	400	250	200	200	200
3	20,900	10,500	7,000	5,300	4,200	2,100	1,100	800	600	500	300	200	150	150	150

Sizes are calculated for detection of 80% power at a significance level of 5 × 10⁻⁸ for Stage 1. Estimates are based on assumptions of 1:1 case:control ratios and a population risk of 0.05. RAF, risk allele frequency.

STAGE 1

Altering significance levels for declaring success can have profound effects on sample size requirements. As a representative example, at a significance level of 1 × 10⁻⁴, approximately 120,000 cases are required to detect a risk variant with RAF 0.02 and allelic OR 1.10. This sample size increases to 150,000 when the significance threshold is decreased to 1 × 10⁻⁵, and to 215,000 when the significance threshold is 5 × 10⁻⁸. Upholding the genome-wide threshold of significance of 5 × 10⁻⁸ requires at least 50,000 cases and an equal number of controls to detect ORs of 1.15 and below even for variants with common allele frequencies (>0.05).

Figure 2A shows that for low RAFs ranging from 0.001 to 0.005, the number of cases required to detect a signal at P<1 × 10⁻⁴ is pragmatic (∼1,300 cases) for large genetic effects (OR 3.0). The identification of more commonly encountered ORs in complex traits (1.20–1.30), however, requires between 58,000 and 125,000 cases.

Large-scale consortia are now starting to surpass combined sample sizes of 100,000. Data sets of this size provide adequate power to detect association with low frequency alleles (RAF down to 0.01) and modest effect sizes of at least 1.20 at genome-wide significance thresholds. However, required sample sizes rapidly approach 200,000 cases for RAFs below 0.01. The detection of association with alleles of MAF<0.05 and a small effect size (allelic OR of 1.05) requires samples sizes of over 340,000 cases and an equal number of controls.

Appropriate sample sizes for the detection of association signals at common variants (MAF>0.05) are more readily achievable (Fig. 2C), as demonstrated by the success of GWAS meta-analyses to date. The reported median RAF of 531 signals from published GWAS studies of common diseases [Hindorff et al., 2009] was 0.36, interquartile range (IQR) 0.21–0.53, and the median OR was 1.33, IQR 1.12–1.61. The sample sizes required to obtain 80% power at genome-wide significance in order to detect these effects would be 4,118 (RAF 0.36, OR 1.33), 1,460 (RAF 0.53, OR 1.61) and 36,132 (RAF 0.21, OR 1.12), respectively (cases and controls combined).

STAGE 2

We estimated replication set sample sizes required to obtain 80% power to detect associations at α = 5 × 10⁻⁸ across the combined Stages 1 and 2 analysis, i.e. the discovery set and the replication set. When Stage 1 threshold is set to 1 × 10⁻⁴, an equal number of samples are required in Stage 2 to replicate Stage 1 signals for a final genome-wide threshold of 5 × 10⁻⁸ (Fig. 2B and D) across all ranges of allele frequencies and ORs.

If the significance threshold for Stage 1 is decreased to 1 × 10⁻⁵, the second stage samples sizes required to obtain a final P-value of 5 × 10⁻⁸ are lower. However, the first stage requires a larger sample size to surpass the initial significance threshold (Fig. 3B and D) and the combined sample sizes remain similar for both Stage 1 thresholds. For example, detection of a signal with OR 1.25 and RAF 0.05 at α = 1 × 10⁻⁵ requires 10,700 cases (and an equal number of controls) in Stage 1, and 2,675 cases in Stage 2 for a combined α = 5 × 10⁻⁸. If Stage 1 significance threshold was set to 1 × 10⁻⁴, the respective sample sizes would be 8,700 in Stage 1 and 7,118 in Stage 2.

Importantly, these calculations were based on the assumption of no heterogeneity between the discovery and replication sets, and we assumed that the replication data set was drawn from the same population as Stage 1 samples.

HETEROGENEITY

The relationship between sample size, RAF, and OR in the presence of heterogeneity is consistent with that derived from power calculations carried out in the absence of heterogeneity (Table IV). We observe that, as expected, a larger sample size is required as heterogeneity increases. Interestingly, however, the incremental increase needed does not appear to be substantial. For example, in the frequently encountered scenario of a common variant with a modest effect size (RAF 0.30 and OR 1.25), it is necessary to increase the case sample size by ∼600 individuals when heterogeneity levels are high in comparison with that needed when all studies have similar effects. This number decreases to 400 in the presence of more modest levels of heterogeneity.

Table IV. Total number of samples (across 10 studies with equal sample size, and 1:1 case:control ratio) required at different levels of heterogeneity (measured using I²)

Allele frequency	Meta-analysis OR	Mean I²	Number of casesa
0.30	1.25	0.11	3,121b
		0.21	3,515
		0.45	3,639
		0.76	3,766
0.05	1.25	0.11	13,768b
		0.26	15,253
		0.50	15,674
		0.82	15,971
0.01	3.00	0.05	2,314b
		0.17	2,455
		0.50	2,629
		0.80	2,852

Results are based on 10,000 simulations.
a ^aNumber of cases required for 80% power to detect association at a significance level of 5 × 10⁻⁸.
b ^bAll studies with ORs equal to the meta-analysis OR.

The increase in sample numbers required for lower RAFs (0.05) and the same genetic effect size of 1.25 in the presence of high heterogeneity is more pronounced (additional 2,000 cases needed to achieve the same power). However, this represents a smaller proportional increase in sample size (14%). Overall, the number of additional samples required to overcome loss of power due to heterogeneity is modest.

DISCUSSION

The issue of statistical power and sample size in genetic association studies has previously been addressed in the literature. For example, Fisher and Lewis [2008] found that required sample sizes increase dramatically with decreasing levels of linkage disequilibrium (LD) between the marker and causal variant. Yang et al. [2010a, b] demonstrated that when the disease prevalence is low (<0.10), a case-control study design with a 1:1 ratio of cases and controls is more powerful than a quantitative trait association study. Moonesinghe et al. [2008] showed that meta-analysis sample size requirements increase steeply when small genetic effects were considered and also in the presence of high levels of between-study heterogeneity. Pereira et al. [2009] demonstrated the importance of choosing the correct model for meta-analysis calculations in simulated combinations of several GWAS data sets.

Our findings indicate that, within our simulation framework parameters, the impact of heterogeneity on sample size requirements is not substantial. Although, as expected, increased sample sizes are required to counteract the effect of increasing heterogeneity, even with modest ORs of 1.25 and low allele frequencies of 0.05, only approximately 20% more samples are required to maintain the same statistical power for genome-wide significance. This observation may be underpinned by the fact that high levels of heterogeneity in our study were attained by considering a wide range of ORs among the 10 studies contributing to the meta-analysis, including easily detectable large effect sizes. There are many parameters one can vary in simulation experiments and each can have implications in the downstream results. In our simulations, we assume a meta-analysis of studies with equal sample size. For scenarios in which studies have different sample sizes, results may vary from those presented here. In the presence of high heterogeneity, the number of samples needed is expected to depend on the sample size of the studies with the largest effect sizes. The required sample sizes may increase comparatively to those observed in our simulation study if the sample sizes of the studies with the largest effects decrease.

In the presence of substantial allelic heterogeneity, an alternative approach would be to undertake random-effects analysis which does not make an assumption of homogeneity across studies, in which case the sample sizes required may deviate from the numbers presented here. The level of heterogeneity observed in a meta-analysis should in any case be taken into account, and results interpreted with caution when the observed heterogeneity is high.

Sample size and power have been at the heart of genetic association study design for several decades. Recent advances in the field have made genome-wide scans possible and have ushered in a new era of successful complex disease locus identification. The power constraints of studies conducted thus far have led to the discovery of associations with common-frequency variants of large or modest effect size. Indeed, meta-analyses of GWAS to date have, most probably, identified the vast majority if not all risk variants with moderate effects and common allele frequencies. The formation of large-scale international consortia is now enabling the accrual of sample sizes surpassing 100,000. For example, a recent combined analysis across ∼184,000 individuals identified ∼180 loci affecting human height variation. As with the vast majority of other complex traits, the proportion of phenotypic variation explained by the combined set of established signals is less than 10% [Lango Allen et al., 2010]. This recurrent observation indicates that further genetic determinants of complex traits remain unidentified. Their allele frequency and effect size spectra could be diverse, with multiple common-frequency variants of small effect size, and/or low-frequency and rare variants of modest/larger effect size contributing towards this missing heritability. Using comprehensive power calculations, we find that the detection and replication of signals within these RAF and OR constraints require large sample sizes (currently achievable through large-scale collaborative efforts) in order to reach genome-wide levels of significance, and that effect size heterogeneity across meta-analyzed studies drawn from similar populations does not appear to have a profound effect on sample size requirements.

ACKNOWLEDGMENTS

The authors thank Will Rayner for help with data management and John Ioannidis for commenting on the manuscript. J.A. and E.Z. are supported by the Wellcome Trust (WT088885/Z/09/Z), K.C. is supported by a Botnar Fellowship and by the Wellcome Trust (WT079557MA), and A.M. is supported by the Wellcome Trust (WT081682/Z/06/Z).

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Citing Literature

Volume35, Issue8

December 2011

Pages 781-789

Defining the power limits of genome-wide association scan meta-analyses

Abstract

INTRODUCTION