Increasing the power of identifying gene × gene interactions in genome-wide association studies

Abstract

In this paper we investigate the power to identify gene × gene interactions in genome-wide association studies. In our analysis we focus on two-stage analyses: analyses in which we only test for interactions between SNPs that show some marginal effect. We give two algorithms to compute significance levels for such an analyses. One involves a Bonferoni correction on the number of interactions that are actually tested, and one is a resampling procedure similar to the one proposed by Lin [2006]. We also give an algorithm to carry out approximate power calculations for studies that plan to use a two-stage analysis. We find that for most plausible interaction effects a two-stage analysis can dramatically increase the power to identify interactions compared to a single-stage analysis based on simulation studies using known genetic models and data from existing genome-wide association studies.

1 Introduction

Genome-wide association studies (GWAs), which genotype hundreds of thousands of Single Nucleotide Polymorphisms (SNPs) on thousands of participants are, fueled by decreasing prices of genotyping, now carried out. Some initial successes have been reported [e.g. Tomlinson et al. 2007; Scott et al. 2007; Easton et al. 2007; WTCCC 2007]. While the primary interest in GWAs is typically the identification of SNPs that are marginally associated with a disease, it is typically also of interest to identify SNPs that jointly have an epistatic (interaction) effect on the disease of interest. Such interactions may shed light on potential disease associated pathways, and they may identify people that are at extreme high risks [e.g. Manolio and Collins, 2007].

It is easy to see that the potential number of interaction to be tested is enormous. When 500,000 SNPs are genotyped there are $\left(\begin{array}{c}\mathrm{500,000}\\ 2\end{array}\right)\approx {10}^{11}$ two SNP combinations and $\left(\begin{array}{c}\mathrm{500,000}\\ 3\end{array}\right)\approx {10}^{16}$ three SNP combinations. This creates both computational and multiple comparisons problems: it is virtually impossible to evaluate each possible model, and, as a multiple comparisons correction needs to be made for very many possible tests, there is limited power for the identification of any of the interactions that are tested.

In this paper we investigate strategies to improve the power in genome-wide association studies, while reducing the computational expense considerably. Our main tool is a two-stage analysis: we only investigate interactions between SNPs that show some (modest) marginal effect. Initially two-stage strategies were proposed as study designs to reduce the expense of a GWA, without a substantial reduction of the power to identify the (marginal) effect of SNPs on a disease [e.g. Lin 2006, Skol et al. 2006, 2007, Thomas et al. 2004]. More recently, there have been proposals to use two-stage analysis strategies to possibly improve the power of identifying interaction effects in GWAs [Evans et al. 2006; Macgregor and Kahn 2006]. A thorough discussion on how power and type 1 error are influenced by such a two-stage design is missing, however.

Marchini et al. [2005] and Evans et al. [2006] investigated whether a two-stage analysis was a viable approach to improve the power to identify SNPs that are marginally associated with a disease. They found, that, possibly because they carried out a multiple comparisons correction for all possible associations, their two-stage analysis did not improve the power for identifying SNPs that jointly have an epistatic effect in GWAs. One of the main goals of this paper is to better correct for multiple comparisons in two-stage analyses. Macgregor and Khan [2006] argued that one only needs to correct for the number of interactions tested for. Their paper, however, gives neither a justification, nor a simulation study validating this assertion. In the current paper we attempt to give a more solid foundation for two-stage analyses, and provide extensive simulation studies to back up our results. We also provide an analytic algorithm to approximate power for the detection of interactions using a two-stage analysis.

The goal in this paper is to develop methods to identify interacting SNPs in (genome-wide) association studies. Several authors [Chapman and Clayton, 2007, Chatterjee et al., 2006, Marchini et al., 2005] recently have proposed methods that make use of interactions in developing powerful tests to determine whether SNPs are marginally associated with a disease outcome, which is a slightly different objective from ours. For example, in the situation where there is in fact no interaction, we would hope that these procedures would still identify SNPs that are associated with the disease, while our procedure would not be expected to identify such combinations. On the other hand, when there is an interaction, the bar is higher for the procedure that we proposing: as we want to identify the SNPs in combination.

2 Methods

An elementary independence result

Let Y_i, i = 1, …, n be independent identically distributed (iid) random variables, and let x_1i and x_2i, i = 1, …, n be predictor variables. Consider the three linear regression models

$$Y_i={\gamma }_{10}+{\gamma }_{11}{x}_{1i}+{\epsilon }_i^1,$$ (1)

$$Y_i={\gamma }_{20}+{\gamma }_{21}{x}_{2i}+{\epsilon }_i^2,$$ (2)

and

$$Y_i={\beta }_0+{\beta }_1{x}_{1i}+{\beta }_2{x}_{2i}+{\beta }_3{x}_{1i}{x}_{2i}+{\epsilon }_i.$$ (3)

Let γ̂₁₀, γ̂₁₁, γ̂₂₀, γ̂₂₁, β̂₀, β̂₁, β̂₂, and β̂₃ be the ordinary least squares (OLS) estimates of the parameters in (1), (2), and (3). Then β̂₃ ⊥ γ̂₁₁ and β̂₃ ⊥ γ̂₂₁.

Proof. Note that OLS estimates are linear in the response. Let γ̂_j1 = Σ_i a_jiY_i, and β̂₁ = Σ_ib_iY_i. Expressions for the a_i and b_i can be found in any elementary linear regression text. Because the Y_i are iid, cov(γ̂_j1, β̂₁) = Σ_i a_jib_ivar(Y_i) = var(Y₁) Σ_i a_jib_i. Algebra yields that for models (1) and (3) Σ_ia_jib_i = 0, thus cov(γ̂_j1,β̂₃) = 0.

The implication of this result is that for a cohort study with a continuous outcome we can carry out a hypothesis test for interactions in a two-stage approach:

• test only interactions between those predictors (SNPs) that are marginally significant at a pre-specified level α₁; and

• control the global (family-wise) type 1 error by controlling for the number of interactions that are actually being tested for (for example, in a Bonferoni fashion), rather than the one that could have been tested for.

Case-control sampling

Most genome-wide association studies employ a case-control study design. Unlike for cohort studies with a continuous outcome the samples are not iid, and likely the study will be analyzed using logistic, rather than linear regression. In particular, we would now estimate using the models

$$\text{logit}(P(Y_i=1|x_{1i},x_{2i}))={\gamma }_{10}+{\gamma }_{11}{x}_{1i},$$ (4)

$$\text{logit}(P(Y_i=1|x_{1i},x_{2i}))={\gamma }_{20}+{\gamma }_{21}{x}_{2i},$$ (5)

and

$$\text{logit}(P(Y_i=1|x_{1i},x_{2i}))={\beta }_0+{\beta }_1{x}_{1i}+{\beta }_2{x}_{2i}+{\beta }_3{x}_{1i}{x}_{2i}.$$ (6)

We believe that in general, in models (4)–(6) the maximum likelihood estimate β̂₃ is not independent of the estimates γ ^₁₁ and γ̂₂₁. However, in simulation studies, including those reported below, we have found that this dependence is sufficiently small that an inference using a Bonferoni approach on the number of tested interactions is still valid. Alternatively, inference can be carried out using a permutation approach described below.

Score statistics based “permutation” tests

A common approach to controlling the type 1 error in situations where the exact distribution of the test statistic is not known is a permutation test. However, permutation tests for interactions are generally not possible, as permutations do not just remove the interaction effect, but they also remove the main effect [Pesarin, 2001]. Inference on an interaction should be conditional on the main effect, but in fact a straightforward permutation test tests for a combination of main effects and interactions. For linear models, for a given permutation, it is possible to permute the residuals or the fitted interaction component (similar to the parametric bootstrap [Efron and Tibshirani, 1993]). For logistic regression, the usual regression model employed for case-control studies, such an approach does not work.

Lin [2006] proposes an approach to obtaining test statistics in two-stage genome-wide association studies that can be adapted to the testing for interaction effects. In our situation the null-hypothesis to be tested is that there is no interaction, although there may be main effects. First, consider the logistic regression model

$$\text{logit}(P(Y_i=1|X_{\mathit{\text{ji}}},X_{\mathit{\text{ki}}}))={\beta }_0+{\beta }_1{X}_{\mathit{\text{ji}}}+{\beta }_2{X}_{\mathit{\text{ki}}}+{\beta }_3{X}_{\mathit{\text{ji}}}{X}_{\mathit{\text{ki}}}.$$ (7)

Set Z_ijk = X_ijX_ik. Then the efficient score for β₃ is

$$U_{\mathit{\text{jk}}}=\sum _{i=1}^n{U}_{\mathit{\text{ijk}}}=\sum _{i=1}^n(Y_i-p_{\mathit{\text{ijk}}})(X_{\mathit{\text{ij}}}{X}_{\mathit{\text{ik}}}-{\mu }_{\mathit{\text{ijk}}}),$$

where p_ijk is the fitted probability for the ith subject from the logistic regression model that includes X_ij and X_ik, but not Z_ijk, and μ_ijk is the fitted value for the ith subject from the linear regression model Z_ijk = ρ₀ + ρ₁X_ij + ρ₂X_ik. Under the null-hypothesis of no interaction effect, U_jk has approximately a normal distribution with mean 0 and variance $V_{\mathit{\text{jk}}}={\sum }_{i=1}^n{U}_{\mathit{\text{ijk}}}^2$. Set $T_{jk}=U_{jk}^2/V_{\mathit{\text{jk}}}$. Let G₁, …, G_n be independent normal random variables. Set Ũ_jk = Σ_i U_ijkG_i. Then the Ũ_jk have mean 0 and approximately the same covariance matrix as the U_jk. Set ${\stackrel{\sim }{T}}_{\mathit{\text{jk}}}={\stackrel{\sim }{U}}_{\mathit{\text{jk}}}^2/V_{\mathit{\text{jk}}}$. Thus, a strategy to get a sample from the T_jk under the null-hypothesis of no association is to generate repeated samples of G₁, …, G_n and compute the T̃_jk. Note that this approach of controlling the type 1 error does not assume independence of the stage one and two tests, as the Bonferoni approach does, but rather the “permutations” for stage two are carried out conditional on the results of stage one, as the permutations are only carried out for the selected SNPs (j and k) and the p_ijk depend on the marginal models.

A strategy to identify interactions in GWAs

Our strategy to test for the presence of an interaction is:

• 1 Fix the first-stage significance level α₁.

• 2 Test every SNP marginally at level α₁ using a logistic regression model. Say that n* SNPs were significant at level α₁.

• 3 Test the interactions between all SNPs that pass Step 2, using the logistic model (7).

• 4-Bonferoni Correct the P-value from Step 3 using a Bonferoni correction for $\left(\begin{array}{c}n\ast \\ 2\end{array}\right)$ tests; or

• 4-Score Compute T_jk for each of the tests from Step 3. Let T* = max_1≤j,k≤n* T_jk. Also generate L independent sets of ${\stackrel{\sim }{T}}_{\mathit{\text{jk}}}^l$, l = 1, …, L, and set ${\stackrel{\sim }{T}}^{\ast l}={max}_{1\le j,k\le n\ast }{\stackrel{\sim }{T}}_{\mathit{\text{jk}}}^l$. Compare T* to the T̃*^l.

Generalizations of this procedure that use the False Discovery Rate, or that can identify more than one interactions are straightforward.

Approximations for power calculations

When the data is generated from model (7) the logistic regression estimate of β̂₃ has approximately a normal distribution with mean β₃ and variance

$$\sum _{l_1{l}_2{l}_3\in \{0,1\}}\frac{1}{n_{l_1{l}_2{l}_3}},$$

where n_l1l2l3 is the number of observations for which Y_i = l₁, X_ij = l₂, and X_ik = l₃. This can be used to compute a useful approximation to the power of identifying an interaction, under the assumption that all SNPs and all interactions between SNPs are independent. Details are given in the appendix.

3 Results

Simulation set-up

We generate 10,000 binary SNPs X₁, …, X_10,000 as a first order Markov chain such that cor(X_i,X_i+1) = ρ. We generate a response Y according to

$$\text{logit}(P(Y=1|X))={\beta }_0+{\beta }_1{X}_{2500}+{\beta }_2{X}_{7500}+{\beta }_3{X}_{2500}{X}_{7500}.$$

The minor allele frequency P(X_i = 1) = p is taken constant for all SNPs i. Note that a minor allele frequency p for this binary SNP model corresponds to a minor allele frequency of $q=1-\sqrt{1-p}$ for a dominant genetic model with bi-allelic SNPs.

We generate data until we have c cases (Y = 1) and controls (Y = 0). For computational reasons we take c = 500 and relatively large effect sizes. For the power analysis approximations below we consider larger sample sizes and smaller effect sizes. In all our simulations we took β₀ = −2, generating outcomes for a moderately rare outcome, and β₁ = β₂, identical main effects for both SNPs.

In our analysis we consider selection on the marginal effects using α₁ = 0.002, 0.005, 0.01, 0.02, and 1.0. We compute the power when significant SNP × SNP interactions are identified using the modified approach of Lin [2006] described above with 1000 permutations, using a Bonferoni correction for the number of interactions that are actually tested, and the analytic approximation. Note that for the analytic approximation the correlation between SNPs is ignored. For α₁ = 1 we do not compute the power using the approach of Lin [2006], as computing 1000 permutations for $\left(\begin{array}{c}10000\\ 2\end{array}\right)\approx 5\times {10}^7$ interactions for a single simulation is not well feasible. We control the overall type 1 error at a global (family-wise) level α = 0.05 using the Bonferoni method.

Simulation results

In Figure 1 we show the results for β₃ = 0, p = 0.3 (q = 0.173), ρ = 0.7, β₂ = β₁, for various values of α₁ as a function of the main effect β₁ = β₂. In this situation there are no interactions, and we would want any procedure to yield the designed type 1 error. Based on 1000 simulations (so that the standard error of the power estimates is about 0.007), it appears that the type 1 error is well controlled by both the Bonferoni and the Lin approach, and that the type 1 error is not inflated by using a two-stage testing procedure for interactions. The power approximation yields exactly a type 1 error of 0.05 in this situation. The Bonferoni correction on all interactions (α₁ = 1) appears somewhat conservative, and the Lin approach is well within the range what would be expected based on the standard error. We repeated these simulations for various other parameter settings, and got consistently the same results.

Figure 1.

Type 1 error for identifying an interaction when there is in fact no interaction effect, but there is a dominant main effect for two SNPs each with minor allele frequency 0.173 for various levels of two-stage testing (α₁ < 1) and one-stage testing (α₁ = 1).

In the remaining simulations we include interactions, and thus prefer approaches in which the power to identify interactions is large. In particular, in Figure 2 we show the results for β₁ = β₂ = 0, β₃ = 0, 1, 1.5, and 2, α₁ = 0.005 and p = 0.3 (q = 0.173) as a function of the correlation ρ between SNPs. We note that all three methods to compute the power provide similar results, with the power computed using the Lin [2006] approach to suggest slightly larger power. We also note that the power is slightly larger when the correlation is 0.9 than when the correlation is smaller, but that the difference is very small. Because of this small effect of the correlation we show further results only for ρ = 0.7.

Figure 2.

Power for identifying an interaction when there is no main effect, for various interaction effects between two SNPs each with minor allele frequency 0.173 for two-stage testing at α₁ = 0.05 for various correlations between the SNPs.

In Figures 3 and 4 we show the results for β₁ = β₂ = 0, ρ = 0.7, p = 0.3 (q = 0.173, Figure 3) and p = 0.4 (q = 0.235, Figure 4) for a variety of choices of α₁ as a function of the interaction effect β₃. We note that the power to identify an interaction is substantially larger when we first filter on main effects, even though there is in fact no main effect when β₁ = β₂ = 0. The power gain of filtering in the first stage over a global Bonferoni correction can be as much as 40%. These power gains are universal, and were observed in many other situations. In Figure 4, where the minor allele frequency was larger it appears that most power is gained when taking the first stage α₁ small. The difference in power between the various choices of α₁ is small compared to the difference between two-stage selection, and overall testing with α₁ = 1.

Figure 3.

Power for identifying an interaction when there is no main effect, for various interaction effects between two SNPs each with minor allele frequency 0.173 for various levels of two-stage testing (α₁ < 1) and one-stage testing (α₁ = 1).

Figure 4.

Power for identifying an interaction when there is no main effect, for various interaction effects between two SNPs each with minor allele frequency 0.235 for various levels of two-stage testing (α₁ < 1) and one-stage testing (α₁ = 1).

The only situation which we identified where a two-stage selection does not help is when the main effect goes in the opposite direction of the interaction effect. In Figure 5 we show the results for β₁ = β₂ = −1, ρ = 0.7, and p = 0.4 (q = 0.235). For some values of β₃ testing all interactions now yields more power than two-stage testing. This corresponds to the situation where the magnitude of the interaction leads to approximately canceling out the marginal association of the two genes to the outcome. Therefore the non-monotonic shape of the power curves can be explained by low power to detect the interaction at the second stage for β₃ small, increasing as the interaction effect increases, but decreasing as β₃ → 2, as the chance that the relevant genes are selected at the first stage is dramatically reduced as the marginal association is weakened.

Figure 5.

Power for identifying an interaction when the main effect goes in the opposite direction of the interaction effect, for various interaction effects between two SNPs each with minor allele frequency 0.235 for various levels of two-stage testing (α₁ < 1) and one-stage testing (α₁ = 1).

Clearly some more unusual interaction patterns like those in Marchini [2005] may also yield more power for overall testing. The main question that a researcher will have to ask before testing is what she/he thinks to be more likely: if an epistatic effect where one SNP enhances the effect of another gene is more likely than a more complicated interaction effect, a two-stage testing procedure should be employed,

Simulations using real GWA data

From Illumina iControlDB (http://www.illumina.com) we retrieved 610 arrays of human array 317K data. The Illumina iControlDB contains user submitted data of Illumina array data that was submitted to be used as “controls in case-control association studies, in which risk factors of individuals with a certain disease (cases) are compared to individuals without the disease (controls)”, as well as methodological studies like the current one. From this data we removed SNPs with a minor allele frequency of under 10%, and selected the remaining 10,321 SNPs on chromosome 13 for further simulations. We phased these 610 arrays using fastPhase (Scheet and Stephens 2006) to obtain 1220 haploid copies of chromosome 13. We then generated a huge population by randomly paring two of these haploid copies, and generated disease status using the model

$$\text{logit}(P(Y=1|\mathit{\text{SNPs}}))={\beta }_0+{\beta }_1{U}_1+{\beta }_2{U}_2+{\beta }_3{U}_1{U}_2,$$

where U₁ = 1 is at least one of the haploids for rs1751871 was a copy of the minor allele and U₂ = 1 is at least one of the haploids for rs9523716 was a copy of the minor allele. These two SNPs are not in LD (r² = 0.01); the minor allele frequencies are 0.13 (rs1751871) and 0.25 (rs9523716).

In Figure 6 we show the results for 1000 simulations with β₀ = −2, β₃ = 0, and β₂ = β₁, for various values of α₁. The results in this figure suggests, that for the data from actual GWAs, just as for the simulated data, the type 1 error is well controlled. In Figure 7 we show the results for 1000 simulations with β₀ = −2, β₁ = β₂ = 0, for various values of β₃ and α₁. The results in this figure suggests that the two-stage procedure indeed improves the power to identify the interactions dramatically compared to a one-stage procedure, and that the score approach yields almost the same results as the (cheaper) Bonferoni correction.

Figure 6.

Type 1 error for identifying an interaction when there is in fact no interaction effect, but there is a dominant main effect for two SNPs in data generated from Illumina HapMap 317K arrays, for various levels of two-stage testing (α₁ < 1) and one-stage testing (α₁ = 1).

Figure 7.

Power for identifying an interaction when there is no main effect, for various interaction effects between two SNPs for two SNPs in data generated from Illumina HapMap 317K arrays for various levels of two-stage testing (α₁ < 1) and one-stage testing (α₁ = 1).

Approximate power calculations

Using the algorithm in Appendix A we can approximate the power of identifying interactions in genome-wide association studies. For example, in Table 1 we show the power for identifying the specific interaction, associated with the parameter β₃ in model (7) where both involved SNPs have at least one variant allele for 40% of the samples (corresponding to a minor allele frequency of 0.225) or 20% of the samples (corresponding to a minor allele frequency of 0.106) for a case-control study with 2000 cases and 2000 controls, measuring 500,000 SNP, for a variety of two-stage procedures, allowing 3 false positives under independence.

Table 1.

Power of identifying a specific interaction, associated with the parameter β₃ in model (7) where both involved SNPs have at least one variant allele for 40% or 20% of the samples (corresponding to a minor allele frequency of 0.225 and 0.106, respectively) for a case-control study with 2000 cases and 2000 controls, measuring 500,000 SNPs, for a variety of two-stage procedures, allowing 3 false positives.

β3	Odds ratio	First stage significance level α1
		0.0001	0.001	0.002	0.005	0.01	0.02	0.05	1
P(SNP > 0) = 0.4 ⇒ minor allele frequency 0.225
0.5	1.65	0.09	0.11	0.10	0.08	0.06	0.04	0.02	0.00
0.6	1.82	0.37	0.44	0.40	0.32	0.26	0.19	0.12	0.02
0.7	2.01	0.75	0.80	0.75	0.66	0.56	0.47	0.35	0.10
0.8	2.23	0.95	0.96	0.94	0.88	0.82	0.75	0.64	0.31
0.9	2.46	0.99	0.99	0.99	0.97	0.95	0.92	0.87	0.60
1.0	2.72	1.00	1.00	1.00	1.00	0.99	0.98	0.97	0.84
P(SNP > 0) = 0.2 ⇒ minor allele frequency 0.106
1.0	2.72	0.09	0.24	0.29	0.34	0.35	0.34	0.29	0.09
1.1	3.00	0.21	0.45	0.52	0.58	0.59	0.58	0.51	0.21
1.2	3.32	0.39	0.66	0.73	0.79	0.80	0.78	0.72	0.38
1.3	3.67	0.60	0.83	0.88	0.91	0.92	0.91	0.87	0.58
1.4	4.06	0.78	0.93	0.95	0.97	0.98	0.97	0.95	0.76

We note from this table that a two-stage procedure can considerably increase the power of identifying an interaction. In particular, the optimal fraction of SNPs to consider for testing for interactions appears to be about 0.001 for the higher minor allele frequency and 0.01 for the lower minor allele frequency in this example. If the sample size is increased to 5000 pairs of cases and controls there is even some power to identify interactions associated with odds ratios of 1.5 (Table 2). It appears that the minor allele frequency is the dominant factor in determining what the “optimal” value of α₁ is, while in particular the odds ratio does not have much influence. Based on a more extensive set of simulations, we feel that values in the order of α₁ ∼ 0.005 are usually fairly close to the optimal value. Power calculations can be carried out in a straightforward fashion to optimize α₁ for any design and hypothesized effect size using our code, available from http://bear.fhcrc.org/∼clk/soft.html.

Table 2.

Power of identifying a specific interaction, associated with the parameter β₃ in model (7) where both involved SNPs have at least one variant allele for 40% or 20% of the samples (corresponding to a minor allele frequency of 0.225 and 0.106, respectively) for a case-control study with 5000 cases and 5000 controls, measuring 500,000 SNPs, for a variety of two-stage procedures, allowing 3 false positives.

β3	Odds ratio	First stage significance level α1
		0.0001	0.001	0.002	0.005	0.01	0.02	0.05	1
P(SNP > 0) = 0.4 ⇒ minor allele frequency 0.225
0.4	1.49	0.41	0.52	0.49	0.41	0.34	0.27	0.18	0.03
0.5	1.65	0.89	0.94	0.92	0.87	0.81	0.74	0.62	0.27
0.6	1.82	0.99	1.00	1.00	0.99	0.98	0.97	0.94	0.72
0.7	2.01	1.00	1.00	1.00	1.00	1.00	1.00	1.00	0.96
P(SNP > 0) = 0.2 ⇒ minor allele frequency 0.106
0.6	1.82	0.02	0.09	0.12	0.15	0.17	0.18	0.16	0.05
0.7	2.01	0.11	0.30	0.38	0.46	0.50	0.51	0.49	0.21
0.8	2.23	0.32	0.60	0.68	0.77	0.82	0.83	0.81	0.52
0.9	2.46	0.61	0.84	0.89	0.94	0.96	0.97	0.96	0.82
1.0	2.72	0.85	0.96	0.97	0.99	0.99	1.00	1.00	0.96

4 Discussion

In this paper we investigate the power for identifying interactions using a two-stage analysis. We found that the power of identifying interactions can be greatly improved using such an analysis. The significance for such an analysis can be controlled using a Bonferoni correction on the number of interactions actually tested, or a resampling approach like the one proposed by Lin [2006]. Approximate power calculations for such an analysis can be computed explicitly.

Our results are somewhat contradictory to those of Marchini et al. [2005] and Evans et al. [2006]. In these papers a two-stage analysis is used to marginally identify SNPs that may have epistatic effects on a disease outcome. In their analysis they find that a two-stage analysis does not improve the power to identify SNPs marginally. Besides that the goal in our analysis is slightly different from that in these two papers, we believe that there are two reasons why we do reach the conclusion that a two-stage analysis is useful.

The main reason is that we focused on what we feel are plausible interactions, interactions where the effects are monotone in the number of minor alleles of both SNPs that are involved. We believe that it is often reasonable to make such an assumption about the type of interaction. Clearly, some assumption is necessary: without such an assumption the number of possible interactions is enormous, and the power to identify them is substantially reduced. In presentations we have sometimes compared this with a cake: if we want to divide the “power” over all possible interactions, nobody will get more than a crumb, and no-one will taste how good the cake is; we are better off dividing the cake among those people we believe to enjoy it. We believe that not all possible interactions are likely, in contrast Evans et al. [2006] consider all sorts of interactions (some of which they themselves label as “exotic”, see their Figure 4).

The second reason is that it is critical that in a two-stage procedure we control the type 1 error for our testing strategy. Correcting for all possible tests in a Bonferoni manner in a two-stage procedure is unnecessarily conservative. In our two-stage procedure, where the first and second stage testing are (virtually) independent, this can be done with a Bonferoni correction on the number of tests actually carried out, or with a permutation based score approach. This later approach could potentially be adapted to situations where the two stages are not independent.

In the current paper we show that a two-stage procedure which takes those issues into account, dramatically increases the power to identify interactions over a one-stage approach. The power to identify them is still limited, but with the increased size of some of the planned GWAs, finding interactions is no longer out of the question. It remains an open question whether a two-stage analysis that take these issues into account would improve the power over a one-stage analysis for identifying SNPs that are associated with a disease.

Appendix: Algorithm for a power calculation for two-stage testing of interactions

Assume model (7), assume that there are n_ca cases and n_co controls.

1. Given model (7) compute P(Y = 1|X_j = 1) and P(Y = 1|X_k = 1).

2. Compute the power z_j (z_k) that X_j (X_k) is marginally significant in a case-control study with n_ca cases and n_co controls at the significance level α₁.

3. Compute q(m) = P(Z = m), where Z has a binomial distribution with n the number of SNPs minus 2, and p = α₁. Thus q(m) is the probability distribution of the number of other SNPs that are significant at level α₁, and that thus go on to stage 2.

4. Compute the expected variance of β̂₃, and then the probability p(ρ) that β̂₃ is significant at the level ρ, using a normal approximation.

5. Let ${\alpha }_2(m)=\frac{\alpha }{\left(\begin{array}{c}m+2\\ 2\end{array}\right)}$, the Bonferoni corrected significance level for the second stage of the analysis when the two SNPs involved with the interaction and m other SNPs are significant at level α₁.

6. We approximate the power of identifying an interaction at a global significance level α by

   $$z_k{z}_l\sum _{m}q(m)p({\alpha }_2(m))+z_k{z}_l\sum _{m}q(m)(1-p({\alpha }_2(m)){\alpha }_2(m)\left(\left(\begin{array}{c}m+2\\ 2\end{array}\right)-1\right)+\left(1-z_k{z}_l\right)\alpha .$$ (8)

   The first term in (8) is the power of identifying the X_kX_l interaction (where we sum over the number of other SNPs that go to the second stage). The second term in (8) is the power of identifying another interaction if the X_kX_l interaction is not identified, but X_k and X_l both were marginally significant at level α₁, again summing over the number of other SNPs that go to the second stage, also noting that one of the $\left(\begin{array}{c}m+2\\ 2\end{array}\right)$ is X_kX_l and is already taken care of in the first term. The third term is the power of identifying an interaction if either X_k and X_l was not marginally significant at level α₁.

7. Similarly, we can approximate the power of identifying an interaction between X_k and X_l (ignoring other interactions that may be identified) by

   $$z_k{z}_l\sum _{m}q(m)p({\alpha }_2(m)).$$ (9)

   The scenario where we allow for, say, F false positives can be approximated by using ${\alpha }_2(m)=F/\left(\begin{array}{c}m+2\\ 2\end{array}\right)$ in (9), so that when $\left(\begin{array}{c}m+2\\ 2\end{array}\right)$ interactions are tested F of them are significant just by chance.

The results of equation (8) are shown in the figures; those of equation (9) are shown in the tables.