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Ann Hum Genet. Author manuscript; available in PMC 2009 November 1.

Published in final edited form as:

Published online 2008 July 24. doi: 10.1111/j.1469-1809.2008.00467.x

PMCID: PMC2574571

NIHMSID: NIHMS58336

*Correspondence to Mitchell H. Gail, Fax:1-301-402-0081, E-mail: gailm/at/mail.nih.gov

The publisher's final edited version of this article is available at Ann Hum Genet

See other articles in PMC that cite the published article.

Large two-stage genome-wide association studies (GWASs) have been shown to reduce required genotyping with little loss of power, compared to a one-stage design, provided a substantial fraction of cases and controls,*π _{sample}*, is included in stage 1. However, a number of recent GWASs have used

Two-stage genome-wide association studies (GWASs) have been advocated because they can reduce required genotyping with little loss in power(Satagopan & Elston, 2003, Satagopan et al., 2004, Skol et al., 2006). In stage 1 of these designs, all available SNPs are analyzed among a proportion, *π _{sample}*, of the available cases and controls; only the SNPs judged to be promising in stage 1 are studied in the remaining cases and controls (stage 2). If the cost per genotype is lower in stage 1, larger

Detection probabilities of two-stage designs with joint analysis versus one-stage designs for random effects disease models with 12.5% of cases and controls in stage 1 (*π*_{sample} =0.125). Other parameters include *T*_{0} =500,000 SNPs, numbers of SNPs **...**

The most studied type of two-stage design is based on an hypothesis testing paradigm: a SNP is said to be disease-associated if its test statistic exceeds a fixed critical value *c _{1}* at stage 1 and if its test statistic at stage 2 (usually based on the combined stage 1 and stage 2 data) exceeds a fixed critical value

However, many studies are not conducted this way. Instead, a SNP is selected for further study in stage 1 if its p-value ranks among the *T*_{1} smallest p-values. Thus, selection of promising SNPs is not based on a fixed critical value but on ranking the SNPs against each other. This strategy was used in the Cancer Genetic Markers of Susceptibility Study (CGEMS) for prostate cancer (http://cgems.cancer.gov/about/replication_strategy.asp) and in studies of various other diseases(Broderick et al., 2007, Buch et al., 2007, van Es et al., 2007). In the CGEMS prostate cancer study, about *n*_{1} = 1200 cases and controls were studied in stage 1 (i.e. *π _{sample}* = 0.14), and roughly

One approach following stage 1 is to continue ranking and selecting the SNPs based on data in stage 2. In particular, one selects a SNP as highly promising at the end of stage 2 if its p-value, which may be based on stage 2 data alone or on some combination of stage 1 and stage 2 data, ranks among the lowest *T*_{2} p-values from the *T*_{1} SNPs studied in stage 2. In this paper we describe the properties of this two-stage selection procedure, and in particular, we calculate the probability that a disease-associated SNP will be selected at the end of stage 2 (the “detection probability”) and the proportion of selected SNPs that are expected to be disease-associated (the “proportion positive”). If a two-stage procedure is designed to have a high detection probability and proportion positive, then one can expect that most disease-associated SNPs will be included in the *T*_{2} selected SNPs and that independent epidemiologic studies will confirm the association with disease in a good proportion of the *T*_{2} selected SNPs.

A second approach is to test for associations with disease among the SNPs selected in stage 1. In a study of gallstone disease(Buch et al., 2007), the authors used the independent stage 2 data alone to produce a p-value. In a study of amyotrophic lateral sclerosis(van Es et al., 2007), the authors ranked SNPs in stage 1 and culled SNPs further by requiring a nominal p-value <0.1 in stage 2 before assigning a p-value based on independent stage 3 data. To study genetic associations with colorectal cancer, investigators combined data from stages 1 and 2 to produce an overall p-value(Broderick et al., 2007). The statistical properties of these hybrid approaches have not been evaluated, although p-values that depend only on an independent final stage should have nominal significance levels.

In previous work(Gail et al., 2008), we studied ranking procedures for one-stage designs and defined the “detection probability” (DP) as the probability that a disease associated SNP would rank among the *T*_{1} largest test statistics (or *T*_{1} smallest p-values). For a two-stage design, we now define DP as the probability that a disease-associated SNP will rank among the *T*_{1} largest statistics (or *T*_{1} smallest p-values) at stage 1 and among the *T*_{2} largest statistics (or *T*_{2} smallest p-values) at stage 2. As in(Gail et al., 2008), we can also compute the expected proportion of the *T*_{2} selected SNPs that are truly disease-associated, namely the proportion positive (PP), under the assumption that the true number of disease-associated SNPs is known. We provide methods for assessing DP in the two-stage design, not only for a “replication analysis” that is based on the rankings in stage 2 data alone, but also for a final “joint” analysis(Skol et al., 2006) that combines data from stages 1 and 2. We show that for magnitudes of odds ratios commonly found in GWASs, the DP can be much lower for a two-stage design with *π _{sample}* ≤0.25 than for a one-stage design with the same number of cases and controls, and that DP is hardly increased by an optimal joint analysis of stage 1 and stage 2 data combined when

A two-stage GWAS analyzes *T*_{0} (say 500,000) tagging SNPs in *n*_{1} cases and *n*_{1} controls (stage 1) and selects *T*_{1} promising SNPs for study in stage 2 in independent cases ( *n*_{2}) and controls ( *n*_{2}). Following stage 2, *T*_{2} SNPs are selected in the hope that they are associated with disease. The proportion of cases and controls in stage 1 is *π _{sample}*

Assuming that the *T*_{0} tagging SNPs are in linkage equilibrium as in(Skol et al., 2006), we proved that SNP genotypes are independent not only in controls, but also in cases for a rare disease(Gail et al., 2008). We used these ideas and asymptotic theory for the Wald and score tests to develop efficient procedures for simulating the chi-square test statistics for the *M*_{0} disease-associated SNPs (“disease SNPs”) and the *T*_{0}−*M*_{0} non-disease SNPs in stage 1. For independent stage 2 data, we now extend these methods to simulate independent chi-square tests for *T*_{1} SNPs, of which a random number, *M*_{1}, are disease SNPs that were selected at stage 1, and the remaining *T*_{1}−*M*_{1} are non-disease SNPs. We study odds ratios per allele of 1.1, 1.2, 1.3, and 1.5, but we use 1.2 in the following description. We consider two models(Gail et al., 2008) for disease SNPs. In the fixed effects model, the log odds ratio per disease allele is fixed at *β* = log (1.2) for each disease SNP. Thus the relative odds is 1.44 for a homozygote and 1.2 for a heterozygote. In the random effects model, *β* is drawn independently for each disease SNP from a normal distribution with mean zero and standard deviation *τ* = (*π*/2)^{1/2} log(1.2) ≈1.253log(1.2). This value of *τ* yields an expected absolute value of *β* of log(1.2). Under both models, *β* =0 for non-disease SNPs. One chi-square test is the squared Wald statistic, ^{2}/*ar* (), for testing *β* =0 in a model for log odds of disease that equals an intercept plus *β* times the number of minor alleles(Gail et al., 2008). We also studied the corresponding squared score test(Armitage, 1955, Sasieni, 1997). Each of these chi-square tests has the same value whether the major or minor allele confers risk.

At stage 1, the *T*_{1} SNPs with the largest chi-square tests (or smallest p-values) are selected. We study two analytical approaches, “replication analysis” and “joint analysis,” similar to (Skol et al., 2006). For a given SNP whose test statistic was in the critical region in stage 1, Skol et al. called an hypothesis test based only on the stage 2 data a “replication analysis.” They used the term “joint analysis” for a final hypothesis test based a linear combination of the test statistics for stage 1 and stage 2 data. Analogously, we use the term “replication analysis” if final selection of the *T*_{2} SNPs from among the *T*_{1} SNPs selected in stage 1 depends only on their rankings derived from the independent stage 2 data. In “joint analysis”, for each of the *T*_{1} SNPs selected in stage 1, we compute *λ C*_{1} + (1−*λ*)*C*_{2} for *λ* = 0, 0.05, 0.10,…,1.0, where *C*_{1} and *C*_{2} are the chi-square statistics observed in stages 1 and 2 respectively. For each *λ* we estimate DP, and we present the maximal P over the 21 values of*λ*, together with the corresponding *λ _{opt}*. If several values maximize DP, we define

For the simulations, we assume that 8000 cases and 8000 controls are available to be apportioned between stages 1 and 2 with equal numbers of cases and controls in each stage. We present data for *M*_{0} = 1 and 10 and for *π _{sample}* = 0.125, 0.25, 0.50, and 1.0, both for fixed effects and random effects models. For each odds ratio, we conducted 12 independent simulation studies, each with 10,000 replications, to estimate DP for each combination of

(1)

We note from the exchangeability of the disease SNPs that DP can be interpreted either as the probability that a particular disease SNP will be selected at stage 2 or as the proportion of disease SNPs selected at stage 2(Gail et al., 2008). The proportion positive, PP, can be estimated(Gail et al., 2008) as

(2)

We performed these simulations in GAUSS(Aptec Systems, 2005).

The following results are for the chi-square test based on the Wald statistic. Very similar results were found for the chi-square version of the score test (data not shown). We present detailed information for the fixed effects model with odds ratio 1.2 per allele in Table 1. With *M*_{0} =1, P was 0.882 or higher for the one-stage design (*π _{sample}* =1.0), for various values of

Probability of detecting a disease SNP (DP) and optimal stage1 weight for joint analysis, λ_{opt}, for the fixed effects model with log odds ratio per allele β= log(1.2)

The proportion positive, PP, can also be estimated for *β* = log(1.2) from Table 1 and equation (2) assuming *M*_{0} is known. For *M* _{0} =1, *T*_{1} =25,000 and *T*_{2} =10, P =0.065 for *π _{sample}* =0.125 and 0.093 for the one-stage design. If

For the random effects model with *τ* = (*π*/2)^{1/2} log(1.2) ≈ 0.2284 and with *M*_{0} =1(Table 2), P ranged from 0.550 to 0.646 for the one-stage design. With *π _{sample}* = 0.5, P was modestly reduced for the replication analysis, but only slightly reduced for the joint analysis.

Probability of detecting a disease SNP (DP) and optimal stage1 weight for joint analysis, λ_{opt}, for the random effects model with standard deviation of log odds ratio per allele τ = (π/2)1/2 log(1.2)

To examine how much the DP of the two-stage design is reduced compared to the one-stage design for fixed effects models with odds ratios per allele of 1.1, 1.2, 1.3 and 1.5, we plotted P for the two-stage design and joint analysis against the P for the corresponding one-stage design with 8000 cases and 8000 controls for *π _{sample}* =0.125 and

Detection probabilities of two-stage designs with joint analysis versus one-stage designs for fixed effects disease models with 12.5% of cases and controls in stage 1 (*π*_{sample} =0.125). Other parameters include *T*_{0} =500,000 SNPs, numbers of SNPs **...**

For the random effects model with *M*_{0} =1 and *π _{sample}* =0.125, decreases in P for the two-stage design are appreciable for standard deviations of log odds ratios of (

For the fixed effects model with *M*_{0} =1 and *π _{sample}* =0.25, the decreases in P for the two-stage design are smaller than for

Detection probabilities of two-stage designs with joint analysis versus one-stage designs for fixed effects disease models with 25% of cases and controls in stage 1 (*π*_{sample} =0.25). Other parameters include *T*_{0} =500,000 SNPs, numbers of SNPs selected **...**

For the random effects model with *M*_{0} =1 and *π _{sample}* =0.25, the decreases in P for the two-stage model are smaller than for

Detection probabilities of two-stage designs with joint analysis versus one-stage designs for random effects disease models with 25% of cases and controls in stage 1 (*π*_{sample} =0.25). Other parameters include *T*_{0} =500,000 SNPs, numbers of SNPs selected **...**

An executable pre-complied GAUSS program is available from the first author to estimate DP and PP for two-stage GWASs.

We studied the detection probability (DP) of a two-stage GWAS design, that is, the chance that a given disease-associated SNP will have among the lowest ranks of p-values (or highest ranks of chi-square statistics) at stages 1 and 2. Our data for fixed effects models indicate that the DP from a two-stage design with *π _{sample}* ≤0.25 and 8000 cases and controls can be substantially less than that of the corresponding one-stage design with the same numbers of cases and controls for odds ratios per allele of 1.1, 1.2, and 1.3, which are typical of statistically significant odds ratios found in recent large GWASs. For the range of values

Our data suggest that additional stage 1 genotyping in most previous studies with *π _{sample}* ≤0.25 will yield additional promising SNPs and that future multistage designs should not use

The two-stage ranking and selection procedure analyzed in this paper differs from two-stage procedures that apply the same fixed critical values to data from each SNP and are designed to select promising SNPs in stage 1 and provide a final p-value for testing an association following stage 2, as in (Skol et al., 2006). In particular, the two-stage ranking and selection procedure does not attempt to control the overall p-value, but only to obtain a very promising set of *T _{2}* SNPs at the end of stage 2. Despite these different goals and methods, Figure 2 in (Skol et al., 2006) shows that power diminishes appreciably, and cannot be retrieved by joint analysis, if

In some circumstances, power calculations can be used to approximate DP. For a one-stage design with *M*_{0} = 1, equation (2.6) in (Gail et al., 2008) shows that DP can be approximated by the power that corresponds to an hypothesis test with size *α* = *T*_{1}/*T*_{0}. Although power calculations performed in this way and extended to the two-stage design may approximate the DP under certain conditions, the results in (Skol et al., 2006) were not based on such significance levels and critical values. We illustrate these differences using the program provided by (Skol et al., 2006) at http://csg.sph.umich.edu. For *π _{sample}* =0.125,

It is worthwhile to recount some differences between power and detection probability. Power is the probability that the test statistic for a given SNP will fall into the pre-determined critical region for a one- or two-stage design that is chosen to control a genome-wide significance level, as for example in (Skol et al., 2006). Power thus depends on the chosen significance level; DP depends, instead, on *T*_{0}, *T*_{1}, and *T*_{2}. The power to reject the null hypothesis for a given SNP does not depend on the test statistics for any other SNP; DP depends on the test results for all SNPs. Power does not depend on the number of disease-associated SNPs, *M* _{0}; DP can be sharply reduced by competition among disease-associated SNPs, especially if *T _{2}* is less than

Satagopan and colleagues (Satagopan et al., 2004, Satagopan et al., 2002) also studied ranking procedures to identify disease-susceptibility SNPs in two-stage designs, but used different rank-based criteria from DP and also assumed that the disease allele was known. The two-sided versions of a Wald test or a score test that we used have the same value whether one counts major or minor alleles, and hence are particularly appropriate for GWASs(Devlin & Roeder, 1999, Pfeiffer & Gail, 2003).

The ranking and selection methods used in this paper depend on the assumption that tagging SNPs are independent (Gail et al., 2008), an assumption that is also widely used in power calculations, e.g. (Skol et al., 2006). Zaykin and Zhivotovsky (Zaykin & Zhivotovsky, 2005) analyzed different ranking criteria for one-stage designs and found that selection probabilities were little affected by correlations of p-values within linkage disequilibrium blocks or among such blocks. In unreported simulations in which non-disease associated SNPs were paired and each member of the pair assigned the same chi-square value (perfect correlation within pairs), we found almost no effect on the estimates of DP and PP, compared to the situation in which SNP gentotypes are independent. Thus it is likely that our estimates of DP are robust to local correlations among tagging SNPs.

In view of the potential losses in DP in multistage designs and trends in costs favoring large values of *π _{sample}*, the one-stage design becomes increasingly attractive. Another advantage of the one-stage design is that it yields data that can readily be used in meta-analyses. For example, if a preliminary study identifies a particular SNP as associated with disease, data from independent one-stage studies can be used to test the association and provide an unbiased estimate of the corresponding odds ratio. Later stages in a multistage design would provide no information if that SNP had not been tested in the later stages.

The Intramural Research Program of the Division of Cancer Epidemiology and Genetics, National Cancer Institute supported this work. We thank the reviewers for comments that improved the paper.

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