Genome-Wide Thresholds in Sample-Size Planning

Why the Threshold Matters Before Recruitment

A genome-wide association study can have many opportunities to detect an association, so its planned significance threshold is often much smaller than the conventional 0.05 used for a single test. A smaller threshold makes the evidence required for a discovery more demanding. If the study is planned around a particular effect size and power, that stricter threshold generally means recruiting more participants.

This tutorial treats the genome-wide threshold as a chosen design input. It shows how to combine that threshold with a target power, allele frequency, and effect size to calculate a sample size. It does not derive a threshold from the number or dependence of tests, nor does it adjust an observed result. Those are distinct tasks: the calculation here begins with a threshold already selected for the planned analysis.

From a Threshold to a Required Signal

Suppose the planned association test is two-sided and its null statistic is approximately standard normal. Write \(\Phi\) for the cumulative distribution function of a standard normal random variable. The critical value corresponding to the chosen threshold is

A test rejects when the absolute value of its statistic exceeds \(z_{\mathrm{crit}}\). For example, \(\alpha_{\mathrm{GW}}=5\times10^{-8}\) gives \(z_{\mathrm{crit}}\approx5.4513\). Under a specified alternative, suppose the test statistic is approximately normal with variance 1 and mean \(\lambda>0\), where \(\lambda\) is the noncentrality parameter. Its power is

For a positive effect and a stringent two-sided threshold, the lower-tail term is usually negligible. A convenient planning approximation for target power \(1-\beta\) is therefore

This approximation chooses the alternative mean so that the upper-tail rejection probability is approximately the target power. For more exact planning, solve the full two-sided power equation for \(\lambda\); the formulas below then use that solved value in place of \(z_{\mathrm{crit}}+z_{1-\beta}\). The simple sum is typically adequate for standard GWAS planning when the effect is in the specified direction and the threshold is very stringent.

Quantitative-Trait Sample Size

Consider an additive association model for a quantitative trait. Let \(G\) be allele dosage, taking values 0, 1, or 2, and let \(f\) be the frequency of the allele whose dosage is counted. Under Hardy–Weinberg equilibrium, the dosage variance is \(2f(1-f)\). Let \(\beta\) denote the change in the trait mean per additional allele, and let \(\sigma\) denote the residual standard deviation after the planned covariate adjustment. Under the usual large-sample normal approximation, the association statistic has noncentrality approximately

where \(N\) is the number of participants with usable phenotype and genotype data. Equating this to the required noncentrality and solving for \(N\) gives

Here \(\beta_{\mathrm{type\,II}}\) is the type II error probability, not the genetic effect \(\beta\); the distinct notation in the formula indicates their different roles. Equivalently, if the trait is standardized so that the residual standard deviation is 1, the effect is expressed in residual standard-deviation units per allele. The estimate assumes a well-calibrated association test, independent participants, the stated allele frequency and effect, and an adequate normal approximation. It also treats \(N\) as the analyzable sample size, not necessarily the number initially recruited.

Case-Control Sample Size

For a case-control study, let \(n_{\mathrm{case}}\) and \(n_{\mathrm{control}}\) be the numbers of cases and controls, and let \(\beta_{\mathrm{logOR}}\) be the log odds ratio per additional allele. For a modest effect and a well-calibrated large-sample test, the approximate information for the genetic effect is \(2f(1-f)n_{\mathrm{eff}}\beta_{\mathrm{logOR}}^2/4\), where the effective sample size is

This definition makes \(n_{\mathrm{eff}}\) equal to the total sample size when cases and controls are equal in number. With a case-control ratio other than 1:1, it is smaller than the total. The corresponding approximate noncentrality is

Solving for the effective sample size gives

This approximation treats the allele frequency as the same in cases and controls under the null and uses the planned log odds ratio as the alternative. It is most useful for large samples and modest genetic effects. It does not replace a more detailed calculation when the model, ascertainment, covariates, or sampling design makes these approximations unsuitable.

Reading the Calculation as a Design Tool

The formulas make several design trade-offs explicit. Holding other inputs fixed, required sample size is proportional to the square of the required noncentrality. It is inversely proportional to the squared effect size and to genotype variance. Thus, halving the effect size requires roughly four times the sample size under this approximation. Alleles with frequencies near 0.5 have greater dosage variance than very rare alleles, so they provide more information per participant for the same per-allele effect.

The chosen threshold is also consequential. A smaller \(\alpha_{\mathrm{GW}}\) increases \(z_{\mathrm{crit}}\), increasing the required noncentrality and sample size. In a planning document, state the threshold itself and its rationale rather than presenting the sample-size result as if the threshold were an intrinsic property of the association statistic. The threshold used here is an input; the calculation does not determine how a testing family should be defined or how an observed p-value should be adjusted.

The result is usually an analyzable sample size. If some recruited participants are expected to lack usable genotype or phenotype data, recruitment must be larger. For example, under a simple assumption that a proportion \(r\) of recruited participants will be usable, a target of \(N_{\mathrm{req}}\) usable participants calls for approximately \(N_{\mathrm{req}}/r\) recruits. This adjustment is only as good as the retention assumption, and it does not account for other departures from the model.

Assumptions, Limitations, and a Common Pitfall

These calculations are useful when the normal approximation is adequate and the planning inputs are defensible. They can be misleading when the assumed allele frequency or effect does not represent the target population, when the phenotype variance differs from the planning value, or when the analysis has substantially less information than the simple model assumes. The test must also be appropriately calibrated under the null; the earlier material on association calibration and regression diagnostics is relevant to evaluating that condition.

A further limitation is that a calculation for one variant does not automatically describe the power of an entire study. Different variants may have different frequencies, effects, and genotype quality. The calculated sample size is specific to the stated alternative and test. It is not the probability of finding any association in the study, and it does not guarantee discovery of variants whose effects are smaller or frequencies rarer than assumed.

• Wrong reasoning: “The study has a genome-wide threshold, so I can use 0.05 in the power formula and adjust the result afterward.”
• Correction: Use the actual planned threshold in the prospective power calculation. A threshold change alters the critical value and therefore the required sample size; a later adjustment of an observed result is not a substitute for planning at the intended threshold.
• Wrong reasoning: “The required total case-control sample size is the effective sample size.”
• Correction: Convert the required effective sample size into cases and controls using the planned allocation. Imbalance means the total recruited number can be appreciably larger than \(n_{\mathrm{eff}}\).