Allele Frequency and Detectable Effect-Size Curves

Why Allele Frequency Shapes Power

A genetic association study can have the same sample size and significance threshold at every tested variant, yet have very different power across variants. One reason is allele frequency: when the minor allele is rare, relatively few participants carry it, so the genotype provides less information about an additive genetic effect. A power curve makes this relationship visible by showing either power across possible effects or the effect size needed to attain a chosen power.

The focus here is a fixed design: sample size, significance threshold, trait scale, and target power are held constant while allele frequency varies. The resulting curve is a planning approximation, not a guarantee for any particular variant. It depends on assumptions about genotype frequencies and the association model, and it does not account for every complication that can alter standard errors in practice.

The variance is central because it describes how much genotype dosage varies in the sample. If nearly everyone has dosage zero, the data contain little information for distinguishing an effect per allele from no effect. At \(f=0.5\), dosage variance reaches its maximum, \(2(0.5)(0.5)=0.5\). The formula assumes Hardy–Weinberg proportions in the population being modeled; departures, relatedness, ancestry structure, or genotype error can change the information available.

From Genotype Variance to a Detectable Effect

For a quantitative trait, consider the additive linear model \(Y=\mu+\beta G+\varepsilon\), where \(\beta\) is the mean trait change per additional minor allele and the residual errors have variance \(\sigma^2\). Assume independent participants, a correctly specified model, genotype dosage measured without error, and residual variance that does not depend on genotype. Under these conditions, the approximate standard error of the estimated effect is \(\sigma/\sqrt{N\,2f(1-f)}\), where \(N\) is the sample size.

For a case-control association, write \(\beta\) for the log odds ratio per additional minor allele. In a logistic regression with no additional covariates, under the null approximation the information for \(\beta\) is approximately \(N\pi(1-\pi)2f(1-f)\), where \(\pi\) is the proportion of sampled participants who are cases. Therefore the approximate standard error is the inverse square root of that information. The worked case-control calculation below uses a balanced sample, \(\pi=0.5\); it does not extend the discussion to imbalanced designs.

In either setting, a test statistic is approximately standard normal under the null and has an approximately normal distribution under a specified alternative. For a two-sided test at significance level \(\alpha\), write \(c=z_{1-\alpha/2}\), the standard-normal quantile with cumulative probability \(1-\alpha/2\). If the effect is positive, the test statistic has approximate mean \(\lambda\), its noncentrality parameter. For the quantitative-trait model, \(\lambda=\beta\sqrt{N\,2f(1-f)}/\sigma\). For the case-control model, \(\lambda=\beta\sqrt{N\pi(1-\pi)2f(1-f)}\).

Here \(\Phi\) is the standard normal cumulative distribution function. The first tail is the probability of rejecting in the direction of the positive effect; the second is the probability of rejecting in the opposite direction. For a desired power \(1-\gamma\), a useful approximation for a positive effect is \(\lambda\approx z_{1-\alpha/2}+z_{1-\gamma}\). It follows by requiring the upper-tail probability \(1-\Phi(c-\lambda)\) to be about \(1-\gamma\). The omitted lower tail is usually very small for a positive effect and a stringent two-sided threshold. For high-precision planning, solve the full power equation rather than relying on this approximation.

Reading the Curves

At fixed \(N\), \(\alpha\), and target power, the quantitative-trait detectable-effect curve is proportional to \(1/\sqrt{2f(1-f)}\). The same frequency dependence appears for the case-control log odds ratio when the case fraction is fixed. Thus, as minor-allele frequency rises from near zero toward 0.5, genotype variance increases and the smallest detectable effect decreases. A rare variant requires a larger effect to reach the same power, all else equal.

If the horizontal axis is allele frequency \(p\in(0,1)\), the dosage variance \(2p(1-p)\) is symmetric around \(p=0.5\): the values at \(p\) and \(1-p\) are identical. On the minor-allele-frequency scale, the domain is only \(0<f\leq0.5\); the detectable-effect curve falls toward \(f=0.5\) on that domain. It is not a curve extending on both sides of 0.5 when plotted against minor-allele frequency.

A power curve can instead hold the effect size fixed and plot power against frequency. Then power rises as \(f\) moves toward 0.5, because the noncentrality parameter increases with the square root of genotype variance. These two views express the same relationship: a frequency with more genotype variation supports greater power for a given effect, or a smaller detectable effect at a given target power.

Case-Control Effects on the Log-Odds Scale

For case-control studies, the frequency curve has the same general shape, but its vertical scale is different: the effect is measured as a log odds ratio, and information depends on the case fraction as well as total sample size. A log odds ratio \(\beta\) corresponds to an odds ratio \(\exp(\beta)\) for each additional minor allele under the additive logistic model. Because odds ratios are multiplicative, it is useful to report both the log-scale calculation and the corresponding odds ratio.

How to Construct and Use a Curve

A smooth line between calculated frequencies is a visual aid, not evidence that the approximation is equally reliable at every frequency. At very low \(f\), expected counts of minor-allele homozygotes and even heterozygotes can be small. The normal approximation may then be poor, and the formula’s implication of an extremely large detectable effect should not be treated as a precise prediction. For example, under Hardy–Weinberg equilibrium the expected number of minor-allele homozygotes in \(N\) participants is \(Nf^2\); if that number is small, sparse genotype data deserve particular attention.

Common Mistakes and Practical Interpretation

• Reading a frequency curve as an effect curve. A falling detectable-effect curve does not mean the true effect gets smaller as frequency rises. It means a smaller effect can be detected at the same target power under the model.
• Confusing allele frequency with genotype variance. The information term is \(2f(1-f)\), not simply \(f\). It rises toward \(f=0.5\) on the minor-allele scale and is symmetric around 0.5 when written using allele frequency \(p\in(0,1)\).
• Reporting an effect without its scale. A quantitative-trait coefficient has trait units per allele, while a case-control coefficient is a log odds ratio per allele. Exponentiating the latter gives the per-allele odds ratio.
• Treating an approximation as an exact guarantee. The formulas rely on model assumptions and normal approximations. Sparse genotypes, dependence among participants, measurement error, or model misspecification can change actual power.
• Calling a detectable effect the expected result. A detectable effect is the effect size associated with a chosen power under specified assumptions. It is not an estimate of the effect that a future study will observe.