Power Trade-Offs in Family-Based Association

Why Power Depends on More Than the Number of Families

A family-based association study can enroll many families yet have limited power if few parents are heterozygous at the variant being tested. Conversely, a smaller study can be informative when the allele is common enough to generate many informative transmissions and the genetic effect is appreciable. Power comparisons therefore require more than counting participants: they must account for the design’s informative observations, the score’s expected shift under the alternative, and its variability.

This tutorial develops a planning framework for trio, sibling, and population-based association designs. The calculations are approximations under stated models, not universal power formulas for every family-based test. In particular, matching two designs on their null variances does not by itself show that they have the same power: their alternative score means must also be specified.

A Power Approximation That Tracks Both Mean and Variance

Suppose a two-sided test rejects when \(|Z|>c\), where \(c\) is the appropriate null critical value (for a large-sample test at \(\alpha=0.05\), \(c\) is approximately 1.96). If \(Z\) is approximately normal under the alternative, then \(Z\approx N(\lambda,s_1^2)\), where \(\lambda=\mu_1/\sqrt{V_0}\) and \(s_1^2=V_1/V_0\). Consequently, approximate power is

Here \(\Phi\) is the standard normal cumulative distribution function. Under local alternatives, or when the alternative variance is close to the null variance, \(s_1\) is often approximated by 1. The familiar approximation \(\Phi(\lambda-c)+\Phi(-\lambda-c)\) follows in that case. If \(s_1\) is not close to 1, omitting it can distort the power calculation.

Trio Power: Effect Size and Informative Transmissions

For a simple planning model of the transmission disequilibrium test (TDT), consider a heterozygous parent and one transmission to an affected child. Let \(B=1\) if the parent transmits the designated risk allele and \(B=0\) otherwise. Assume that, conditional on being an informative heterozygous-parent transmission, \(B\) is Bernoulli with probability \(q\). Under the conditional null of no preferential transmission, \(q=0.5\). For this planning model, write the transmission odds ratio as \(\theta=q/(1-q)\). This is a parameter of the specified conditional transmission model, not a general identity equating every disease odds ratio with a transmission odds ratio.

For \(N\) independent informative transmissions, the TDT count \(T=\sum B_i\) has null mean \(N/2\) and null variance \(N/4\). The null-standardized score is \(Z=(T-N/2)/\sqrt{N/4}\). Under the alternative, its mean and variance are

These expressions follow because \(E_1(T)=Nq\) and \(\operatorname{Var}_1(T)=Nq(1-q)\). Thus the standardized mean shift increases with both the effect \(2q-1\) and the square root of the number of informative transmissions. The alternative variance is generally not exactly 1.

Under Hardy–Weinberg equilibrium, \(2p(1-p)\) is largest at \(p=0.5\), so common alleles generally produce more heterozygous parents than very rare alleles, all else equal. This is not a claim that power always increases monotonically with allele frequency: effect size, genotype model, ascertainment, missing genotypes, and the actual number of independent informative units also matter. Moreover, if both parents are heterozygous, treating their transmissions as independent is a modeling assumption, not an automatic consequence of observing two transmissions.

Sibling Information Requires Its Own Alternative Model

Sibling-based designs may be useful when parental genotypes are unavailable, but the amount of information depends on which siblings are observed, their phenotypes, and the conditional score used. There is no single conversion from “one sibling pair” to a fixed number of independent TDT transmissions. To calculate power, specify the score and its conditional distribution under both null and alternative.

Population-Based Comparison: A Different Information Structure

A population-based case-control association test uses allele-frequency differences between cases and controls rather than within-family transmission contrasts. For a simple comparison, assume a rare disease, a multiplicative per-allele disease odds ratio \(\theta\), baseline allele frequency \(p\), and approximately independent allele copies. Under this planning approximation, the case allele frequency is \(p_1=p\theta/(1-p+p\theta)\), while the control allele frequency is approximated by \(p_0=p\). This model is useful for illustrating efficiency; it does not remove the need to address population structure or other sources of bias in an actual study.

With \(n_1\) cases and \(n_0\) controls, compare the observed allele proportions. Under the null, the approximate variance of their difference is \(p(1-p)[1/(2n_1)+1/(2n_0)]\). Under the alternative, the difference in expected proportions is \(p_1-p_0\). The ratio of this difference to the null standard error gives the approximate alternative mean of the null-standardized statistic; the alternative variance can also be computed by replacing each group’s null allele variance with its group-specific value.

How to Make a Fair Design Comparison

A useful comparison keeps the target effect and significance threshold clear, then evaluates each design under a defensible model for its own score. The quantities that matter are not interchangeable: family structures change the number and dependence of informative contrasts, while population studies change the sampling variance of allele-frequency estimates. For each candidate design, state the following:

• What observation contributes to the score, and which families or individuals are informative?
• What are the score’s null variance and its alternative mean?
• What is its alternative variance, or why is using the null variance as an approximation reasonable?
• How do allele frequency, effect size, missing data, and the sampling plan change those quantities?

In trios, the heterozygote probability \(2p(1-p)\) is a useful first approximation to transmission availability under Hardy–Weinberg equilibrium. In sibling designs, informative phenotype and genotype configurations determine how many pairs contribute and how their scores behave. In population designs, case and control counts and allele-frequency variance determine precision. The practical efficiency comparison should also account for genotyping cost, recruitment, family availability, and the study’s susceptibility to bias.

Common Mistakes and a Planning Workflow

A frequent error is to say, “This sibling design has the same null variance as 300 independent transmissions, so its noncentrality is \(\sqrt{300}\delta\).” That conclusion does not follow from null variance alone. To justify it, one must show that the sibling score’s alternative mean is \(300\delta\) (or derive an appropriate local-alternative relationship), and also consider its alternative variance. Without that information, a power value cannot be inferred from the variance match.

Another mistake is to treat every enrolled family as informative. The number recruited is not necessarily the number of independent score contributions. A robust planning sequence is:

Power calculations are conditional on the model used to produce them. Ascertainment can alter which families enter a study and the conditional distribution relevant to a test; the assumptions used in a planning calculation should match the intended analysis. Likewise, population-based estimates can be more precise in a numerical model yet remain vulnerable to population structure, whereas family-based tests address a different set of design concerns. Power alone is not a complete measure of study quality.