Power and Design for Family QTL Studies

Why Family QTL Power Depends on Design

A family QTL study seeks evidence that relatives’ phenotypes share covariance attributable to a genomic region. Its power is not determined by the number of people alone. Relatives differ in how much genetic information they share; markers differ in how well they reveal that sharing; and a QTL effect can be difficult to distinguish from other sources of familial resemblance. Families selected because of their phenotypes also require an analysis that accounts for selection.

Design calculations connect these features to a test’s ability to detect a nonzero QTL variance component. The calculations below use a local, score-based approximation to make the role of information explicit. It is a planning tool, not a substitute for a model that matches the proposed sampling and analysis.

The entries of \(\Pi\) depend on the family relationships and on the IBD information used by the model. With uncertain IBD, the likelihood accounts for that uncertainty rather than treating an unobserved sharing state as known. In a design calculation, marker informativeness therefore affects how much covariance information is available, not merely how many markers are counted.

Information and a Local Power Approximation

First suppose the null covariance \(V_0\) is specified and positive definite. Let \(P=V_0^{-1}-V_0^{-1}X(X^\mathsf{T}V_0^{-1}X)^{-1}X^\mathsf{T}V_0^{-1}\), assuming the columns of \(X\) are linearly independent. This matrix removes the fitted mean directions: \(PX=0\), and \(PV_0P=P\). The restricted-likelihood score for \(\tau\) at zero is

The quadratic term is effectively based on the mean-adjusted phenotype: because \(PX=0\), \(Py=P(y-X\beta)\). This matters when the mean is not zero; using an uncentered quadratic form without removing the fitted mean would generally produce a nonzero expected score under the null.

A short justification

Under \(H_0\), \(E(y)=X\beta\) and \(\operatorname{Cov}(y)=V_0\). Since \(PX\beta=0\),

Thus \(E(U)=0\); under the Gaussian model its null variance is \(I_{\tau\tau}\). Under the alternative with the same mean and covariance \(V_0+\tau\Pi\), the mean term still vanishes after projection, and

Consequently, \(E_\tau(U)=\tau I_{\tau\tau}\). For small alternatives and enough independent family units for a normal approximation, \(U/\sqrt{I_{\tau\tau}}\) has approximate mean \(\tau\sqrt{I_{\tau\tau}}\) and variance near one. For a one-sided score test at significance level \(\alpha\), approximate power is

Here \(\Phi_N\) is the standard normal cumulative distribution function, \(z_c\) is its \(c\)-quantile, \(1-\beta\) is target power, and \(n\) is the number of independent families when their per-family information is equal. The sample-size expression rearranges the approximation; it is not exact, especially for small numbers of pedigrees or alternatives far from zero.

In practice, nuisance covariance components are often estimated too. Their score can carry information that overlaps with the QTL score. For covariance derivatives \(A_j\) of nuisance parameters \(\lambda_j\), define score information \(I_{ab}=\tfrac12\operatorname{tr}(P A_a P A_b)\), with \(A_\tau=\Pi\). The information for \(\tau\) after accounting for nuisance covariance parameters is the efficient information \(I_{\tau\cdot\lambda}=I_{\tau\tau}-I_{\tau\lambda}I_{\lambda\lambda}^{-1}I_{\lambda\tau}\), when \(I_{\lambda\lambda}\) is invertible. Use this adjusted information in planning rather than treating a confounded component as known.

Marker Information and Target Sample Size

Marker informativeness affects the precision of the IBD information used to construct the linkage likelihood. Better information can sharpen the distinction between the covariance under linkage and the null, but its effect is model- and pedigree-dependent. A scalar “reliability” is useful only as a transparent approximation; it does not replace integrating uncertain IBD probabilities in the actual analysis.

QTL Effect Size, Ascertainment, and Practical Design

Because the local noncentrality is \(\tau\sqrt{I}\), effect size and information have distinct roles. For a fixed design, halving \(\tau\) approximately halves the noncentrality; maintaining the same power then requires about four times as much information. Increasing family count generally increases information nearly linearly for independent families, while changing relationship structure can alter information per family. The latter depends on the full covariance pattern, including how well the QTL component is separated from polygenic, environmental, and residual components.

Ascertainment can change the distribution of observed families. If families are selected based on their phenotypes, an ordinary likelihood for an unselected sample may not describe the sampling process. Power should then be evaluated using an ascertainment-corrected or conditional likelihood consistent with the selection rule. A design calculation that ignores selection can misstate both the null calibration and the alternative distribution. When selection is complex, simulate the full process: generate pedigrees and phenotypes, apply the selection rule, generate or condition on marker information, and analyze each simulated sample with the intended corrected procedure.

The same principle applies to incomplete phenotypes, departures from the assumed covariance model, and unequal pedigree sizes. A simple information formula is useful for screening candidate designs, but final planning should reflect the expected number of usable relatives, actual marker informativeness, nuisance parameters, and the exact test calibration. Parametric simulation is especially valuable when the boundary at \(\tau=0\), small family counts, or ascertainment makes a normal approximation unreliable.

Common Pitfalls and Design Checks

• Counting people instead of covariance information. The same number of participants can yield different power when pedigree relationships differ. Compare covariance information after nuisance components are accounted for.
• Treating marker uncertainty as if IBD were observed. Use the information implied by marker data and the planned IBD treatment; do not assume perfect sharing information unless the design supports it.
• Ignoring the fitted mean. A score quadratic must remove fixed-effect means. With \(PX=0\), \(Py=P(y-X\beta)\); an uncentered score without this projection can have a nonzero null expectation.
• Using raw information when nuisance components are estimated. Overlap between QTL and baseline covariance patterns reduces efficient information and can substantially lower power.
• Applying an unselected-sample calculation to selected families. Phenotype-based ascertainment must be reflected in the likelihood and in power evaluation.

A defensible design starts with a plausible QTL variance, the family structures that can be recruited, expected marker information, and a covariance model that includes relevant nuisance components. Use local information calculations to compare options, then verify the selected design by simulation under the actual ascertainment and analysis procedures when those features are consequential.