Sampling Variation and Confidence Intervals for LD

Why Linkage-Disequilibrium Estimates Vary

Haplotype frequencies and linkage-disequilibrium measures are estimated from a finite sample. A different sample of chromosomes from the same population would usually produce different estimated frequencies, and therefore different values of \(D\) and \(r^2\). A confidence interval describes this sampling uncertainty; it does not remove uncertainty about how the sample was collected, whether its chromosomes represent the target population, or how phase was inferred.

The calculations below use two biallelic loci and phased chromosomes. Write the four haplotype frequencies as \(p_{AB}, p_{Ab}, p_{aB}, p_{ab}\), with corresponding counts \(n_{AB}, n_{Ab}, n_{aB}, n_{ab}\). The total number of sampled chromosomes is \(n\), and each estimated haplotype frequency is its count divided by \(n\).

The Sampling Model and Delta-Method Uncertainty

If the sampled chromosomes are independent draws from a population with fixed haplotype frequencies, the four counts follow a multinomial distribution with size \(n\) and category probabilities \(p_{AB},p_{Ab},p_{aB},p_{ab}\). This model assumes that haplotypes are observed correctly and that the sampled chromosomes are representative and independent. If the two chromosomes from one person are not independent observations, uncertainty should instead be estimated by resampling people as units.

A standard large-sample method for estimating uncertainty is the delta method. It approximates the variance of a smooth function of estimated frequencies by applying a first-order Taylor expansion. For \(D\), use the first three frequencies as coordinates: \(q=(p_{AB},p_{Ab},p_{aB})\), so \(p_{ab}=1-p_{AB}-p_{Ab}-p_{aB}\). The gradient of \(D\) with respect to these coordinates is:

For a single multinomial draw, the covariance matrix of the first three category indicators is \(\Sigma=\operatorname{diag}(q)-qq^\mathsf{T}\). The estimated frequency vector has covariance \(\Sigma/n\). Consequently, the delta-method estimate of the variance of \(\hat{D}\) is \(\nabla D^\mathsf{T}\Sigma\nabla D/n\), evaluated at the sample frequencies. The square root is the estimated standard error, or \(SE\).

The same delta-method principle can be applied to \(r^2\), which is a nonlinear function of \(D\) and the two allele frequencies. Let \(V=p_A(1-p_A)p_B(1-p_B)\). For \(V>0\), its gradient is:

Its estimated variance is \(\nabla r^2{}^\mathsf{T}\Sigma\nabla r^2/n\). The resulting Wald interval can be a poor choice when \(r^2\) is near zero or one: \(r^2\) is constrained to the range from zero to one, while a normal interval is not, and the first derivative of \(r^2\) with respect to the frequencies is zero when \(D=0\). In that case a first-order approximation can understate uncertainty.

Resampling to Estimate Uncertainty

A bootstrap estimates sampling variation by repeatedly making new samples from the observed data, recalculating the statistic each time, and summarizing the resulting values. For phased, independent chromosomes, a nonparametric bootstrap draws \(n\) chromosomes with replacement from the observed \(n\). A parametric bootstrap instead draws haplotype counts from a multinomial distribution using the estimated frequencies. Both methods preserve the four-haplotype structure; neither automatically solves problems caused by a misspecified sampling design.

When chromosomes are paired within sampled individuals, resample whole individuals, retaining both observed chromosomes and any associated genotype data. This preserves dependence within a person. If phase is uncertain and haplotype frequencies were estimated from unphased genotypes, the resampling procedure should repeat the frequency-estimation process in each resample. Treating estimated haplotypes as if their phase were known can omit a source of uncertainty.

Small Counts and Boundary Problems

With small haplotype counts, normal approximations can be inaccurate and bootstrap distributions can be discrete or strongly skewed. A zero observed count does not establish that the population frequency is zero. Yet an ordinary nonparametric bootstrap cannot draw a haplotype that was absent from the original sample, so it cannot represent uncertainty about unobserved haplotypes. A parametric bootstrap has a related limitation when a fitted frequency is exactly zero: it will also generate no copies of that haplotype.

Small samples can also produce resamples in which one locus is monomorphic. The denominator of \(r^2\) is then zero, so the replicate \(r^2\) is undefined. Silently dropping such replicates changes the resampling distribution and may distort an interval. The frequency of undefined replicates should be reported and treated as evidence that a routine bootstrap interval may not be reliable. For sparse counts, a carefully constructed likelihood-based or exact method, or a justified model that accounts for the design, may be more appropriate. Adding a pseudocount changes the estimator; it does not by itself provide a confidence interval with a guaranteed coverage level.

Derivation: Why the Delta-Method Variance Has This Form

The delta-method result follows from a first-order expansion around the true frequency vector. Let \(\hat{q}\) be the vector of the first three estimated haplotype frequencies and \(g(q)=D\). For estimates near \(q\), a Taylor expansion gives \(g(\hat{q})\approx g(q)+\nabla g(q)^\mathsf{T}(\hat{q}-q)\). The constant term has no variance, so the variance of the linear part is:

The final equality uses the multinomial covariance \(\operatorname{Var}(\hat{q})=\Sigma/n\). Replacing unknown population frequencies with their sample estimates yields the estimated standard error used above. This approximation relies on the estimate being sufficiently far from problematic boundaries and the sample being large enough for the linear approximation to be informative; it is not a small-count exact result.

Common Mistakes and Practical Interpretation

• Reading the interval as certainty about the parameter. A 95% confidence procedure has 95% long-run coverage under its assumptions; the parameter is fixed, not randomly inside this realized interval.
• Using chromosome resampling when the independent units are people. If paired chromosomes are dependent, resample people and retain their paired data.
• Treating zero observed counts as known zero frequencies. A sample may miss a rare haplotype; methods that cannot generate an unobserved category can understate uncertainty.
• Trusting a normal interval near a boundary. A Wald interval can extend outside the feasible range for \(D\) or the range \([0,1]\) for \(r^2\). Clipping its endpoints does not repair the approximation.
• Interpreting an interval as a causal result. An interval quantifies sampling uncertainty in an LD measure; it does not show that one variant causes another or that the sample represents every population.

The choice of interval method should follow the sampling design and the observed counts. For reasonably large, well-behaved samples, a delta-method interval offers a transparent approximation. Resampling can reveal asymmetry and non-normality, but it does not guarantee good coverage when counts are sparse. Report the sample size, the haplotype counts, the method, and any boundary or undefined-resample issue so that the interval can be interpreted in context.