Linkage Disequilibrium And Effective Marker Counts

Why the Raw Marker Count Can Mislead

A relatedness estimate may combine information across thousands of SNPs. It is tempting to think that multiplying the number of markers by four will reduce its standard error by a factor of two. That conclusion relies on the added markers contributing independent information. When nearby SNPs are in linkage disequilibrium (LD), they tend to carry overlapping information, so the gain in precision can be much smaller.

The practical issue is not simply whether SNP genotypes are correlated. It is how dependence across markers affects the marker-level contributions to the relatedness estimator. LD pruning removes some correlated markers; marker weighting reduces or redistributes their influence. Both can affect precision and the stability of estimates, but neither automatically improves every estimate.

For relatedness estimation, let \(Z_l\) denote the contribution from marker \(l\), with common expected value \(\theta\) and common variance \(\tau^2\). Here \(\theta\) is the relatedness quantity the estimator targets, and the \(Z_l\) are understood to be on the estimator's scale. The correlations among these contributions are written \(\rho_{lm}=\operatorname{Corr}(Z_l,Z_m)\). In general, \(\rho_{lm}\) is not automatically equal to genotype \(r^2\): its value depends on how marker genotypes enter the estimator and on the individuals being compared. LD measures can be useful proxies, but the estimator's contribution covariance is the quantity that determines variance.

Variance and Effective Marker Count

Suppose the estimate is an equally weighted average of \(M\) marker contributions, each with variance \(\tau^2\). Its variance is the sum of all individual variances and all pairwise covariances, divided by \(M^2\):

If the markers were independent, every \(\rho_{lm}\) would be zero, and the variance would be \(\tau^2/M\). Define \(M_{\mathrm{eff}}\) by setting that independent-marker variance equal to the variance with dependence. This gives the following result.

If all distinct pairs have the same contribution correlation \(\rho\), there are \(M(M-1)/2\) such pairs. Substituting this number yields \(M_{\mathrm{eff}}=M/[1+(M-1)\rho]\). Positive correlations therefore reduce the effective count. The formula also shows why simply dividing the raw marker count by an informal “LD factor” can be unreliable: the dependence pattern and the estimator matter.

Worked Example: Four Correlated Marker Contributions

Weighted Contributions and LD Pruning

The same reasoning applies when markers have unequal weights. For fixed weights \(w_l\), define the estimate as \(\hat{\theta}_w=\sum_l w_lZ_l/\sum_lw_l\). If the contributions have common variance \(\tau^2\), its variance-equivalent effective count is

The double sum includes the diagonal terms, for which \(\rho_{ll}=1\). With independent markers, this reduces to \((\sum_lw_l)^2/\sum_lw_l^2\), which is no greater than the number of nonzero weights. Correlation can reduce it further. This calculation describes variance under the assumed contribution covariance; it does not establish that any particular weighting rule is optimal.

LD pruning usually removes one or more markers from groups that exceed a chosen LD threshold. It reduces redundancy and computation, and can make an analysis less dominated by dense clusters of correlated SNPs. However, pruning also discards data. A threshold such as a maximum pairwise \(r^2\) is a practical selection rule, not a direct calculation of \(M_{\mathrm{eff}}\) or a guarantee of a particular standard error. The result depends on the threshold, the region used, and the sample in which LD is assessed.

Weighting can instead limit the combined influence of dense regions. For example, assigning each region a fixed total weight means that a region with many typed SNPs does not automatically dominate one with fewer SNPs. This changes how the estimator combines its marker contributions. The weights should be stated and justified; a weighted estimate need not equal the unweighted estimate, and a different weighting scheme can have a different variance.

A Short Derivation and Its Limits

The effective-count result follows directly from covariance addition. For the equally weighted estimate, each variance contributes \(\tau^2\). Each distinct pair contributes \(\operatorname{Cov}(Z_l,Z_m)=\rho_{lm}\tau^2\), and each pair appears twice when the variance of a sum is expanded. Hence

Equating this to \(\tau^2/M_{\mathrm{eff}}\) and solving for \(M_{\mathrm{eff}}\) gives the stated formula. Its assumptions matter: the contributions must have the common variance used in the calculation, and their covariance pattern must be represented adequately. If variances differ, replace the common-\(\tau^2\) covariance model with the actual variances and covariances. If marker weights are used, use the weighted formula rather than the equal-weight formula.

Genotype LD summaries such as \(r^2\) can help identify redundant markers, but they are not universally interchangeable with \(\rho_{lm}\) for relatedness contributions. In particular, dependence can differ across populations and estimator constructions. An effective count calculated from an LD summary is therefore best described as an approximation under a stated convention, not a universal property of a marker panel.

Why Precision and Stability Both Matter

Precision concerns the variability of an estimate under repeated sampling or comparable data. Redundant markers contribute less additional precision than independent ones, even if they make the raw marker count appear large. Stability concerns how sensitive results are to reasonable changes in marker selection, weighting, or sample composition. A dense LD block can have disproportionate influence on an unweighted estimate; pruning or block weighting can reveal whether that influence matters.

These procedures can also change results for reasons other than variance. Pruning selects a subset, and weighting changes the contribution of genomic regions. If marker contributions are not all centered on the same target, either choice can shift the estimate as well as its precision. Thus, compare estimates and uncertainty under clearly described choices; do not interpret agreement or disagreement as proof of a particular biological relationship by itself.

Common Mistake

• Wrong reasoning: “There are 10,000 SNPs, so the standard error is the single-marker standard deviation divided by \(\sqrt{10{,}000}\).”
• Correction: That calculation assumes independent, equally variable contributions. LD can make the effective marker count much smaller than 10,000, so the standard error must reflect covariance among contributions or a clearly stated approximation to it.
• Wrong reasoning: “Pruning always improves precision because it removes correlated SNPs.”
• Correction: Pruning removes redundant information but also discards markers. It can improve balance or stability without reducing the variance of a particular estimate. Calculate or compare uncertainty under the actual procedure.