Block Jackknife Uncertainty for Heritability Partitions

Why Block-Level Uncertainty Matters

Partitioned heritability analyses estimate how genetic variation is distributed across annotations. In a stratified LD score regression analysis, the fitted annotation coefficients and quantities derived from them depend on many SNP-level regression observations. Those observations are not independent: linkage disequilibrium creates dependence among nearby genomic regions. Treating every SNP row as an independent replicate can therefore make uncertainty look too small.

A block jackknife addresses this dependence by repeatedly omitting a genomic region, refitting the same model, and measuring how much the estimate changes. The resulting standard error reflects variation across large genomic units rather than treating each SNP as independent. The procedure can be applied directly to a coefficient or to a derived quantity such as a heritability proportion or enrichment, provided that the quantity is recalculated after every deletion.

Setting Up the Block Jackknife

In a weighted stratified LD score regression, the model relates association statistics to LD scores for the annotations and estimates an annotation coefficient vector along with nuisance parameters such as an intercept. The block jackknife does not change that model. It changes which regression observations are used for each replicate: for deletion \(b\), remove the observations assigned to block \(b\), then refit all model parameters using the remaining observations.

Blocks are usually contiguous genomic regions. Their size should be large enough that much of the dependence induced by local linkage disequilibrium is contained within blocks. They should also be numerous enough to provide a useful set of deletions. The standard formula above assumes blocks are reasonably comparable in their contribution to the estimator. A design with highly unequal blocks may require a method that accounts for unequal group contributions; the ordinary formula should not be applied automatically in that situation.

The analysis also requires that every deletion can be fit in a comparable way. In particular, the remaining design must identify the parameters being reported, and the same model specification and fitting rules should be used in every replicate. If a coefficient is not estimable after deleting a block, its jackknife standard error is not obtained simply by substituting a different model for that replicate.

Why the Jackknife Variance Has This Form

The factor \((B-1)/B\) adjusts the spread of the deletion estimates to estimate the sampling variance of the full-data estimator. A useful way to see the relationship is through the jackknife pseudovalues. For block \(b\), define

These values express how the full estimate compares with each block-deleted estimate. Their mean is

This mean is the jackknife bias-corrected estimate; it is not generally equal to the full-data estimate \(\hat{\theta}\). The centered pseudovalues satisfy

If \(s_p^2=(B-1)^{-1}\sum_b(p_b-\bar{p})^2\) is the sample variance of the pseudovalues, then

Thus the pseudovalue variance divided by \(B\) equals the delete-one-block jackknife variance. This identity explains the formula; it does not imply that the pseudovalue mean equals the full-data estimate.

Apply the Deletions to the Estimand

The jackknife target need not be a regression coefficient. Let \(H_A\) denote the estimated heritability attributed to annotation \(A\), and let \(H_{\mathrm{total}}\) denote the estimated total heritability on the same scale. A heritability proportion is

For each block deletion, the model-derived \(H_A\) and \(H_{\mathrm{total}}\) can change. Calculate their ratio within that deletion to obtain \(P_{A,(-b)}\), then use those \(B\) ratio estimates in the jackknife formula. This automatically reflects the joint changes in numerator and denominator. Calculating separate standard errors for \(H_A\) and \(H_{\mathrm{total}}\), then combining them as though they were independent, generally fails to account for their covariance.

Enrichment compares the proportion of heritability attributed to an annotation with the proportion of SNPs (or annotation mass, under the chosen encoding) in that annotation. If \(M_A/M_{\mathrm{total}}\) is that annotation’s SNP-mass proportion, define

When annotation mass is fixed across the deletion fits, calculate enrichment from each deletion’s heritability proportion and use those deletion-specific enrichments in the jackknife formula. If the definition or estimation of annotation mass itself varies across fits, that part must also be handled consistently in each replicate.

Interpretation, Checks, and Pitfalls

A jackknife standard error is an estimate of sampling variability under the block-deletion approximation. It can be used to summarize uncertainty and, with suitable inferential assumptions, as an input to intervals or tests. It is not an exact guarantee that all dependence has been handled. Blocks that are too short may leave substantial LD dependence across blocks; too few blocks may make the estimated variability unstable. Results should be interpreted in light of the block construction and the data contributing to each refit.

• Refit, do not merely remove a term. A deletion replicate omits a block’s observations and refits the specified model, including nuisance parameters and all annotation coefficients.
• Recalculate the target quantity. For a proportion or enrichment, use the deletion-specific numerator and denominator (and annotation mass if it varies), rather than applying a coefficient’s standard error to the full-data ratio.
• Check estimability. Confirm that every deletion fit converges and that the reported parameter is identifiable in each replicate.
• Inspect block design. The ordinary formula relies on blocks that are reasonably comparable as jackknife units and sufficiently large to capture relevant local dependence.

A particularly important mistake is to claim that the mean pseudovalue equals the full-data estimate. In the coefficient example, the full estimate is \(1.08\times10^{-4}\), while the mean pseudovalue is \(1.02\times10^{-4}\). The correct identity is \(\bar p=B\hat{\theta}-(B-1)\bar{\theta}_{(-)}\), which is generally a bias-corrected estimate rather than \(\hat{\theta}\). The variance formula uses centered pseudovalues, so the distinction does not invalidate the jackknife standard error.

For a reported confidence interval, a common large-sample approximation is \(\hat{\theta}\pm 1.96SE_{\mathrm{JK}}(\hat{\theta})\) when a normal approximation is justified. This interval is an approximation, not a direct consequence of the jackknife formula alone. In particular, a small number of blocks, skewed ratio estimates, or estimates near a boundary can make a symmetric normal interval misleading.