The Clumping-and-Thresholding Algorithm

Why Combine Clumping and Thresholding?

A genome-wide association study (GWAS) can report many variants associated with a phenotype. Some may carry useful predictive information, while others may be correlated with nearby variants through linkage disequilibrium (LD). If correlated variants are all included with their marginal GWAS effects, the score can give repeated weight to overlapping genetic information. Clumping-and-thresholding (C+T) is a simple construction strategy that addresses this by first pruning correlated variants and then filtering the remaining variants by their GWAS p-values.

The method has two distinct decisions. Clumping uses LD to retain a representative variant from a group of correlated candidates. Thresholding keeps only retained variants whose GWAS p-values meet a chosen cutoff. The surviving variants contribute to a weighted sum of effect-allele dosages in each target individual. The method is transparent and straightforward to implement, but it is a heuristic: it does not guarantee that the resulting score is optimal for prediction.

Required Data and Notation

C+T requires three kinds of information. First, GWAS summary statistics must identify each variant and give its association p-value and an effect estimate, such as a regression coefficient. The summary statistics also need allele labels so the effect estimate can be matched to the allele counted in the target genotypes. Second, the target sample needs genotype data at the variants that may enter the score. Third, an LD reference sample needs genotypes at candidate variants so pairwise correlations can be estimated. The GWAS, target, and reference data need compatible variant identifiers and allele coding.

For variant \(j\), write its GWAS p-value as \(p_j\), its effect estimate as \(\hat{\beta}_j\), and the target individual's dosage of the GWAS effect allele as \(G_{ij}\). A dosage is the number of copies of that allele, usually 0, 1, or 2 for a diploid genotype; imputed data may use a value between 0 and 2. If \(S(P_T)\) is the set of variants that survive clumping and the p-value threshold, the score for target individual \(i\) is

This formula assumes that the dosage and weight refer to the same effect allele. When weights are log-odds coefficients, the score is on a log-odds scale before any further model interpretation; with linear-regression coefficients, the score uses the outcome units per dosage unit specified by that GWAS model. The score's scale therefore depends on the phenotype and weight definition.

Before clumping, analysts also specify settings that define which variants can compete and how much LD is considered sufficient for pruning. For example, a procedure may use an LD threshold \(r^2_{\mathrm{clump}}\) and an LD window. The next tutorial addresses how to choose windows and reference panels. Here, treat the settings as fixed inputs to the algorithm, not as universal constants.

The Two Selection Stages

In a common conceptual formulation, clumping is applied to a quality-controlled candidate set. Variants are ranked by GWAS p-value, with the smallest p-value considered first. A variant selected as an index variant forms a clump with other candidates that meet the specified proximity and LD criteria. Those correlated candidates are removed from further consideration, and the procedure continues with the strongest remaining candidate. After pruning, the p-value threshold is applied to the retained variants.

Implementations can combine the stages differently. For instance, software may use p-value settings to limit which variants can become index variants or enter a clump. Because such settings can alter the retained set, a reproducible analysis should record the exact software options and the order in which they are applied. The conceptual distinction remains: LD determines redundancy among candidates, whereas the GWAS p-value cutoff determines which retained associations are eligible for the score.

Worked Examples

Why Clumping Helps, and What It Does Not Do

Clumping is motivated by the dependence among genotypes in LD. To see the role of that dependence, let \(G_j\) and \(G_k\) denote two genotype dosages and consider their weighted contribution \(G_j\hat{\beta}_j+G_k\hat{\beta}_k\). The variance of this two-variant sum is

This follows by expanding the variance of a sum: for random variables \(A\) and \(B\), \(\operatorname{Var}(A+B)=\operatorname{Var}(A)+\operatorname{Var}(B)+2\operatorname{Cov}(A,B)\); multiplying a random variable by a constant multiplies its variance by the square of that constant and its covariance by the product of the constants. When genotypes are correlated, the covariance term is not zero. Including both variants can therefore make their contributions partly redundant, depending on the signs and magnitudes of their weights. Clumping reduces this redundancy by selecting fewer correlated variants.

Pruning is not the same as estimating jointly adjusted effects, and it does not establish that an index variant is causal. The index is selected by a ranking rule, not by proof that it is the biological source of an association. A pruned variant may contain predictive information not fully represented by the index, while a retained variant may have a weak or non-replicating association. C+T trades a simple, interpretable selection rule for the possibility of discarding useful signal.

Assumptions and Interpretation

The procedure depends on several practical conditions. Variant identifiers and allele labels must be aligned across files, and the effect allele used to define \(\hat{\beta}_j\) must match the allele counted in \(G_{ij}\). Otherwise, the sign or magnitude of a contribution can be wrong. The LD reference must provide usable genotypes for candidate variants, and its estimated correlations are used as a proxy for the relevant LD structure. The chosen reference and settings are therefore part of the score definition, not incidental software details.

The GWAS effect estimates are used as weights, but the resulting score is not automatically calibrated as a predicted phenotype. Its predictive performance must be assessed in a suitable target or validation sample. In particular, a p-value cutoff is a selection rule, not a probability that a variant is causal or a guarantee that the selected variant will improve prediction. A score built from a different threshold can contain a different set of variants and have a different scale.

C+T is useful as a transparent baseline and can be evaluated across prespecified p-value thresholds. Its simplicity makes the construction easy to audit: one can report the candidate data, the LD pruning rule, the p-value cutoff, the retained variants, and the weights. The method does not, however, account for all LD structure through a joint model, nor does it shrink the retained GWAS effects as a regularized estimator would.

Common Mistakes

• Using p-values as score weights. A p-value ranks evidence against a null association; it is not an effect estimate. The score uses genotype dosage multiplied by a compatible GWAS effect estimate.
• Confusing clumping with thresholding. LD pruning removes correlated candidates according to an LD rule. The p-value threshold selects retained variants according to GWAS evidence. These are separate criteria.
• Ignoring effect-allele alignment. A dosage counted for the other allele cannot be multiplied by the original effect estimate without recoding. If the dosage is changed to count the opposite allele, the effect estimate's sign must also be changed.
• Calling the index variant causal. The smallest-p-value variant in a clump is the representative chosen by the algorithm. That choice alone does not identify a causal variant.
• Treating a chosen threshold as universally correct. A cutoff defines a particular score; it does not establish that its included variants are all predictive or that excluded variants have no effect.