Deriving the Clumping-and-Thresholding Score

From GWAS Results to a Score

A polygenic risk score combines genetic variants into one weighted quantity. In clumping-and-thresholding (C+T), the weights come from GWAS estimates, while two filters determine which variants contribute: clumping removes some variants that are strongly correlated through linkage disequilibrium (LD), and thresholding retains variants whose GWAS p-values are at or below a chosen cutoff. The resulting score is simple to compute, but its weights are not generally the causal effects of the retained variants.

Use $j=1,\ldots,m$ to index variants and $i$ to index individuals. Let $z_{ij}$ be individual $i$'s centered, standardized genotype at variant $j$, and let $\beta_j$ be its additive effect in outcome units per one-standard-deviation increase in genotype. If the outcome was standardized, these effects and the score are on that standardized outcome scale; standardizing genotypes alone does not standardize the outcome.

Definition: Let $C$ be the set of variants retained by a specified clumping procedure, and let $p_j$ be the GWAS p-value for variant $j$. For a p-value threshold $t$, the C+T weight is $w_j=\hat b_j$ if $j\in C$ and $p_j\leq t$, and $w_j=0$ otherwise. The score for individual $i$ is $$ \mathrm{PRS}_i(t)=\sum_{j=1}^{m} z_{ij}w_j =\sum_{j\in C:\,p_j\leq t}z_{ij}\hat b_j. $$ Here $\hat b_j$ is the GWAS effect estimate for the specified allele and genotype scale. The allele coding, genotype scale, and outcome scale must match between the GWAS estimates and the genotypes used to calculate the score.

A common clumping rule orders variants by GWAS p-value, retains the most significant available variant, and removes nearby variants whose LD with that lead variant exceeds a chosen $r^2$ cutoff. The procedure then considers the remaining variants in order. The genomic window, LD reference, $r^2$ cutoff, and tie-handling rule are part of the definition of $C$; different choices can produce different scores. Thresholding is applied to the GWAS p-values, usually after clumping. Clumping does not mean that all retained variants are independent.

Model and Interpretation of the GWAS Weights

To see what the weights represent, consider an idealized additive model in a discovery population:

Y_i=\sum_{j=1}^{m}z_{ij}\beta_j+\varepsilon_i.

Assume the genotypes and outcome have been centered, effects are additive on the chosen outcome scale, and the residual $\varepsilon_i$ has conditional mean zero given the genotypes. For the simplest derivations, also assume the discovery and target populations have the same relevant LD structure, there is no population structure or other confounding of the GWAS associations, and the discovery estimates are obtained independently of the target individuals. Real analyses may adjust for covariates; the idealized equations below then describe the appropriately adjusted or residualized quantities.

Let $R$ denote the genotype correlation matrix, so $R_{jk}=\operatorname{Cov}(z_j,z_k)$ for standardized genotypes. In a marginal GWAS, the population regression coefficient for variant $j$, when other variants are not included in that regression, is

b_j^{\mathrm{marg}}=\operatorname{Cov}(z_j,Y) =\sum_{k=1}^{m}R_{jk}\beta_k, \qquad \mathbf b^{\mathrm{marg}}=R\boldsymbol\beta.

The equality uses $\operatorname{Var}(z_j)=1$, the additive model, and a residual uncorrelated with the genotypes. Thus, with LD, a marginal estimate can reflect effects at correlated variants as well as the effect at its own site. The observed estimate $\hat b_j$ also contains sampling error; its p-value is calculated from the observed estimate and its standard error under the GWAS test, not from the unknown true effect.

Worked Example: Applying Clumping and a P-Value Cutoff

Suppose a hypothetical GWAS reports four variants. Assume a two-sided normal test, and suppose the clumping rule scans the variants from smallest to largest p-value, removing a later variant if its $r^2$ with an already retained lead exceeds $0.20$. Variant A and B have $r^2=0.64$; C and D have $r^2<0.20$ with all retained variants. Use $t=0.05$.

Variant	$\hat b_j$	$SE$	$\hat b_j/SE$	Two-sided p-value
A	0.40	0.08	5.00	about $5.73\times10^{-7}$
B	0.32	0.10	3.20	about 0.00137
C	0.20	0.10	2.00	about 0.0455
D	-0.15	0.10	-1.50	about 0.134

For example, A's test statistic is $0.40/0.08=5.00$, giving $p=2P(Z\geq5.00)$, where $Z\sim N(0,1)$. The other p-values are calculated from their displayed observed estimate-to-standard-error ratios in the same way. A is considered first and retained. B is then removed because $r^2=0.64>0.20$ with A. C and D are not removed by this LD rule, so the clumped set is $C=\{A,C,D\}$. Applying $p_j\leq0.05$ retains A and C but not D. The score is therefore $$ \mathrm{PRS}_i=0.40z_{iA}+0.20z_{iC}. $$

If a hypothetical target individual has $z_{iA}=0.5$ and $z_{iC}=1.2$, the score is $0.40(0.5)+0.20(1.2)=0.20+0.24=0.44$ outcome units. These units are those represented by the GWAS weights; they are standardized outcome units only if the GWAS outcome was standardized.

Why the Filters Affect Bias and Variance

The central trade-off can be stated precisely under the additive model. For a target individual, write the true modeled genetic component as $G_i=\mathbf z_i^\mathsf{T}\boldsymbol\beta$, and the score as $S_i=\mathbf z_i^\mathsf{T}\mathbf w$. If target genotypes are centered and have covariance matrix $R$, then, conditional on the fitted weights,

E\{(S_i-G_i)^2\mid\mathbf w\} =E\{[\mathbf z_i^\mathsf{T}(\mathbf w-\boldsymbol\beta)]^2\mid\mathbf w\} =(\mathbf w-\boldsymbol\beta)^\mathsf{T}R(\mathbf w-\boldsymbol\beta).

The last equality follows by expanding the square and using $E(\mathbf z_i\mathbf z_i^\mathsf{T})=R$. This is prediction error for the genetic component, not total outcome error; independent residual variation in $Y_i$ adds to total squared prediction error. The expression also shows why the effect of a weight cannot always be assessed variant by variant: when genotypes are correlated, the off-diagonal entries of $R$ contribute.

A simple independent-variant case makes the variance cost of noisy estimates visible. Suppose $R=I$, the selected set is fixed independently of estimation noise, and retained estimates are unbiased for their corresponding $\beta_j$, with variances $\tau_j^2$. For the unselected set $U$ and selected set $A$, the expected squared error in estimating the genetic component is

E\{(S_i-G_i)^2\} =\sum_{j\in U}\beta_j^2+\sum_{j\in A}\tau_j^2.

Indeed, with $R=I$, each genotype has variance one, so squared score error averages to the sum of squared weight errors. For $j\in U$, $w_j=0$ and the squared error is $\beta_j^2$. For $j\in A$, $w_j=\hat b_j$, so $E[(\hat b_j-\beta_j)^2]=\tau_j^2$ by unbiasedness. This derivation relies on the stated fixed-selection and independence idealization. In real C+T, selection uses noisy p-values, so retained estimates are selected partly for looking large; the simple expression is a guide to the trade-off, not an exact risk formula for the data-dependent procedure.

Worked Example: A Fixed-Set Bias–Variance Calculation

Consider four independent standardized variants with hypothetical true effects $(0.20,0.10,0.05,0)$ outcome units. Suppose each GWAS estimate has variance $0.04$ outcome-units squared. For illustration, compare a fixed rule retaining all four variants with a fixed rule retaining only the first two. Assume estimates are unbiased and their errors are independent of target genotypes; this calculation is not a claim that a p-value rule would always select exactly those variants.

With all four retained, no true effect is omitted, but the four estimation variances contribute $4(0.04)=0.16$ to expected squared prediction error. With only the first two retained, their variance contribution is $2(0.04)=0.08$. The omitted effects contribute $0.05^2+0^2=0.0025$. Thus the second rule has expected error $0.08+0.0025=0.0825$ outcome-units squared under these assumptions. It has greater omission bias but lower total error in this particular calculation because it avoids two noisy weights. A different signal size or estimation variance could reverse the comparison.

What Clumping Does Under Linkage Disequilibrium

Clumping is intended to limit redundant and potentially noisy contributions from correlated variants. It does not, however, turn marginal estimates into conditional or causal effects. A lead variant can act as a proxy for an unobserved or unretained causal variant. The choice of which proxy survives also depends on the discovery p-values, the LD reference, and the clumping settings.

Worked Example: A Correlated Proxy and a Lead Variant

Suppose two standardized genotypes have correlation $0.80$, and the true additive effects are $\boldsymbol\beta=(0.20,0)$ outcome units. Then $$ R=\begin{pmatrix}1&0.80\\0.80&1\end{pmatrix}, \qquad \mathbf b^{\mathrm{marg}}=R\boldsymbol\beta =\begin{pmatrix}0.20\\0.16\end{pmatrix}. $$

The second variant has a nonzero population marginal association even though its direct effect in this model is zero: $0.80(0.20)+1(0)=0.16$. If the first variant has the smaller observed p-value and the clumping rule removes the second, the idealized lead-variant score uses weight $0.20$ on variant 1. It matches the modeled genetic component $0.20z_1$. If both population marginal weights were used instead, the score would be $0.20z_1+0.16z_2$; its difference from the modeled genetic component is $0.16z_2$, whose mean squared size is $0.16^2\operatorname{Var}(z_2)=0.0256$ outcome-units squared. This example illustrates one setting where removing a correlated proxy avoids adding a redundant marginal weight. It does not establish that clumping always improves prediction: with different effects, sample estimates, or target LD, the retained lead may be a less useful proxy.

Assumptions, Diagnostics, and Alternatives

A C+T score is interpretable only when its inputs are compatible. The GWAS and score genotypes must use the same effect allele and compatible dosage scale, and the GWAS weights must refer to the stated outcome scale. The LD reference should reasonably represent the population in which clumping is performed; substantial ancestry or LD mismatch can change which variants are treated as redundant. The additive model is an approximation, and the score does not automatically account for interactions, nonlinear effects, or differences in allele frequencies and LD across populations.

Useful checks include confirming that the clumped list follows the documented p-value ordering and LD rule; checking for unexpected duplicate or highly correlated retained variants; verifying that retained weights and allele directions agree with the harmonized GWAS data; and confirming that the score has the intended scale. If many retained variants remain strongly correlated, the LD rule may not have had the intended effect. If a small change in the reference panel or clumping settings changes the retained lead variants substantially, the resulting score may be unstable. Such findings warrant investigation rather than automatic interpretation as evidence that one setting is best.

C+T is computationally transparent and uses only marginal GWAS estimates plus an LD reference. Using all marginal estimates without either filter avoids hard exclusions but can add noise and correlated proxy weights. More LD-aware approaches model or shrink weights jointly rather than relying only on hard pruning; they require additional assumptions and estimation choices. These approaches are alternatives, not a guarantee of better prediction in every sample. In any approach, performance must be assessed in data separate from the data used to choose score settings. Choosing the p-value threshold is a distinct model-selection task; it should not be judged using a final test set.

Common pitfall: “A variant has a small p-value, so its GWAS estimate must be its causal effect; adding every significant variant must improve the score.” The correction is that a p-value measures evidence against a marginal association null under a specified test. In LD, marginal estimates can reflect neighboring effects, and noisy selected estimates can add variance. Clumping and thresholding control which estimates enter the score; they do not prove causality or guarantee improved prediction.

Practical Construction Sequence

Specify compatible inputs.
Confirm allele direction, genotype scale, outcome scale, discovery estimates, and an LD reference appropriate to the clumping population.

Define the clumping rule.
State the genomic window, $r^2$ cutoff, p-value ordering, and reference panel; apply the rule to obtain $C$.

Apply the p-value cutoff.
For the specified threshold $t$, retain only $j\in C$ with $p_j\leq t$; assign each retained variant its compatible GWAS estimate $\hat b_j$.

Form and audit the score.
Compute the weighted genotype sum, verify its scale and allele directions, and evaluate predictive performance in independent data.

Key takeaway: C+T estimates a score by assigning marginal GWAS weights to variants that survive both an LD-based clumping rule and a p-value cutoff. Pruning and thresholding can reduce variance from redundant or weakly supported estimates, but may omit predictive signal. Their consequences depend on effect sizes, estimation noise, LD, and the population where the score will be used.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

If $\hat b_j=0.12$ and $SE=0.04$, what observed test statistic is used for a two-sided normal-test p-value? Does it equal the true effect divided by its standard error?
A variant has $p_j=0.03$, but clumping removes it because it is too correlated with an earlier lead variant. Does it contribute to the C+T score at $t=0.05$?
In the independent-variant calculation, what is the expected squared error from omitting a variant with true effect $0.08$, when its genotype variance is one?
Why can a variant with zero direct effect have a nonzero marginal GWAS coefficient when it is correlated with a causal variant?

Solutions. (1) The observed statistic is $0.12/0.04=3.00$. It is computed from the observed estimate; it is not generally the true effect divided by its standard error, because the true effect is unknown. (2) No. The variant is not in the clumped set, so it receives weight zero regardless of passing the p-value cutoff. (3) The squared omission error is $0.08^2(1)=0.0064$ outcome-units squared. (4) Its genotype carries information about the causal genotype through LD, so its marginal coefficient includes the correlated causal effect, as expressed by $\mathbf b^{\mathrm{marg}}=R\boldsymbol\beta$.