Summary-Statistic Penalization With lassosum

Why Penalize Summary-Statistic Effects?

A marginal GWAS effect describes the association of one variant with a phenotype at a time. When variants are correlated through linkage disequilibrium (LD), these marginal effects are not generally the joint effects that would be obtained by fitting all variants together. Using marginal estimates directly can therefore give redundant or unstable contributions to a polygenic score.

Lassosum addresses this by combining GWAS summary associations with an LD matrix from a reference panel. It estimates joint effects by minimizing a penalized regression objective that includes the LD structure. Its penalty can both shrink effects toward zero and set some effects exactly to zero.

The formulation below uses standardized genotype predictors and a continuous, standardized phenotype. Variants must be aligned to the same effect allele in the summary statistics and reference data. As in the earlier material on choosing and auditing an LD reference panel, the reference should represent the ancestry and variant set of the discovery GWAS well enough for its correlations to be informative.

Definition: Let $R$ be the $m \times m$ LD correlation matrix for $m$ variants, and let $r$ be the vector of their marginal associations with the phenotype, expressed on compatible standardized scales. Lassosum estimates a vector of joint effects $\beta$ by minimizing $$ Q(\beta)=\frac{1}{2}\beta^{T}R\beta-r^{T}\beta +\lambda\left[s\|\beta\|_{1}+\frac{1-s}{2}\|\beta\|_{2}^{2}\right]. $$ Here $\|\beta\|_{1}=\sum_{j=1}^{m}|\beta_j|$, $\|\beta\|_{2}^{2}=\sum_{j=1}^{m}\beta_j^2$, $\lambda \geq 0$ is the overall penalty strength, and $0\leq s\leq1$ sets the balance between the L1 and ridge penalties.

In this convention, $s=1$ gives the lasso penalty, which can set effects exactly to zero; $s=0$ gives a ridge penalty, which shrinks effects but ordinarily does not make them exactly zero. Intermediate values combine the two. The overall penalty $\lambda$ controls how strongly the chosen combination is applied. Thus, $s$ is not itself the overall amount of shrinkage: both parameters matter.

Where the Objective Comes From

Suppose, for motivation, that a discovery sample has standardized genotype columns in a matrix $X$ and a standardized phenotype vector $y$. If $X^{T}X/n=R$ and $X^{T}y/n=r$, the squared-error regression criterion, apart from a constant that does not depend on $\beta$, is

\frac{1}{2n}\|y-X\beta\|_{2}^{2} =\text{constant}+\frac{1}{2}\beta^{T}R\beta-r^{T}\beta.

Adding an elastic-net penalty to this criterion yields the lassosum objective. The key practical change is that the regression part can be constructed from summary associations and an LD matrix rather than requiring the discovery genotypes and phenotypes themselves. This is an approximation when $R$ comes from an external reference: the reference LD must be sufficiently close to the discovery-sample LD for the quadratic term to represent the relevant correlations.

This derivation assumes compatible scaling. In the standardized setup, a marginal association $r_j$ is the covariance between standardized genotype $j$ and the standardized phenotype; it is also the marginal regression coefficient. Raw GWAS effects per allele are not automatically on that scale. They must be converted consistently with genotype and phenotype scales before being used in this form of the objective.

How the Penalty Produces Shrinkage and Sparsity

The quadratic term accounts for LD, while the penalty discourages large or numerous effects. A useful way to see the distinct roles of $\lambda$ and $s$ is to examine the optimality condition for one coefficient. For the lasso case $s=1$, holding the other coefficients fixed, define the residual association for variant $j$ as $c_j=r_j-\sum_{k\ne j}R_{jk}\beta_k$. The part of the objective that depends on $\beta_j$ is $\frac{1}{2}\beta_j^2-c_j\beta_j+\lambda|\beta_j|$. At zero, the absolute-value penalty has a range of possible slopes, so zero is optimal when the residual association is not large enough to overcome the penalty.

\widehat{\beta}_j=0 \quad\Longleftrightarrow\quad |c_j|\leq\lambda \qquad\text{(lasso, conditional on the other coefficients).}

This condition explains why larger $\lambda$ tends to produce more zero coefficients. For a single variant with $R=1$, the minimizer is the soft-thresholded association:

\widehat{\beta} =\operatorname{sign}(r)\max(|r|-\lambda s,0)\, \big/\,[1+\lambda(1-s)].

When $s=1$, this is the familiar lasso threshold. When $s<1$, the L1 part applies a threshold of $\lambda s$, and the ridge part adds a denominator that shrinks the surviving effect. In a multivariant problem, LD couples the coefficients, so this single-variant expression does not give each fitted effect independently; the full objective must be minimized jointly.

For $\lambda>0$ and $s<1$, the ridge component adds positive curvature to the objective, helping make the solution unique even when the LD matrix is singular. With $s=1$, the objective remains convex if $R$ is positive semidefinite, but a unique solution is not guaranteed when predictors are perfectly collinear. In practice, solutions are computed numerically; these properties describe the objective, not a guarantee that the summary statistics or reference LD are appropriate.

Worked Example: Two Correlated Variants With a Lasso Penalty

Consider a hypothetical pair of standardized variants with LD correlation $0.60$ and marginal associations $r_1=0.30$ and $r_2=0.25$. Thus $R=\begin{pmatrix}1&0.60\\0.60&1\end{pmatrix}$. Use $s=1$ and $\lambda=0.18$. The objective is $\frac12\beta^TR\beta-r^T\beta+0.18(|\beta_1|+|\beta_2|)$.

Check the candidate $\widehat{\beta}_1=0.12,\widehat{\beta}_2=0$. For the first variant, the lasso solution with the second coefficient fixed at zero is $r_1-\lambda=0.30-0.18=0.12$, which is positive. For the second variant, the residual association after accounting for the first is $c_2=r_2-R_{21}\widehat{\beta}_1=0.25-(0.60)(0.12)=0.178$. Because $|c_2|=0.178\leq0.18$, the zero-coefficient condition holds. The candidate therefore satisfies the lasso optimality conditions and is a solution.

Although both marginal associations are positive, the fitted effect for variant 2 is exactly zero at this penalty. Its association is largely accounted for by its correlation with variant 1. This is a penalized joint estimate, not evidence that variant 2 has no biological effect.

Worked Example: Changing the L1–Ridge Balance

Use the same hypothetical $R$ and $r$, but take $\lambda=0.20$ and $s=0.50$. The L1 coefficient is $\lambda s=0.10$, and the ridge derivative contributes $\lambda(1-s)\beta_j=0.10\beta_j$. If both effects are positive, the optimum solves

\begin{pmatrix}1.1&0.60\\0.60&1.1\end{pmatrix} \begin{pmatrix}\beta_1\\\beta_2\end{pmatrix} = \begin{pmatrix}0.30-0.10\\0.25-0.10\end{pmatrix} = \begin{pmatrix}0.20\\0.15\end{pmatrix}.

The determinant is $(1.1)(1.1)-(0.60)^2=0.85$. Solving gives $\beta_1=[(1.1)(0.20)-(0.60)(0.15)]/0.85=0.1529$, and $\beta_2=[(1.1)(0.15)-(0.60)(0.20)]/0.85=0.05294$. Both values are positive, as assumed, so this solution is consistent.

Compared with the lasso fit in the previous example, the second variant now has a nonzero effect. The ridge component distributes the fit more continuously across correlated variants, while the smaller L1 coefficient makes the threshold for zeroing a coefficient lower. This illustrates why lowering $s$ generally reduces sparsity; it does not mean the overall penalty has disappeared.

Worked Example: Converting an Effect to a Standardized Association

Suppose a hypothetical GWAS reports a marginal effect of $0.12$ phenotype units per additional effect-allele copy. In the discovery sample, the genotype dosage has standard deviation $0.70$ allele copies and the phenotype has standard deviation $2.0$ units. Under linear scaling, the standardized marginal association is

r_j=0.12\frac{0.70}{2.0}=0.0420.

The units cancel: $0.12$ phenotype units per allele copy, multiplied by $0.70$ allele copies and divided by $2.0$ phenotype units, gives a dimensionless standardized association. This calculation does not itself estimate a joint effect; it shows why the GWAS association and LD matrix must be placed on compatible scales before fitting the penalized objective. For binary phenotypes or other GWAS effect scales, use an appropriate method-specific conversion rather than treating an odds ratio as a standardized linear coefficient.

Assumptions, Diagnostics, and Alternatives

Lassosum’s result depends on both the summary associations and the LD reference. The method is most interpretable when the variants and effect alleles are matched, the association statistics have compatible scaling, and the reference LD adequately represents the discovery population. It also relies on the summary associations and LD matrix being sufficiently coherent to describe one underlying regression problem. These conditions are not guaranteed merely because the optimizer returns coefficients.

Useful checks include confirming that the reference contains the intended variants, verifying allele alignment and effect directions, and checking that the LD matrix has correlations in the expected range and is numerically usable. If results change sharply under small changes in the reference or variant set, that can signal instability or LD mismatch. Such a change is a prompt to investigate, not proof of a particular cause. A severe mismatch can make the fitted joint effects misleading even if the optimization itself converges.

Common failure modes include feeding raw per-allele effects into a standardized objective without conversion, using an LD panel with a different ancestry structure, and interpreting a zero lasso coefficient as a definitive null effect. Correlated variants can substitute for one another, so the identity of the selected variant may be unstable while the aggregate predictive signal is less affected. A nonzero penalized coefficient is likewise not a causal estimate: the objective is designed for prediction and shrinkage, not causal identification.

The methods covered earlier in this unit make different modeling choices. Clumping-and-thresholding removes correlated variants and applies a significance threshold rather than jointly fitting effects through an LD quadratic term. LDpred2, PRS-CS, and SBayesR use Bayesian or prior-based formulations to shrink effects; lassosum instead directly minimizes an elastic-net-penalized summary-statistic objective. These approaches differ in assumptions and tuning, and none removes the need for appropriate data alignment and LD information. The most suitable method depends on the data, modeling goals, and performance assessment.

The parameters have a clear qualitative interpretation but do not have a universal best setting. Larger $\lambda$ generally increases shrinkage and, when $s>0$, tends to increase sparsity. Larger $s$ gives the L1 term more influence relative to the ridge term, tending toward a sparser solution; smaller $s$ favors more diffuse ridge-like shrinkage. Numerical effects depend on the scale of the summary statistics, the LD matrix, and the variant set, so a parameter value cannot be interpreted independently of those choices. Parameter selection must be separated from final performance assessment to avoid optimistic evaluation.

Common pitfall: “The marginal association is small, so the joint effect must be zero.” This ignores LD and the penalty’s conditional role. A coefficient is zero under the lasso when its association after accounting for the other fitted variants lies within the penalty threshold; a small marginal statistic alone does not establish that condition.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

What does $s=0$ imply about the penalty, and does it ordinarily produce exact zeros?
In the two-variant lasso example, calculate the second variant’s residual association at the candidate fit and state why its coefficient can be zero.
For one standardized variant with $r=0.07$, $s=1$, and $\lambda=0.05$, calculate the lassosum estimate.
Why is an LD reference from a poorly matched population a concern even if the optimization converges?
State one distinction between lassosum and clumping-and-thresholding.

Solutions. (1) At $s=0$, the penalty is ridge; it shrinks effects but ordinarily does not set them exactly to zero. (2) $c_2=0.25-(0.60)(0.12)=0.178$. Since $0.178\leq0.18$, the lasso zero-coefficient condition is met. (3) Soft thresholding gives $\widehat{\beta}=\max(0.07-0.05,0)=0.020$. (4) The reference correlations may not represent discovery-sample LD, so the quadratic part of the objective can encode the wrong joint structure; convergence does not validate that structure. (5) Lassosum jointly fits penalized effects using the LD matrix, whereas clumping-and-thresholding removes correlated variants and applies a significance threshold.

Summary-Statistic Penalization With lassosum

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