Opposing Copy-Number Effects In Burden Analyses

Why Direction Matters

A burden analysis compresses information across rare structural variants into one or more subject-level summaries. This can be useful when individual events are uncommon, but compression is consequential: if deletions and duplications affect a phenotype in opposite directions, a single unsigned total can obscure both effects. A weak pooled association therefore does not, by itself, establish that neither class is associated with the phenotype.

The choice of summary should reflect the scientific model. A pooled burden asks whether a unit of deletion burden and a unit of duplication burden have the same effect. Separate directional burdens allow those effects to differ. A signed dosage summary instead assumes that the two effects are equal in magnitude and opposite in sign. These are distinct assumptions, not interchangeable ways of writing the same analysis.

Three Models, Three Questions

For a continuous phenotype, a linear model with separate directional burdens can be written as

Here \(Y_i\) is the phenotype, \(\mathbf{C}_i\) contains included covariates, and \(\varepsilon_i\) is the residual. The coefficients \(\beta_D\) and \(\beta_U\) describe the conditional change in mean phenotype per unit of deletion and duplication burden, respectively, holding the other model terms fixed. Their units are phenotype units per burden unit. Interpreting these coefficients as associations does not establish that the variants cause the phenotype.

An unsigned pooled model uses \(B_i\) in place of the two directional burdens. It assigns the same coefficient to a unit of either class: algebraically, it constrains \(\beta_D=\beta_U\). A signed model uses \(S_i=U_i-D_i\) and assigns the coefficient \(\beta_S\) to that contrast. Written in terms of \(D_i\) and \(U_i\), this constrains \(\beta_D=-\beta_U\), with \(\beta_D=-\beta_S\) and \(\beta_U=\beta_S\). Thus, a signed summary is appropriate only when equal-and-opposite effects per coded unit are a plausible restriction.

For a binary phenotype, analogous models can be fitted on a specified link scale, such as the log-odds scale in logistic regression. The coefficients then refer to that link scale, not directly to a change in probability. Opposing effects may still be obscured by pooling, but the simple linear averaging calculations below should not be transferred directly to a nonlinear link scale.

How an Unsigned Burden Can Cancel Effects

The cancellation is easiest to see when carriers have one burden unit, deletion and duplication carriers are distinct, and there are no covariates. Let \(B_i\) indicate any carrier, \(D_i\) indicate a deletion carrier, and \(U_i\) indicate a duplication carrier, so \(B_i=D_i+U_i\). In a linear model with an intercept, the coefficient for this binary pooled indicator is the difference between the mean phenotype among carriers and the mean among noncarriers.

Suppose, in this simple setting, the deletion and duplication carrier groups have mean differences \(\beta_D\) and \(\beta_U\) from the same noncarrier mean. If a fraction \(\pi\) of carriers have deletions and \(1-\pi\) have duplications, the pooled carrier difference is

This is a mixture of the two directional differences. If the terms have opposite signs and similar weighted magnitudes, the mixture can be close to zero. With more general burdens, covariates, or correlations between predictors, the pooled coefficient need not be this simple weighted average. In the intercept-only linear model with \(B=D+U\), its slope is \(\operatorname{Cov}(B,Y)/\operatorname{Var}(B)\), so the association depends on the distributions and covariances of the burdens as well as on their effects.

What a Signed Summary Assumes

The sign convention in \(S_i=U_i-D_i\) follows copy-number deviation from the reference state: deletion burden is negative and duplication burden is positive. A regression on \(S_i\) can retain a contrast that an unsigned total removes, but it does not allow arbitrary directional effects. It assumes that a one-unit deletion contribution and a one-unit duplication contribution have opposite effects of the same size on the model's outcome scale.

Separate burdens are more flexible: they estimate \(\beta_D\) and \(\beta_U\) independently. The signed model is a constrained alternative that may be more efficient when the equal-and-opposite assumption is appropriate, but can misrepresent effects when the magnitudes differ. A signed coefficient should be interpreted per signed burden unit, not as a separate estimate of the deletion effect or duplication effect.

Worked Comparison With Weighted Burdens

Weights do not remove the need to choose a directional model. They define how much each eligible event contributes to a burden. To interpret a signed summary, deletion and duplication contributions should be placed on a meaningful common scale; otherwise, subtracting one weighted burden from the other may have no coherent dosage interpretation. Even with commensurate units, the signed model still imposes equal-and-opposite effects per unit.

Choosing and Interpreting the Analysis

The model should be chosen to match the effect pattern the analysis is intended to detect. Separate burdens are appropriate when deletion and duplication effects may differ in direction or magnitude. A signed dosage summary is defensible when opposite, equal-magnitude effects per coded unit are scientifically plausible. An unsigned pooled burden is suitable when a common-direction, common-effect-per-unit assumption is meaningful. These choices concern the association model; they do not replace the usual checks on variant eligibility, covariate specification, phenotype definition, or calibration.

In a linear model, a joint test of \(H_0:\beta_D=\beta_U=0\) asks whether either directional burden contributes to the conditional mean under the specified model. It differs from testing a single pooled coefficient or the signed coefficient: each tests a different restriction or contrast. Estimates and uncertainty should be reported on their model scale with burden and phenotype units stated. If many directional or alternative summaries are tested, the analysis should also account for the resulting multiplicity.

Classical linear-regression standard errors and tests require appropriate assumptions, including a correctly specified conditional mean and independent errors; the usual homoskedastic standard errors additionally assume constant residual variance. If those assumptions are not suitable, inference needs a method appropriate to the sampling and variance structure. A small point estimate alone is not evidence of a precise null result: uncertainty matters.

Common Mistakes

• Wrong reasoning: “The pooled burden coefficient is near zero, so neither variant class is associated.” Correction: Opposite directional effects can cancel. Inspect the separate estimates or use a signed contrast only if its restriction is justified.
• Wrong reasoning: “A signed burden estimates the deletion and duplication effects separately.” Correction: It estimates one coefficient under the constraint that the effects are equal in magnitude and opposite in sign per coded unit.
• Wrong reasoning: “The pooled coefficient is always the average of the two class-specific coefficients.” Correction: That simple mixture applies in a restricted binary-carrier setting. With varying burden values, covariates, or correlated predictors, the pooled slope also depends on the predictor distributions and covariances.
• Wrong reasoning: “Opposite association directions show that copy-number changes cause opposite phenotype changes.” Correction: Regression coefficients describe conditional associations under the specified model; causal claims require additional design and assumptions.