Interpreting A Burden Regression Coefficient

What a Burden Coefficient Describes

A burden regression coefficient connects a person’s weighted variant score to an outcome. Its meaning depends on the outcome model and on how the score is defined. A coefficient is not meaningful in isolation: interpretation requires stating the score contrast, the score’s units, the outcome scale, and any covariate conditions.

Definition: For person $i$, a weighted burden score is $$ B_i=\sum_{j=1}^{m} w_jG_{ij}, $$ where the sum is over the $m$ variants in the specified mask, $G_{ij}$ is person $i$'s coded genotype or allele dosage at variant $j$, and $w_j$ is that variant’s fixed weight. The burden score is a single predictor formed by combining the included variants. Its units are determined by the genotype coding and weights; they are not automatically “variants” or “alleles.”

For example, if each $G_{ij}$ is an alternate-allele dosage from 0 to 2 and the weights are dimensionless, $B_i$ is measured in weighted dosage units. A one-unit increase in $B_i$ can arise from different changes in genotype, depending on the weights. If the weights themselves have a particular scale or unit, that also affects the score’s units. Interpretation should therefore refer to a specified change in the defined score, rather than calling it simply “one more variant.”

Coefficient on a Quantitative-Trait Scale

For a quantitative outcome $Y_i$, a common adjusted linear model is

Y_i=\beta_0+\beta_BB_i+\boldsymbol{\gamma}^{\mathsf T}\mathbf{C}_i+\varepsilon_i,

where $\mathbf{C}_i$ contains covariates, $\beta_0$ is the intercept, $\beta_B$ is the burden coefficient, and $\varepsilon_i$ is the residual. For the usual interpretation, the conditional mean of $Y_i$ is linear in the burden and included covariates, with errors having conditional mean zero. Standard errors also rely on appropriate assumptions for the chosen inference method, such as independent observations and, for conventional model-based standard errors, a suitable residual variance model.

Interpretation: Holding the modeled covariates fixed, a one-unit higher weighted burden score is associated with an average difference of $\beta_B$ outcome units in the conditional mean of $Y$. For a change of $\Delta B$ score units, the modeled mean difference is $\beta_B\Delta B$. The coefficient describes an association under the fitted model; it does not by itself establish a causal effect.

Worked Example: Quantitative Outcome Difference

In a hypothetical study, three included variants have weights $1.5$, $0.5$, and $2$, and each genotype is coded as alternate-allele dosage. A fitted covariate-adjusted linear model for a biomarker measured in milligrams per liter gives $\hat{\beta}_B=1.35$ milligrams per liter per burden-score unit. Person A has dosages $(1,0,1)$; person B has dosages $(0,2,0)$. What does the model imply for their adjusted mean difference if their modeled covariates are the same?

First calculate the scores:

B_A=(1.5)(1)+(0.5)(0)+(2)(1)=3.5, \qquad B_B=(1.5)(0)+(0.5)(2)+(2)(0)=1.0.

The score difference is $3.5-1.0=2.5$ burden-score units. The fitted mean difference, A minus B, is

\hat{\beta}_B(B_A-B_B)=(1.35)(2.5)=3.375 \text{ milligrams per liter}.

Thus, under this model and at the same covariate values, A’s predicted conditional mean biomarker level is 3.38 milligrams per liter higher than B’s. This is a comparison of the two weighted scores, not a claim that A has a particular variant whose effect is 3.38 milligrams per liter.

Coefficient on a Log-Odds Scale

For a binary outcome $Y_i$, let $P(Y_i=1\mid B_i,\mathbf{C}_i)$ denote the probability of the outcome given the burden and covariates. A logistic regression model is

\log\left( \frac{P(Y_i=1\mid B_i,\mathbf{C}_i)} {1-P(Y_i=1\mid B_i,\mathbf{C}_i)} \right) = \beta_0+\beta_BB_i+\boldsymbol{\gamma}^{\mathsf T}\mathbf{C}_i.

This model assumes that the log odds are linear in the burden score and included covariates, and that the binary outcomes are appropriately modeled as conditionally independent Bernoulli observations. The coefficient $\beta_B$ is measured in log-odds units per one burden-score unit. Exponentiating it gives the odds ratio for a one-unit score increase, conditional on the modeled covariates. For a change of $\Delta B$, the odds ratio is $\exp(\beta_B\Delta B)$. An odds ratio is not a probability difference or a risk ratio.

Worked Example: Translating a Log-Odds Coefficient

In a hypothetical case-control analysis, two variants have weights 1 and 2, and dosage is coded from 0 to 2. A fitted logistic regression gives $\hat{\beta}_B=-0.28$ log-odds units per burden-score unit. Person L has dosages $(1,0)$, giving $B_L=(1)(1)+(2)(0)=1$. Person H has dosages $(0,2)$, giving $B_H=(1)(0)+(2)(2)=4$. At the same covariate values, what is the modeled odds ratio for H versus L?

The score contrast is $4-1=3$. The change in log odds and corresponding odds ratio are

\hat{\beta}_B(B_H-B_L)=(-0.28)(3)=-0.84, \qquad \exp(-0.84)=0.4317.

The model therefore associates H’s three-unit higher score with odds about 0.432 times L’s odds, or about 56.8% lower odds, conditional on the same covariates. This is not a 56.8% lower probability. For illustration, if L’s modeled odds were 0.20, H’s corresponding odds would be $(0.20)(0.4317)=0.08634$. Those odds correspond to probabilities $0.20/(1+0.20)=0.1667$ and $0.08634/(1+0.08634)=0.0795$, respectively. The probability difference depends on the starting odds.

Why This Is Not an Effect Estimate for Each Variant

The regression uses $B_i$, one combined predictor, rather than estimating a separate coefficient for every included genotype. Consequently, $\beta_B$ describes the modeled association with the aggregate score. It does not report each variant’s own association, nor does it show that all included variants have effects of the same size.

Key distinction: In a model with only the burden predictor, different genotype patterns that produce the same $B_i$ receive the same fitted contribution from that predictor. The burden coefficient is therefore an aggregate, score-dependent quantity, not a list of variant-specific effects.

The restriction can be seen by comparing the burden model with a model containing separate additive genotype terms. If the separate terms have coefficients $\theta_j$, their total contribution is $\sum_j\theta_jG_{ij}$. The burden model’s contribution is $\beta_B\sum_jw_jG_{ij}$. These expressions match under the restriction $\theta_j=\beta_Bw_j$ for every included variant. Fitting the burden model does not separately estimate those $\theta_j$; the proportional relationship is imposed by using the weighted sum.

Worked Example: Same Score, Different Variant Patterns

Consider a hypothetical quantitative-trait model with two included variants, weights $w_1=2$ and $w_2=1$, and fitted burden coefficient $\hat{\beta}_B=0.8$ outcome units per score unit. Person A has dosages $(1,0)$, while person B has dosages $(0,2)$.

Their scores are

B_A=(2)(1)+(1)(0)=2, \qquad B_B=(2)(0)+(1)(2)=2.

Both people have the same burden score, so the model assigns them the same burden contribution, $0.8(2)=1.6$ outcome units relative to a score of zero, when other modeled covariates are held fixed. Yet their genotype patterns differ. The fitted burden coefficient is not an estimated effect of 0.8 outcome units for either variant: one copy of variant 1 changes the score by 2 and the modeled mean by $0.8(2)=1.6$, whereas one copy of variant 2 changes the score by 1 and the modeled mean by $0.8(1)=0.8$. These are consequences of the chosen weights and burden model, not separate variant-effect estimates from this regression.

Score Scaling Changes the Numerical Coefficient

The coefficient’s numerical value depends on the score scale, even when the fitted association and predictions are unchanged. If a score is multiplied by a positive constant $c$, so that $B_i^*=cB_i$, its coefficient becomes $\beta_B^*=\beta_B/c$. The product of coefficient and score remains the same: $\beta_B^*B_i^*=\beta_BB_i$. This is why coefficients from burdens built with differently scaled weights should not be compared as though their units were identical.

Worked Example: Rescaling the Burden

Suppose a hypothetical linear model has a burden coefficient of $1.2$ outcome units per score unit for a score $B$. A contrast from $B=0.2$ to $B=0.7$ is $0.5$ score units, so the modeled mean difference is

(1.2)(0.7-0.2)=(1.2)(0.5)=0.60 \text{ outcome units}.

Now define $B^*=2B$. The same observations have scores $0.4$ and $1.4$, and the coefficient in the rescaled model is $1.2/2=0.60$ outcome units per $B^*$-score unit. The contrast is now 1.0 rescaled unit, giving $(0.60)(1.0)=0.60$ outcome units. The numerical coefficient changed, but the modeled outcome contrast did not.

Common Mistake and Practical Interpretation

A common mistake is to read “the burden coefficient is 1.35” as “each included variant increases the outcome by 1.35,” or to read a logistic coefficient as the same odds change for every possible genotype pattern. The correction is to name the score and its scale, state the score contrast, and translate that contrast using the fitted model. In linear regression, multiply the coefficient by the score difference to obtain a modeled mean difference. In logistic regression, multiply by the score difference on the log-odds scale and exponentiate to obtain an odds ratio.

A complete interpretation also identifies the outcome units or the odds scale and says whether the comparison holds covariates fixed. The association is conditional on the included predictors and the specified mask, genotype coding, and weights. It does not establish that changing a person’s burden would cause the outcome to change, and it does not establish separate effects for the included variants.

State the predictor.
Name the weighted burden, including the genotype coding and the weights or score scale needed to understand its units.

Identify the outcome model.
For a quantitative outcome, use outcome units; for a binary outcome modeled by logistic regression, use log odds and odds ratios.

Specify the contrast.
Say whether the interpretation is per one score unit or for a stated difference in burden, and indicate which covariates are held fixed.

Keep the claim at the aggregate level.
Describe the modeled association with the combined score; do not present the burden coefficient as a separately estimated effect for each variant.

Check Your Understanding

Use the outcome model and the score definition when answering each question.

A quantitative-trait burden coefficient is 0.6 millimeters per score unit. What modeled mean difference corresponds to a 2.5-unit higher score, with covariates held fixed?
A logistic burden coefficient is 0.2 per score unit. Write the odds ratio for a four-unit increase and calculate its approximate value.
Why is it incomplete to say that a burden coefficient is “the effect of the variants” without describing the score?
If every burden score is multiplied by 5, how does the coefficient change in a model with the same fitted predictions?
Two people have different variant genotypes but the same weighted score. What does a regression using only that score predict about their burden contributions, holding covariates fixed?