Dosage Regression For Copy-Number Association

Why Regress on Copy-Number Dosage?

A copy-number variant can leave an individual with fewer or more copies of a genomic segment than the usual reference state. If the scientific question is whether the trait changes systematically as copy count changes, dosage regression represents copy number numerically and estimates a change per copy. This is a specific modeling choice: it summarizes the association with a single slope rather than estimating a separate effect for every copy-number state.

For example, a locus might have observed copy counts of 1, 2, and 3. A linear dosage model assumes that the expected quantitative trait changes by the same amount from 1 to 2 copies as from 2 to 3 copies, after accounting for included covariates. A logistic dosage model makes the corresponding equal-step assumption on the log-odds scale. The data may or may not support that simple pattern; the coefficient should be interpreted as the effect specified by the model, not as proof that biology follows it exactly.

Sometimes dosage is coded as copy count minus a reference count, such as \(d_i=c_i-2\) when two copies are the reference state. This changes the zero point and the intercept, but not the per-copy slope: subtracting the same constant from every predictor value leaves pairwise dosage differences unchanged. The coding direction does matter. A coefficient for copy count describes the change per additional copy; a coefficient for number of copies lost has the opposite direction.

Linear Regression for a Quantitative Trait

For a quantitative outcome, a standard dosage model with covariates is

Here, \(z_{i1},\ldots,z_{iq}\) are included covariates, \(\beta_0\) is the intercept, \(\beta_D\) is the dosage coefficient, and \(\epsilon_i\) is the residual. The conditional mean model is \(E(Y_i\mid d_i,z_{i1},\ldots,z_{iq})=\beta_0+\beta_Dd_i+\sum_{j=1}^{q}\gamma_jz_{ij}\). Thus, holding the included covariates fixed, a one-copy increase corresponds to a modeled change of \(\beta_D\) units in the mean outcome.

Ordinary least squares estimates the coefficients by choosing values that minimize the sum of squared residuals. For the usual coefficient standard errors and tests, observations should be independent, the conditional mean should be correctly specified, and residual variance should be constant across predictor values. Normal residuals are additionally needed for exact small-sample \(t\)-based inference; with larger samples, conventional inference is often justified approximately. In genetic data, relatedness, population structure, and technical differences between batches can violate assumptions or create confounding if not appropriately addressed.

Why the Linear Slope Has a Per-Copy Meaning

The per-copy interpretation follows directly from the model. Consider two individuals with the same covariate values, but dosages \(d\) and \(d+1\). Their fitted conditional means differ by

The intercept and covariate terms cancel, leaving \(\beta_D\). More generally, a contrast between dosage values \(d_a\) and \(d_b\) is \(\beta_D(d_b-d_a)\). This is why subtracting a reference copy count from every dosage changes the intercept but not the per-copy effect: \(d_b-d_a\) is unchanged.

In a simple regression with an intercept and no covariates, the slope formula used in the example can also be obtained by minimizing the sum of squared residuals. For \(S(\beta_0,\beta_D)=\sum_i(Y_i-\beta_0-\beta_Dd_i)^2\), setting the derivative with respect to \(\beta_0\) to zero gives \(\hat{\beta}_0=\bar{Y}-\hat{\beta}_D\bar{d}\). Substituting this into the derivative with respect to \(\beta_D\), setting that derivative to zero, and rearranging gives:

The denominator must be positive: if every participant has the same dosage, there is no dosage variation from which to estimate a slope. With covariates, the fitted coefficient is estimated while accounting for those predictors; the basic interpretation as a conditional per-copy change remains.

Logistic Regression for a Case-Control Outcome

When the outcome is binary, let \(Y_i=1\) indicate a case and \(Y_i=0\) a control. Logistic regression models the conditional probability of being a case, \(P(Y_i=1\mid d_i,z_i)\), using the logit link:

The model assumes that the log odds change linearly with dosage, conditional on the covariates, and that observations are independent Bernoulli outcomes given their predictors. The fitted model also requires enough information to estimate its coefficients; for example, perfect separation can prevent ordinary maximum-likelihood estimates from being finite. In a case-control study, sampling depends on case status. Under standard case-control sampling in which selection within case and control groups does not additionally depend on dosage, the logistic slope estimates the disease odds ratio association, while the fitted intercept generally does not estimate disease prevalence in the source population.

Coefficient Inference and a Covariate-Adjusted Contrast

Regression software reports an estimated coefficient and its standard error. For a linear model, a common test of no dosage association uses \(H_0:\beta_D=0\) against \(H_a:\beta_D\ne0\), with a test statistic formed as \(\hat{\beta}_D/SE(\hat{\beta}_D)\). Under the usual linear-model assumptions, its reference distribution is a \(t\) distribution with residual degrees of freedom; large-sample approximations are also common. For logistic regression, the same null can be assessed with a Wald statistic or a likelihood-based procedure, using the relevant model-based reference distribution. A small p-value measures how unusual the observed statistic would be under the null and the model assumptions; it is not the probability that the null is true.

Model Checks, Interpretation, and Common Pitfalls

Dosage regression is useful because a single coefficient can summarize an association across copy-number states and make contrasts easy to report. That compact summary relies on a trend assumption. A linear quantitative model imposes equal changes in the expected trait for each additional copy. A logistic model imposes equal changes in log odds, or equivalently a constant per-copy odds ratio. If the effect differs sharply among states, one slope may be misleading. The fitted model should therefore be considered alongside observed state frequencies, outcome patterns, and relevant diagnostics.

The copy-number variable must also be coded and described clearly. If the predictor is absolute copy count, the coefficient is per additional copy. If it is coded as a deviation from two copies, the same slope still describes a one-copy increase, but the intercept now refers to the reference count. If the predictor instead counts copies lost, a positive coefficient means something different. Report both the coding and the unit of the outcome so that the direction and scale of the association are unambiguous.

When dosage values are uncertain and represented by expected copy counts, a regression coefficient describes change per unit of that numerical dosage under the fitted model. Treating an expected dosage as an observed predictor does not automatically account for all genotype uncertainty; the interpretation and uncertainty estimate depend on how that dosage was obtained and modeled. Technical artifacts, population structure, or unmodeled dependence among participants can likewise distort association estimates, so a precise regression coefficient is not by itself evidence of a biological effect.

• Mistake: interpreting a logistic coefficient as a probability change. A coefficient of \(0.6931\) is not a 69.31 percentage-point increase in disease probability. Its exponent, \(\exp(0.6931)=2\), is the per-copy odds ratio under the model.
• Mistake: treating the slope as causal. Regression estimates an association conditional on included predictors. Confounding, selection, or technical bias can remain; causal language needs additional design and assumptions.
• Mistake: assuming dosage coding is irrelevant without checking direction. Shifting the zero point preserves the slope, but reversing the predictor from copy count to copies lost reverses its sign.
• Mistake: assuming a per-copy slope fits every copy-number pattern. The model enforces a constant increment on its outcome scale. If that constraint is unsuitable, the single coefficient may conceal important differences between states.