Additive Coding for Quantitative Traits

From Genotype Dosage to a Quantitative-Trait Model

A genotype can be represented by the number of copies it carries of a chosen allele. When the outcome is quantitative, such as a measured concentration or blood pressure, that count can serve as a numerical predictor in a simple linear model. The model’s dosage coefficient summarizes the fitted change in the trait associated with one additional copy of the counted allele.

This is a specific way to represent genotype in a trait model. It keeps the allele-count scale explicit: the coefficient is measured in trait units per allele, rather than being a difference attached to an unspecified genotype label. Its interpretation depends on which allele is counted and on the model being used.

For example, if \(B\) is the counted allele, a genotype with no copies of \(B\) has dosage 0, a genotype with one copy has dosage 1, and a genotype with two copies has dosage 2. This tutorial concerns the fitted linear model for a quantitative outcome; the dosage is an encoding of genotype, not the trait itself.

The Model and Its Coefficient

For \(n\) individuals, the simple additive-coded model is

Here \(Y_i\) is the quantitative trait for individual \(i\), \(G_i\) is that individual’s counted-allele dosage, \(\beta_0\) is the intercept, and \(\beta_1\) is the dosage coefficient. The error \(\varepsilon_i\) is the difference between the observed outcome and the model’s conditional mean. The model specifies

Thus, under this model, \(\beta_1\) describes the change in fitted mean trait value for a one-unit increase in dosage—that is, for one additional counted allele. If \(Y\) is measured in mg/L, then \(\beta_1\) is measured in mg/L per counted allele. The intercept is the fitted mean when dosage is zero.

No Hardy–Weinberg equilibrium assumption is needed to fit this regression. The genotype counts determine which dosage values occur in the sample, but the least-squares calculation itself uses the observed \(G_i\) and \(Y_i\) pairs. A fitted coefficient describes an association in the modeled population or sample; by itself, it does not establish that changing an individual’s genotype would cause a change in the trait.

How Least Squares Estimates the Dosage Coefficient

In simple linear regression with an intercept, least squares chooses \(\hat{\beta}_0\) and \(\hat{\beta}_1\) to minimize the sum of squared residuals, where each residual is \(Y_i-\hat{\beta}_0-\hat{\beta}_1G_i\). Define \(\bar{G}\) as the sample mean dosage and \(\bar{Y}\) as the sample mean trait value. The fitted coefficients are

The numerator is the centered cross-product of dosage and trait; the denominator is the centered sum of squared dosages. The fitted dosage coefficient therefore reflects how the trait varies with dosage relative to the amount of dosage variation. The intercept is then chosen so that the fitted line passes through \((\bar{G},\bar{Y})\).

Worked Examples

Interpreting and Reporting the Result

A clear report identifies the outcome units, the counted allele, the model, and the coefficient’s units. For example: “In the fitted additive-coded linear model, each additional counted \(B\) allele was associated with an estimated 2.5 mg/L higher mean trait value.” If the estimated coefficient were negative, “lower” would replace “higher.” The wording “associated with” is appropriate because the fitted regression alone does not establish causality.

The coefficient depends on the chosen allele-count direction. Counting the other allele reverses the dosage direction in a diploid biallelic setting, so the sign and intercept representation change accordingly; the reported allele must therefore be named. Also, a dosage coefficient is not a genotype label: its units are trait units per one dosage unit. A coefficient without its counted allele and trait units is difficult to interpret.

Common Mistakes

• Calling the coefficient a trait value. The intercept is the fitted value at dosage zero, whereas the slope is a change in fitted mean per additional counted allele.
• Dropping the units. If the outcome is measured in mg/L, the slope is in mg/L per allele, not simply “2.5.”
• Interpreting association as causation. A fitted dosage coefficient describes a model-based association; it does not, on its own, show that the counted allele causes the outcome difference.
• Forgetting which allele was counted. A positive coefficient means higher fitted trait values with increasing dosage of the specified allele, not necessarily with increasing dosage of either allele.
• Assuming every individual changes by exactly the slope. The coefficient describes the fitted conditional mean trend. Individual outcomes can vary around that mean.