Additive Dosage Effects and Per-Allele Slopes

Why Count Alleles?

At a biallelic genetic variant, a diploid individual carries zero, one, or two copies of either allele. If the scientific question is whether the phenotype tends to change with each additional copy of a chosen allele, the genotype can be represented by a single number: its allele-count dosage. This coding gives a direct interpretation to a regression slope: the expected phenotype difference associated with one additional copy of that allele.

The model is called additive because it assumes equal steps in the expected phenotype mean from zero to one copy and from one to two copies. The word describes the assumed pattern of genotype means on the phenotype scale. It does not, by itself, imply that the allele causes the phenotype, nor does it say that every individual’s phenotype changes by exactly the same amount.

Definition: For a biallelic variant in a diploid individual, let $G_i$ be the number of copies of a designated effect allele carried by individual $i$. Then $G_i\in\{0,1,2\}$. The additive dosage model for a quantitative phenotype $Y_i$ is $$ Y_i=\beta_0+\beta_1G_i+\varepsilon_i, $$ where $\beta_0$ is the expected phenotype at dosage zero, $\beta_1$ is the change in expected phenotype per additional effect-allele copy, and $\varepsilon_i$ is the individual-specific deviation from the model mean.

From Genotype to a Per-Allele Slope

The dosage records an allele count, not a ranking of how “good” or “bad” a genotype is. Dosage zero means neither chromosome carries the chosen effect allele; dosage one means one chromosome carries it; dosage two means both do. For example, if the alleles are A and G and G is chosen as the effect allele, the genotypes AA, AG, and GG have dosages 0, 1, and 2, respectively.

The model specifies a conditional mean for each dosage. Its key assumption is that the three means lie on a straight line when plotted against dosage. In particular, each additional copy is associated with the same difference in expected phenotype. Write the three genotype-specific means as $\mu_0$, $\mu_1$, and $\mu_2$, where $\mu_g=E[Y_i\mid G_i=g]$.

\mu_0=\beta_0,\qquad \mu_1=\beta_0+\beta_1,\qquad \mu_2=\beta_0+2\beta_1.

Thus, under this model, the first dosage step has expected difference $\mu_1-\mu_0=\beta_1$, and the second has expected difference $\mu_2-\mu_1=\beta_1$. Comparing the two homozygote dosage groups gives $\mu_2-\mu_0=2\beta_1$. The slope is therefore measured in phenotype units per effect-allele copy.

Model assumption: On the scale used for $Y$, the expected means at dosages 0, 1, and 2 are equally spaced. Equivalently, $$ \mu_1-\mu_0=\mu_2-\mu_1, \qquad\text{or}\qquad \mu_1=\frac{\mu_0+\mu_2}{2}. $$ This is the additive genotype-mean assumption. It is an assumption about expected values, not a claim that individual outcomes are equally spaced or that observed group averages must fall exactly on a line.

A Short Derivation of the Additive Model

The three genotype-specific means can be described without imposing additivity: in general, they are three separate quantities, $\mu_0,\mu_1,\mu_2$. An additive dosage model restricts those quantities by requiring the heterozygote mean to be halfway between the two homozygote means. Once that condition holds, the line through the three means has an intercept and a per-copy slope.

To see this directly, set the dosage-zero mean equal to the intercept, $\beta_0=\mu_0$. The slope from dosage zero to dosage two is the total mean difference divided by the two-copy increase. Therefore,

\beta_1=\frac{\mu_2-\mu_0}{2}.

At dosage one, the model then predicts $\beta_0+\beta_1=\mu_0+(\mu_2-\mu_0)/2=(\mu_0+\mu_2)/2$. That equals $\mu_1$ exactly when the additive mean constraint holds. This establishes both directions: equally spaced genotype means can be written in the dosage model, and the dosage model implies equally spaced genotype means.

The model also has a simple decomposition of the individual outcome: $Y_i$ is its dosage-based expected mean plus $\varepsilon_i$. For the mean interpretation to hold, the deviation must satisfy $E[\varepsilon_i\mid G_i]=0$. This condition says that the model’s mean is correct at each dosage; individual deviations can still be positive or negative. Standard regression-based uncertainty calculations commonly add assumptions such as independent observations and constant conditional error variance, but those concern inference rather than the meaning of the per-allele slope.

Worked Example: Reading the Slope in Context

Suppose a hypothetical study uses a quantitative phenotype measured in millimoles per liter and counts the G allele. The specified additive model is $$ E[Y_i\mid G_i]=4.80-0.12G_i. $$ Find the model mean for each dosage and interpret the slope.

At dosage zero, the expected phenotype is $4.80-0.12(0)=4.80$ mmol/L. At dosage one, it is $4.80-0.12(1)=4.68$ mmol/L. At dosage two, it is $4.80-0.12(2)=4.56$ mmol/L.

The consecutive differences are $4.68-4.80=-0.12$ mmol/L and $4.56-4.68=-0.12$ mmol/L, as required by additivity. The slope is $-0.12$ mmol/L per G-allele copy. In this model, each additional G copy is associated with an expected phenotype lower by $0.12$ mmol/L. This is an association stated on the measured phenotype scale; it is not, on its own, evidence that G causes the difference.

Worked Examples of Dosage Interpretation

Worked Example: Recovering the Slope From Genotype Means

Consider hypothetical genotype-specific expected means, in centimetres, of 160.0 for dosage zero, 160.6 for dosage one, and 161.2 for dosage two. Check whether these means satisfy the additive model and express that model using $\beta_0$ and $\beta_1$.

The two consecutive differences are $160.6-160.0=0.6$ cm and $161.2-160.6=0.6$ cm. They are equal, so the means satisfy the additive constraint. The intercept is the dosage-zero mean, $\beta_0=160.0$ cm. The per-copy slope is $$ \beta_1=\frac{161.2-160.0}{2} =\frac{1.2}{2} =0.6\text{ cm per allele copy}. $$

The resulting mean model is $E[Y_i\mid G_i]=160.0+0.6G_i$, with $Y_i$ measured in centimetres. Substitution gives 160.0 cm at dosage zero, 160.6 cm at dosage one, and 161.2 cm at dosage two, matching the stated means.

Worked Example: Changing Which Allele Is Counted

In a hypothetical analysis, A is initially counted as the effect allele, and the model for a phenotype measured in milligrams per decilitre is $E[Y_i\mid G_i]=100+2G_i$. Rewrite the same model using the number of copies of the other allele, C, as the dosage.

Because each individual has two copies at this diploid biallelic variant, the C dosage $G_i^*$ equals $2-G_i$. Substitute $G_i=2-G_i^*$ into the original mean: $$ E[Y_i\mid G_i^*] =100+2(2-G_i^*) =104-2G_i^*. $$

The recoded intercept is 104 mg/dL, the expected mean when there are zero A copies and therefore two C copies. The recoded slope is $-2$ mg/dL per C-allele copy. The predicted means are unchanged: AA has A dosage 2 and C dosage 0, so its mean is $100+2(2)=104$ mg/dL; AC has dosage 1 for each allele, so its mean is 102 mg/dL under either coding; CC has A dosage 0 and C dosage 2, so its mean is 100 mg/dL under either coding. Changing the counted allele reverses the slope’s sign and shifts the intercept, but it does not change the fitted genotype means.

Worked Example: Comparing One and Two Additional Copies

Suppose a hypothetical additive model for a laboratory phenotype measured in units per litre is $E[Y_i\mid G_i]=35+1.5G_i$, where the counted allele is T. Calculate the expected difference between dosage two and dosage zero, and between dosage two and dosage one.

The predicted means are 35 units/L at dosage zero, $35+1.5(1)=36.5$ units/L at dosage one, and $35+1.5(2)=38.0$ units/L at dosage two. The difference between dosage two and dosage zero is $38.0-35=3.0$ units/L, which is also $2\beta_1=2(1.5)=3.0$ units/L. The difference between dosage two and dosage one is $38.0-36.5=1.5$ units/L, exactly one slope increment. The slope is 1.5 units/L per T-allele copy.

This comparison illustrates why the slope should not be described as the expected difference between the two homozygote groups: that difference is twice the per-copy slope under the additive model.

Interpreting the Model Carefully

The intercept $\beta_0$ has a specific coding-dependent meaning: it is the expected phenotype for dosage zero of the chosen effect allele, at the reference levels of any other included predictors. It is not necessarily the overall population mean, and it can refer to a genotype that is uncommon. The slope $\beta_1$ is the model’s expected change for a one-unit increase in dosage, holding any other predictors fixed if they are included in the model.

Units should be stated with the coefficient. If $Y$ is measured in kilograms, then $\beta_1$ is in kilograms per allele copy; if $Y$ is measured in a standardized score, the slope is in score units per allele copy. Its numerical value depends on the phenotype scale. The allele-count unit is always one copy, not one percentage point of allele frequency and not a change in genotype category of unspecified size.

A dosage model is a model for an association between genotype and phenotype. Other variables, population structure, sampling, and study design can affect that association. Including covariates can make the slope conditional on those covariates, but the interpretation remains an expected phenotype difference per additional counted allele at fixed covariate values. Neither the additive coding nor a nonzero slope establishes a causal effect.

Common Pitfall: Treating the Slope as a Two-Copy Contrast

A common error is to say, “The coefficient is the expected difference between individuals with two copies and individuals with no copies.” That is not the per-allele interpretation. Under the dosage model, moving from zero to two copies changes the predictor by two units, so the expected difference is $2\beta_1$. The coefficient itself describes a one-copy increase: zero to one or one to two.

A second error is to treat additive as a statement about individual outcomes. The model requires equal steps in the conditional means; it does not require every individual with dosage one to have a phenotype exactly halfway between particular individuals with dosages zero and two. The error term allows individual outcomes to vary around their dosage-specific means.

Finally, the sign cannot be interpreted without naming the allele being counted. A positive slope for allele A becomes a negative slope of equal magnitude when the alternative allele is counted instead. Always state the effect allele, phenotype scale, and slope units together.

Key takeaway: With dosage $G_i\in\{0,1,2\}$, the additive model assigns expected means $\beta_0,\ \beta_0+\beta_1,\ \beta_0+2\beta_1$. The coefficient $\beta_1$ is the expected phenotype change per additional copy of the designated effect allele, on the phenotype scale used in the model.

Check Your Understanding

Use the dosage model and its allele-count interpretation to answer these questions.

If the expected means at dosages zero and two are 8 and 10 units, respectively, what is the additive slope, and what mean does the model imply at dosage one?
A slope is $-0.4$ milligrams per decilitre per allele copy. What expected difference does the model imply between dosage two and dosage zero?
If a model counts allele C with dosage $G_i^*$, what is the dosage of the alternative allele in a diploid individual?
Explain why a per-allele slope does not imply that every individual’s phenotype changes by that exact amount.
Under an additive dosage model, what condition must the three genotype-specific expected means satisfy?