Linear Regression for Quantitative Traits

Why Use Linear Regression for a Quantitative Trait?

A quantitative phenotype, such as height or a blood concentration, can be related to genotype by comparing phenotype means across genotype values. Linear regression expresses that relationship as a model, makes the assumed pattern of mean differences explicit, and allows measured covariates to be included. The model describes an association in the sampled population; its coefficient is not automatically a causal genetic effect.

For a single variant, the predictor commonly records the number of copies of a specified effect allele. A linear regression then asks how the expected phenotype changes as that dosage increases, while optionally holding other measured predictors fixed. The interpretation depends on the genotype coding and the assumptions about the mean structure.

The Additive-Dosage Regression Model

Definition: For individual $i$, let $Y_i$ be a continuous phenotype and let $G_i\in\{0,1,2\}$ be the number of copies of the effect allele at one variant. An additive-dosage linear regression with covariates $C_i$ is $$ Y_i=\beta_0+\beta_G G_i+\boldsymbol{\gamma}^{\mathsf T}C_i+\varepsilon_i. $$ Here $\beta_0$ is the intercept, $\beta_G$ is the genotype coefficient, $\boldsymbol{\gamma}$ contains covariate coefficients, and $\varepsilon_i$ is the error for individual $i$. With no covariates, the term involving $C_i$ is omitted.

The additive-dosage coding assigns values 0, 1, and 2 to the three possible allele counts. It does not require that the biological mechanism be literally additive. Rather, the regression’s mean model assumes equal increments in the conditional phenotype mean for the two steps from 0 to 1 and from 1 to 2 copies. Whether that simplification is adequate is a scientific and statistical question.

The covariates $C_i$ might include age, sex, or variables used to account for ancestry differences. A coefficient is interpreted conditional on the included covariates: it compares expected phenotypes at different genotype dosages for individuals with the same values of those covariates. Including covariates changes the comparison being described; it does not guarantee that all confounding has been removed.

What the Genotype Coefficient Means

The central interpretation follows from the conditional mean of the model. If the error has conditional mean zero, then the expected phenotype at a given genotype and covariate value is obtained by dropping the error term. Increasing genotype dosage from $g$ to $g+1$, while holding covariates fixed, changes that expectation by $\beta_G$.

E(Y_i\mid G_i=g,C_i=c) =\beta_0+\beta_G g+\boldsymbol{\gamma}^{\mathsf T}c

Thus, $\beta_G$ has the units of the phenotype per additional effect allele. If $Y_i$ is measured in millimoles per litre, then $\beta_G$ is in millimoles per litre per allele. It is a difference in conditional means, not a change that must occur in every individual. With no covariates, the model implies that the conditional means at dosages 0, 1, and 2 lie on a straight line. With covariates, it implies the same genotype-related increment at each fixed covariate value.

The intercept is the model’s expected phenotype when $G_i=0$ and all numerical covariates equal zero (and categorical covariates are at their reference levels). That combination may be scientifically unusual or outside the observed data. Centering a covariate can make the intercept more interpretable without changing the genotype slope.

Worked Example: An Additive Dosage Mean Pattern

Suppose, for illustration, a hypothetical analysis models fasting glucose in millimoles per litre using genotype dosage alone. The specified model is $E(Y_i\mid G_i)=5.00+0.18G_i$. Calculate the model-implied conditional means at each dosage and interpret the genotype coefficient.

For $G_i=0$, the conditional mean is $5.00+0.18(0)=5.00$ mmol/L. For $G_i=1$, it is $5.00+0.18(1)=5.18$ mmol/L. For $G_i=2$, it is $5.00+0.18(2)=5.36$ mmol/L. The adjacent differences are $5.18-5.00=0.18$ mmol/L and $5.36-5.18=0.18$ mmol/L.

Under this model, each additional effect allele is associated with a 0.18 mmol/L higher expected fasting glucose. These values are model-implied means, not three observed measurements or evidence that every person with two copies has glucose exactly 5.36 mmol/L. The equal increments follow from the assumed additive mean structure.

The numerical pattern is internally consistent with that structure. In real data, one would need to assess whether the mean pattern and error assumptions are plausible; a set of predicted means cannot establish them. The example is hypothetical and does not describe a real study.

Adding Covariates

Covariate adjustment is often important in genetic association analysis because phenotype levels can vary with characteristics that are also distributed differently across genotype groups. For example, ancestry-related differences can induce an association between genotype and phenotype even when there is no direct relationship of interest. A regression can include measured covariates to describe the genotype association conditional on those variables.

Worked Example: Interpreting an Adjusted Coefficient

Consider a hypothetical model for systolic blood pressure, measured in millimetres of mercury: $E(Y_i\mid G_i,A_i,F_i)=120+1.5G_i+0.6(A_i-50)+4F_i$. Here $A_i$ is age in years, and $F_i=1$ for the specified female category and $F_i=0$ for the reference category. Calculate the predicted mean for a 60-year-old in the $F_i=1$ category at genotype dosages 0 and 2.

At dosage 0, substitute $G_i=0$, $A_i=60$, and $F_i=1$: $120+1.5(0)+0.6(60-50)+4(1)=120+0+6+4=130$ mm Hg.

At dosage 2, substitute the same covariates and $G_i=2$: $120+1.5(2)+0.6(60-50)+4(1)=120+3+6+4=133$ mm Hg. The conditional mean difference is $133-130=3$ mm Hg, or $1.5$ mm Hg per additional effect allele.

This interpretation holds age and the specified category fixed. It does not say that changing a person’s genotype would change their blood pressure by exactly 1.5 mm Hg, and it does not establish a causal effect. The hypothetical calculations illustrate the model’s conditional means; they do not verify that its assumptions hold in a dataset.

When the Dosage Line Is Not the Intended Mean Model

An additive-dosage model is one choice of genotype mean structure. If the scientific question or observed pattern calls for separate means for the three genotype categories, a linear model can instead use indicator variables. This makes no equal-increment restriction between the three category means, although it still models a quantitative phenotype using a linear model.

Y_i=\beta_0+\beta_1 I(G_i=1)+\beta_2 I(G_i=2)+\varepsilon_i

Here $I(\cdot)$ equals 1 when the condition is true and 0 otherwise. The mean for dosage 0 is $\beta_0$, for dosage 1 it is $\beta_0+\beta_1$, and for dosage 2 it is $\beta_0+\beta_2$. Unlike additive dosage, this model permits the two adjacent mean differences to be unequal. The model choice determines which patterns the regression can represent; it should not be selected solely because one coding produces a smaller p-value.

Worked Example: Separate Genotype-Category Means

For a hypothetical analysis of an LDL-related quantitative measure in millimoles per litre, suppose the genotype-category model has $\beta_0=3.40$, $\beta_1=0.20$, and $\beta_2=0.70$. Calculate the three fitted conditional means and compare the increments.

At dosage 0, both indicators are zero, so the mean is $3.40$ mmol/L. At dosage 1, $I(G_i=1)=1$ and $I(G_i=2)=0$, giving $3.40+0.20(1)+0.70(0)=3.60$ mmol/L. At dosage 2, the indicators are 0 and 1, giving $3.40+0.20(0)+0.70(1)=4.10$ mmol/L.

The dosage 0-to-1 difference is $3.60-3.40=0.20$ mmol/L, while the dosage 1-to-2 difference is $4.10-3.60=0.50$ mmol/L. The two increments are unequal, so these category means do not follow the equal-increment pattern required by an additive-dosage mean model. This example defines a model pattern; it does not claim that these values were measured in a real sample.

Assumptions Behind the Model and Its Usual Inference

A regression equation is not just a way to summarize data. Its interpretation and the uncertainty calculations built on it rely on assumptions. It is useful to distinguish assumptions about the conditional mean from assumptions used to justify standard inference.

Correct conditional mean form. The model’s expected phenotype, given the predictors, follows the stated relationship. In the additive model, the conditional mean is linear in genotype dosage and the genotype increment is the same across covariate values. If that restriction is wrong, the genotype coefficient may not describe the intended pattern.
Zero conditional mean error. For each predictor pattern, $E(\varepsilon_i\mid G_i,C_i)=0$. This is what allows the regression’s systematic part to represent the conditional mean rather than leaving a systematic difference in the error.
Independent observations. Conditional on the predictors, errors for different individuals are independent under the ordinary regression model. Related individuals, repeated measures, or other clustering can violate this assumption if dependence is not modeled.
Constant conditional error variance. The usual homoscedastic model assumes $\operatorname{Var}(\varepsilon_i\mid G_i,C_i)=\sigma^2$ for all predictor values. If variability changes across genotype or covariate groups, the ordinary standard-error calculation may not be appropriate.
Normal errors for exact small-sample inference. Exact finite-sample t- and F-based inference in the usual linear model additionally assumes normally distributed errors conditional on predictors. Normality is not needed to define the regression mean, but it matters for those exact reference distributions.
Identifiable predictors. The data must contain enough variation to distinguish the model coefficients. For example, if every participant has the same genotype dosage, the genotype slope cannot be separated from the intercept using those data.

These assumptions do not mean genotype itself must be normally distributed. Genotype dosage is a discrete predictor. Normality in the classical model concerns the conditional errors, not the predictor or the observed phenotype pooled across all genotypes.

A Short Justification of the Conditional-Mean Interpretation

The coefficient interpretation follows directly from the model and the zero conditional mean assumption. Start with the additive model and condition on the observed genotype and covariates. The predictor values are fixed within that conditional expectation, and the error contributes its conditional mean.

Take the conditional expectation.
$E(Y_i\mid G_i=g,C_i=c)=E(\beta_0+\beta_GG_i+\boldsymbol{\gamma}^{\mathsf T}C_i+\varepsilon_i\mid G_i=g,C_i=c)$.

Use fixed predictor values and zero-mean error.
By linearity of expectation, this equals $\beta_0+\beta_Gg+\boldsymbol{\gamma}^{\mathsf T}c+E(\varepsilon_i\mid G_i=g,C_i=c)$, and the last term is 0 by assumption.

Compare adjacent dosages.
The difference between the means at $g+1$ and $g$, for the same $c$, is $[\beta_0+\beta_G(g+1)+\boldsymbol{\gamma}^{\mathsf T}c]-[\beta_0+\beta_Gg+\boldsymbol{\gamma}^{\mathsf T}c]=\beta_G$.

This proves the conditional mean difference under the stated model. It does not establish that the model is correct or that the observed coefficient is precisely known. Estimation and the uncertainty around estimates require additional calculations.

Why the Assumptions Matter

In a genetic association analysis, population structure is a particularly important concern. If ancestry is related both to allele frequencies and to the phenotype, an unadjusted genotype–phenotype association can reflect those population differences. Including ancestry-related covariates may address measured structure, but it cannot automatically remove bias from unmeasured variables or an inappropriate mean model. The resulting coefficient remains an association conditional on the predictors included.

The independence assumption also deserves attention: a dataset containing multiple related participants may have correlated phenotypes. Treating those observations as independent can misstate uncertainty. Likewise, strong genotype imbalance or sparse genotype categories can make a fitted pattern depend heavily on limited data. Such issues do not change the definition of $\beta_G$, but they affect whether the model and its usual inferential procedures are defensible.

Common Mistakes

Calling the slope a guaranteed individual change. The coefficient describes a difference in conditional means per allele, not a fixed change for every person.
Forgetting the coding. A coefficient of $\beta_G$ per allele applies when genotype is coded as the number of effect-allele copies. A different contrast has a different interpretation.
Assuming additive coding proves biological additivity. The coding imposes equal increments in the modelled means; it does not demonstrate the underlying biological mechanism.
Treating adjustment as a guarantee against confounding. The coefficient is conditional on the included covariates. Omitted structure or other misspecification can still affect its interpretation.
Confusing residual assumptions with genotype assumptions. The usual normal-error assumption concerns errors conditional on predictors, not the distribution of genotype dosage.
Interpreting association as causation. A regression association alone does not establish that the variant causes the phenotype.

Key takeaway: A single-variant linear regression relates a continuous phenotype to genotype dosage through a specified conditional mean. Under additive dosage coding, the genotype coefficient is the conditional mean difference per additional effect allele. Its interpretation and usual inference depend on the model and error assumptions being appropriate.

Check Your Understanding

Answer each question using the model assumptions and coefficient interpretations stated above.

In a dosage model, what units does the genotype coefficient have if the phenotype is measured in grams per litre?
What equal-increment restriction does additive dosage impose on the conditional means at genotype dosages 0, 1, and 2?
Why does including age in a regression make the genotype coefficient a conditional rather than an unadjusted mean comparison?
Does normality of genotype dosage need to hold for the usual linear regression model? State what the normal-error assumption concerns.
Why can a regression coefficient for a genetic variant not, by itself, be interpreted as a causal effect?