The Additive Linear Regression Model

From Genotype Dosage to a Regression Model

In a single-variant GWAS of a quantitative phenotype, the additive linear regression model connects each participant’s phenotype to the number of copies of a selected effect allele. The model provides a compact way to describe how the conditional mean phenotype changes across genotype dosage, while allowing other measured variables to enter as covariates.

The model is a specification, not a claim that an association is causal. Its meaning depends on what allele is counted, how the phenotype is measured, which covariates are included, and whether the assumptions are reasonable for the data. The dosage and allele labels follow the earlier material on Allele Dosage Coding and Effect Alleles And Reference Alleles.

For a biallelic SNP, \(G_i\) is typically coded as 0, 1, or 2 copies of the selected effect allele. Under the additive model, the genotype contribution to the conditional mean is \(\beta_G G_i\). Thus the model constrains the conditional means at dosages 0, 1, and 2 to lie at equally spaced values when the covariates are held fixed.

The residual \(\varepsilon_i\) is not the same as the model’s fitted mean or the observed phenotype. It is the difference between the observed value and the conditional mean specified by the model. In the population model, the residual is not directly observed because the coefficients are unknown; after fitting a model, the corresponding observed-minus-fitted quantity is called a fitted residual.

What “Additive” Specifies

The word additive refers to the assumed relationship between allele dosage and the conditional mean, not to a claim that all biological processes are additive. With covariates held fixed, increasing dosage by one changes the model’s conditional mean by \(\beta_G\). Increasing dosage from 0 to 2 therefore changes that mean by \(2\beta_G\). This is a linear restriction on the three genotype-specific means.

The model does not require the raw phenotype to equal its conditional mean. Individuals with the same dosage and covariate values can have different phenotypes because their residuals differ. The model describes the systematic part of the conditional mean and a probability model for the remaining variation.

Assumptions and Conditions

A statistical model is useful only when its assumptions are stated. For the standard single-variant linear model, treat the genotypes and covariates as the observed design values and describe the phenotype conditionally on them. The core mean assumption is that the conditional mean is linear in the specified dosage and covariate terms. The error assumptions commonly used for the classical model are:

• Zero conditional mean: \(E(\varepsilon_i\mid G_i,C_{i1},\ldots,C_{iq})=0\). Consequently, the stated regression expression is the conditional mean of \(Y_i\).
• Constant conditional variance: \(\operatorname{Var}(\varepsilon_i\mid G_i,C_{i1},\ldots,C_{iq})=\sigma^2\), the same for all design values.
• Independence across individuals: conditional on the design, the residuals for distinct individuals are independent in the standard model.
• Normality when exact small-sample normal-theory inference is required: conditionally, the residuals are normally distributed. Normality is not required to define the linear mean model itself.

The predictors must also contain enough distinct information to distinguish their coefficients. For example, if a covariate is exactly a linear combination of the dosage and other included covariates, the individual coefficients cannot be uniquely identified from the design. This is a design limitation, not something repaired by the additive assumption.

These conditions have distinct roles. The linear conditional-mean form defines the target model. Zero conditional mean supports interpretation of its coefficients as describing that conditional mean. Constant variance and independence are part of the usual classical error model; departures can affect the validity of standard uncertainty calculations. Conditional normality is an additional assumption for exact classical finite-sample inference, not a requirement that the observed phenotype itself look normally distributed.

Intercept, Coefficients, and Residual in Context

The intercept \(\beta_0\) is the model’s conditional mean when \(G_i=0\) and every included covariate equals zero. That combination may or may not be common or scientifically meaningful. If zero is an implausible value for a covariate such as age, centering that covariate can make the intercept correspond to a more relevant covariate value. Centering changes the intercept’s reference point, not the fitted linear relationship.

The SNP coefficient \(\beta_G\) is attached to a one-unit increase in \(G_i\), which is one additional copy of the chosen effect allele. Its units are the phenotype units per allele copy. This is a definition of the model coefficient; estimation, uncertainty, and inferential interpretation are separate questions. Each \(\gamma_k\) is the coefficient attached to a one-unit increase in the corresponding covariate, with the other model terms held fixed.

Allele orientation matters because dosage is defined relative to a selected allele. If the alternative allele is counted instead, its dosage is \(G_i^*=2-G_i\). In the model with the same covariates, substituting \(G_i=2-G_i^*\) gives \(\beta_0+2\beta_G-\beta_G G_i^*+\sum_k\gamma_k C_{ik}+\varepsilon_i\). The coefficient for the reversed dosage is therefore \(-\beta_G\), and the intercept becomes \(\beta_0+2\beta_G\). A change of allele coding changes the coefficient’s sign, so the counted allele must be reported alongside it.

Checking a Model Specification

Before interpreting a fitted model, verify that its terms match the question and the data. The phenotype should be quantitative for this linear-mean specification. Confirm that the dosage counts the intended allele, that the dosage scale is consistent, and that every covariate is defined for the analyzed individuals. Check that the model matrix is not redundant and that the conditional mean form is scientifically plausible for the dosage range being used.

Residual patterns can reveal that the usual error model is not a good description. For example, residual spread that increases systematically with fitted mean is evidence against constant conditional variance; clustering by family or other sampling unit raises concern about independence. These patterns matter because standard errors and tests based on the classical model can be unreliable under violations. A model specification alone cannot remove dependence, measurement error, or unmeasured structure.

A linear dosage term also imposes a restriction: the conditional means at dosage 0, 1, and 2 are equally spaced, after accounting for the included covariates. If that restriction is not appropriate, the additive model may fail to describe the genotype-phenotype pattern. A model with separate genotype indicators can relax the restriction, but it answers a different modeling question and requires additional parameters. The choice is not decided by the word “GWAS”; it depends on the scientific model and the available genotype data.

Common Mistakes and Alternatives

• Confusing an additive model with identical individual outcomes. The model constrains conditional means; residual variation allows observed phenotypes to differ among individuals with the same predictors.
• Calling \(\beta_G\) an effect per genotype group. It is the change in the modeled conditional mean per one-copy dosage increment, with covariates held fixed.
• Ignoring the allele counted. Reversing allele dosage reverses the SNP coefficient’s sign. Report the allele whose copies define \(G_i\).
• Assuming the intercept is always the overall mean. It is the conditional mean at dosage zero and covariate values zero, which may be an unrepresentative combination.
• Treating covariate inclusion as a guarantee of valid adjustment. The model only conditions on the covariates actually included and assumes its stated form; it cannot guarantee that relevant structure has been handled.

The main alternative for genotype coding is to replace the single dosage term with separate genotype indicators, allowing the three genotype means not to be equally spaced. The additive model uses fewer genotype parameters and expresses a per-copy linear trend; the indicator model is more flexible but does not impose that trend. Neither coding is universally preferable. Also, a model without covariates and one with covariates describe different conditional means, so coefficients from them need not represent the same specification.