Why Look Beyond Additive Genetic Variation?
Genetic variation is often summarized by an additive component because it is useful for describing how genetic differences predict phenotypic differences in a population. But a genotype’s effect need not be the sum of separate contributions from its alleles. The two alleles at one locus can jointly affect the phenotype, and effects at different loci can interact. These patterns contribute to dominance variance and epistatic variance, respectively.
These components are statistical descriptions of variation in a specified population and on a specified phenotype scale. They do not, by themselves, identify a biological mechanism or prove that a genetic factor causes a phenotype. To interpret a component, first define the population, the phenotype, and the model used to divide genetic values into parts.
Variance Components and Their Conditions
A useful starting point is a phenotype model \(Y=\mu+G+E\), where \(\mu\) is the population mean, \(G\) is the genetic value, and \(E\) collects non-genetic influences. With centered genetic and environmental deviations, the variance is
Here \(V_P=\operatorname{Var}(Y)\), \(V_G=\operatorname{Var}(G)\), and \(V_E=\operatorname{Var}(E)\). The frequently used sum \(V_P=V_G+V_E\) requires \(\operatorname{Cov}(G,E)=0\). If genetic and environmental deviations covary, that covariance term cannot be omitted.
To divide genetic variance further, represent the centered genetic value as \(G=A+D+I\), where \(A\), \(D\), and \(I\) denote additive, dominance, and epistatic contributions. A variance partition into just the three component variances requires these contributions to be mutually uncorrelated in the population under study. More generally, expanding the variance gives
Thus, \(V_G=V_A+V_D+V_I\) holds when all three covariance terms are zero. A statistical construction that makes components orthogonal can provide such a partition, but it is tied to the population distribution and the terms included in the construction. Without orthogonality, reporting component variances as if they sum to \(V_G\) is not justified.
One way to understand the construction is as a sequence of projections. The additive part is the best linear prediction of genetic value from the chosen allele-count predictors in the population. The dominance part captures within-locus patterns left beyond additive prediction. The epistatic part captures joint-locus patterns beyond the lower-order terms. The precise predictors, projections, and resulting residuals matter: different population genotype distributions can produce different projections, even if the genotype-to-phenotype mapping itself is unchanged.
Worked Examples of Genetic Components
Worked Example: A Purely Additive Genotype Pattern
Suppose a hypothetical population has genotypes \(AA\), \(Aa\), and \(aa\) at one locus, with respective frequencies \(0.25\), \(0.50\), and \(0.25\). Their genetic values are 10, 14, and 18 phenotype units. Let \(x\) count the number of \(a\) alleles, so \(x\) is 0, 1, or 2.
The mean genetic value is
The variance is
The values follow \(G=10+4x\): each additional \(a\) allele corresponds to a 4-unit increase, with no within-locus departure from that additive pattern. The variance in \(x\) is \(0.25(0-1)^2+0.50(1-1)^2+0.25(2-1)^2=0.50\). Therefore the variance in the additive predictor \(10+4x\) is \(4^2(0.50)=8\). In this example, \(V_A=V_G=8\), and \(V_D=0\).
Worked Example: A Dominance Pattern With No Additive Projection
In another hypothetical population, the three genotypes occur at frequencies \(0.25\), \(0.50\), and \(0.25\), and their genetic values are 0 for \(AA\), 8 for \(Aa\), and 0 for \(aa\). Let \(x\) again count \(a\) alleles.
The mean is \(0.25(0)+0.50(8)+0.25(0)=4\). Each genotype is 4 units from the mean, so
To check the additive projection, calculate the covariance of \(x\) and \(G\). Since \(E[x]=1\) and \(E[G]=4\),
The linear additive prediction from \(x\) is therefore constant, so its variance is zero. All the genetic variation here is a within-locus departure from that prediction: \(V_D=16\). This calculation is specific to the stated genotype frequencies and values; it does not imply that dominance is generally uncorrelated with additive effects in every population.
Worked Example: A Joint-Locus Contribution
Consider two hypothetical binary genotype indicators, \(X_1\) and \(X_2\), each taking values 0 and 1 with equal probability. Assume they are independent, so the four combinations are equally frequent. Let the centered genetic value be \(G=3(2X_1-1)(2X_2-1)\). The value is 3 when the indicators match and \(-3\) when they differ.
Each locus alone has no average association with \(G\): for either value of \(X_1\), averaging over the equally frequent values of \(X_2\) gives one value of 3 and one of \(-3\), hence \(E[G\mid X_1]=0\). The same calculation gives \(E[G\mid X_2]=0\). There is therefore no variation in the single-locus mean predictions. But \(G\) varies across the four combinations, with mean 0 and squared value 9 in every combination:
In this construction, the variation is entirely in the joint-locus term, so \(V_I=9\). The example shows how averaging over the other locus can hide a joint pattern. It does not establish that a joint statistical contribution necessarily corresponds to a particular molecular mechanism.
Population and Phenotype-Scale Dependence
A variance component is defined by a distribution, not just by a list of possible genotypes. Changing genotype frequencies changes how often each genetic value occurs, and can change both the amount of genetic variation and its division into additive, dominance, and epistatic parts. In the dominance example, changing the frequencies could change the additive projection, even if the three genotype values stayed the same. Linkage and other dependencies among loci can likewise change the joint distribution on which interaction components are defined.
The phenotype scale matters too. If a phenotype is rescaled linearly as \(Y^*=cY+b\), with \(c\ne0\), then every variance component measured on that scale is multiplied by \(c^2\). A nonlinear transformation can change the relative patterns of genotype values, and therefore alter the component decomposition as well. For example, a component defined for a continuous measurement need not have the same interpretation after transforming that measurement or categorizing it.
Accordingly, statements such as “the additive component is 20 units squared” are incomplete without the population and phenotype scale. Comparisons across populations or analyses using different phenotype definitions should not assume that component values are directly interchangeable.
Variance Explained Is Not the Same as Causation
A genetic predictor can explain a large share of phenotypic variance because it tracks differences among people in the population. That fact alone does not show that changing the predictor would change the phenotype. Variance explained is a property of observed variation and a model’s predictions; a causal effect describes how outcomes would differ under a defined intervention or change, under suitable causal assumptions.
Worked Example: High Explained Variance Without a Causal Marker Effect
Suppose a hypothetical sample contains equal numbers from two ancestry groups. A marker perfectly identifies group membership. The phenotype has group means 0 and 10 units, with within-group variance 4 units squared in each group. Suppose, by construction, the phenotype difference is due to an environmental difference between groups, and the marker itself has no effect on the phenotype.
The overall mean is \(0.5(0)+0.5(10)=5\). The variance of the group means around 5 is \(0.5(0-5)^2+0.5(10-5)^2=25\). Adding the average within-group variance gives
Because the marker identifies the groups, the variance explained by the marker’s group-based predictions is 25, and its explained fraction is \(25/29=0.8621\), or about 86.2%. Yet changing the marker while holding group membership and environment fixed would not change the phenotype in this hypothetical setup. The marker is predictive because it tracks group membership, not because its own variation causes the outcome. This illustrates why a large explained fraction cannot, on its own, establish a causal genetic effect.
Why the Distinction Matters
Variance components are useful summaries of genetic architecture: they describe how genetic values vary in a population under a chosen decomposition. They can also inform prediction and expectations about resemblance among relatives. But a component is not a direct count of causal variants, a measure of the biological importance of a pathway, or a guarantee that the same prediction will work in another population.
Causal questions require more than a variance partition. They require a clearly specified causal contrast and assumptions or study designs that address confounding and other sources of bias. Conversely, a genetic effect can be causal while contributing little to population variance if the relevant allele is rare or has little variation in the studied population. A small component does not prove that a mechanism is unimportant; a large component does not prove that its predictors are causal.
Common Mistakes
- Adding components automatically. Writing \(V_G=V_A+V_D+V_I\) without checking orthogonality ignores possible covariance terms. State the conditions for the sum.
- Treating a component as population-invariant. Component values depend on genotype frequencies and dependencies in the target population, as well as on how the components are defined.
- Calling explained variance a causal effect. A predictor can capture ancestry, environment, or other correlated differences. Predictive fit alone does not establish what would happen under an intervention.
- Comparing values on different phenotype scales as if they were identical. Linear rescaling changes variances by a squared factor, and nonlinear transformations can change the decomposition itself.
Check Your Understanding
Try these questions without looking back.
- What distinguishes a dominance contribution from an epistatic contribution?
- Which covariance terms must be zero for \(V_G=V_A+V_D+V_I\) to hold?
- Why can changing genotype frequencies change the additive variance even when genotype-specific values do not change?
- If a phenotype is multiplied by 2, how do its variance components change? What additional caution applies to a nonlinear transformation?
- Give one reason a genetic marker might explain substantial phenotypic variance without having a causal effect on the phenotype.