Quantitative-Trait Epistasis On The Mean Scale

Why the Mean Scale Matters

For a quantitative trait, two loci may jointly relate to the trait in a way that is not captured by adding their separate dosage effects. A linear regression product term makes this departure explicit on the mean scale: it describes how the mean-trait difference associated with one locus changes as the dosage at the other locus changes. Its interpretation depends on the coding, the reference dosage, and the trait’s measurement units.

A product-term coefficient is not, by itself, a universal measure of biological interaction. It is a contrast in conditional means under a specified regression model. The same data can also be summarized by marginal means that average over the other locus, but those summaries depend on which genotype distribution is used for averaging.

Definition: Let $Y$ be a quantitative trait, and let $A$ and $B$ denote effect-allele dosages at two loci, each coded $0,1,$ or $2$. The two-locus additive-dosage interaction model is $$ E(Y\mid A=a,B=b)=\beta_0+\beta_A a+\beta_B b+\beta_{AB}ab. $$ Here $\beta_0$ is the mean at dosages $A=0,B=0$; $\beta_A$ and $\beta_B$ are dosage effects at the other locus’s zero dosage; and $\beta_{AB}$ is the change in one locus’s dosage effect for a one-unit increase in the other locus’s dosage. Coefficients are measured in trait units per corresponding dosage contrast.

This model represents the conditional mean, not every individual’s observed trait value. A corresponding individual-level model writes $Y_i$ as the fitted mean plus a residual. For interpretation of regression coefficients, assume the conditional residual mean is zero given the genotypes (and any included covariates), and that the model’s mean form is appropriate. Independent observations and adequate variation in genotype combinations are also needed for ordinary regression estimation. Usual standard errors require additional variance assumptions or a method that accommodates departures from them.

Dosage coding imposes structure. It treats the mean changes across 0-to-1 and 1-to-2 dosage steps as equal at each fixed dosage of the other locus, and represents the interaction through the product $ab$. It does not automatically allow every one of the nine two-locus genotype means to vary freely.

Reading the Coefficients as Conditional Contrasts

The main-effect coefficient $\beta_A$ is the mean difference for a one-unit increase in $A$ specifically when $B=0$. It is not generally the mean difference at every value of $B$. At a fixed dosage $b$, increasing $A$ by one changes the conditional mean by $\beta_A+\beta_{AB}b$. Likewise, at fixed dosage $a$, a one-unit increase in $B$ changes the mean by $\beta_B+\beta_{AB}a$.

\begin{aligned} E(Y\mid A=a+1,B=b)-E(Y\mid A=a,B=b) &=\beta_A+\beta_{AB}b,\\ E(Y\mid A=a,B=b+1)-E(Y\mid A=a,B=b) &=\beta_B+\beta_{AB}a. \end{aligned}

The interaction coefficient has a direct contrast interpretation: it is the change in the one-unit $A$ mean difference for a one-unit increase in $B$. Equivalently, it is the change in the one-unit $B$ mean difference for a one-unit increase in $A$. This is a difference-in-differences on the trait’s original mean scale.

\begin{aligned} \beta_{AB} ={}&\big\{E(Y\mid A=a+1,B=b+1)-E(Y\mid A=a,B=b+1)\big\}\\ &-\big\{E(Y\mid A=a+1,B=b)-E(Y\mid A=a,B=b)\big\}. \end{aligned}

The expression does not depend on which adjacent dosage values $a$ and $b$ are chosen, provided the stated model applies. A positive $\beta_{AB}$ means the conditional $A$-associated mean difference becomes more positive as $B$ increases. It does not necessarily mean that either conditional difference is positive: their signs depend on $\beta_A$, $\beta_{AB}$, and $b$.

Worked Example: Interpreting a Product-Term Model

Suppose a hypothetical study measures a trait in millimetres and fits the following conditional mean model:

E(Y\mid A=a,B=b)=12+1.4a-0.6b+0.9ab.

The intercept is $12$ mm, the modelled mean when both dosages are zero. At $B=0$, a one-unit increase in $A$ corresponds to a $1.4$ mm higher conditional mean. At $B=2$, the same dosage contrast is $1.4+(0.9)(2)=3.2$ mm. Thus the $A$-associated difference is $1.8$ mm larger at $B=2$ than at $B=0$, because $2(0.9)=1.8$ mm.

For a direct check, the four means at dosages $(A,B)=(0,0),(1,0),(0,1),(1,1)$ are $12,\ 13.4,\ 11.4,$ and $13.7$ mm. The difference in $A$ means is $13.4-12=1.4$ mm at $B=0$, and $13.7-11.4=2.3$ mm at $B=1$. Their difference is $2.3-1.4=0.9$ mm, exactly the product-term coefficient.

Conditional Effects and Marginal Effects

A conditional contrast compares mean outcomes at specified dosages of the other locus. A marginal contrast averages over that other locus. The averaging distribution must be stated, because it determines the result. For example, standardize both $A$ groups to a common distribution $q_b$ over $B$ dosages. Under the dosage-product model, the standardized difference for a one-unit increase in $A$ is

\sum_b q_b\{\beta_A+\beta_{AB}b\} =\beta_A+\beta_{AB}\sum_b q_b b.

Thus, the standardized mean difference is the $A$ effect at the chosen distribution’s mean $B$ dosage. A raw difference between observed $A$ groups need not equal this quantity: if the groups have different distributions of $B$, their means combine conditional effects at different weights. Such a raw contrast can reflect genotype-distribution differences as well as the conditional mean pattern.

Worked Example: Counts Change the Raw Marginal Contrast

Consider hypothetical data with two coded genotype indicators, $A,B\in\{0,1\}$, and trait values measured in trait units. The fitted cell means follow $E(Y\mid A,B)=10+2A+3B+4AB$. The table gives the genotype counts and corresponding cell means.

A	B	Count	Cell mean
0	0	20	10
0	1	10	13
1	0	10	12
1	1	20	19

At $B=0$, the conditional difference for $A=1$ versus $A=0$ is $12-10=2$ units. At $B=1$, it is $19-13=6$ units. Their difference is $6-2=4$ units, the product-term coefficient. This is the conditional interaction contrast.

Now calculate the raw mean in each observed $A$ group using its actual $B$ counts. For $A=0$, the mean is $\{20(10)+10(13)\}/30=330/30=11$ units. For $A=1$, it is $\{10(12)+20(19)\}/30=500/30=16.67$ units. The raw difference is $16.67-11=5.67$ units. It is not the conditional difference at either fixed $B$ value, because the $B=1$ proportion is $10/30$ in the $A=0$ group but $20/30$ in the $A=1$ group.

For comparison, standardize both groups to the pooled $B$ distribution: $B=0$ has proportion $30/60=0.5$, and $B=1$ has proportion $30/60=0.5$. The standardized mean at $A=0$ is $0.5(10)+0.5(13)=11.5$; at $A=1$, it is $0.5(12)+0.5(19)=15.5$. The standardized difference is $15.5-11.5=4$ units, the average of the conditional differences $2$ and $6$ with equal weights. The raw and standardized contrasts answer different questions.

Interpreting a Dosage-by-Dosage Model Across Genotypes

When dosages take values $0,1,2$, the product model can be used to calculate conditional contrasts across the genotype table. Because dosage increments are assumed equal in the mean model, the same one-unit contrast formula applies to both the $0$-to-$1$ and $1$-to-$2$ steps. The model predicts means for all nine dosage pairs, but it constrains those means; it is not a separate estimate for every cell.

Worked Example: Calculating a Standardized Dosage Effect

Suppose hypothetical cell means for a quantitative trait in millimetres follow

E(Y\mid A=a,B=b)=50+1.5a-2b+0.8ab.

The mean at $(0,0)$ is $50$ mm. At $(1,0)$, it is $50+1.5=51.5$ mm, and at $(0,1)$, it is $50-2=48$ mm. At $(1,1)$, it is $50+1.5-2+0.8=50.3$ mm. The difference-in-differences is $(50.3-48)-(51.5-50)=2.3-1.5=0.8$ mm per one-unit increase in each dosage, matching $\beta_{AB}$.

At fixed $B=2$, the conditional difference for a one-unit increase in $A$ is $1.5+(0.8)(2)=3.1$ mm. Suppose the target distribution for standardization has $B=0,1,2$ weights $0.2,0.5,0.3$, respectively. Its mean dosage is $0(0.2)+1(0.5)+2(0.3)=1.1$. The standardized one-unit $A$ difference is therefore $1.5+(0.8)(1.1)=2.38$ mm. This is an average over the specified target distribution, not a universal $A$ effect.

The model’s dosage assumptions matter: for example, it predicts an $A$ difference of $1.5$ mm for $B=0$ at both the $A=0$-to-$1$ and $A=1$-to-$2$ steps. If observed mean patterns show unequal steps or additional departures from the product model, these four coefficients may not adequately describe all nine genotype means.

Justification: Why the Product Coefficient Is a Difference-in-Differences

The contrast interpretation follows directly by substituting the mean model, so it does not require an additional biological assumption. First calculate the difference for a one-unit increase in $A$ at fixed $B=b$:

\begin{aligned} &(\beta_0+\beta_A(a+1)+\beta_Bb+\beta_{AB}(a+1)b)\\ &\quad-(\beta_0+\beta_Aa+\beta_Bb+\beta_{AB}ab) =\beta_A+\beta_{AB}b. \end{aligned}

Now compare that difference at $b+1$ with the difference at $b$:

\{\beta_A+\beta_{AB}(b+1)\}-\{\beta_A+\beta_{AB}b\}=\beta_{AB}.

The intercept and the $B$ main effect cancel, as does the $A$ effect that does not depend on $B$. What remains is the change in the conditional $A$ difference associated with a one-unit increase in $B$. This derivation also explains why $\beta_{AB}$ is measured in trait units for a one-unit-by-one-unit dosage contrast.

Assumptions, Scope, and a Common Mistake

The interpretation is tied to the modeled scale: here it is the conditional arithmetic mean of the quantitative trait in its stated units. The regression describes an association conditional on included variables; it does not establish that changing a genotype would cause the predicted mean difference. Ancestry structure, relatedness, selection, and other sources of confounding can affect genetic association estimates, so study design and appropriate adjustment remain important.

A common mistake is to read $\beta_A$ as “the effect of locus $A$ for everyone,” or to report the raw $A$-group mean difference as the interaction-adjusted effect. In this model, $\beta_A$ is the $A$ difference only at $B=0$; at dosage $b$, it is $\beta_A+\beta_{AB}b$. A marginal contrast requires an explicit averaging distribution, and the observed raw contrast can also reflect different $B$ distributions across $A$ groups.

A second practical caution is that a product term does not remove the need to consider the main-effect terms or to check whether the chosen dosage form describes the mean pattern. Nor does a nonzero product coefficient alone establish a biological mechanism. It indicates a departure from additivity in this particular conditional mean model and coding.

Key takeaway: On the quantitative-trait mean scale, $\beta_{AB}$ is a difference-in-differences: it quantifies how much the conditional mean difference for one dosage changes per one-unit increase in the other dosage. Conditional contrasts fix the other genotype; marginal contrasts average over a stated genotype distribution.

Check Your Understanding

Use the dosage-product model and distinguish carefully between conditional and marginal statements.

In $E(Y\mid A=a,B=b)=8+2a-3b+0.5ab$, what is the one-unit $A$ mean difference when $B=2$, in trait units?
Explain why the coefficient of $A$ in a product-term model is not generally the $A$ difference at every value of $B$.
What contrast does the product-term coefficient represent in a two-by-two table of conditional means?
Why can a raw difference between observed $A$ groups differ from a standardized difference using a common distribution of $B$?
Name one mean-pattern assumption imposed by using dosage $0,1,2$ with a single dosage-by-dosage product term.