Binary-Trait Epistasis On Odds And Risk Scales

Why the Scale Matters for a Binary Trait

For a binary trait, each person either has the outcome or does not. A two-locus analysis can describe how the outcome risk varies across the four combinations of two binary genotype indicators. But the meaning of “interaction” depends on the scale used to compare those risks. On the risk-difference scale, interaction concerns departures from additivity of risks. In logistic regression, the product-term coefficient concerns departures from additivity of log odds, equivalently a ratio of odds ratios.

These are different comparisons. A pair of loci can have no interaction on the logistic odds scale and still have a nonzero risk-difference interaction. It can also have no risk-difference interaction and a nonzero logistic interaction. Neither result, by itself, establishes a biological mechanism or a causal effect.

Definition: Let $A$ and $B$ be binary indicators for two genotype contrasts, each coded 0 or 1. Write $p_{ab}=P(Y=1\mid A=a,B=b)$ for the outcome risk in the genotype cell $(a,b)$. The risk-difference interaction contrast is $$ \Delta_{\mathrm{RD}}=p_{11}-p_{10}-p_{01}+p_{00}. $$ There is no risk-difference interaction when $\Delta_{\mathrm{RD}}=0$. This means the risk increase for changing $A$ from 0 to 1 is the same at $B=0$ and $B=1$.

The four risks refer to the same outcome and the same target population. If covariates are part of the analysis, these risks must also be defined at the same covariate values or under the same covariate-standardization procedure. Comparing risks from different populations or different covariate conditions would mix interaction with other differences.

Interaction in a Logistic Regression Model

For the two binary indicators, a logistic regression model can be written as

\operatorname{logit}\{P(Y=1\mid A,B)\} =\beta_0+\beta_A A+\beta_B B+\beta_{AB}AB,

where $\operatorname{logit}(p)=\log\{p/(1-p)\}$. The model assumes that the log odds in each genotype cell follow this linear predictor. With only these two binary predictors and their product, the model assigns a log odds to each of the four cells. The coefficient $\beta_{AB}$ describes the departure from additivity of the two indicators on the log-odds scale.

Definition: In this model, $\beta_{AB}$ is the difference between the log odds contrast for $A$ at $B=1$ and the log odds contrast for $A$ at $B=0$. Equivalently, $$ \beta_{AB} =\log\left(\frac{\operatorname{odds}_{11}\operatorname{odds}_{00}} {\operatorname{odds}_{10}\operatorname{odds}_{01}}\right), \qquad \operatorname{odds}_{ab}=\frac{p_{ab}}{1-p_{ab}}. $$ Thus $\exp(\beta_{AB})$ is a ratio of odds ratios. There is no logistic interaction when $\beta_{AB}=0$, or equivalently when this ratio equals 1.

The main-effect coefficients are conditional contrasts under the chosen coding. Specifically, $\beta_A$ is the log odds ratio comparing $A=1$ with $A=0$ when $B=0$, and $\beta_B$ compares $B=1$ with $B=0$ when $A=0$. When the product term is present, neither coefficient is an overall effect across the other locus.

Deriving the Scale-Specific Contrasts

The logistic definition follows by evaluating the model in each cell. The four linear predictors are $\beta_0$, $\beta_0+\beta_A$, $\beta_0+\beta_B$, and $\beta_0+\beta_A+\beta_B+\beta_{AB}$. Because the linear predictor is the log odds, take the fourth-cell value minus the $A=1,B=0$ value, minus the $A=0,B=1$ value, plus the reference-cell value:

(\beta_0+\beta_A+\beta_B+\beta_{AB}) -(\beta_0+\beta_A) -(\beta_0+\beta_B) +\beta_0 =\beta_{AB}.

Exponentiating this equality gives the ratio of odds ratios in the definition. For the risk scale, the analogous four-cell contrast is $p_{11}-p_{10}-p_{01}+p_{00}$. It is zero precisely when the risk increase for $A$ is unchanged across the two values of $B$:

(p_{11}-p_{01})-(p_{10}-p_{00}) =\Delta_{\mathrm{RD}}.

The logit function is nonlinear, so the risk contrast is not generally equal to $\beta_{AB}$, nor is a zero value of one contrast guaranteed to imply a zero value of the other. The logistic model specifies a pattern on log odds; probabilities obtained by applying the inverse-logit function can have a different pattern on the risk scale.

Worked Example: A Nonzero Interaction on Both Scales

Suppose a hypothetical dataset has 100 people in each genotype cell, with the following outcome counts: 10 when $(A,B)=(0,0)$, 20 when $(1,0)$, 30 when $(0,1)$, and 60 when $(1,1)$. Treat the cell proportions as the risks for this illustration.

The four risks are $p_{00}=0.10$, $p_{10}=0.20$, $p_{01}=0.30$, and $p_{11}=0.60$. The risk-difference interaction is

\Delta_{\mathrm{RD}} =0.60-0.20-0.30+0.10 =0.20.

The risk increase for $A$ is $0.20-0.10=0.10$ at $B=0$, but $0.60-0.30=0.30$ at $B=1$. Their difference is 0.20, or 20 percentage points.

The cell odds are $1/9$, $1/4$, $3/7$, and $3/2$, respectively. Therefore the logistic interaction ratio and its log are

\exp(\beta_{AB}) =\frac{(3/2)(1/9)}{(1/4)(3/7)} =\frac{14}{9} \approx 1.556, \qquad \beta_{AB}=\log(14/9)\approx 0.4418.

The ratio of odds ratios is greater than 1, so the odds ratio for $A$ is larger at $B=1$ than at $B=0$. Here the risks show interaction on both scales, but the numerical contrasts describe different quantities: a 20-percentage-point risk contrast and a ratio of odds ratios of about 1.556.

Why a Zero Logistic Interaction Can Coexist With Risk Interaction

A zero product-term coefficient means the log odds are additive across $A$ and $B$; it does not mean that risks add. The inverse-logit transformation changes the size of probability differences as the underlying log odds change. Consequently, equal log-odds effects can translate into unequal risk differences.

Worked Example: No Logistic Interaction, but a Risk-Difference Interaction

Consider the hypothetical logistic model

\operatorname{logit}\{P(Y=1\mid A,B)\} =0-A-B.

This model has $\beta_{AB}=0$, so it has no interaction on the log-odds scale. Its four predicted risks are obtained by applying $p=\exp(\eta)/(1+\exp(\eta))$ to each linear predictor $\eta$:

At $(0,0)$, $\eta=0$, so $p_{00}=0.5$.
At $(1,0)$ and $(0,1)$, $\eta=-1$, so each risk is $\exp(-1)/(1+\exp(-1))\approx0.2689414$.
At $(1,1)$, $\eta=-2$, so $p_{11}=\exp(-2)/(1+\exp(-2))\approx0.1192029$.

Using the unrounded model-predicted risks, the risk-difference interaction is

\Delta_{\mathrm{RD}} =0.1192029-0.2689414-0.2689414+0.5000000 \approx0.0813201.

Thus the model has no logistic interaction but has a risk-difference interaction of about 0.0813, or 8.13 percentage points. This is not a contradiction: the two statements concern different scales. The calculation uses the unrounded risks; subtracting risks rounded early can change the last reported digit.

Why Zero Risk Interaction Can Coexist With Logistic Interaction

The converse also occurs. If the risk for $A$ increases by the same amount at each value of $B$, then the risk-difference contrast is zero. But the odds corresponding to those risks need not have the same relative change, so the ratio of odds ratios can differ from 1.

Worked Example: No Risk-Difference Interaction, but a Logistic Interaction

Suppose the hypothetical risks are $p_{00}=0.10$, $p_{10}=0.20$, $p_{01}=0.30$, and $p_{11}=0.40$. The risk increase for $A$ is $0.20-0.10=0.10$ at $B=0$ and $0.40-0.30=0.10$ at $B=1$, giving

\Delta_{\mathrm{RD}} =0.40-0.20-0.30+0.10 =0.

The odds for the four cells are $1/9$, $1/4$, $3/7$, and $2/3$. Therefore,

\exp(\beta_{AB}) =\frac{(2/3)(1/9)}{(1/4)(3/7)} =\frac{56}{81} \approx0.6914, \qquad \beta_{AB}=\log(56/81)\approx-0.3691.

The ratio of odds ratios is below 1, indicating that the odds ratio for $A$ is smaller at $B=1$ than at $B=0$. That logistic interaction coexists with exactly additive risks. In particular, the logarithm of the exact ratio $56/81$ is approximately $-0.3691$.

Choosing and Reporting an Interaction Scale

Neither scale is universally correct for every question. A risk-difference interaction directly describes departure from additive absolute risks, which can be useful when the question concerns how many additional outcomes are associated with a genotype combination. A logistic interaction describes a departure from multiplicativity of odds, conditional on the model’s predictors and coding. Odds ratios and risk differences are not interchangeable, especially when outcome risks are not small.

A clear report should name the scale, state the genotype coding and reference cells, and provide predicted risks or cell-specific contrasts when interpretation depends on them. If the analysis includes covariates, specify whether the reported risks are conditional on covariates or standardized over a population distribution. A coefficient alone may not convey the absolute risk pattern.

These are statistical descriptions of outcome distributions. Genotype is not randomly assigned in the usual experimental sense, and population structure, linkage disequilibrium, selection, or other factors may affect observed associations. A nonzero statistical interaction therefore does not, by itself, establish that the loci interact biologically or causally.

Common Mistake

A frequent error is to conclude that “there is no interaction” simply because the logistic product-term coefficient is zero, or because a risk-difference contrast is zero. That conclusion omits the scale. The correction is to report the scale-specific statement: for example, “there is no interaction on the log-odds scale, although the risk-difference contrast is 8.13 percentage points.” If the scientific question concerns another scale, calculate and interpret its contrast directly rather than treating one scale’s result as universal.

Key takeaway: Two-locus interaction for a binary trait is scale-dependent. The risk-difference contrast measures departure from additivity of risks; the logistic product term measures departure from additivity of log odds. Calculate and name the scale that answers the question being asked.

Check Your Understanding

Answer each question by naming the scale and showing the relevant contrast where appropriate.

For risks $p_{00}=0.05$, $p_{10}=0.10$, $p_{01}=0.15$, and $p_{11}=0.20$, calculate the risk-difference interaction. What does its value say about additive risks?
In the logistic model with binary $A$ and $B$, what does $\exp(\beta_{AB})$ compare?
Can a logistic model with $\beta_{AB}=0$ have a nonzero risk-difference interaction? Explain why.
For $p_{00}=0.10$, $p_{10}=0.20$, $p_{01}=0.30$, and $p_{11}=0.40$, explain why zero risk-difference interaction does not imply zero logistic interaction.
Why should a reported interaction state the scale and genotype coding, and why does a statistical interaction alone not establish biological causation?