Why Sensitivity to Definitions Matters
A gene–environment interaction is not a single number independent of analysis choices. The estimated interaction can change when an environmental exposure is divided at a different threshold, represented by a different transformation, or analyzed on a different outcome scale. Those choices can alter the contrast being estimated, not merely the way the same contrast is calculated.
A sensitivity analysis repeats an analysis under a set of scientifically defensible alternatives and asks which conclusions remain stable. The goal is not to search for a preferred result. It is to show how dependent the findings are on choices that could reasonably have been made differently. Comparisons are most informative when the population, genetic coding, adjustment set, and interpretation of each contrast are made explicit.
Two distinctions are essential. First, changing a binary exposure threshold changes which individuals are classified as exposed and therefore generally changes the contrast itself. Second, changing an exposure transformation changes how a contrast is parameterized; coefficients on different exposure scales need not be directly comparable. Changing the outcome scale can change the definition of interaction even when the underlying cell-specific outcomes are the same.
Interaction Contrasts and Exposure Functions
For a quantitative outcome, consider a model with a binary genotype \(G\), a binary exposure \(E\), and their product:
Here \(E(Y\mid G,E)\) denotes the conditional mean outcome. The coefficient \(\beta_{GE}\) measures departure from additivity of the two main effects on the modeled mean scale. This interpretation follows directly from the cell means.
The interaction contrast is the change in the exposure difference between genotype groups:
This identity is a useful check when interpreting alternative analyses: ask which four means, risks, or predicted outcomes define the reported interaction. With a continuous exposure represented by a function \(f(E)\), the model instead includes \(\beta_{GE}Gf(E)\). Comparing genotypes across two exposure values \(e_0\) and \(e_1\), the difference-in-differences is then \(\beta_{GE}\{f(e_1)-f(e_0)\}\). The coefficient alone is tied to the chosen units and function.
Worked Example: Changing an Exposure Threshold
Worked Example: Changing an Exposure Threshold
Suppose a hypothetical study measures a continuous environmental exposure in units per day and records a quantitative health score. Investigators compare two prespecified ways to define “high exposure”: at least 10 units per day, and at least 12 units per day. Genotype is coded \(G=0\) or \(G=1\). For each threshold, the table gives the mean health score in each genotype–exposure cell. Assume each cell contains \(n=50\) independent participants and the pooled within-cell standard deviation is \(s=4\) score points. These data are hypothetical.
| Exposure definition | \(G=0\), low | \(G=0\), high | \(G=1\), low | \(G=1\), high |
|---|---|---|---|---|
| High = at least 10 units/day | 20 | 24 | 21 | 28 |
| High = at least 12 units/day | 20.5 | 23 | 21.5 | 26 |
For the threshold of 10, the exposure difference is \(28-21=7\) score points for \(G=1\), and \(24-20=4\) points for \(G=0\). Thus the interaction contrast is \(7-4=3\) score points. Equivalently, \(28-21-24+20=3\).
For the threshold of 12, the corresponding differences are \(26-21.5=4.5\) points and \(23-20.5=2.5\) points. The interaction contrast is \(4.5-2.5=2\) score points. Both estimates are positive, but the estimated contrast is smaller under the higher threshold.
Under independent cell observations and a common within-cell standard deviation, the standard error of a four-cell difference-in-differences is \(s\sqrt{4/n}\). For either threshold, this gives \(4\sqrt{4/50}=1.131\) score points. Using a large-sample normal approximation, the approximate 95% interval for the threshold-10 contrast is \(3\pm1.96(1.131)=[0.783,5.217]\); for threshold 12 it is \(2\pm1.96(1.131)=[-0.217,4.217]\). These intervals are approximate and rely on the stated variance and sampling assumptions.
The direction of the estimated interaction is stable, while its magnitude and interval relative to zero are not identical. Because the same people may be classified differently under the two thresholds, the estimates are dependent; their difference cannot be assessed by treating the two estimates as independent. These results alone also do not show that the exposure causes the health score to change.
Changing the Exposure Transformation
A threshold discards information about differences within the “low” and “high” groups. When the exposure is measured continuously, one sensitivity analysis is to compare reasonable exposure functions while retaining a clearly specified comparison. For example, a linear function \(f(E)=E\) assumes a constant change in the modeled mean per exposure unit. A transformed function such as \(f(E)=\log(1+E)\), for nonnegative \(E\), allows the mean to change nonlinearly with the original exposure.
The product-term coefficient has different units in these models. With \(f(E)=E\), it is the change in the genotype-specific mean difference per one exposure unit. With \(f(E)=\log(1+E)\), it is the change per one unit of the transformed exposure. A fairer comparison reports the interaction contrast between the same meaningful exposure values in both models.
Worked Example: Comparing Linear and Log-Transformed Exposure
Consider a hypothetical quantitative biomarker, measured in biomarker units, modeled using the same participants, genotype coding, and adjustment variables in two analyses. One model uses \(f(E)=E\); the other uses \(f(E)=\log(1+E)\). Suppose the estimated product-term coefficients and standard errors are \(0.070\) and \(0.040\) for the linear model, and \(0.180\) and \(0.080\) for the log-transformed model. Assume the exposure is measured in units per day and compare \(E=0\) with \(E=4\) units per day.
In the linear model, the interaction contrast is the coefficient multiplied by the exposure difference:
Its standard error is \(0.040(4)=0.160\). The approximate 95% interval is \(0.280\pm1.96(0.160)=[-0.034,0.594]\) biomarker units.
In the transformed model, the relevant transformed exposure difference is \(\log(1+4)-\log(1+0)=\log(5)\). Therefore the interaction contrast is \(0.180\log(5)=0.290\) biomarker units, and its standard error is \(0.080\log(5)=0.129\). The approximate 95% interval is \(0.290\pm1.96(0.129)=[0.037,0.542]\) biomarker units.
The two point estimates for the same endpoint comparison are close and positive. The intervals differ, including in whether they cross zero. That distinction is important: the estimated direction is stable here, but the strength of evidence under a conventional 95% interval is not identical. The coefficient values \(0.070\) and \(0.180\) should not be compared as if they represented the same exposure increment.
These calculations assume the stated conditional-mean models are appropriate over the exposure range being compared. A transformed model is not automatically better because it yields a larger coefficient or a narrower interval. The shape should be scientifically plausible, supported by the observed exposure range, and not selected solely because its result looks more favorable.
Changing the Outcome Scale
For a binary outcome, interaction may be defined on a risk-difference scale or on a multiplicative odds scale, among other choices. These are different estimands. Additive interaction in risks compares differences in probabilities; a logistic product term describes departure from multiplicativity of odds. Neither scale is universally “the” interaction scale. The scientific question determines which is most relevant.
Worked Example: Additive Risk and Multiplicative Odds Interaction
Suppose a hypothetical cohort yields the following outcome risks for four genotype–exposure groups. The values are illustrative proportions, not findings from a real study.
| Genotype and exposure | Risk |
|---|---|
| \(G=0,\ E=0\) | 0.05 |
| \(G=0,\ E=1\) | 0.10 |
| \(G=1,\ E=0\) | 0.08 |
| \(G=1,\ E=1\) | 0.20 |
The additive risk interaction is the difference between exposure-associated risk differences across genotype groups:
Thus, in these hypothetical data, the joint \(G=1,E=1\) group has a risk that departs from additivity by 0.07, or 7 percentage points, relative to the three other risks in this contrast.
For a saturated logistic model with binary \(G\) and \(E\), the product-term coefficient is the logarithm of the ratio of odds ratios. The odds in the four cells are \(0.05/0.95=0.05263\), \(0.10/0.90=0.1111\), \(0.08/0.92=0.0870\), and \(0.20/0.80=0.2500\). The multiplicative odds interaction is therefore
The positive additive interaction and the odds interaction ratio above 1 both indicate departure from their respective null values in the positive direction. Their magnitudes are not interchangeable: 0.07 is a risk-scale difference, while 1.362 is a ratio of odds ratios. Point estimates without uncertainty do not establish that either departure is distinguishable from sampling variation.
How to Summarize Stability
A useful report does more than list alternative \(p\)-values. For each analysis, state the exposure definition or transformation, outcome scale, target contrast, estimate, uncertainty, and relevant units. Then separate conclusions that are stable from those that depend on the choice. For example, the sign of an estimate may persist while its magnitude, interval, or practical interpretation changes.
Name the genotype contrast, exposure comparison, outcome, and scale before presenting alternatives.
Use exposure thresholds or transformations justified by measurement, subject-matter knowledge, or plausible model forms, rather than selecting them after inspecting which gives the strongest result.
For transformed exposures, calculate predicted interaction contrasts at the same original exposure values. For different outcome scales, label the distinct estimands instead of treating their coefficients as competing versions of one number.
Report whether direction, magnitude, uncertainty, and practical interpretation persist. If one changes, say which choice drives the difference and avoid reducing the comparison to whether an interval crosses zero.
Common Mistake: Treating Every Coefficient as the Same Quantity
A common error is to say that an interaction “replicated” because two product-term coefficients are both positive, or that it “disappeared” because one interval includes zero. Coefficient signs are meaningful only after checking how genotype, exposure, and outcome are coded. Coefficient magnitudes across exposure transformations or outcome scales may represent different units and estimands. Also, an interval crossing zero is not proof of no interaction; it indicates that the data and model do not rule out the null value at that interval level.
The correction is to define the contrast first, compute or report it on the relevant scale, and compare its estimate and uncertainty across analyses. When a choice changes the estimand, explain that change rather than presenting it as a mere robustness check of an identical quantity. Sensitivity analysis clarifies dependence on modeling choices; it does not remove bias from unmeasured confounding, measurement error, selection, or incorrect model assumptions.
Check Your Understanding
Answer each question by identifying the contrast or assumption that matters.
- Why does changing a binary exposure threshold generally change the target interaction contrast, rather than only changing its precision?
- In a model containing \(Gf(E)\), what exposure comparison corresponds to the product-term coefficient, and why can the coefficient not be compared directly across transformations?
- For a quantitative outcome, explain why the product-term coefficient in a linear mean model equals a difference-in-differences of four cell means.
- Why can additive risk interaction and multiplicative odds interaction have different magnitudes without being contradictory?
- If two defensible models have positive interaction estimates but only one approximate 95% interval excludes zero, which parts of the conclusion are stable and which are sensitive?