An interaction model is a statistical model that tests whether the relationship between one variable and an outcome changes depending on a third variable. In a standard regression model, each predictor has its own independent effect on the outcome. An interaction model adds a step: it asks whether two predictors combine in a way that’s different from simply adding their individual effects together. This distinction matters in medicine, genetics, social science, and any field where the effect of one factor might depend on the presence or level of another.
How Interactions Differ From Main Effects
In a basic regression, you might find that both exercise and age independently predict blood pressure. Those are “main effects,” each variable contributing its own piece to the prediction. An interaction model goes further by asking: does the effect of exercise on blood pressure differ for younger versus older people? If it does, there’s an interaction between exercise and age.
More formally, an interaction is a type of third-variable effect that assesses whether the relationship between a predictor and an outcome is modified by another variable. The third variable doesn’t just have its own separate influence. It changes the strength, direction, or both of the relationship you’re already studying. Without testing for this, you might report a single average effect of exercise on blood pressure that’s actually misleading, because it’s strong in one age group and weak in another.
Additive and Multiplicative Scales
Researchers measure interactions on two different scales, and the choice affects what they find. On the additive scale, the question is whether the combined effect of two exposures is greater or smaller than the sum of their individual effects. On the multiplicative scale, the question is whether the combined effect is greater or smaller than the product of their individual effects. It’s possible to find an interaction on one scale but not the other, which is why a tutorial from the Harvard T.H. Chan School of Public Health recommends presenting both measures whenever possible.
The additive scale tends to be more useful for public health decisions. If an intervention prevents more disease in one subgroup than another on the additive scale, that tells you where to target resources for the greatest impact. The multiplicative scale is more commonly reported in practice because standard software tools like logistic regression naturally produce multiplicative results, making it the easier model to fit. Neither scale is inherently “correct.” They answer slightly different questions about the same underlying data.
What Interactions Look Like in Practice
One of the most studied interaction models in genetics involves a variant in the serotonin transporter gene and life stress. A landmark study using the Dunedin Longitudinal Study, a birth cohort of over 1,000 children followed for two decades in New Zealand, found that people carrying the short variant of this gene were more susceptible to depression after stressful life events than people carrying the long variant. Stress alone increased depression risk. The gene variant alone had a modest effect. But the combination produced a risk greater than you’d expect from adding those two effects together. That’s a gene-environment interaction: the effect of stress on depression depends on your genotype.
A similar logic applies to retinoblastoma, a type of eye cancer. People with a mutation that impairs DNA repair are far more susceptible to UV radiation damage than people without it. The environmental exposure (UV light) and the genetic factor (impaired repair) interact, meaning the radiation’s effect on eyesight depends heavily on whether the mutation is present.
Drug combinations offer another clear example. Some drug pairs are synergistic, meaning their combined effect exceeds what you’d predict from each drug alone. Targeting two closely related signaling pathways in cancer cells, for instance, often produces synergistic results. Other combinations are antagonistic: the antibiotic combination of a DNA replication inhibitor and a ribosome inhibitor, which might seem like it should work well, actually produces weaker results than expected. Whether a drug combination is synergistic or antagonistic often depends on the biological network the drugs are acting on, not just their individual potency.
Reading an Interaction Plot
Interaction effects are often displayed as line graphs where two lines represent different groups or levels of the modifying variable. The key feature to look for is whether the lines are parallel. Parallel lines mean no interaction: the effect of the main predictor is the same regardless of the modifier. Non-parallel lines suggest an interaction.
When the lines diverge but don’t cross, researchers call it an ordinal interaction, because the ranking of the groups stays the same across all values. When the lines actually cross within the observed data range, it’s called a disordinal or crossover interaction, meaning one group has a higher outcome at low values of the predictor, but the other group takes over at high values. This crossover point can be calculated directly from the regression coefficients. One important caution: all interactions will appear to cross if you extend the graph far enough beyond your actual data. A crossover that falls outside the range of your observed data shouldn’t be interpreted as meaningful.
Interpreting the Interaction Coefficient
In a regression output, the interaction term has its own coefficient, separate from the main effects. This coefficient tells you how much the slope of one predictor changes per unit of the other predictor. If you’re modeling test scores as a function of study hours and teaching method, the interaction coefficient tells you how much the benefit of each additional study hour differs between teaching methods. A positive interaction coefficient means the combined effect is larger than the sum of the parts. A negative one means it’s smaller.
When both variables in the interaction are categorical (like treatment group and sex), the interaction coefficient represents the additional deviation from the overall average that comes from being in both categories simultaneously. For example, if a drug works differently in men versus women, the interaction coefficient captures that extra bump, or dip, that you only see when you look at a specific combination of drug and sex together.
Why Interactions Are Hard to Detect
Interaction effects are notoriously difficult to find, not because they’re rare in the real world, but because detecting them requires much larger studies than detecting main effects. In a standard two-by-two design, the sample size needed to detect an interaction is four times the sample size needed to detect a main effect of the same magnitude. This has been verified both mathematically and through simulation studies in mixed-effects regression models.
This fourfold requirement means many studies are underpowered for interaction testing. A clinical trial designed to detect whether a drug works (a main effect) will often lack the statistical power to determine whether the drug works differently in specific subgroups (an interaction). This is one reason the updated CONSORT 2025 guidelines for reporting clinical trials emphasize that subgroup analyses looking for interactions should be prespecified in the trial protocol, not explored after the fact. Post-hoc interaction tests are far more likely to produce false findings. The guidelines also require that interaction results be reported as estimated differences with confidence intervals, not just as yes-or-no p-values.
When Interaction Models Matter Most
Interaction models become essential whenever you suspect that a one-size-fits-all conclusion is hiding important differences. In precision medicine, the goal is to match treatments to patients based on individual characteristics. That’s fundamentally an interaction question: does the treatment effect vary by patient profile? In epidemiology, understanding which populations are most vulnerable to an environmental hazard requires testing whether the hazard interacts with demographic or genetic factors.
Ignoring interactions can lead to real consequences. A drug that appears moderately effective on average might be highly effective in one subgroup and useless, or even harmful, in another. A public health intervention that seems to have no overall effect might work powerfully in a specific community. The interaction model is the statistical tool that reveals these hidden patterns, but only when studies are designed with enough power to find them.