Interaction Contrasts on Continuous Response Scales

Why the Response Scale Matters

When two drugs are studied together, the combination response can differ from what would be expected by combining their separate effects. For a continuous outcome, that comparison can be expressed as a factorial interaction contrast. The contrast is not a scale-free property of the drugs: it asks whether effects combine additively on a specified response scale. A conclusion on the original scale may therefore differ from a conclusion on the log scale.

Consider a two-by-two factorial design. Drug A is either absent or present, as is drug B. Let \(\mu_{ab}\) be the mean outcome for the cell with A status \(a\) and B status \(b\), where \(a,b\in\{0,1\}\). The outcome could be a continuous biomarker, symptom score, or another measured endpoint. The four means must refer to the same outcome definition, measurement time, and target population.

The sign has a direct interpretation only after defining what larger outcomes mean. If larger values indicate greater benefit, a positive contrast means the combination exceeds the additive prediction; if larger values indicate greater harm, its clinical meaning reverses. A nonzero contrast is evidence of nonadditivity on the chosen scale, not by itself proof of a biological mechanism or a universal claim of synergy.

From the Four Cell Means to a Product Term

A linear model with binary indicators \(a\) and \(b\) represents the four cell means as

The product term \(ab\) is zero in three cells and one only when both drugs are present. Substituting each pair of indicator values gives \(\mu_{00}=\beta_0\), \(\mu_{10}=\beta_0+\beta_A\), \(\mu_{01}=\beta_0+\beta_B\), and \(\mu_{11}=\beta_0+\beta_A+\beta_B+\beta_{AB}\). Subtracting the cell means in the interaction contrast yields

Thus the product coefficient is exactly the factorial interaction contrast when the model uses 0/1 indicators and is saturated for the four cell means. The main-effect coefficients are conditional: \(\beta_A\) is A's mean effect when B is absent, and \(\beta_B\) is B's mean effect when A is absent. They are not necessarily average effects across both levels of the other drug.

Worked Example: Estimating an Additive Interaction

In an unbalanced study, the same contrast of estimated cell means remains the estimand, but its standard error must reflect the cell-specific sample sizes and variance estimates. A model-based standard error can be used if its variance assumptions are suitable; robust or design-based methods may be preferable when they are not. Randomization supports a causal interpretation of cell-mean differences for the assigned treatments when treatment implementation, outcome measurement, and follow-up preserve the intended comparison. Missing outcomes, nonadherence, or interference between participants can complicate that interpretation.

Interaction on an Additive or Multiplicative Scale

For a positive outcome, a common alternative is to model its logarithm. Let \(\nu_{ab}=E(\log Y\mid a,b)\), and define the interaction contrast on that transformed scale as \(\Delta_{\log Y}=\nu_{11}-\nu_{10}-\nu_{01}+\nu_{00}\). Exponentiating gives a ratio of ratios:

Here \(\exp(\nu_{ab})\) is the cell's geometric mean, when that quantity is defined. A ratio of ratios equal to 1 corresponds to no interaction on the log scale: the relative effect of A is the same with and without B. This is different from a zero additive contrast in the original outcome units. A log model requires positive outcomes unless a different transformation or model is justified; adding an arbitrary constant to accommodate zero values changes the estimand and needs substantive justification.

Affine transformations behave differently from nonlinear ones. If \(Y^*=c+dY\), then

The constant cancels, and a positive change of units preserves the sign and zero status of the contrast. A nonlinear transformation such as a logarithm does not generally preserve additivity, so both the magnitude and whether the contrast is zero can change. The scale should therefore be chosen to match the scientific question and the outcome's measurement properties, rather than selected only because it produces a preferred interaction result.

Worked Example: Recovering the Contrast From a Regression

Assumptions, Diagnostics, and Alternatives

The contrast is defined by four means and can be calculated without assuming normality. Statistical uncertainty and tests require an appropriate sampling or modeling framework. For the pooled \(t\) analysis illustrated above, observations should be independent, errors should be reasonably normal within cells for small-sample inference, and the cell variances should be sufficiently similar. With larger samples, inference on means is often less sensitive to moderate nonnormality, but severe outliers, dependence, or strong variance differences can still undermine a simple pooled analysis.

Check the design and data before interpreting the result. Confirm that all four treatment combinations are represented, the same endpoint and assessment time are used in every cell, and the outcome coding has the intended direction. Inspect cell-specific distributions, residuals, and variances; look for influential observations, ceiling or floor effects, and uneven missingness. A missing or very small cell makes the contrast unstable or impossible to estimate precisely. A model can return a number despite poor support in the data, so successful computation is not evidence that assumptions hold.

The contrast is useful because it is transparent: it compares observed combination response with an additive prediction on a stated scale. Simple-effect comparisons can show which conditional drug effect differs, but several separate tests do not replace a direct test of the prespecified interaction contrast. A log-scale contrast instead targets multiplicative combination of geometric means. Bliss, Loewe, and Highest-Single-Agent additivity use different reference expectations for combination response; their conclusions should not be treated as interchangeable with a factorial contrast on raw or log-transformed continuous outcomes. None of these comparisons alone establishes a molecular mechanism.

A four-cell contrast also has limits. It summarizes interaction at the particular doses, population, endpoint, and time studied; it does not describe how interaction varies across dose combinations. That broader dose-response question requires a model appropriate to the design and is a separate modeling task. Finally, an estimated contrast near zero does not prove exact additivity: its interval may still include effects large enough to matter clinically. Precision and a clinically meaningful threshold should be considered alongside a p-value.

Common Mistake