Population Variability in Drug-Interaction Effects

Why Population Variability Matters

A combination may not have the same interaction effect for every individual. Differences in physiology, disease severity, concomitant treatment, or other measured characteristics can make the response to the same pair of exposures stronger for some people and weaker for others. A single interaction estimate can therefore conceal clinically relevant variation: it may describe an average tendency while poorly predicting responses for particular patients.

The key modeling task is to distinguish two sources of systematic differences in interaction effects. Covariates explain variation associated with measured characteristics. Between-subject variability represents remaining differences among individuals after accounting for those covariates. The resulting model can predict both a subject-specific response and the distribution of responses in a target population.

A Random Interaction Coefficient

Consider a response measured on a scale from 0 to 1, where \(m_i\) is its expected value for individual \(i\). Let \(x_A\) and \(x_B\) be the exposures to drugs A and B, each scaled by a prespecified reference concentration. One possible response model is

The first three terms describe the baseline and the individual-drug contributions on the logit scale. The final term is the combination interaction term. In this model, \(\theta_i=0\) means there is no explicit cross-product term on the logit scale; it does not mean that all possible definitions of additivity are satisfied. A positive \(\theta_i\) increases the modeled response relative to the model without that term when both exposures are positive, while a negative value decreases it. This interpretation is conditional on the response coding and the specified model.

A simple population model for the interaction coefficient is

Here \(\theta_0\) is the interaction coefficient at the reference covariate values, \(\mathbf{z}_i\) is a vector of covariates (often centered), and \(\boldsymbol{\gamma}\) contains their effects on the interaction coefficient. The individual random effect \(\eta_i\) has mean zero and variance \(\omega^2\). Thus, \(\omega\) measures residual between-subject variability in the interaction coefficient on its modeled scale. If the coefficient has units, those units are carried by \(\theta_0\), \(\boldsymbol{\gamma}\), and \(\omega\); with dimensionless exposures, they are on the logit scale per unit of \(x_Ax_B\).

This formulation assumes that, conditional on the covariates and the individual random effect, the response mean follows the stated model. It also assumes the specified distribution for \(\eta_i\), and that its mean-zero component is independent of the included covariates. Other model parameters are treated as fixed here for clarity; they can also vary between individuals, but adding random effects requires enough information to estimate them. The observation model must be appropriate for the measured response and must describe variation around \(m_i\); it is not interchangeable with the distribution of \(\theta_i\).

From Individual Predictions to a Population Prediction

For a specified individual, insert that person's covariates and interaction coefficient into the response equation. A population-average prediction instead averages the response over the distribution of covariates and random effects in the target population. In general, these are not the same calculation: because the response function is nonlinear, applying it to the mean interaction coefficient need not give the mean response.

A brief derivation clarifies why the population average is not generally the response at the average coefficient. At fixed exposures and covariates, write the response as \(f(\theta)=\operatorname{logit}^{-1}(c+\theta x_Ax_B)\), where \(c\) collects the other terms. A second-order Taylor expansion around the mean coefficient \(\mu_\theta\) gives

Taking expectations, the linear term vanishes because \(E(\theta-\mu_\theta)=0\), while the squared term has expectation \(\operatorname{Var}(\theta)\). Consequently,

Unless the response is locally linear or the variability is negligible, this second term need not be zero. The approximation is a justification, not a replacement for calculating the population average from the full distribution when accurate predictions are needed. The direction and size of the difference depend on the exposures and the curvature of the response function.

Population Averaging in a Hypothetical Mixture

A simple finite mixture makes the distinction between individual and population predictions explicit. Suppose a hypothetical target population has three interaction-coefficient values with stated proportions. Its average response is obtained by predicting for each group and weighting those predictions by their proportions.

Estimation, Diagnostics, and Practical Choices

Estimating a random interaction coefficient requires data that inform individual differences in the combination effect. Repeated observations across combination exposures can help distinguish a subject-specific interaction pattern from observation-level noise. Data concentrated at single-drug exposures provide little direct information about a cross-product interaction, because \(x_Ax_B=0\) when either exposure is zero. A design with adequate variation in both exposures and in their combinations is therefore important.

Covariate effects also require informative variation across subjects. If a covariate is nearly constant, or if it is strongly associated with another covariate, its effect on \(\theta_i\) may be difficult to distinguish. A random-effect distribution that is too flexible for the amount of data can likewise produce unstable estimates. These are information limitations, not problems that can be solved merely by reporting more decimal places.

Useful checks include examining whether the estimated between-subject variability is near zero or implausibly large, whether covariate estimates are highly uncertain, and whether predicted subject-specific interaction effects are dominated by model assumptions rather than observations. Compare observed combination responses with predictions across exposure ranges and covariate groups. If systematic discrepancies remain, the population-variability model may be inadequate. Such checks do not by themselves identify the cause; an incorrect mean response surface, an unsuitable distribution for random effects, or unmodeled covariates can produce similar patterns.

Several approaches are possible, and each answers a different question:

A random coefficient is useful when the aim includes predicting heterogeneity, but it does not automatically make individual predictions precise. With limited repeated information per subject, the model may estimate population variability more reliably than any one person's interaction effect. Also, \(\omega\) describes variability in the interaction parameter, not variability in observed responses; the latter includes the observation-level error as well.

Common Pitfall

A common mistake is to compute the response for the average interaction coefficient and call it the population-average response. This is generally incorrect for a nonlinear response function: \(E\{f(\theta)\}\) is not generally equal to \(f\{E(\theta)\}\). The correction is to average predicted responses over the modeled distribution of coefficients and covariates in the target population. Another mistake is to interpret a covariate effect on \(\theta_i\) as a fixed response-scale difference; the response-scale impact depends on exposures and the other model terms.

Population predictions also depend on which population is intended. Changing the distribution of covariates can change the average response even when the model coefficients remain the same. State the target population and the exposure combination whenever reporting an averaged prediction. Finally, the random-effect distribution is a model assumption, not a direct observation of every person's interaction effect; predictions outside the exposures and covariate range represented in the data need particular caution.