Diagnosing Interaction-Parameter Identifiability

Why an Interaction Estimate Can Be Uncertain

A fitted combination model can return a numerical interaction estimate even when the observed data do not pin that estimate down. In that situation, the reported value may depend strongly on small changes in the data, the dose grid or the model assumptions. A useful diagnosis asks not only whether an optimizer converged, but whether substantially different interaction values can also explain the data almost as well.

This tutorial focuses on practical identifiability: how strongly a particular data set constrains an interaction parameter within a specified model. It is conditional on the chosen model, dose levels, response measurements and error assumptions. It is different from asking whether a parameter could be uniquely recovered from ideal, noise-free observations; that broader question requires a separate analysis.

For a concrete starting point, consider a continuous response measured at fixed times under a two-drug dose matrix. Let \(a\) and \(b\) be normalized doses, and let \(y\) be the response in response units. A simple local model is

Here, \(\beta_0\) is the baseline, \(\beta_A\) and \(\beta_B\) are the dose effects in the model, and \(\gamma\) is the interaction coefficient. In this model, the interaction contribution is \(\gamma ab\); its units are response units per product of normalized doses. This equation is a diagnostic example, not a claim that every combination response is linear in dose. Conclusions about identifiability always depend on the response model being appropriate for the data being analyzed.

A diagnosis should account for the design and the response data together. Dose combinations determine which model effects can be separated; replication and measurement precision determine how clearly their response differences can be seen. The standard errors from a local covariance approximation are useful, but profile likelihood can reveal shapes that a single standard error hides.

Profile Likelihood: Vary the Interaction, Refit the Rest

Suppose the responses are independent normal observations with common variance, and the dose values are treated as known. For each fixed candidate value \(g\) of \(\gamma\), maximize the likelihood over the remaining parameters, such as \(\beta_0,\beta_A,\beta_B\), and possibly \(\sigma^2\). The resulting function is the profile likelihood for \(\gamma\). Profiling allows nuisance parameters to adjust as \(\gamma\) changes, so it directly asks how well the best-fitting model can explain the data at each candidate interaction value.

A profile is often displayed as a likelihood-ratio statistic relative to the best fit:

Small \(D(g)\) means that fixing \(\gamma=g\) costs little fit after the other parameters are re-estimated. Under regular likelihood assumptions, an approximate 95% confidence set for one parameter contains values with \(D(g)\leq 3.84\), the 95th percentile of a chi-square distribution with one degree of freedom. This approximation can be unreliable in small samples, near parameter boundaries, or when the likelihood is irregular. A confidence set is a procedure with a long-run coverage interpretation; it does not assign a 95% probability to a fixed parameter being in the particular interval computed.

For a linear Gaussian model with known \(\sigma\), profiling gives a particularly clear justification. Write the design matrix columns for the nuisance terms as \(Z\), and the interaction column as \(w\), with entries \(w_i=a_i b_i\). Project \(w\) onto the part not explained by the nuisance columns, giving \(w_\perp\). Once the nuisance parameters have been re-estimated, a change \(\gamma-\hat{\gamma}\) leaves a residual change proportional to \(w_\perp\). The increase in squared error is therefore \((\gamma-\hat{\gamma})^2 w_\perp^\mathsf{T}w_\perp\), and the likelihood-ratio statistic is

This result shows why dose geometry matters: if the interaction column is almost explained by the other model columns, \(w_\perp^\mathsf{T}w_\perp\) is small, so the profile is shallow and many interaction values fit similarly. If the variance is estimated rather than known, the profile also re-estimates it; the exact likelihood calculation changes, but the same practical diagnosis applies.

Parameter Correlations and Dose-Design Geometry

A second diagnostic is the correlation matrix of parameter estimates, usually obtained from the inverse of the local information matrix or from an appropriate covariance estimate. A large absolute correlation between \(\hat{\gamma}\) and a main-effect estimate means that changes in those parameter estimates can compensate for each other locally. Correlation close to \(1\) or \(-1\) is a warning, not a universal pass/fail threshold: it should be interpreted alongside the profile, parameter uncertainty, model fit and intended use of the estimate.

Correlations among dose-design columns can help explain parameter correlations, but they are not always identical to them. In a linear model, the covariance depends on the design columns after the other terms have been accounted for. In nonlinear models, it also depends on the fitted parameter values and response sensitivity. Therefore, inspect parameter correlations from the fitted model where possible, and use the dose grid to understand why a problem may be occurring.

This example deliberately conditions on other model terms being known. In a full analysis, uncertainty in those terms must also be included; conditioning on estimated values can understate uncertainty. The example isolates one design problem: if \(b\) barely varies among the combination observations, the columns \(a\) and \(ab\) have nearly the same pattern.

Choosing Dose Levels and Response Measurements

Improving an interaction estimate usually requires making its response pattern distinguishable from the patterns of the baseline and main effects. In the bilinear example, the interaction column is \(ab\), so useful designs include combination doses that vary both drugs, rather than only increasing one drug while holding the other close to a single level. Including suitable single-agent and baseline conditions also helps anchor the main effects and intercept. The precise best design depends on the model and scientific question; simply adding the highest doses is not a general solution.

Dose selection is only one way to improve precision. Replicate measurements can reduce uncertainty when error is predominantly random, while improved assay precision can reduce \(\sigma\). Measuring a response that is sensitive to the proposed interaction may be more informative than collecting many precise measurements of a response that barely changes across the dose matrix. In dynamic combination models, measurements at informative times may also matter; the schedule and time course must match the model being estimated.

A useful design comparison evaluates expected information or the anticipated profile for \(\gamma\) under candidate designs, not just the number of dose groups. Check that dose levels span the region in which the model is meaningful, that the response is measurable there, and that the chosen observations help distinguish \(\gamma\) from nuisance parameters. Predictions outside the observed dose range can be especially sensitive to the assumed model, even if a fitted profile is narrow within the range.

Diagnostics, Alternatives, and Limitations

A practical workflow is to fit the stated model, inspect the interaction profile over a scientifically plausible range, and examine parameter correlations and uncertainty estimates. Then check that each profile point was successfully optimized, that the likelihood calculation uses the same data and assumptions as the main fit, and that the fitted responses are compatible with the observations. A failed optimizer can create an artificial kink or apparent second optimum; rerun from multiple starting values before interpreting such features.

• Flat or very broad profile: a wide range of interaction values costs little fit, indicating weak practical information in the data.
• Asymmetric or irregular profile: the symmetric standard error approximation may be misleading; report the profile-based uncertainty and investigate boundaries or multiple optima.
• Strong parameter correlation: related coefficients trade off in the fitted model. Check which dose patterns or response measurements could separate them.
• Unexpectedly narrow profile: verify that nuisance parameters were genuinely re-estimated and that constraints or fixed parameters have not created unjustified certainty.
• Systematic residual patterns: precision calculations cannot rescue a model that misses the response behavior. Reconsider whether the model form and error assumptions are adequate before interpreting the interaction estimate.

Profile likelihood has an advantage over a local standard error: it can show asymmetry, flat regions and multiple plausible values. A covariance matrix is quicker and useful for detecting local trade-offs, but it summarizes uncertainty near the fitted optimum and can be unreliable when the likelihood is far from quadratic. Bootstrap resampling can assess sampling variability under a specified resampling scheme, but it does not repair a poor dose design or an unsuitable model. No one diagnostic replaces checking the data, fit and assumptions.

These conclusions are conditional. The simple Gaussian examples assume independent observations, constant variance, known doses, and an adequate mean model. Real response measurements can be correlated, heteroscedastic, bounded or subject to uncertainty in dose and exposure. Such features should be represented in the likelihood when relevant; treating them as if they were absent can distort both the profile and parameter correlations. Also, an identifiable fitted interaction is not by itself evidence of a particular biological mechanism: it is an estimate within the chosen model, data and response definition.