Identifiability of Combination-Response Parameters

Why Identifiability Matters

A combination-response model can fit observed data closely and still fail to determine its parameters. For example, a weak response at the doses studied could reflect low maximum effect, low potency at those concentrations, or an interaction that offsets the individual drug effects. These explanations are scientifically different, even if they predict nearly the same responses in the observed dose range.

Identifiability asks whether the available response data can distinguish parameter values under a specified model. It is not the same as goodness of fit: a small residual error does not establish that individual estimates are meaningful. Here the focus is on how coverage of each drug alone and in combination affects estimation of potency, maximum effect, and interaction in a two-drug Emax response surface.

A Combination-Response Model and Its Assumptions

To make the confounding problem concrete, consider a bounded empirical response surface. Let \(C_1\) and \(C_2\) be the concentrations of drugs 1 and 2 at the response measurement time. Define each drug’s fractional Emax response as \(f_i=C_i/(K_i+C_i)\), where \(K_i>0\) is its concentration producing half of its own modeled maximum effect. The response model is

Here \(E_0\) is baseline response, \(M_i\) is the drug \(i\) monotherapy effect increment at saturating concentration, and \(\gamma\) is the combination interaction increment on the response scale. A positive \(\gamma\) raises response above the additive prediction of the two modeled monotherapy increments; a negative \(\gamma\) lowers it. Its units are the same as the response \(Y\). This is a particular empirical interaction model, not a universal definition of synergy or antagonism.

The model assumes that the concentrations used are appropriate predictors of the response, the response has the stated mean structure, and the error \(\varepsilon\) has mean zero. For the ordinary least-squares examples below, errors are also taken to be independent with constant variance. The model treats concentrations as known and does not account for population variability. It also assumes that the monotherapy response follows the stated Emax form and that the interaction contribution is proportional to \(f_1f_2\). Identifiability conclusions are conditional on these assumptions.

Monotherapy observations have a special role. When \(C_2=0\), \(f_2=0\), so the response reduces to \(E_0+M_1f_1\); neither \(M_2\) nor \(\gamma\) contributes. When \(C_1=0\), the corresponding reduction is \(E_0+M_2f_2\). Thus, adequate single-drug concentration coverage can inform baseline, potency, and maximum effect separately from the interaction term. Combination observations are then needed to inform \(\gamma\).

Why Dose Coverage Determines Separation

The model’s sensitivities show how each observation informs the parameters. For one predicted response, the sensitivities to \(M_1\), \(M_2\), and \(\gamma\) are \(f_1\), \(f_2\), and \(f_1f_2\), respectively. The sensitivity to \(K_1\) is

If all observed \(C_1\) values are far below \(K_1\), then \(f_1\) is approximately \(C_1/K_1\). In that range, the monotherapy contribution is approximately \(M_1C_1/K_1\), so the data mainly inform the ratio \(M_1/K_1\), not each parameter separately. If all \(C_1\) values are far above \(K_1\), then \(f_1\) is close to one: the responses can inform \(M_1\), but have little sensitivity to the exact value of \(K_1\). Concentrations across the transition region are important for estimating potency.

The interaction term is similarly difficult to estimate if either drug is rarely present at concentrations giving appreciable \(f_i\). When both \(f_1\) and \(f_2\) are small, their product is smaller still. In addition, a narrow combination range may make the interaction contribution resemble a change in one or both monotherapy effects. A full set of monotherapy observations and combinations that cover a meaningful range of both drugs helps distinguish these patterns.

A Short Identifiability Derivation

The single-dose confounding in the first example follows directly from the Emax equation. For a baseline-corrected response increment \(\Delta\) at one positive concentration \(C\),

For every positive \(K\), this equation supplies an \(M\) that gives the same \(\Delta\). Thus the mapping from the two parameters to this one predicted response is not one-to-one. With observations over multiple concentrations, the local information about the parameters depends on whether their sensitivity patterns differ across those observations. If sensitivity patterns are nearly proportional, changing one parameter can be offset by changing another with little change in predicted responses. In that case, the model may be structurally identifiable over a richer set of concentrations but practically weakly identified by the available data.

The same logic extends to the whole surface. Form the sensitivity matrix whose rows contain the derivatives of predicted responses with respect to the parameters. If columns are linearly dependent, the observations cannot distinguish the corresponding parameter changes locally. If columns are nearly dependent, estimates can be very imprecise. This is a local diagnostic: it does not prove that the model is correct, and it does not rule out other parameter combinations that fit equally well elsewhere.

Profiles and Simulated Surfaces as Diagnostics

A parameter profile makes a trade-off visible. To profile \(\gamma\), fix it at a sequence of candidate values; at each value, re-estimate the other parameters and record the best fit. A sharply defined optimum with fit worsening on both sides indicates that the data constrain \(\gamma\). A broad, nearly flat profile indicates weak information. A long ridge in a two-parameter profile, such as \(M_1\) against \(K_1\), indicates that combinations of those parameters fit similarly well. Profile likelihoods provide an analogous diagnostic when parameters are estimated by likelihood rather than least squares.

Simulated response surfaces offer a complementary check. Using the fitted parameters, predict response over a grid of \(C_1\) and \(C_2\), and compare the surface with the observed dose locations. If observations occupy only a narrow band of that grid, very different parameter sets may make nearly indistinguishable predictions on the band while diverging elsewhere. That disagreement is evidence that estimates depend strongly on extrapolation rather than on observed information. Simulations here diagnose what the fitted model implies; they do not replace data in the poorly covered regions.

Comparing Modeling Approaches and Recognizing Limits

The empirical product-term model separates two sources of information: monotherapy curves describe individual potency and maximum effects, while combination departures from those curves inform \(\gamma\). A mechanistic response-surface model may instead encode a particular interaction mechanism through shared or compound-specific potency terms. Such a model can be more interpretable if its assumptions are scientifically justified, but it does not automatically solve identifiability: its parameters still require dose conditions that make their effects distinguishable.

A simpler alternative is to set \(\gamma=0\) and use an additive response prediction. This removes one parameter and may be appropriate when supported by the scientific question and data, but it cannot estimate an interaction. Another option is to fix a poorly informed parameter using independent evidence. This can stabilize estimation, but the resulting estimates are conditional on that fixed value. Reparameterization can clarify which combinations of parameters the data actually estimate; it cannot create information absent from the observations.

An important limitation is that a well-identified parameter under this equation is not automatically a biologically universal interaction effect. The meaning of \(\gamma\) depends on the chosen response scale, Emax structure, and concentration definition. A different response-surface model can assign a different numerical meaning to interaction. Identifiability also does not establish model adequacy: a precise estimate under a misspecified surface remains misleading.

Common Pitfall

• Wrong reasoning: “The optimizer converged and the fitted surface passes near every observation, so all six parameters are identified.” Convergence and fit describe the optimization and predictions, not whether alternative parameter values fit nearly as well.
• Correction: Inspect dose coverage, parameter uncertainty and correlation, sensitivity patterns, and profiles. Check whether observations include informative monotherapy concentrations and combinations where both fractional effects are appreciable. Treat flat profiles or large changes in estimates under small data perturbations as evidence of weak practical identifiability.
• Wrong reasoning: “A positive fitted \(\gamma\) proves a general biological synergy.” The parameter is a departure from additivity on this model’s response scale, and its estimate may be confounded or model-dependent.
• Correction: Describe the result in the context of the specified model and dose range, and separate estimation from broader mechanistic claims.