Why Diagnose a Combination-Response Model?
A combination-response model is useful only to the extent that its predictions describe the responses observed across the dose combinations of interest. A model can fit the average response reasonably well and still miss a systematic feature: it may predict too little benefit at combinations, miss saturation at high doses, or leave residuals that vary with drug exposure. These failures matter because estimated interaction effects and predictions at unobserved combinations depend on the model’s structure.
This tutorial focuses on diagnosing those failures for continuous responses. The goal is not to declare a combination synergistic or antagonistic from a plot. It is to find where a specified model is inadequate, describe the pattern and its uncertainty, and identify which parts of the model need scrutiny.
Before interpreting residuals, specify the response scale, the model’s predictors, and the direction of benefit. For example, if larger values mean greater symptom relief, a positive residual means more relief than predicted. If larger values mean greater symptom severity, the same positive residual means worse symptoms than predicted. Residuals diagnose predictions on the scale actually modeled; changing the scale can change the apparent pattern.
What Residual Patterns Can Tell You
Suppose a null model \(m_0(D_1,D_2;\theta)\) predicts the mean response from doses \(D_1\) and \(D_2\), with parameters \(\theta\). It might encode an additivity assumption on a prespecified response scale. The diagnostic question is whether observed responses fluctuate around those predictions without a systematic pattern. Under a correctly specified conditional-mean model with true parameters \(\theta_0\),
where \(Z_i\) represents any other covariates included in the model. This follows by subtracting the model prediction from the conditional mean: if \(E(Y_i\mid D_{1i},D_{2i},Z_i)=m_0(D_{1i},D_{2i};\theta_0)\), their difference is zero. Thus, smooth or repeated residual patterns across dose conditions are evidence to investigate, not proof that a particular biological mechanism is present.
In practice, parameters are estimated, so fitted residuals are not independent of the fitting process and may look artificially small in influential regions. Replicate observations, predictive checks on data not used for fitting, and uncertainty intervals around predictions help assess whether a visible pattern is larger than expected from noise and estimation uncertainty. For repeated measurements within a participant, residuals are also correlated; treat the participant as the relevant clustering unit rather than counting every measurement as independent evidence.
Three views are especially useful. Use a dose-surface residual plot to locate regions of over- or underprediction; an observed-versus-predicted plot to assess calibration across the full response range; and stratified checks to see whether residual patterns differ by exposure, time, or other relevant covariates. These views complement rather than replace one another.
Dose-Surface Residuals and Additivity
A dose-surface plot places the two drug doses on its axes and represents residual size and sign by color, labels, or vertical height. Plot single-agent conditions as well as combinations, and show placebo or baseline conditions where relevant. If replicated observations are available, display individual residuals or summarize them within dose cells with uncertainty; a surface built only from averages can conceal substantial variability.
A broad region of positive residuals at combination doses means the model tends to underpredict the response there; a broad negative region means it tends to overpredict. A pattern concentrated at combinations, while single-agent residuals remain centered near zero, is consistent with a failure in the model’s combination structure. It does not by itself establish a specific interaction, since another omitted feature could generate the same pattern.
Worked Example: Residuals Concentrated at Combination Doses
Suppose a hypothetical continuous endpoint is coded so that larger values mean greater benefit. A prespecified additive model predicts response in endpoint units as \(\hat{m}=10+3x+2y\), where \(x\) and \(y\) are normalized doses of drugs A and B. The single-agent conditions fit closely. The table gives hypothetical mean responses and the resulting residuals at four combination conditions.
| \(x\) | \(y\) | Prediction \(10+3x+2y\) | Observed mean | Residual |
|---|---|---|---|---|
| 1 | 1 | 15.0 | 17.0 | 2.0 |
| 1 | 2 | 17.0 | 19.1 | 2.1 |
| 2 | 1 | 18.0 | 19.0 | 1.0 |
| 2 | 2 | 20.0 | 22.0 | 2.0 |
For example, at \(x=1,y=2\), the prediction is \(10+3(1)+2(2)=17.0\), so the residual is \(19.1-17.0=2.1\) endpoint units. The mean residual across these four combinations is \((2.0+2.1+1.0+2.0)/4=1.775\) units. All four residuals are positive, whereas the single-agent conditions fit closely. This is a localized pattern of underprediction at combinations, and it motivates closer examination of the combination-response structure. Without replicate-level variability or predictive uncertainty, these numbers do not establish how unlikely the pattern is under the model, and they do not by themselves justify a claim of synergy.
When the outcome is measured repeatedly, or when the model has unequal residual variance across the response range, raw residuals may not be comparable across conditions. A standardized residual divides a residual by an estimate of its appropriate standard deviation. That denominator must reflect the observation’s error variance and, for a prediction, relevant parameter uncertainty; a single common denominator is not automatically appropriate. In nonlinear models, use diagnostics suited to the fitted likelihood and design rather than applying a linear-model rule without checking its assumptions.
Observed Versus Predicted Responses and Curvature
Plot observed responses against predicted responses, add the identity line \(Y=\hat{m}\), and look for systematic departures. A well-calibrated model should not consistently fall above or below that line in a region. Curvature in this plot may indicate a missing nonlinear response feature, although a plot against predicted response alone cannot show which dose or covariate is responsible. Color or facet the points by dose combination, and pair this plot with residuals against each dose and against predicted response.
Worked Example: Detecting High-Dose Curvature
Consider a hypothetical response coded so that larger values indicate greater benefit. A candidate model assumes a linear increase of 3 endpoint units per unit of total normalized dose \(t=x+y\), predicting \(\hat{m}=3t\). The observations below are hypothetical means from single-agent and combination conditions selected to span the same range of \(t\).
| Total normalized dose \(t\) | Prediction \(3t\) | Observed mean | Residual |
|---|---|---|---|
| 0 | 0 | 0.0 | 0.0 |
| 1 | 3 | 3.2 | 0.2 |
| 2 | 6 | 5.8 | -0.2 |
| 3 | 9 | 7.2 | -1.8 |
| 4 | 12 | 7.8 | -4.2 |
At \(t=3\), the prediction is \(3(3)=9\), giving residual \(7.2-9=-1.8\) units. At \(t=4\), the residual is \(7.8-12=-4.2\) units. The residuals change from near zero at low doses to increasingly negative values at high doses, indicating overprediction at the upper end. If that pattern appears in single-agent conditions as well as combinations, dose-response curvature is a more immediate concern than a combination-specific departure. A saturating alternative may be worth checking, but this diagnostic alone does not determine the correct nonlinear model.
A useful check is to plot residuals separately against \(D_1\) and \(D_2\), including the single-agent arms. Curvature that appears along one drug’s axis may be obscured by a plot against total dose. Conversely, residuals that are small on each single-agent axis but patterned only in the interior of the dose surface point toward a combination-structure issue. These are clues for model revision, not mechanical classification rules.
Stratifying Checks by Exposure
Dose is not always an adequate proxy for pharmacologically relevant exposure. If a drug interaction changes the victim drug’s concentration, participants at the same administered dose may have different exposures and hence different expected responses. A dose-based model can then leave residuals associated with concentration even if the concentration-response relation is otherwise plausible. Stratifying or coloring residual plots by measured or estimated exposure can reveal this pattern.
Worked Example: Residuals Track Exposure at the Same Dose
In a hypothetical study, participants receive the same nominal combination dose. A dose-only model predicts a benefit of 6 endpoint units for everyone. Four participants have different estimated victim-drug exposures \(C\), in concentration units, and the observed benefits shown below.
| Exposure \(C\) | Dose-only prediction | Observed benefit | Dose-only residual | Exposure-model prediction \(2+C\) | Exposure-model residual |
|---|---|---|---|---|---|
| 2 | 6 | 4 | -2 | 4 | 0 |
| 3 | 6 | 5 | -1 | 5 | 0 |
| 5 | 6 | 8 | 2 | 7 | 1 |
| 6 | 6 | 9 | 3 | 8 | 1 |
For \(C=5\), the dose-only residual is \(8-6=2\) endpoint units. Across the four observations, dose-only residuals are \(-2,-1,2,3\): they are negative at lower exposures and positive at higher exposures. The illustrative exposure model predicts \(2+C\), producing predictions \(4,5,7,8\) and residuals \(0,0,1,1\). The exposure-based pattern is smaller here, but this tiny example cannot establish that the model is adequate or that exposure causes the response difference. In real analyses, account for uncertainty in estimated concentrations and assess whether exposure is confounded with participant characteristics, sampling time, or other predictors.
A practical stratified check is to divide observations into prespecified exposure ranges and compare residual distributions or observed-versus-predicted calibration across those ranges. Avoid choosing cut points solely because they make a pattern look strongest. When exposure estimates are uncertain, treating them as exact can make residual-versus-exposure patterns misleading; diagnostics should reflect that uncertainty rather than silently ignoring it.
How to Conduct and Interpret the Checks
Specify the response scale, the direction of benefit, the doses or exposures used, and whether predictions are conditional on participant-level effects or covariates.
Confirm that dose labels, units, baseline conditions, and response times are correct. Mark dose combinations with few or no observations; a smooth-looking surface cannot supply evidence in unsupported regions.
Plot residuals over the dose surface, observed against predicted responses, and residuals against each dose and relevant exposure. Show uncertainty or replicate-level variation where available.
Compare model predictions with replicate, held-out, or otherwise predictive checks when the design permits. Look for a pattern that persists beyond one influential observation or one sparsely sampled cell.
Check whether curvature, exposure dependence, or combination structure accounts for the pattern, and compare predictive performance as well as fit. Preserve the original model as a reference rather than selecting a more flexible model solely because it fits the same data better.
The main alternatives answer different questions. Residual plots are flexible visual checks that can locate patterns but depend on judgment and can be noisy. A formal comparison of nested models can assess a specified extension under its statistical assumptions, but it tests that extension rather than every possible form of misspecification. A more flexible surface can reveal departures yet may overfit sparse dose grids and produce unstable predictions between observed combinations. Predictive checks assess performance on new or held-out observations when such data are available, but they may have low precision in small studies. These approaches are complementary, not substitutes for sound design.
Residual diagnostics also have limitations. A flat residual plot does not prove that the model is correct: different errors can cancel, and sparse dose coverage may hide important failures. Conversely, random variation can create an apparent patch on a small dose surface. A misspecified variance model can distort the apparent prominence of residuals, while an incorrect mean model can be hard to distinguish from omitted covariates, time effects, or concentration error. Interpret plots alongside the design, uncertainty, and scientific plausibility.
Common Pitfall: Calling Every Residual Pattern “Synergy”
Wrong reasoning: “Combination residuals are positive, so the drugs are synergistic.” This jumps from a model’s underprediction to a mechanistic or comparative claim. The pattern could instead arise from unmodeled saturation in the single-agent effects, an incorrect response scale, exposure differences, baseline imbalance, or a small number of influential observations.
Correction: State the narrower diagnostic conclusion first: for example, “the prespecified model underpredicts the observed benefit in these combination-dose cells.” Then check whether the pattern remains after examining single-agent curvature, exposure, response coding, uncertainty, and design support. A residual plot identifies where predictions fail; it does not, by itself, identify why they fail or establish a causal mechanism.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A response is coded so higher values mean more symptom relief. A dose-surface region has mostly negative residuals. What does that indicate, and what does it not establish?
- Residuals are near zero in single-agent conditions but mostly positive in a well-sampled combination region. Name a reasonable diagnostic interpretation and one alternative explanation to check.
- A residual-versus-dose plot is near zero at low doses and increasingly negative at high doses in both single-agent arms. What model feature should be investigated first?
- At a fixed administered dose, residuals increase with estimated victim-drug concentration. Give one interpretation and one reason not to treat the pattern as proof of an exposure effect.
Solutions. 1. The model tends to overpredict symptom relief in that region because observed values are below predictions. This alone does not establish antagonism or identify the reason for the failure. 2. The pattern is consistent with underprediction by the model’s combination structure. Check, for example, whether exposure differs systematically in that region or whether a single-agent curvature feature was missed. 3. Investigate dose-response curvature, since the systematic high-dose overprediction appears in single-agent arms as well as combinations. 4. The pattern may indicate that dose-only predictions omit exposure-related variation. It is not proof because exposure estimates are uncertain and may be associated with other participant or sampling-time differences that also relate to response.