Why Dose-Response Structure Matters
A dose recommendation depends not only on estimated parameter values but also on the structure chosen to describe how response changes with dose. A model that assumes a continuing linear increase can predict a different target dose from one that levels off toward a plateau or rises steeply only after a threshold-like region. These differences can be modest where data are dense and consequential where the target lies near or beyond the observed dose range.
The purpose of a structural comparison is not to declare one curve correct from its shape alone. It is to determine whether plausible candidate structures lead to materially different predictions of target attainment and therefore different dose choices. Parameter uncertainty within each model and uncertainty about which model structure is appropriate are distinct sources of uncertainty; the comparison should keep them visible.
A common set of candidate structures includes linear response, Emax response, and sigmoid Emax response. For example, with baseline response \(E_0\), maximum increase \(E_{\max}\), half-maximal dose \(ED_{50}\), and shape parameter \(h>0\), their mean-response forms can be written as follows.
These forms encode different assumptions about shape. The linear model has a constant slope over the range where it is used. The Emax model increases rapidly at lower doses and approaches a plateau. The sigmoid form can have a relatively shallow low-dose region, a steeper middle region, and a plateau. Each form requires appropriate parameter values and a dose range over which its use is scientifically defensible. In particular, a mathematically defined curve is not automatically reliable when used to extrapolate far beyond the doses informing the data.
Compare the Target-Relevant Predictions
For a fixed target, the key comparison is not whether curves have different equations in the abstract. It is how their predictions differ near the target and across feasible candidate doses. If the decision rule is to choose the lowest dose whose predicted mean reaches \(T\), one can solve \(E_j(d)\geq T\) separately for each model. This is a descriptive comparison of fitted curves; it does not account for parameter uncertainty unless that uncertainty is added explicitly.
Worked Example: Different Shapes, Different Threshold Doses
Hypothetical illustration. Suppose response is measured in response units, the target is \(T=60\) units, baseline response is zero, and three plausible mean-response curves are under consideration:
For the linear curve, solve \(0.60d\geq 60\). Dividing both sides by \(0.60\) gives \(d\geq 100\). Thus its threshold dose is 100 dose units.
For the Emax curve, the denominator is positive for \(d\geq0\), so multiplying the inequality by \(50+d\) preserves its direction:
For the sigmoid curve, \(10000+d^2\) is also positive. Therefore,
For nonnegative dose, this gives \(d\geq\sqrt{15000}\approx122.5\). The three continuous threshold doses are therefore 100, 75, and 122.5 dose units, respectively. If available doses are restricted to multiples of 25, the lowest target-reaching choices are 100 for the linear model, 75 for Emax, and 125 for sigmoid Emax. These are model-based mean-response thresholds, not estimates of target-attainment probabilities. The calculation isolates the effect of shape; a real analysis would estimate the model parameters from data and quantify their uncertainty.
A useful justification links target attainment across doses to the distribution of a threshold dose. Suppose that, for each parameter value \(\theta_j\), response is nondecreasing in dose and, whenever the target is reached, the set of target-reaching doses has a minimum. Define \(D_{T,j}(\theta_j)\) as that minimum, setting it to infinity if the target is never reached. Then, for any candidate dose \(d\), these conditions give the equivalence \(R_j(d;\theta_j)\geq T\) if and only if \(D_{T,j}(\theta_j)\leq d\). Taking probabilities conditional on the data and model yields:
Thus, under these conditions, the target-attainment probability at dose \(d\) is the cumulative probability that the uncertain threshold dose is no greater than \(d\). This gives a direct interpretation of how parameter uncertainty translates into attainment uncertainty. The equivalence need not hold in this form for nonmonotone dose-response relationships, because a response could reach the target at one dose and then fall below it at a higher dose.
Attainment Probabilities and Model-Specific Choices
To compare uncertainty-aware decisions, calculate or simulate \(q_j(d)\) at the same candidate doses under each structure. For example, parameter draws can be generated from the fitted uncertainty distribution conditional on each model. At each draw, evaluate the model response at dose \(d\), record whether it reaches the target, and estimate \(q_j(d)\) by the fraction of draws that do so. If the target concerns an individual outcome rather than a model-predicted mean, the simulation must also include the relevant individual-level variability; parameter draws alone do not supply that variability.
Worked Example: Comparing Simulated Attainment by Structure
Hypothetical illustration. For each of three fitted structures, suppose 100 parameter draws were used to predict mean response at four candidate doses. In each draw, success means the predicted mean reaches a prespecified target. The following counts are hypothetical simulation results, not observations from a clinical study.
| Dose (dose units) | Linear successes out of 100 | Emax successes out of 100 | Sigmoid Emax successes out of 100 |
|---|---|---|---|
| 50 | 25 | 18 | 8 |
| 75 | 62 | 54 | 37 |
| 100 | 88 | 82 | 75 |
| 125 | 97 | 94 | 91 |
At 100 dose units, for example, the estimated probabilities are \(88/100=0.88\) for the linear model, \(82/100=0.82\) for Emax, and \(75/100=0.75\) for sigmoid Emax. Suppose the illustrative decision rule is to choose the lowest listed dose with estimated attainment probability at least 0.80. Under the linear model, 75 units gives 0.62, below 0.80, while 100 gives 0.88, so the selected dose is 100. Under Emax, the corresponding values are 0.54 and 0.82, so the selected dose is also 100. Under sigmoid Emax, 100 gives 0.75, below the rule, while 125 gives 0.91, so its selected dose is 125.
The difference is decision-relevant: sigmoid Emax requires the higher candidate dose under this rule. The probabilities are estimated from only 100 draws, so they also have Monte Carlo error; they should not be treated as exact. More simulation draws can reduce simulation error, but cannot repair a misspecified structural model or an unsuitable target definition.
Combining Structures Without Hiding Disagreement
When model weights are defensible, a model-averaged attainment probability at dose \(d\) is the weighted sum of the conditional probabilities: \(q_{\mathrm{avg}}(d)=\sum_j w_jq_j(d)\), where \(w_j\geq0\) and \(\sum_jw_j=1\). This is the law of total probability applied across the candidate models. It provides a summary under the specified weights, but it does not make the structural uncertainty disappear. The individual \(q_j(d)\) values should still be inspected, especially when they imply different actions.
Worked Example: A Weighted Attainment Probability
Hypothetical illustration. At 100 dose units, suppose the model-specific attainment estimates are 0.88 for linear, 0.82 for Emax, and 0.75 for sigmoid Emax, as in the previous example. For illustration, assign weights 0.25, 0.40, and 0.35, respectively. These weights are assumed for this calculation; they are not established simply by comparing the three probabilities.
Under these illustrative weights, the model-averaged attainment probability at 100 dose units is 0.8105, or about 81.1%. This number is above a decision threshold of 0.80, but the sigmoid model’s estimate at the same dose is 0.75. Reporting only the weighted value would hide that disagreement. A transparent decision summary would report the weights, the weighted result, and the model-specific results, then explain whether the recommendation is sensitive to the structural assumptions or weights.
If no defensible model weights are available, it is preferable to show model-specific predictions and recommendations rather than silently assigning equal weights. One can also describe the range of model-specific probabilities at each dose or identify the lowest dose that meets a chosen rule under every candidate structure. The latter is a conservative comparison across the specified set of models; it does not guarantee protection against structures that were not considered.
Assumptions, Diagnostics, and Alternatives
A structural comparison is informative only when the candidate models are fit and evaluated on a meaningful common basis. In practice, check that the response and target have the same definition and units across models, the candidate doses are within a clinically relevant range, and each model's parameterization allows the stated curve shape. The data should support the parameters needed by each structure, and uncertainty calculations should reflect the fitting procedure. If the models are fit to different data subsets or use different target definitions, their attainment estimates are not a clean comparison of structure.
Several patterns deserve attention. Large differences between curves near the target, widely separated threshold doses, or attainment estimates on opposite sides of the decision cutoff indicate structural sensitivity. A model that predicts continued benefit while the alternatives approach a plateau should be examined particularly carefully at high doses. Sparse observations in the steep or plateau regions can leave the relevant shape weakly informed. Plotting fitted curves with data and uncertainty summaries across the decision-relevant dose range can reveal such issues; numerical fit criteria alone do not establish that extrapolated target predictions are reliable.
| Approach | What it contributes | Main caution |
|---|---|---|
| Select one candidate structure | A direct recommendation conditional on that model | Does not show how the recommendation changes under other plausible shapes |
| Report model-specific attainment | Makes structural disagreement and conditional decisions visible | Leaves the decision-maker to interpret the set of results |
| Average over structures | Summarizes attainment under stated model weights | The answer depends on those weights and can conceal disagreement |
| Require a rule to hold across candidate structures | Shows a dose satisfying the rule throughout the specified model set | May be conservative and covers only the models considered |
These approaches answer different decision questions. A single-model analysis is useful when one structure is strongly supported and scientifically appropriate. Model averaging is useful when weights have a justified interpretation and an overall probability is desired. A cross-model requirement is useful when the decision rule explicitly prioritizes performance across the plausible set. None removes the need to assess whether the candidate structures and their dose ranges are credible.
Common Mistakes and Practical Limits
- Treating a fitted curve as certainty. A threshold calculated from estimated mean-response parameters is not an attainment probability. Propagate relevant parameter uncertainty to estimate \(q_j(d)\).
- Confusing parameter and structural uncertainty. Many parameter draws under one model describe uncertainty conditional on that structure. They do not represent uncertainty about whether the structure is appropriate.
- Calling a model-averaged result model-independent. A weighted probability depends on the candidate models and their weights. Report the model-specific estimates alongside it when they differ meaningfully.
- Assuming a target for the mean is an individual target. If the clinical decision concerns individual patients, include appropriate individual variability in the prediction rather than interpreting a parameter-only probability as patient-level attainment.
- Choosing the preferred curve because it gives the preferred dose. A convenient recommendation is not evidence for a model. Structural assessment should rely on scientific plausibility, data support, and prediction in the relevant range.
Structural comparisons remain conditional on the candidate set, fitted data, target definition, and dose range. They do not resolve uncertainty in a clinical target itself, population composition, or dependence among different outcomes. Their practical value is narrower and important: they show whether the dose decision changes when plausible assumptions about the dose-response shape change.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A model has target-attainment probability 0.84 at a dose based on parameter draws for predicted mean response. Does this directly mean that 84% of individual patients attain the target? Explain.
- For a nondecreasing response model whose target-reaching dose set has a minimum whenever it is nonempty, explain why the event of reaching the target by dose \(d\) can be expressed using the model-specific threshold dose.
- At one dose, two candidate structures give attainment estimates 0.92 and 0.63. What should be reported if model weights have not been justified?
- Using weights 0.50 and 0.50 with model-specific probabilities 0.70 and 0.90, calculate the weighted attainment probability.
- Why can increasing the number of parameter simulations fail to resolve disagreement between candidate model structures?
Solutions. (1) No. The 0.84 probability concerns uncertainty in predicted mean response under the stated simulation. Individual-level attainment requires including relevant individual variability and defining the patient-level event. (2) If response is nondecreasing and the target-reaching dose set has a minimum whenever it is nonempty, then \(R(d)\geq T\) exactly when \(D_T\leq d\); taking probabilities gives \(q(d)=P(D_T\leq d)\). (3) Report both model-specific estimates and explain that they imply different attainment predictions; do not invent weights or present a weighted value as established. (4) The weighted probability is \((0.50)(0.70)+(0.50)(0.90)=0.35+0.45=0.80\). (5) More draws reduce Monte Carlo error within each model, but the draws remain conditional on their respective structures. They cannot determine which structure is appropriate or eliminate the difference between the structures’ predictions.