Why Model Choice Is a Dose-Justification Decision
A dose-response model is not just a curve drawn through study results. It encodes assumptions about how response changes between tested doses, what happens near the ends of the dose range, and which contrasts can be estimated with useful precision. Those assumptions matter when the evidence is used to justify a dose that was not tested, compare a candidate dose with placebo, or assess whether increasing dose is likely to add meaningful benefit.
The appropriate model therefore depends on three things together: the dose range represented in the data, the endpoint and its scale, and the decision question. A simple model may be preferable when its assumptions are credible and the decision is local to the observed range. A more flexible curve may be needed to represent a threshold-like transition or plateau, but additional flexibility can be difficult to estimate from sparse data.
The equations below first describe the expected value of a continuous endpoint using the identity scale. For binary, count, or time-to-event endpoints, the same general dose-shape choices can be used on a suitable model scale with an appropriate likelihood and link. For instance, a binary endpoint may be modelled on the log-odds scale, so a linear dose effect on that scale does not mean a linear change in event probability. The endpoint model and scale must be stated rather than assumed.
As in the earlier material on dose-response estimands, the target should be clear before fitting a curve: for example, a contrast with placebo at a specified dose, or a decision about whether a higher dose offers enough additional benefit to justify its risks. In a randomized dose-ranging study, assigned dose supports interpretation of dose-group contrasts under the trial’s design and conduct. In nonrandomized data, dose assignment and other factors may be associated; a dose-response curve alone does not remove confounding.
Four Common Dose-Response Structures
For a continuous endpoint, write \(\mu(D)\) for its expected value at dose \(D\). The models below are alternative descriptions of this dose relationship, not interchangeable parameterizations with identical assumptions.
Here \(\beta_0\) is the expected response at dose zero and \(\beta_1\) is the change in expected response per unit of dose. The model is often useful for a limited range over which a straight-line approximation is plausible. It does not, by itself, impose a plateau or a biological maximum. Extrapolating it far beyond the observed range can predict implausible values, especially for a bounded endpoint.
In this form, \(E_0\) is the expected response at zero dose, \(E_{\max}\) is the asymptotic change from \(E_0\), and \(ED_{50}\) is the dose at which half of that change is reached. The sign of \(E_{\max}\) determines whether the modelled response increases or decreases as dose rises. The structure assumes a smooth, monotonic approach to an asymptote; it does not describe a threshold or a downturn.
At \(D=ED_{50}\), the fraction of the asymptotic change is one-half, because the numerator and the two terms in the denominator are then both \(ED_{50}^h\). When \(h=1\), this reduces to the Emax model. Values of \(h\) different from one allow a sharper or less sharp transition around \(ED_{50}\), while retaining a monotonic curve that approaches an asymptote. This flexibility is useful only if the dose range and data contain enough information to support it.
The categorical model estimates a response at each dose \(D_j\). It can represent irregular patterns among the tested levels without forcing them into a curve. It does not specify what the response would be at an untested dose: interpolation or extrapolation requires an additional assumption or model. With a small sample at each dose, estimating many separate means can also be imprecise.
Assumptions and a Practical Selection Sequence
Every candidate model needs assumptions about the endpoint distribution as well as the dose-response shape. For example, an identity-scale model for a continuous endpoint may be paired with residual assumptions about variability; a binary endpoint needs a binary-data likelihood and a link. Those choices affect interpretation and predictions. In all cases, define the population, endpoint, analysis time, and dose contrast consistently with the decision being made.
Specify the response, time point, population, and dose contrast of interest. A decision about a local contrast between two tested doses may not need a model intended to describe an asymptotic maximum.
Ask whether the trial covers a narrow range, a broad range that may include a plateau, or levels around a suspected transition. Check whether a proposed response-scale model respects the endpoint’s possible values.
Use a linear form for a plausible local straight-line trend, an Emax form for a plausible smooth saturation, a sigmoid Emax form when a sharper transition is supported, or categorical doses when imposing a curve is not justified for the immediate comparison.
Compare observed dose-group summaries with fitted predictions, examine uncertainty and residual patterns, and consider whether estimates depend heavily on unobserved parts of the dose range. A good numerical fit alone does not establish that a model’s shape assumptions are true.
These are selection-stage checks, not proof that a model is correct. A visibly poor fit, boundary-violating prediction, or strong dependence on one dose group is a reason to reconsider the structure, endpoint scale, or data limitations. Conversely, a small study can fail to reveal a real lack of fit. The absence of an obvious warning is not confirmation that interpolation or extrapolation is reliable.
Worked Examples
Worked Example: A Linear Approximation Across a Limited Range
Suppose a hypothetical randomized study measures symptom improvement on a 0-to-10-point scale at doses 0, 50, 100, and 150 mg. The hypothetical group means are 1, 3, 5, and 7 points, respectively. The immediate question is whether the trend over these tested doses is adequately represented by a constant change per milligram.
The slope from the two endpoint means is \(\beta_1=(7-1)/(150-0)=6/150=0.0400\) points per mg. The intercept is the mean at zero dose, \(\beta_0=1\) point. Thus the candidate line is \(\mu(D)=1+0.0400D\).
At 100 mg, the predicted mean is \(1+(0.0400)(100)=5.00\) points, matching the hypothetical group mean. At 150 mg, it predicts \(1+(0.0400)(150)=7.00\) points. The means lie exactly on this line, so these summaries show no lack of linearity over the tested range. They do not establish that the relationship remains linear beyond 150 mg. At 250 mg the same line predicts \(1+(0.0400)(250)=11.0\) points, above the scale maximum of 10. That prediction illustrates why this line is a local approximation, not a credible unlimited-dose model for this bounded endpoint. The exact alignment of four means also says nothing by itself about sampling uncertainty or individual-level variability.
Worked Example: Saturation Across a Broad Dose Range
In a second hypothetical study, the endpoint is mean symptom improvement in points, with larger values indicating greater improvement. Group means at 0, 50, 100, 200, and 400 mg are 0, 5.0, 6.7, 8.0, and 8.9 points. Consider an illustrative Emax curve with \(E_0=0\), \(E_{\max}=10\) points, and \(ED_{50}=50\) mg.
At 50 mg, the predicted response is \(\mu(50)=0+10(50)/(50+50)=500/100=5.00\) points. At 100 mg, \(\mu(100)=10(100)/(50+100)=1000/150=6.67\) points. At 200 mg, \(\mu(200)=10(200)/(50+200)=2000/250=8.00\) points. At 400 mg, \(\mu(400)=10(400)/(50+400)=4000/450=8.89\) points. These predictions match the hypothetical means to the displayed precision.
The diminishing increments in the observed means are consistent with a saturating shape: the increase from 50 to 100 mg is 1.7 points, whereas the increase from 200 to 400 mg is 0.9 points over a larger dose increment. This pattern makes an Emax model a plausible candidate for describing these data, not a proven description of the underlying response. The model assumes smooth monotonic saturation, and the means alone do not establish how precisely the asymptote or \(ED_{50}\) can be estimated.
Worked Example: A Sharper Transition With a Sigmoid Emax Curve
Consider a hypothetical endpoint with an expected baseline response of 0 points and an asymptotic improvement of 10 points. Suppose responses are recorded at 30, 60, and 120 mg. Compare a sigmoid curve with \(h=2\) and \(ED_{50}=60\) mg with an Emax curve using the same \(E_0\), \(E_{\max}\), and \(ED_{50}\), which has \(h=1\).
For \(h=2\), at 30 mg the response is \(\mu(30)=10(30^2)/(60^2+30^2)=9000/4500=2.00\) points. At 60 mg it is \(\mu(60)=10(60^2)/(60^2+60^2)=36000/7200=5.00\) points. At 120 mg it is \(\mu(120)=10(120^2)/(60^2+120^2)=144000/18000=8.00\) points.
For \(h=1\), the corresponding predictions are \(10(30)/(60+30)=3.33\), \(10(60)/(60+60)=5.00\), and \(10(120)/(60+120)=6.67\) points. The \(h=2\) curve predicts a lower response below \(ED_{50}\) and a sharper transition around \(ED_{50}\). In particular, it rises by 3.00 points from 30 to 60 mg, while the \(h=1\) curve rises by 1.67 points over that same interval. The comparison does not imply that the sigmoid form is better: data must support the extra shape flexibility, especially around the transition.
Worked Example: Retaining Tested-Dose Contrasts With a Categorical Model
Suppose a hypothetical trial reports mean improvement of 0, 4, 7, and 6 points at 0, 25, 50, and 100 mg. The endpoint is continuous and measured on the same scale and at the same time in all groups. The observed decline from 50 to 100 mg could reflect sampling variability, or it could signal that a monotonic curve is inappropriate; the group means alone cannot decide between these explanations.
A categorical model assigns \(\mu_0=0\), \(\mu_{25}=4\), \(\mu_{50}=7\), and \(\mu_{100}=6\) points. The estimated contrast between 50 and 100 mg is \(7-6=1\) point in favor of 50 mg in these hypothetical means. This model represents the observed group pattern without imposing monotonicity. It does not estimate a response at, for example, 75 mg, nor does it establish that the one-point difference is meaningful or precise; uncertainty estimates and the decision context are also needed.
Comparing Models, Diagnostics, and Limitations
| Model | Useful when | Main shape assumption | Important limitation |
|---|---|---|---|
| Linear | The decision concerns a limited range with a plausible near-linear trend. | Constant change in expected response per unit dose over the modelled range. | Can misrepresent curvature and give implausible extrapolations. |
| Emax | A smooth, monotonic approach to a plateau is plausible and the range informs curvature. | One saturating transition with a fixed shape corresponding to \(h=1\). | Cannot represent a sharp transition or nonmonotonic pattern. |
| Sigmoid Emax | The data and decision warrant a steeper or less steep transition than the Emax form allows. | A smooth, monotonic Hill-shaped transition with \(h>0\) and an asymptote. | Extra flexibility can be poorly identified when dose levels are sparse or do not span the transition. |
| Categorical | Inference is chiefly about tested doses and a smooth trend is not justified. | No shape constraint between dose levels. | Uses more dose-specific estimates and does not supply a curve for untested doses. |
At the selection stage, useful checks include plotting observed summaries and uncertainty against dose, comparing them with model predictions, and checking whether the endpoint’s allowed range is respected. Look for systematic departures such as curvature left unexplained by a line, an apparent plateau that a line misses, or a dose pattern inconsistent with the monotonicity of an Emax-family model. For noncontinuous outcomes, also check that the chosen likelihood and link are suitable; a reasonable dose shape cannot rescue an inappropriate endpoint model.
Candidate models can be compared using scientific plausibility, fit, uncertainty, and whether they answer the same estimand. A categorical model is a useful reference for observed-dose patterns, but its estimates may be noisy. A parsimonious curve can improve precision by sharing information across doses, but that gain depends on its shape assumptions. Information criteria or fit statistics may inform comparison, yet they do not decide whether extrapolation is credible or whether a dose is clinically worthwhile. Model averaging and formal model-selection procedures are separate topics; here the central task is to make the structural assumptions explicit before using a model to justify a dose.
There are important limits to what these four structures cover. The linear and Emax-family models above impose monotonic dose-response relationships; they do not describe a U-shaped or inverted-U-shaped response. Categorical models can retain irregular observed-dose means but may be inefficient and cannot support predictions between tested levels without further assumptions. Sparse dose-ranging studies may not distinguish a plateau from the highest tested dose, or a transition from an apparently linear segment. In such cases, conclusions should be restricted to what the data and design support.
A Common Model-Choice Pitfall
A common mistake is to select the most flexible curve because it follows the observed group means most closely, then treat its untested-dose predictions as established facts. The reasoning is incomplete: added flexibility can fit random variation, and a model’s predictions depend on its assumptions even when its in-sample fit looks good. The correction is to state the decision and dose range first, assess whether the candidate shape is biologically and statistically plausible, compare it with simpler or less shape-constrained descriptions, and communicate uncertainty and extrapolation limits. A fitted curve is evidence conditional on its structure, not direct observation at every dose.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- What response-shape assumption distinguishes a linear dose model from an Emax model?
- For a sigmoid Emax model with \(h=2\), \(ED_{50}=40\) mg, \(E_0=0\), and \(E_{\max}=12\) points, what is the predicted response at 40 mg?
- A study has only placebo and one active dose, and its decision concerns those two tested groups. Why might a categorical comparison be preferable to claiming an estimated plateau?
- What does a categorical dose model imply about the response at a dose not tested?
- Why does a close fit of a sigmoid curve to observed means not, by itself, justify a dose recommendation based on extrapolation?
Solutions. (1) A linear model assumes a constant change per unit dose over its modelled range; an Emax model assumes a smooth monotonic approach to an asymptote. (2) At \(D=ED_{50}=40\) mg, the fraction is \(40^2/(40^2+40^2)=1/2\), so the predicted response is \(0+12(1/2)=6.00\) points. (3) The categorical comparison directly estimates the two observed group responses without imposing a saturating curve whose plateau cannot be informed by only two dose levels. (4) It makes no prediction there unless an additional interpolation or curve assumption is introduced. (5) The extrapolated value depends on the assumed curve shape and may be weakly informed or unsupported by the observed dose range; fit alone does not establish that the shape remains valid beyond the data.