Why Endpoint Scale Changes the Dose Contrast
A dose-response analysis does not produce a scale-free statement about treatment effect. If the endpoint is continuous, a contrast may be expressed as a difference in mean score units. If it is binary, the contrast may be a difference in response probabilities, a risk ratio, or an odds ratio. If it is ordinal, the effect may be described using probabilities of reaching or exceeding ordered categories. These are different summaries of the same clinical outcomes, and they need not tell the same story numerically.
Scale choice begins with what was measured and what distinction matters clinically. It is not a license to try several summaries and report whichever looks most favorable. For dose \(d\), let \(Y(d)\) denote the endpoint under that dose. A dose contrast compares a feature of the distribution of \(Y(d_2)\) with the same feature of \(Y(d_1)\). The feature must be stated: for example, a mean, a responder probability, or the probability of being at least in a specified ordinal category.
The scale is also a modeling choice. A model may describe the outcome directly, or it may describe a transformed quantity such as the log odds of response. Consequently, a dose-response curve that is simple on one scale can be curved on another. A fitted dose-response model must be interpreted on the scale on which its parameters and predictions are defined.
Continuous Endpoints: Differences in Units
For a continuous endpoint, a common contrast is the difference between dose-specific mean outcomes. With lower values indicating benefit, a negative mean difference favors the higher dose; with higher values indicating benefit, the direction is reversed. The value retains the endpoint's units, which often makes it directly interpretable. A percent change or standardized difference can be useful in some settings, but it answers a different question and can obscure the original measurement scale.
Worked Example: Comparing Mean Symptom Scores
In a hypothetical randomized trial, the week-12 symptom score is measured on a 0-to-100 scale, with lower scores indicating fewer symptoms. Suppose 100 participants are assigned to each of 0 mg, 10 mg, and 30 mg, and their observed group means are 100, 88, and 76 points, respectively. Assume the same score definition and assessment time in all groups, treatment assignment is randomized, and the group means are used to estimate the corresponding dose-specific population means. No missing-data adjustment is needed for this illustration.
The estimated mean contrast for 30 mg versus 10 mg is
Thus, in this hypothetical data set, the 30 mg group has a mean score 12 points lower than the 10 mg group at week 12. The corresponding 0-to-30 mg difference is \(76-100=-24\) points. Relative to the 0 mg mean, the 30 mg mean is 24 points, or \(24/100=24\%\), lower. The mean difference remains 24 points; the percentage is a secondary description with the 0 mg mean as its denominator. These summaries do not establish that every participant benefits or that the difference is clinically important.
A continuous model should be checked for whether its mean structure describes the dose pattern adequately and whether the assumed residual distribution and variance structure are reasonable for the intended estimation procedure. Strong floor or ceiling effects, markedly changing variability across doses, or influential observations may make a simple mean-based account incomplete. A transformation can help, but it changes the scale of the modeled effect and requires predictions to be translated back to the endpoint units when those are the intended interpretation.
Binary Endpoints: Probability and Odds Contrasts
A binary endpoint records whether a specified event occurred, such as meeting a prespecified responder definition by a fixed assessment time. At dose \(d\), write \(p(d)=P(Y=1\mid d)\), where \(Y=1\) denotes the event of interest. The risk difference \(p(d_2)-p(d_1)\) is in probability units; the risk ratio \(p(d_2)/p(d_1)\) compares probabilities multiplicatively; and the odds ratio compares \(p/(1-p)\) between doses. These are not interchangeable.
Worked Example: Three Contrasts for a Responder Endpoint
In a hypothetical trial, response is defined in advance as at least a 30% improvement by week 12. There are 100 participants at each of 10 mg and 30 mg. Suppose 10 of 100 participants respond at 10 mg and 22 of 100 respond at 30 mg. Assume the response definition and follow-up are identical across groups, participants are independently sampled within randomized dose groups, and the observed proportions estimate the dose-specific response probabilities.
The estimated probabilities are \(p(10)=10/100=0.10\) and \(p(30)=22/100=0.22\). The risk difference is
That is an increase of 12 percentage points in the observed response probability. The risk ratio is \(0.22/0.10=2.20\), so the observed response probability at 30 mg is 2.20 times the probability at 10 mg. For the odds ratio, use the original proportions without rounding intermediate odds:
The estimated odds of response are about 2.538 times as high at 30 mg as at 10 mg. This does not mean the response probability is 2.538 times as high; that probability ratio is 2.20. Nor does an odds ratio of 2.538 mean an increase of 153.8 percentage points. Each number answers a distinct question.
The responder threshold is consequential. Participants just below and just above the 30% improvement cutoff are classified differently, while a large improvement and a marginal improvement above the cutoff count equally. The binary endpoint can be useful when crossing a threshold has a clear interpretation, but it discards information in the underlying continuous measurement. Results can also depend on the chosen threshold, so the definition must be clinically justified and fixed independently of seeing which threshold favors a dose.
Ordinal Endpoints: Ordered Categories and Cumulative Probabilities
An ordinal endpoint has categories with a meaningful order but without an assumption that the gaps between adjacent categories are equal. For example, a global assessment might be poor, fair, good, or very good. Assigning scores 1, 2, 3, and 4 does not by itself make the distance from poor to fair equal to the distance from good to very good. Therefore, a mean of the assigned codes requires an additional interpretation that may not be warranted.
A transparent dose contrast is often the change in probability of being at or above a named category. An ordinal regression can instead summarize shifts across all category thresholds. One common model is the cumulative logit proportional-odds model:
where each threshold has its own intercept \(\alpha_k\), and the same dose function \(f(d)\) applies at every threshold. A positive contrast \(f(d_2)-f(d_1)\) means higher cumulative odds of being at or above every modeled threshold at \(d_2\), under this model. The common dose effect across thresholds is the proportional-odds assumption; it is not guaranteed by the fact that categories are ordered.
Worked Example: Reading an Ordinal Dose Pattern
In a hypothetical study, participants rate their outcome at week 12 as poor, fair, good, or very good. The table gives the observed percentages in each category, with 100 participants per dose. Assume categories are assessed identically at all doses, follow-up is common, and dose groups are randomized. For this descriptive comparison, the percentages estimate each dose's category probabilities.
| Dose | Poor | Fair | Good | Very good |
|---|---|---|---|---|
| 0 mg | 40% | 35% | 20% | 5% |
| 10 mg | 25% | 35% | 30% | 10% |
| 30 mg | 15% | 25% | 40% | 20% |
The probability of being at least good is 40% at 10 mg and 60% at 30 mg. Thus, the estimated 30 mg minus 10 mg contrast for this threshold is \(0.60-0.40=0.20\), or 20 percentage points. The median category is fair at 10 mg because 25% are poor and 60% are poor or fair, so the cumulative percentage first reaches at least 50% in fair. At 30 mg, 15% are poor and 40% are poor or fair; adding the 40% in good gives 80%, so the median category is good.
The proportional-odds assumption can be examined by comparing cumulative odds at different thresholds. For being at least fair, the estimated 30 mg versus 10 mg odds ratio is
For being at least good, it is
These empirical odds ratios differ, which cautions against assuming one common odds ratio across thresholds. The pattern may reflect sampling variation as well as departure from proportional odds; a small hypothetical table cannot decide that question by itself. Inspecting threshold-specific estimates and their uncertainty, along with model diagnostics, is more informative than treating the assumption as automatically true. If it is not reasonable, report category probabilities or use a model that allows threshold-specific dose effects.
Why the Modeling Scale Matters
The way a dose effect changes with the endpoint's baseline level can be seen directly in a binary model. Suppose the logit of response probability is \(\eta(d)\), so that \(p(d)=\exp(\eta(d))/(1+\exp(\eta(d)))\). Differentiating with respect to dose gives
Thus, a constant change in log odds per unit dose does not produce a constant change in response probability per unit dose. The probability-scale slope depends on \(p(d)\). This is one reason an odds-ratio model and a risk-difference model can give different-looking dose effects without contradiction: they describe dose response on different scales.
The same general issue applies to continuous endpoints. A dose model for the mean in original units answers a different question from a model for the log mean or another transformed response. In model-informed dose evaluation, state whether the dose function is defined on the response scale or on a link scale, and present predictions on a scale that can be interpreted for the endpoint. A parameter or curve shape on a link scale should not be casually described as a constant change in the observed outcome.
Choosing and Checking a Scale
The main alternatives are not simply competing statistical techniques; they often define distinct clinical summaries. The appropriate choice depends on the endpoint's measurement properties and the contrast that needs interpretation.
| Endpoint scale | Useful dose contrast | Key caution |
|---|---|---|
| Continuous | Difference in mean endpoint units; sometimes a ratio or percent change | Mean summaries can hide skewness, floor or ceiling effects, and heterogeneous variability |
| Binary | Risk difference, risk ratio, or odds ratio | A threshold discards within-category information; relative contrasts depend on the chosen measure |
| Ordinal | Category probabilities or probability of reaching a specified category; cumulative odds under a model | Numeric category codes do not guarantee equal spacing; proportional odds may fail |
Several checks help establish whether the chosen scale is serving the question:
- Verify endpoint definition and direction. Confirm the measurement time, responder cutoff if used, category ordering, and which direction represents benefit.
- Inspect the data on the original scale. For continuous outcomes, examine distributions by dose and look for extreme observations, skew, floor or ceiling accumulation, and changing spread. For binary outcomes, report event counts as well as proportions and identify sparse dose groups. For ordinal outcomes, inspect the full category distribution and cumulative proportions rather than only a coded mean.
- Check model assumptions that determine interpretation. For a proportional-odds model, assess whether the dose effect is plausibly similar at each threshold. For any selected model, compare observed and predicted endpoint summaries across doses and look for systematic discrepancies.
- Check robustness of the contrast. A responder analysis can be accompanied by the underlying continuous endpoint when available; an ordinal result can be displayed as category probabilities. Agreement across reasonable summaries supports interpretation, while disagreement identifies dependence on scale or threshold.
No scale removes limitations in the data. Sparse binary events can make odds-ratio estimates unstable, and ordinal models can be difficult to interpret when some categories are rare. A continuous mean may be clinically familiar but insufficient if the distribution has strong floor effects or if only crossing a threshold matters. Moreover, none of these endpoint-scale contrasts alone accounts for safety, duration, or the uncertainty required for a dose decision. The purpose here is to make the efficacy contrast explicit, not to turn one scale into a complete decision rule.
Common Pitfalls
- Calling an odds ratio a probability ratio. The odds are \(p/(1-p)\), not \(p\). Report the measure by name and, when useful, give the corresponding dose-specific probabilities.
- Assuming ordinal category numbers are equally spaced. A category code is an ordering device unless equal intervals have been justified. Do not interpret its mean as an endpoint-unit mean automatically.
- Selecting a responder cutoff after inspecting results. This makes the binary endpoint data-dependent and can exaggerate apparent dose differences. Define and justify the threshold in advance.
- Assuming one model scale is neutral. A dose effect can be nearly constant on a log-odds scale and vary on a probability scale. State the scale before interpreting the effect.
- Reporting only a relative contrast. A risk ratio or odds ratio may sound large while the absolute probability difference remains modest. Present dose-specific probabilities or an absolute contrast when these are relevant to interpretation.
Check Your Understanding
Answer each question using the scale and endpoint definition stated. Short solutions follow the questions.
- In the responder example, what is the risk difference for 30 mg versus 10 mg, and how should it be interpreted?
- Why is the odds ratio in that example approximately 2.538 rather than a probability ratio of 2.538?
- Using the ordinal table, calculate the difference between 30 mg and 10 mg in probability of being at least good.
- What does the proportional-odds assumption require about the dose effect across ordinal thresholds?
- Give one reason a responder endpoint may not preserve all the information in an underlying continuous outcome.
Solutions. (1) \(0.22-0.10=0.12\): the observed response probability is 12 percentage points higher at 30 mg. (2) The odds ratio compares \(0.22/0.78\) with \(0.10/0.90\), giving \(33/13\approx2.538\); the probability ratio is \(0.22/0.10=2.20\). (3) \(0.60-0.40=0.20\), a 20-percentage-point increase. (4) The model uses the same dose effect on cumulative log odds at every category threshold. (5) People just below and above the cutoff receive different binary labels even if their underlying outcomes are very similar, while outcomes well above the cutoff are all counted as responders.