1. What Is an Ordinal Response?
An ordinal response is an outcome whose possible values have a meaningful order but where the distances between adjacent categories are not necessarily equal.
Examples in pharmacodynamic studies include symptom severity graded as none, mild, moderate, or severe; clinical response categorized as poor, partial, or complete; and global assessments recorded on ordered scales.
| Category | Example interpretation | Order |
|---|---|---|
| 0 | No response | Lowest |
| 1 | Mild response | ↑ |
| 2 | Moderate response | ↑ |
| 3 | Marked response | Highest |
The ordering contains useful information. However, treating categories 0, 1, 2, and 3 as if they were equally spaced numerical measurements can impose an assumption that may not be justified.
2. Why Use an Ordinal PK/PD Model?
A conventional exposure-response analysis may be straightforward when the pharmacodynamic endpoint is continuous or binary. Ordinal endpoints require a different treatment because several ordered response levels are observed.
A PK/PD model can connect concentration to the probability distribution across those categories:
This framework allows questions such as:
- How does increasing concentration shift the response distribution?
- What concentration is associated with a specified probability of achieving a clinically relevant response?
- How does response probability change over time after a dose?
- How do patient characteristics modify the exposure-response relationship?
- What response distribution is predicted under a proposed dosing regimen?
3. The Basic Ordinal PK/PD Structure
Suppose the observed response \(Y\) has \(K\) ordered categories:
The PK model produces a concentration \(C(t)\), which becomes the exposure input to the PD model.
One common approach is a cumulative logit model, also called a proportional-odds model. For category threshold \(j\), the model describes the cumulative probability:
Here, \(\alpha_j\) is a threshold-specific intercept and \(\beta\) describes the exposure-response effect shared across thresholds under the proportional-odds assumption.
An ordinal PK/PD model uses the PK-predicted concentration as an exposure input to a categorical response model.
4. Why Model Cumulative Probabilities?
With several ordered categories, it is often useful to model the probability of achieving at least a particular response level.
For example, with four categories \(0,1,2,3\), there are three meaningful cumulative thresholds:
| Threshold | Cumulative event |
|---|---|
| 1 | \(Y\ge1\): at least mild response |
| 2 | \(Y\ge2\): at least moderate response |
| 3 | \(Y\ge3\): marked response |
The corresponding cumulative probabilities are:
These probabilities are connected because they describe the same underlying ordered response distribution. Modeling them jointly avoids fitting completely unrelated models to each threshold.
5. The Proportional-Odds Model
The proportional-odds model assumes that the exposure effect is the same on the log-odds scale across cumulative thresholds.
A common parameterization is:
There is a separate threshold parameter \(\alpha_j\) for each cumulative boundary, but one common exposure coefficient \(\beta\).
If \(\beta>0\), increasing exposure increases the odds of being in a higher response category. If \(\beta<0\), increasing exposure shifts probability toward lower categories, assuming the response categories are ordered from lower to higher response.
The proportional-odds assumption is therefore an important structural assumption, not simply a mathematical convenience.
6. Interpreting the Exposure Effect
For a one-unit increase in concentration, the cumulative odds ratio is:
Under the proportional-odds assumption, this odds ratio applies to each cumulative threshold.
For example, suppose:
Then:
This means that a one-unit increase in the exposure variable multiplies the cumulative odds of being at or above a given response threshold by approximately 1.65, under the model assumptions.
The interpretation depends on the exposure scale. If concentration is measured in mg/L, the coefficient describes a one-mg/L increase. If the model instead uses log concentration, the interpretation changes accordingly.
7. From Cumulative Probabilities to Category Probabilities
The model can be used to obtain the probability of each individual response category.
For four categories, define:
Then:
Thus, the ordinal model provides a complete probability distribution rather than only a single predicted response.
| Response | Predicted probability |
|---|---|
| No response | \(P(Y=0)\) |
| Mild | \(P(Y=1)\) |
| Moderate | \(P(Y=2)\) |
| Marked | \(P(Y=3)\) |
This probability distribution is often more informative for PK/PD interpretation than reporting only an odds ratio.
8. Linking Ordinal Response to an Emax Exposure Model
Drug effects are frequently nonlinear with respect to concentration. Instead of putting concentration directly into the ordinal model, the exposure component can therefore be represented using an \(E_{\max}\)-type function.
A simple Emax exposure function is:
The ordinal model can then use this exposure-response quantity:
This separates two concepts:
- PK: what concentration occurs over time?
- Exposure-response: how does concentration influence the latent pharmacodynamic effect?
- Ordinal observation model: how does that latent effect translate into ordered response categories?
This structure can be useful when the response approaches a plateau as concentration increases.
9. Thinking of Ordinal Response as a Latent Continuous Effect
One useful interpretation of an ordinal model is that the observed categories arise from an underlying continuous or latent pharmacodynamic response.
Conceptually:
Suppose a latent response \(E^*\) increases with drug exposure. The observed category depends on where \(E^*\) falls relative to thresholds:
This interpretation explains why threshold parameters naturally appear in ordinal PK/PD models.
10. Modeling Ordinal Response Over Time
Because PK models generate concentration as a function of time, the ordinal response probabilities can also be predicted over time.
For example:
After a dose, concentration may rise and fall. The predicted probabilities of mild, moderate, and marked response can therefore rise and fall with exposure.
Illustrative cumulative response probabilities changing over time as drug concentration changes.
This allows PK/PD simulations to describe not just an expected response, but the probability of reaching clinically relevant response thresholds at different times after dosing.
11. Adding Patient Characteristics and Covariates
Ordinal PK/PD models can incorporate covariates that influence either exposure or the pharmacodynamic response.
For example, a covariate \(X\) could enter the ordinal model as:
An exposure-covariate interaction can also be considered:
Here, the interaction term allows the exposure-response relationship to differ according to the covariate.
In population PK/PD analysis, covariates may also act through the PK model. For example, body size or renal function might alter predicted concentration, which then indirectly changes predicted response probability.
12. Population PK/PD Models for Ordinal Data
Clinical PK/PD datasets often contain repeated observations from multiple individuals. A population model can represent typical response behavior while accounting for between-subject variability.
A simplified latent formulation might be:
where \(\eta_i\) represents an individual-specific deviation from the typical exposure-response relationship.
More elaborate models can allow random effects on parameters such as:
- baseline response tendency;
- exposure-response sensitivity;
- maximum effect;
- threshold locations;
- other clinically meaningful PD parameters.
Repeated observations from the same individual also require appropriate treatment of within-subject dependence.
13. The Proportional-Odds Assumption
The proportional-odds model assumes that the exposure coefficient is common across all cumulative response thresholds.
For three thresholds:
The intercepts differ, but the exposure slope is the same.
This assumption can be attractive because it gives a parsimonious description of how exposure shifts the entire ordinal distribution. However, it may not adequately describe every drug-response relationship.
If exposure has a substantially different effect on mild versus severe response thresholds, a more flexible ordinal model may be needed.
14. Alternatives to Proportional Odds
Several approaches can be considered when the proportional-odds assumption is not appropriate or when a different scientific interpretation is desired.
| Approach | General idea |
|---|---|
| Proportional odds | One common exposure effect across cumulative thresholds |
| Partial proportional odds | Allows selected predictors to have threshold-specific effects |
| Adjacent-category models | Models odds comparing neighboring response categories |
| Continuation-ratio models | Models progression through ordered stages or conditional category probabilities |
| Multinomial models | Models categories separately without using their ordering; useful when ordering assumptions are unsuitable |
| Latent-variable models | Explicitly represents an underlying continuous response with category thresholds |
The appropriate choice depends on the endpoint, scientific question, data structure, and assumptions that can reasonably be supported.
15. Baseline and Placebo Response
Ordinal PD endpoints often have substantial baseline structure. Subjects may have a meaningful probability of each response category even when drug exposure is negligible.
A baseline term can be incorporated into the model:
The threshold parameters establish the baseline probability distribution when \(C=0\), depending on the chosen parameterization.
Placebo effects can also be represented explicitly when response changes over time independently of drug concentration. In longitudinal PK/PD models, this may require a time-varying baseline or placebo component.
16. Worked Example: Predicting an Ordinal Response
Consider a hypothetical four-level PD endpoint:
| Category | Description |
|---|---|
| 0 | No response |
| 1 | Mild response |
| 2 | Moderate response |
| 3 | Marked response |
Suppose the cumulative-logit PK/PD model is:
with:
Step 1: Specify the concentration
Suppose the PK model predicts:
Step 2: Calculate the cumulative logits
For the first threshold:
For the second threshold:
For the third threshold:
Step 3: Convert logits to probabilities
Using the logistic transformation:
gives approximately:
| Threshold | Logit | Probability |
|---|---|---|
| \(Y\ge1\) | 0.40 | 0.599 |
| \(Y\ge2\) | -0.40 | 0.401 |
| \(Y\ge3\) | -1.40 | 0.198 |
Step 4: Obtain individual category probabilities
The probability of no response is:
The probability of a mild response is:
The probability of a moderate response is:
The probability of a marked response is:
The resulting predicted distribution is therefore approximately:
| Response | Predicted probability |
|---|---|
| No response | 40.1% |
| Mild | 19.8% |
| Moderate | 20.3% |
| Marked | 19.8% |
The important result is not simply the value of \(\beta\). The PK/PD model produces an interpretable prediction of the entire response distribution at the specified concentration.
17. Using the Model for Dose and Exposure Simulation
Once the PK and ordinal PD components have been estimated, the model can be used to simulate response probabilities under different dosing conditions.
A typical simulation sequence is:
- Specify a dose or dosing regimen.
- Use the PK model to predict \(C(t)\).
- Pass \(C(t)\) into the ordinal PD model.
- Calculate cumulative response probabilities.
- Convert cumulative probabilities into category probabilities.
- Summarize the predicted response distribution over time.
This allows questions such as whether a proposed regimen maintains a sufficiently high probability of achieving a target response while avoiding unnecessary exposure.
Simulation can also incorporate between-subject variability so that the output represents a distribution of possible response trajectories rather than only a single typical subject.
18. Targeting a Clinically Relevant Response
One advantage of ordinal response modeling is that clinically relevant targets can often be expressed as cumulative probabilities.
For example, a target might be:
meaning that the probability of achieving at least a moderate response should be at least 70%.
The model can then be used to determine the concentration associated with a desired target probability.
Under the proportional-odds model:
so the concentration corresponding to probability \(p\) is:
This provides a direct bridge from a clinically interpretable response target to an exposure target, which can then be connected back to dosing through the PK model.
19. Evaluating an Ordinal PK/PD Model
Model evaluation should address both the PK and PD components.
For the PK component, useful checks include:
- observed versus predicted concentrations;
- residual diagnostics;
- individual prediction performance;
- parameter plausibility;
- visual predictive checks or related simulation-based diagnostics.
For the ordinal PD component, useful checks include:
- observed versus predicted category frequencies;
- observed versus predicted cumulative probabilities;
- response probability across exposure ranges;
- threshold-specific model behavior;
- assessment of the proportional-odds assumption;
- individual-level predictive performance when repeated observations are available.
20. What Ordinal PK/PD Models Do Not Tell Us Automatically
Ordinal PK/PD models are powerful, but their interpretation depends on assumptions about both exposure and response.
- Ordinal categories are not necessarily equally spaced. The model uses their ordering rather than assuming equal numerical distances.
- Proportional odds is an assumption. A common exposure effect across thresholds may not always be appropriate.
- Odds ratios are not probability differences. Absolute response changes depend on the baseline threshold probabilities.
- PK uncertainty propagates into PD predictions. Incorrect concentration predictions can affect exposure-response conclusions.
- Threshold parameters are model-dependent. Their numerical values depend on the parameterization and response coding.
- Extrapolation requires caution. Predictions at concentrations outside the observed range depend strongly on the assumed exposure-response structure.
- Association is not automatically causation. Confounding, placebo effects, disease progression, and other mechanisms may influence the observed ordinal endpoint.
21. A Practical Workflow for Ordinal PK/PD Modeling
- Define the ordinal endpoint. Confirm that the categories have a meaningful ordering.
- Understand the clinical meaning of each category. Avoid treating arbitrary numerical codes as continuous measurements.
- Develop the PK model. Obtain reliable individual or population concentration predictions.
- Explore response versus exposure. Examine cumulative response probabilities across concentration ranges.
- Choose an ordinal PD structure. A proportional-odds model may be a useful starting point.
- Specify baseline and threshold parameters. Make the response-category structure explicit.
- Add nonlinear exposure effects when appropriate. Emax, sigmoid Emax, or other functions may describe saturation.
- Evaluate proportionality assumptions. Determine whether the exposure effect is reasonably consistent across thresholds.
- Consider covariates and random effects. Separate exposure-mediated effects from pharmacodynamic sensitivity effects.
- Evaluate predictive performance. Compare observed and predicted ordinal distributions.
- Simulate relevant dosing scenarios. Translate dose into concentration and then into response probabilities.
- Interpret clinically. Focus on predicted probabilities of meaningful response categories rather than coefficients alone.
22. Key Takeaways
- Ordinal responses have meaningful ordering but do not necessarily represent equally spaced numerical measurements.
- PK/PD models can connect drug concentration to the probability distribution across ordered response categories.
- Cumulative-logit or proportional-odds models are common ways to model ordinal exposure-response relationships.
- The proportional-odds model uses threshold-specific intercepts with a common exposure effect across cumulative response thresholds.
- The exposure coefficient can be transformed into a cumulative odds ratio through \(OR=e^\beta\).
- Category-specific probabilities can be recovered from cumulative probabilities.
- Nonlinear exposure functions such as Emax models can be incorporated when the pharmacodynamic response saturates.
- Ordinal response can be viewed as arising from a latent pharmacodynamic effect separated by response thresholds.
- PK predictions determine the concentration-time input to the ordinal PD model, allowing response probabilities to be predicted over time.
- Covariates can influence response through pharmacokinetics, pharmacodynamic sensitivity, or both.
- The proportional-odds assumption should be evaluated rather than accepted automatically.
- Model evaluation should consider both the PK component and the predicted ordinal response distribution.
- Simulation can translate proposed dosing regimens into predicted probabilities of clinically meaningful response.
- The most useful output is often the predicted probability of achieving a target response rather than the exposure coefficient alone.
Where to Go Next
A natural progression is to study PK/PD models for continuous endpoints, followed by count and binary response models, before moving to more advanced categorical and longitudinal PK/PD models.
From there, useful extensions include nonlinear mixed-effects ordinal models, time-varying covariates, Markov models for transitions between response states, joint PK/PD models, and simulation-based dose optimization.