Conditional Versus Marginal PD Predictions

Why the Prediction Target Matters

A PD model can be used to predict the response of a particular person or to describe the average response across a population. Those are different targets when individuals have different PK parameters. A prediction that inserts one person's estimated clearance or exposure into the PD model is conditional on that estimate. A population-marginal prediction averages the PD response over the distribution of PK parameters in the target population.

The difference matters especially when the concentration-response relationship is nonlinear. In general, calculating a response at an average concentration does not give the average response across people. The distinction also affects uncertainty: a prediction for a person with PK parameters treated as known does not include the between-person PK variation that contributes to uncertainty for a new, unspecified person.

These predictions answer different questions. The conditional prediction asks what the model predicts for a person whose PK parameters are represented by \(\hat{\phi}_i\). The marginal prediction asks what response is expected for a randomly selected person from the specified population. Neither is automatically the right answer for every application: the intended decision and target population determine which quantity is relevant.

Nonlinearity Makes Averaging Order Matter

Consider an Emax concentration-response model with no baseline response:

Here \(E_{\max}\) is the maximum modeled response, and \(EC_{50}\) is the concentration producing half of that response. The population-marginal mean response is the average of \(m(C)\) over the concentration distribution. It is not generally \(m\) evaluated at the average concentration.

For positive \(E_{\max}\) and \(EC_{50}\), the response curve is concave for nonnegative concentrations. Differentiating twice gives:

For a concave function, Jensen's inequality states that the average function value is no greater than the function evaluated at the average input. Thus, for this model and a population concentration variable with a finite mean:

The inequality is strict when concentration varies over values where the curve is nonlinear. It is not a universal direction for every possible PD function: the direction depends on the curvature of the particular response function over the relevant range.

What Each Prediction Includes

A conditional prediction based on an estimated individual PK parameter is often called a plug-in prediction. If \(\hat{\phi}_i\) is held fixed, the prediction includes the consequences of that individual's estimated PK profile, but it does not automatically include uncertainty in \(\hat{\phi}_i\). Nor does it include residual variation unless the aim is to predict an observed response rather than the model's mean response.

A population-marginal mean accounts for between-person PK variability by integrating over the specified PK distribution. That integral changes the mean prediction, but calculating the mean alone does not describe the full range of individual responses. For a predictive distribution, residual variation and the variation in responses induced by PK differences must also be represented.

The distinction can be written using the law of total variance. Let \(Y\) be an observed response, let \(C\) denote the concentration arising from an individual's PK parameters, and suppose the conditional response has mean \(m(C)\) and residual variance \(\sigma^2(C)\). Then:

The first term averages the residual variance across the population. The second is the between-person variance in the conditional mean response induced by PK variability. A conditional predictive distribution at a fixed, known \(C\) has the first type of variation but not the second. Parameter-estimation uncertainty can add further uncertainty to either prediction and must be handled separately.

Parameter Uncertainty Is a Separate Layer

In practice, PD parameters and the distribution of PK parameters are estimated rather than known. A single marginal prediction computed at point estimates, \(\hat{\theta}_{PD}\) and \(\hat{\Omega}\), is still a plug-in estimate of the population-marginal mean. It accounts for the PK variability represented by the fitted distribution, but it does not by itself show how uncertain that estimated mean is.

One way to represent parameter uncertainty is to calculate the marginal prediction for each parameter draw from a bootstrap or another fitted inferential procedure. The spread across those marginal predictions describes uncertainty in the estimated marginal mean. To predict a new person's observed response, each calculation must additionally represent that person's PK variability and residual response variation. These are distinct sources of variation, not interchangeable labels for one standard error.

Calculation, Diagnostics, and Alternatives

For simple PK distributions and response functions, the marginal prediction may be calculated analytically. More often, it is evaluated numerically. A direct simulation approach draws PK parameters from the fitted population distribution, calculates the concentration profile for each draw, applies the PD model, and averages the resulting responses. If the target is a distribution of observed responses rather than a mean response, the simulation also draws residual error. Repeating the calculation over parameter draws can incorporate estimation uncertainty.

A reliable calculation should preserve the target population and conditioning information. For example, a prediction for a specified covariate group should average over the PK distribution appropriate to that group, not over an unrelated overall distribution. If relevant PK parameters and PD parameters are modeled jointly, the integration must respect their modeled dependence rather than treating them as independent without justification.

Check the calculation by verifying that the sampled PK values reproduce the intended distribution, that concentrations and responses have the correct units, and that the conditional predictions behave sensibly over the concentration range. Compare numerical integration or simulation results across larger sample sizes or quadrature settings; material changes indicate inadequate numerical accuracy. Also compare a marginal prediction with a direct average of conditional predictions in a small test calculation. Disagreement beyond numerical error can signal that weights, transformations, or the conditioning set have been applied incorrectly.

A common alternative is to predict at a typical or mean PK parameter value. This is computationally simple and can be useful for describing a representative profile, but it is not generally equivalent to integrating over PK variability. Another alternative is a linear approximation around typical parameters. It can be efficient when variability is small and the response is nearly linear over the relevant range, but it can be inaccurate with substantial variability, strong curvature, or concentrations near a steep part of the response curve.

Common Pitfalls and Limitations

• Calling a typical-person prediction a population average. A prediction evaluated at typical PK parameters is conditional on those parameters; the population average requires averaging the responses over the target PK distribution.
• Averaging concentrations before applying a nonlinear model. This calculates \(m(E[C])\), not \(E[m(C)]\). For a concave Emax curve, it will be at least as large as the marginal mean response under the stated conditions.
• Reporting a marginal mean as if it described every individual. The mean does not show the spread of responses across PK profiles. A predictive distribution is needed for individual-level uncertainty.
• Ignoring uncertainty in fitted inputs. Integrating over estimated PK variability does not automatically account for uncertainty in PD parameters, PK parameters, or the PK variability distribution itself.

Marginal predictions are only as relevant as the population distribution used in the calculation. A fitted distribution may be poorly supported in regions with sparse PK sampling, and misspecification of the PK model or distribution can distort the resulting PD average. The integral also describes an average over the specified population, not necessarily a treatment effect or a causal contrast. The prediction target, model assumptions, and population should therefore accompany any reported result.