Defining Sequential and Simultaneous PK/PD Estimands

Why the Target Must Be Stated

A PK/PD analysis can report a PD parameter, an individual effect, or an average response over a population. These quantities are related, but they are not interchangeable. For example, a PD effect calculated at a subject’s estimated concentration is conditional on that concentration value. An effect averaged over uncertainty in the subject’s PK parameters is a different target, especially when the PD response is nonlinear in concentration.

The distinction matters when comparing sequential and simultaneous estimation. These are estimation strategies, not estimands by themselves. Either strategy can be used with a clearly specified target, but a sequential analysis that fixes PK estimates naturally produces conditional PD estimates unless the target is explicitly averaged over PK uncertainty. A simultaneous model can estimate shared population parameters while also defining individual or marginal response targets.

A Notation for the Main Targets

For subject \(i\), let \(C_i(t;\theta_i)\) be the concentration at time \(t\) implied by that subject’s PK parameters \(\theta_i\). Let \(m\{C_i(t;\theta_i),\beta,b_i\}\) be the model’s expected PD response. Here, \(\beta\) denotes population PD parameters and \(b_i\) denotes subject-specific PD deviations when the model includes them. A PD effect is often a contrast between two conditions, such as an exposure condition and a reference condition:

The reference might be zero concentration, placebo, or another specified exposure. The contrast and time point must be stated; a parameter such as an Emax or slope is not itself automatically the treatment effect at every concentration.

• Population PD parameter: a shared parameter such as \(\beta\), defined by the chosen population model. It describes a feature of that model; it is not necessarily an average individual response.
• Individual conditional effect: \(\Delta_i(\hat\theta_i,b_i)\), evaluated at a specified PK value, often a fitted or estimated value \(\hat\theta_i\). The target is conditional on that value being fixed.
• PK-uncertainty-marginal effect: an average of the individual effect over a distribution of plausible PK values, such as \(E\{\Delta_i(\theta_i,b_i)\mid D_{\mathrm{PK}}\}\). Here \(D_{\mathrm{PK}}\) denotes the subject’s PK data.
• Population-average effect: an average over the target population’s distribution of PK and, if present, PD subject effects. The averaging distribution and population must be specified.

A conditional effect and a marginal effect answer different questions. “What is the model-predicted effect at this fixed PK estimate?” is conditional. “What effect is expected when PK uncertainty is accounted for?” is marginal over the specified uncertainty distribution. “What is the average effect in the target population?” additionally requires a population averaging rule.

Marginalizing Over PK Uncertainty

Suppose the PK data do not determine a subject’s PK parameters exactly. A marginal target averages the response contrast over a distribution representing that uncertainty. In a Bayesian analysis this may be a posterior distribution conditional on PK data; in another framework it may be a specified sampling or uncertainty distribution. The distribution used must be stated because it defines the average.

The distinction follows directly from the definition of an expectation. A plug-in result substitutes one value, such as a PK estimate, into the response function. A marginal result first evaluates the response across the uncertainty distribution and then averages those responses. In general,

Equality holds in special cases, including when the effect is linear in the uncertain quantity being averaged. With nonlinear concentration-response functions, the two sides can differ even when the expected concentration is known exactly.

Population Parameters and Population-Average Effects

A population PD parameter and an average effect in a population are also distinct. A shared parameter may describe the typical response function, while a population-average effect averages individual responses over the population’s PK and PD variability. Those individual values need not be independent: subjects with different exposures may also have different PD effects. Consequently, the population-average response depends on the joint distribution specified by the model, not just on a typical parameter and a typical concentration.

For example, if individual effects follow \(\Delta_i=(\beta+b_i)C_i\), where the population mean of \(b_i\) is zero, then \(\beta\) is the population mean slope. The population-average effect is \(E\{(\beta+b_i)C_i\}\). It equals \(\beta E(C_i)\) only under additional conditions, such as zero covariance between \(b_i\) and \(C_i\). This is a definition-based distinction, not a claim that either target is always the desired one.

Sequential and Simultaneous Estimation: What Changes?

In sequential estimation, PK parameters or concentrations are first estimated and then supplied to a PD analysis. If those values are treated as fixed, the PD target is conditional on them. A simultaneous analysis estimates PK and PD quantities within a joint model, but that fact alone does not define whether the reported result is a population parameter, a subject-specific conditional effect, or a marginal effect. The analysis must still specify the target.

If two approaches use the same correctly specified model and target the same parameter, their estimators may differ while the estimand remains the same. Conversely, comparing a conditional plug-in effect from one analysis with a population-average effect from another compares different quantities, even if both are labelled “the PD effect.” A joint model can also report a parameter such as \(\beta\) without that parameter being a marginal clinical response contrast.

Assumptions, Diagnostics, and Limitations

Each target is meaningful only relative to its assumptions. A conditional effect assumes the fixed PK values and the PD model are suitable for the stated subject and time. A PK-marginal effect requires a defensible uncertainty distribution for PK parameters and a response model that is adequate across the values receiving weight. A population-average effect additionally requires a defined target population and an appropriate distribution for its subject-level characteristics.

Before interpreting a result, check whether the report identifies the exposure contrast, time point, target population, and averaging rule. Check whether a quantity described as marginal was actually averaged over the relevant distribution, rather than evaluated at a mean estimate. For a population average, consider whether the model’s assumed relationship between PK and PD variability is plausible. Predictions that rely heavily on exposure values outside the observed data also deserve caution: the mathematical target may be defined, but the data may provide weak support for it.

A common mistake is to say, “The simultaneous fit gives the population effect, while the sequential fit gives the individual effect.” That confuses estimation strategy with estimand. Either strategy can target different quantities depending on the model and the reported functional. The correction is to state the estimand explicitly and then explain how the fitted analysis estimates it. Another mistake is to substitute mean exposure into a nonlinear PD function and call the result the mean effect. As the Emax example showed, the mean of predicted effects need not equal the prediction at mean exposure.

No single target is always best. A conditional effect can answer a subject-specific prediction question but omit uncertainty from PK estimation. A marginal effect accounts for a specified distribution of PK uncertainty but depends on that distribution. A population-average effect can support population-level interpretation, yet may conceal important subject heterogeneity. The target should follow the scientific question, and conclusions should be limited to the population, contrasts, and conditions used to define it.