Constructing Scientifically Defensible PK/PD Candidate Sets

Candidate Sets Are Sets of Scientific Hypotheses

Model selection begins before a criterion or test is calculated: the analyst must decide which models are eligible to be compared. A candidate model is more than an equation that can be fit. It is a specified hypothesis about how exposure, time, baseline, covariates, and unexplained variation relate to the pharmacodynamic (PD) outcome. A comparison can only be scientifically informative if the candidate set represents plausible explanations of the observed data.

For example, a concentration-response curve may appear to describe a drug effect, but the observed response could also reflect baseline differences, disease progression, placebo change, or a delayed effect of exposure. If all candidates assume an immediate drug effect and a constant baseline, the comparison cannot show that this entire class of assumptions is inadequate. It can only distinguish the included immediate-effect models.

To make a candidate explicit, state its response variable and analysis population, its predictor and exposure history, its structural form, and its observation assumptions. In repeated-measures PK/PD data, also specify how subject-level differences and within-subject variation are represented. Parameter values may differ across candidates; the scientific meaning of a parameter, however, should be clear within each model.

Define the Shared Data and Likelihood Basis

A likelihood assigns a probability density or probability mass to the observed data under a model. In a PK/PD analysis, let \(y_{ij}\) denote the response for subject \(i\) at observation \(j\), and let \(\mu_{ij,m}\) be its model-predicted mean under candidate \(m\). One possible observation model is

with conditionally independent residuals. Under these assumptions, the likelihood is the product of the normal densities for the analyzed observations. A different model might use a lognormal distribution, a binary response likelihood, or a serial correlation structure. Such choices can be scientifically appropriate, but they are part of the hypothesis, not harmless implementation details.

For a focused comparison of exposure-response structures, candidates should ordinarily use the same response records, analysis population, response definition, and handling of missing or censored observations. If one model is fit to fewer records, its likelihood reflects a different data set. If one candidate uses observed concentrations and another uses a different exposure source or PK assumption, a difference in fit may reflect that change as well as the PD structure. Make such differences explicit; do not describe the result as a clean comparison of PD shapes.

The likelihood assumptions also need to fit the data. Conditional independence is questionable if residuals remain serially correlated. A constant residual variance is questionable if variability changes systematically with the mean. For population models, the likelihood must account for the subject-level effects as specified by the model; treating estimated subject effects as known values can give a different likelihood basis. The candidate set should not conceal these choices behind a shared label such as “same data.”

State the Exposure-Response Hypotheses Explicitly

A useful candidate set starts with the scientific question and translates each plausible answer into a model. For example, an immediate, saturable effect can be written as

where \(C_{ij}\) is concentration at the response time, \(E_0\) is the baseline mean, \(E_{\max}\) is the maximum change from baseline, and \(EC_{50}\) is the concentration producing half that maximum change under this parameterization. This candidate assumes that the response at a given time depends on the current concentration through this curve, with no additional time trend or delayed exposure state.

A linear immediate-effect candidate instead might use \(\mu_{ij}=E_0+\beta_C C_{ij}\). These are distinct shape hypotheses, but neither automatically represents placebo change, disease progression, tolerance, rebound, or an effect lag. Whether to include those structures depends on the scientific setting, design, and diagnostic evidence. The earlier material on PD diagnostics provides ways to recognize residual patterns that can motivate revising the set; it does not make an omitted structure part of the current candidate set.

Candidates should be neither arbitrary variations nor a long list of every imaginable equation. Each should have a rationale, such as a known mechanism, established pharmacology, design feature, or plausible alternative explanation. Keep parameterizations interpretable, state constraints where required, and avoid candidates that cannot be meaningfully estimated from the available exposure range or sampling schedule.

Omitted Structure Can Misattribute an Effect

If exposure changes alongside another cause of response, an exposure-only candidate can assign their combined pattern to exposure. This is not merely a risk of choosing the wrong model after comparison. If the relevant alternative is absent, the candidate set cannot express it. The resulting exposure coefficient may describe the observed association under the imposed structure without isolating a drug effect.

This example is an identification issue: multiple explanations yield the same fitted values for the available observations. It is not repaired simply by using a more flexible exposure curve. Before interpreting a fitted exposure-response relation, examine whether concentration is entangled with time, dose group, baseline, concomitant treatment, or other predictors. A model comparison cannot resolve an ambiguity that the data and candidate set do not resolve.

Justification: Why a Candidate Set Cannot Validate Itself

A model comparison ranks or tests models that were supplied to it. It does not establish that the supplied set contains the process that generated the data. The logic can be stated directly. Let \(M_1,\ldots,M_K\) be the candidate set and let \(D\) be the observed data. Any selection rule based only on these candidates returns an element of \(\{M_1,\ldots,M_K\}\). If a scientifically plausible model \(M^*\) is not in that set, then

This is a simple but important limitation: a favorable relative comparison is conditional on the candidates and assumptions under consideration. It is not proof of correct mechanism, adequate prediction, or causal interpretation. Predictive and residual diagnostics remain important for detecting patterns that the candidate structures fail to capture.

Diagnostics, Alternatives, and Limitations

Before fitting, check whether the design can distinguish the proposed structures. For a saturation model, ask whether observed concentrations span the region around \(EC_{50}\); observations only far below or above it may weakly identify the curve shape. For a lag model, check whether response sampling is frequent enough to reveal a delay. For time trends and exposure effects, ask whether exposure varies independently enough of time to distinguish their contributions. If key parameters are poorly identified, a nominally rich candidate set may yield unstable or uninterpretable comparisons.

After fitting, examine whether residuals show systematic patterns by concentration, time, dose, baseline, or subject, and whether the assumed residual distribution and dependence structure remain plausible. A common pattern of delayed residuals, for instance, suggests that a direct-effect candidate may be missing exposure history. A systematic trend shared across dose groups may point toward baseline or disease progression structure rather than an exposure curve. These are prompts to revisit the scientific hypotheses, not automatic instructions to add parameters until residuals disappear.

The main alternative to a narrow candidate set is a broader, scientifically justified set that includes relevant baseline, time, lag, and shape hypotheses. Broader is not always better: adding many weakly identified candidates can make results difficult to interpret and can obscure the question of interest. Another approach is to treat different likelihood assumptions as separate, explicitly labeled modeling packages, rather than claiming they isolate a single structural difference. In either case, document what varies and why.

A candidate set has unavoidable limits. It is based on current knowledge, available measurements, and design support. Some mechanisms cannot be distinguished with the data at hand; some plausible models may be computationally infeasible; and unmeasured confounding cannot be eliminated by adding a mathematical term without justification. The defensible conclusion is therefore conditional: among the stated candidates, under their stated assumptions, the data favor one representation according to the chosen comparison procedure. The procedure itself should be selected and interpreted separately.

Common Mistake

• Wrong reasoning: “The exposure-response model fits better than the other candidates, so exposure caused the response change.”
• Correction: The result concerns relative fit within the specified candidate set. If baseline change, time trend, exposure lag, or another plausible structure was omitted, the comparison cannot rule it out. Check the design, assumptions, and diagnostics before making a mechanistic or causal interpretation.