Parameter Uncertainty in Local Design

Why Parameter Uncertainty Changes a Local Design

A design calculation for a nonlinear pharmacokinetic (PK) model needs parameter values. For example, predicted concentrations and their sensitivities to clearance depend on the clearance value used. A local design calculation evaluates information at a specified parameter vector, often called the nominal value. The result is conditional on that choice: if the true parameters differ, the information available from the same sampling schedule can differ too.

That dependence is a design issue, not a computational error. Before comparing designs, it is useful to be precise about how uncertain parameters are being represented. A single nominal value, a probability distribution, and a plausible range answer different questions. They also lead to different information summaries. Here the focus is on how those representations enter the calculation, rather than on choosing an optimality criterion for combining or comparing the resulting information.

A nominal value is often a point estimate or a scientifically reasonable working value. A probability distribution represents uncertainty through probabilities assigned across possible parameter values. A plausible range specifies values regarded as possible or credible without necessarily assigning probabilities within the range. These representations are not interchangeable: a range alone does not say which values are more likely, and a probability distribution contains more information than its mean.

A Nonlinear PK Model for the Calculations

Consider a hypothetical one-compartment model after an intravenous bolus dose. Let \(D\) be dose, \(V\) be volume of distribution, and \(CL\) be clearance. The predicted concentration at time \(t\) is

For the worked calculations, assume \(D\) and \(V\) are known and fixed, observations have independent additive normal errors with known variance \(\sigma^2\), and the model is correctly specified. Thus \(Y(t)=C(t;CL)+\epsilon\), where \(\epsilon\sim N(0,\sigma^2)\). These assumptions make the information calculation transparent; real PK analyses may require a different residual-error model and may estimate additional parameters.

It is convenient to express sensitivity to clearance on a log scale. Define \(q=\log(CL/(1\text{ L/h}))\), so that \(CL=\exp(q)\text{ L/h}\). For one observation, the Fisher information for \(q\), conditional on its value, is the squared concentration sensitivity divided by the known error variance:

The derivative follows from the chain rule: \(dCL/dq=CL\), and differentiating the exponential concentration model with respect to \(CL\) gives \(-Ct/V\). The quantity \(CLt/V\) is dimensionless, so the sensitivity has concentration units per unit change in \(q\). Under the stated independent-error assumptions, information from multiple observations adds across observation times.

What a Distribution Adds

A probability distribution over parameter values allows calculations to account for more than one possible value. For instance, one may calculate the information at each value and then average it using the distribution’s probabilities. This is an average of conditional information values. It is generally not the same as evaluating information at the distribution’s mean, because the model and its sensitivity are nonlinear functions of the parameters.

The distinction does not depend on a particular interpretation of the distribution. It could represent uncertainty from a fitted parameter estimate or be an externally specified probability model. In either case, the distribution must be stated, including its scale and any dependence among parameters. A distribution for \(CL\), for example, is not the same as a distribution for \(\log(CL)\), even if both are described using a similar central value.

What a Plausible Range Adds

A plausible range identifies values that should not be ignored, but it does not assign probabilities to them. It is useful for asking how a local calculation changes across the values considered possible. For example, a range can reveal that a sampling time is informative near one clearance value but less informative near another. Without probabilities, however, it does not determine an average information value.

The nonlinear model also shows why the location of informative sampling can move. Write \(A=D/V\) and \(x=CLt/V\). For the log-clearance parameter, information at one time is proportional to \(x^2e^{-2x}\). For positive \(x\), differentiating gives

The derivative is positive for \(0<x<1\) and negative for \(x>1\), so this single-observation information is maximized at \(x=1\). Since \(x=CLt/V\), the locally most informative time for log clearance in this simplified model is \(t=V/CL\). This result assumes the model and error structure above and concerns a single parameter; it is not a general prescription for a multi-parameter sampling design.

Assumptions, Diagnostics, and Common Pitfalls

The calculations above rely on a correct one-compartment model, fixed dose and volume, known homoscedastic residual variance, and independent observations. They also assume that the chosen parameterization is appropriate for the question: information for \(q=\log(CL/(1\text{ L/h}))\) is not numerically identical to information for clearance on its original scale. If volume, bioavailability, or error parameters are also uncertain or estimated, the relevant information is a matrix, and parameter correlations can affect the precision of individual estimates.

Useful checks include verifying units in the exponent, confirming that the parameter derivative matches the chosen parameter scale, and recalculating information at several parameter values rather than only at the nominal value. In this model, concentration must decrease with time and clearance, and the log-clearance sensitivity is negative. A positive derivative, or an exponent with units, signals an error. For actual designs, also check whether the assumed residual variance and structural model are credible over the sampling times being considered.

A frequent mistake is to say, “Clearance is uncertain, so use its mean and proceed as if it were known.” That produces a valid local calculation at the mean, but it does not represent the distribution or range of possible clearances. The correction is to state explicitly whether the calculation is conditional on one nominal value, averages conditional quantities over a probability distribution, or examines values across a plausible range.

These approaches have complementary roles. A nominal value is simple and transparent, but can conceal sensitivity to misspecification. A probability distribution quantifies relative likelihoods and supports probability-weighted summaries, but its result depends on the distribution being defensible. A plausible range avoids assigning unsupported probabilities, but cannot by itself yield a probability-weighted average. All three remain limited by the model: uncertainty representations cannot compensate for an incorrect structural or residual-error model.