Bias From Nonlinear Exposure Plug-In

Why Unbiased Concentrations Can Still Produce Biased PD Predictions

In sequential PK/PD estimation, a PD model may be fitted using concentrations predicted by an earlier PK fit. Even if those predicted concentrations are unbiased across repeated samples, substituting them directly into a nonlinear exposure-response function does not generally preserve unbiasedness. The issue is curvature: averaging the response at uncertain concentrations need not equal the response at the average concentration.

This tutorial isolates that mechanism. It distinguishes bias in a predicted PD response from bias in an estimated PD parameter, since the latter also depends on the observation design and fitting procedure. The calculations assume the PK model is correctly specified and focus on concentration uncertainty rather than other sources of error.

The distinction is between applying a function to an average concentration and averaging the function over uncertain concentrations. Those operations coincide for a linear function, but generally not for a curved one. The direction of the difference depends on the curvature in the range of concentrations with appreciable uncertainty.

Curvature and a Local Approximation to Bias

Suppose \(\hat C=C+U\), where \(E(U)=0\), \(\operatorname{Var}(U)=\sigma_C^2\), and the uncertainty is small enough that a Taylor approximation around \(C\) is informative. Expanding \(f(C+U)\) gives

Taking expectations, using \(E(U)=0\) and \(E(U^2)=\sigma_C^2\), yields the second-order approximation

The first-order term vanishes because concentration error has mean zero. The leading remaining term depends on both concentration variance and the second derivative of the exposure-response curve. This approximation neglects higher-order terms and is most useful when uncertainty is modest and \(f\) is smooth over the relevant concentration range. For large or asymmetric uncertainty, the exact expectation or numerical integration is preferable.

For a standard increasing Emax relationship,

where \(E_0\) is baseline response, \(E_{\max}>0\) is the maximum increase above baseline, and \(EC_{50}>0\) is the concentration producing half of that increase. Its second derivative is

Thus this increasing Emax curve is concave for nonnegative concentrations. If the concentration uncertainty stays in that domain, the average plug-in response is no greater than the response at the unbiased mean concentration. The approximation also shows that, locally, the downward bias grows with concentration variance and with the magnitude of the curve's curvature.

The sigmoid Emax relationship can change curvature:

When \(h>1\), its inflection concentration is \(EC_{50}((h-1)/(h+1))^{1/h}\). Below this point the curve is convex; above it the curve is concave. Consequently, plug-in bias can be upward at low concentrations and downward at high concentrations. If uncertainty spans both regions, neither a single local curvature value nor a simple directional rule is sufficient.

Worked Examples

Response Bias Is Not the Same as Parameter Bias

A biased plug-in response does not translate into a universal direction of bias for every fitted PD parameter. Parameter estimates are obtained by fitting a model across the study's observations; the direction and size of their bias can depend on the concentration design, which parameters are fixed or estimated, the residual model, and the fitting criterion. The following deliberately simple example isolates one parameter-estimation mechanism.

Assumptions, Diagnostics, and Practical Choices

The derivations concern uncertainty in concentration estimates passed through a correctly specified nonlinear response function. They presume that the concentration estimates target the true concentrations at the times used for PD modeling. In an actual analysis, systematic PK misspecification, time-dependent errors, or dependence between concentration errors and PD observations may add other distortions; the curvature argument alone does not describe those effects.

Useful checks focus on the uncertainty and the local shape of the exposure-response curve:

• Plot the fitted PD curve and the distribution or uncertainty range of predicted concentrations at the PD observation times. Identify whether that range is nearly linear, mostly convex, mostly concave, or spans an inflection point.
• Use the local approximation \(\tfrac{1}{2}f''(C)\sigma_C^2\) as a scale check, not as a substitute for exact integration when uncertainty is large or curvature changes across the range.
• Compare plug-in predictions with predictions averaged over plausible PK parameter or concentration uncertainty. A material difference indicates that evaluating the PD curve only at a point estimate may be consequential.
• Check whether the plug-in fit leaves systematic residual patterns across concentration or time. Such patterns can be a warning, though residual checks alone cannot establish that curvature-induced bias is the cause.

The main alternatives differ in how they handle PK uncertainty. A sequential plug-in fit treats the estimated concentrations as fixed and is simple, but can omit uncertainty and create nonlinear bias. A delta-method adjustment can approximate local bias or uncertainty when the response is smooth and concentration uncertainty is modest. Integrating the PD response over the distribution of concentration uncertainty more directly targets \(E[f(\hat C)]\), while simultaneous estimation models PK and PD data together and can account for shared parameter uncertainty under a suitable joint model. As discussed in the earlier material on when simultaneous estimation does not improve precision, joint estimation is not automatically more precise; nor does it guarantee protection from a misspecified model.

Common Mistakes and Limitations

• Wrong reasoning: “The concentration estimates average to the true concentration, so the average predicted response must also be correct.” Correction: This conclusion holds for a linear response function, not generally for a nonlinear one. Curvature makes \(E[f(\hat C)]\) differ from \(f(E[\hat C])\).
• Assuming the bias is always downward. The standard increasing Emax curve is concave, but a sigmoid Emax curve with \(h>1\) is convex below its inflection point and concave above it. The bias direction may change with concentration and uncertainty range.
• Equating response bias with parameter bias. The plug-in response difference is a property of evaluating the function under concentration uncertainty. Parameter bias also depends on the data design and fitting procedure, as the Emax example illustrates.
• Treating the second-order formula as exact. It omits higher-order contributions. When uncertainty is wide, asymmetric, or crosses a curvature change, evaluate the response over the uncertainty distribution instead.

These checks do not identify every source of error in a sequential analysis. They isolate a particular risk: even unbiased concentration estimates can be inadequate inputs to nonlinear PD functions when their uncertainty is consequential. The importance of that risk must be judged in the concentration range and design of the analysis at hand.