Selecting PD Parameters For Between-Subject Variability

Why the Choice of Random-Effects Parameters Matters

A population PD model can allow individuals to differ in one or more parameters. Choosing which parameters vary is a scientific decision: the choice determines which kinds of subject-level response differences the model can represent. A random effect on baseline describes a different pattern from a random effect on maximum drug effect, potency, or response-curve steepness. If the chosen parameters do not match the observed patterns, the model may fit poorly or assign variation to the wrong source.

The question is not simply whether subjects differ. They often do. The question is which features of their concentration-response curves differ, and whether the observations contain enough information to distinguish those features. Subject-level plots, knowledge of the measurement process, and the concentration range observed all contribute to that decision.

A Model for Comparing Candidate Parameters

For illustration, consider a stimulatory Emax model. Let \(R_{ij}\) be the response for subject \(i\) at observation \(j\), and let \(C_{ij}\) be the relevant drug concentration. The conditional mean response is

R_{ij}=B_i+E_{\max,i}\frac{C_{ij}^{h_i}}{EC_{50,i}^{h_i}+C_{ij}^{h_i}}+\epsilon_{ij}.

Here \(B_i\) is subject \(i\)'s baseline response, \(E_{\max,i}\) is the maximum increase above baseline, \(EC_{50,i}\) is the concentration producing half of that maximum increase, and \(h_i\) is the Hill coefficient, which controls the steepness of the concentration-response curve. The examples use a positive, increasing drug effect; inhibitory models have corresponding parameter interpretations but a different direction of effect.

Definition: A between-subject random effect represents a subject-specific departure from a typical model parameter. For positive parameters, one common specification is \(B_i=B\exp(\eta_{B,i})\), \(E_{\max,i}=E_{\max}\exp(\eta_{E,i})\), \(EC_{50,i}=EC_{50}\exp(\eta_{50,i})\), and \(h_i=h\exp(\eta_{h,i})\). The \(\eta\) terms have mean zero on the log scale; their variances and any modeled correlations describe between-subject variability. A scale-appropriate additive specification may be used when positivity is not required.

The parameterization must reflect the scientific meaning and valid range of each parameter. For example, log-scale effects keep \(E_{\max,i}\), \(EC_{50,i}\), and \(h_i\) positive. The choice of random-effects distribution is a separate modeling decision; the earlier material on choosing random-effects distributions and correlated random effects addresses those considerations.

A basic working assumption for the examples is that, conditional on the subject-specific parameters and concentrations, the residuals have mean zero, constant variance, and are independent: \(\epsilon_{ij}\sim N(0,\sigma^2)\). Real analyses may require a different residual or serial-dependence model. If that model is inadequate, apparent subject-level parameter differences can instead reflect unmodeled observation error or time structure.

What Each Random Effect Looks Like

The concentration-response function is \(f(C)=C^h/(EC_{50}^h+C^h)\). Changing one parameter at a time gives useful qualitative predictions:

Baseline \(B\): changes response before drug and shifts the curve vertically. With the same drug-response parameters, a baseline difference produces approximately the same response difference at every concentration.
\(E_{\max}\): changes the size of the maximum drug-related response. It has little influence near zero concentration and is most apparent where the curve approaches its plateau.
\(EC_{50}\): changes potency, shifting the curve along the concentration axis. At a fixed concentration, a larger \(EC_{50}\) generally gives a smaller response increment in this stimulatory model. The curves approach the same plateau if their baselines and \(E_{\max}\) values are shared.
Hill coefficient \(h\): changes steepness. Its influence depends on how far concentrations lie from \(EC_{50}\); curves with different \(h\) meet at \(C=EC_{50}\), where the drug-response fraction is one half.

These distinctions can be made more precise using local sensitivities, which describe how a small parameter change affects the predicted response. For the Emax model, the sensitivity to baseline is constant. On the log scale for \(EC_{50}\) and \(h\), the key derivatives are:

\frac{\partial R}{\partial \log EC_{50}}=-E_{\max}h f(C)\{1-f(C)\}, \qquad \frac{\partial R}{\partial \log h}=E_{\max}h f(C)\{1-f(C)\}\log\!\left(\frac{C}{EC_{50}}\right).

The first derivative has its largest magnitude near \(f(C)=1/2\), rather than at very low or very high concentrations. The second derivative is zero at \(C=EC_{50}\); it changes sign on either side. Thus, observations only at the midpoint are poorly suited to distinguishing Hill-coefficient differences, while observations spanning concentrations below and above the midpoint can reveal differences in steepness. These are guides to interpretation, not proof that a parameter is estimable from any particular data set.

Worked Example: A Vertical Difference Points to Baseline

Suppose two hypothetical subjects have the same \(E_{\max}=20\) response units, \(EC_{50}=10\) concentration units, and \(h=1\), but different baselines: \(B_A=10\) and \(B_B=12\) response units. Responses are observed at \(C=0\), \(10\), and \(100\) concentration units. For \(h=1\), the response fraction is \(f(C)=C/(EC_{50}+C)\).

At \(C=0\), \(f=0/(10+0)=0\). At \(C=10\), \(f=10/(10+10)=0.5\). At \(C=100\), \(f=100/(10+100)=0.9091\). The predicted responses are therefore:

R_A=(10,\ 10+20(0.5),\ 10+20(0.9091))=(10,\ 20,\ 28.18),

R_B=(12,\ 12+20(0.5),\ 12+20(0.9091))=(12,\ 22,\ 30.18).

The subject difference is \(R_B-R_A=(2,2,2)\) response units at all three concentrations. In this constructed example, a baseline random effect represents the pattern directly; adding an \(E_{\max}\) effect would instead predict a difference that grows with concentration. In observed data, baseline measurements and repeated observations are needed to check whether a roughly constant offset is credible rather than an artifact of noise or time trends.

Worked Example: A Plateau Difference Points to Emax

Consider a hypothetical pair with shared \(B=10\) response units, \(EC_{50}=10\) concentration units, and \(h=1\). Subject A has \(E_{\max,A}=20\), while subject B has \(E_{\max,B}=30\) response units. At \(C=10\), \(f=10/(10+10)=0.5\), so the predicted responses are \(10+20(0.5)=20\) and \(10+30(0.5)=25\) response units. The difference is 5 response units.

At \(C=100\), \(f=100/(10+100)=0.9091\). The predictions are \(10+20(0.9091)=28.18\) and \(10+30(0.9091)=37.27\) response units, a difference of 9.09 response units. As concentration increases further, the predictions approach 30 and 40 response units, respectively, so their plateau difference approaches 10 response units.

A difference that grows toward a distinct plateau is consistent with \(E_{\max}\) variability, provided observations actually approach the plateau. By contrast, if responses differ mainly around the rising part of the curve but converge at high concentrations, an \(EC_{50}\) difference may be a more plausible explanation. Without high-concentration observations, the plateau interpretation remains an extrapolation, not a direct observation.

Worked Example: A Steepness Difference Requires a Wide Concentration Range

Suppose two hypothetical subjects share \(B=10\) response units, \(E_{\max}=20\) response units, and \(EC_{50}=10\) concentration units. Subject A has \(h_A=1\); subject B has \(h_B=2\). At \(C=1\), subject A has \(f_A=1/(10+1)=0.09091\), giving \(R_A=10+20(0.09091)=11.82\). Subject B has \(f_B=1^2/(10^2+1^2)=1/101=0.009901\), giving \(R_B=10+20(0.009901)=10.20\) response units.

At \(C=10=EC_{50}\), both fractions equal \(0.5\), so both predict \(10+20(0.5)=20\) response units. At \(C=100\), subject A has \(f_A=100/(10+100)=0.9091\), giving \(R_A=28.18\). Subject B has \(f_B=100^2/(10^2+100^2)=10000/10100=0.9901\), giving \(R_B=29.80\) response units.

The curves meet at the midpoint and diverge on either side, consistent with a steepness difference. In a data set concentrated near \(C=10\), this distinction would be hard to see. A Hill-coefficient random effect is more scientifically defensible when the concentration range includes observations below and above the midpoint and the response pattern supports different steepness, rather than merely reflecting residual scatter.

A Practical Selection Strategy

Begin with the scientific question and design, not with a search over every possible random-effects combination. Ask which subject-level differences are plausible: do subjects have different untreated response levels, different maximum capacities, different potency, or different response steepness? Then examine observed response patterns against concentration, including baseline or placebo observations where available. A useful sequence is:

Specify the response mechanism.
Choose a concentration-response structure and define what each parameter means in the clinical and biological context. Confirm that baseline, maximum effect, potency, and steepness are distinguishable concepts for the response being modeled.

Map observed patterns to parameters.
Look for vertical offsets, differing plateaus, horizontal shifts, or differing steepness. Use subject-level displays and the parameter sensitivities to check which patterns each candidate random effect can produce.

Check data coverage and measurement assumptions.
Assess whether observations cover the concentrations where the proposed effect is visible, and whether baseline observations or repeated measurements are available. Check whether residual and time-course structure could explain the same pattern.

Fit a small set of scientifically motivated candidates.
Compare models that differ in a clearly stated random effect, retaining the same fixed structure and residual model where possible. Evaluate fit, plausibility, parameter uncertainty, and predictive behavior rather than relying on a single fit statistic.

Retain only interpretable, supported variation.
Prefer the simpler model when a more complex model adds little credible improvement or yields unstable estimates. Record the scientific reason for each retained random effect and the evidence that supports it.

Diagnostics, Alternatives, and Limitations

Useful diagnostics include plots of individual observed responses and predictions against concentration, plots over time, and residuals stratified by concentration and subject. If the model includes baseline variability, inspect whether it reproduces pre-dose differences. If it includes \(E_{\max}\) variability, check behavior near the plateau. For \(EC_{50}\) and \(h\), inspect the rising portions of individual curves and whether observations span both sides of the typical midpoint. A candidate random effect is suspect when the fitted model assigns substantial variation to it but the design has little coverage in the region where that parameter changes the response.

Compare the random-effects model with reasonable alternatives. A fixed-parameter model may be adequate when subject-level differences are small relative to residual variability. A model with random effects on a different parameter may better match the shape of the observed differences. A more appropriate residual model or a time-varying PD structure may explain patterns that otherwise appear to be parameter variability. Covariate effects are another possible explanation for systematic differences across subjects, but covariate modeling is a distinct question and should not be substituted for random effects without scientific justification.

A lower objective function or improved fit alone does not establish that a random effect belongs on a particular parameter. Flexible models can absorb residual misspecification, and estimates may be unstable when the data contain little information about a parameter. Assess uncertainty, convergence, and sensitivity to reasonable modeling choices. The next concern in model building is whether random effects on the selected parameters can be separately identified; the present selection step does not establish that they can.

Common pitfall: “Subjects have different responses, so every PD parameter should have a random effect.” This reasoning confuses the existence of variability with evidence about its location. Baseline, \(E_{\max}\), \(EC_{50}\), and \(h\) predict different patterns, and some patterns cannot be distinguished in a limited design. Start with a scientific hypothesis, check the observed pattern and concentration coverage, and add only the random effects the data can meaningfully support.

Check Your Understanding

For each question, identify the most directly suggested random effect and state one limitation of that interpretation.

Subjects have different pre-dose responses, and their response differences remain about the same at low and high concentrations. Which parameter is a natural candidate?
Two subject curves have similar baselines and rising portions but appear to approach different high-concentration plateaus. Which parameter is suggested, and what data are important?
Curves appear to share a plateau but reach intermediate response levels at different concentrations. Which parameter is suggested?
Why do observations only at \(C=EC_{50}\) provide little direct information about between-subject variation in \(h\)?
Observed responses are more dispersed at high concentrations, but there are no individual concentration-response patterns and the residual model has not been checked. Why is this not sufficient evidence for random \(E_{\max}\)?

Solutions. (1) A baseline random effect is natural because it represents a vertical offset that persists across concentrations. The offset could still reflect time trends, residual structure, or measurement differences, so pre-dose and repeated observations matter. (2) \(E_{\max}\) is suggested because it controls the plateau height. Observations near the plateau are important; otherwise the apparent difference may be an extrapolation. (3) \(EC_{50}\) is suggested because it shifts the curve along the concentration axis. The conclusion depends on adequate observations around the rising portion. (4) At \(C=EC_{50}\), \(f=0.5\) for every positive \(h\), and \(\partial R/\partial\log h=E_{\max}h f(1-f)\log(C/EC_{50})=0\). (5) High-concentration dispersion can arise from residual heteroscedasticity, measurement error, or other model misspecification. Subject-level patterns and a checked residual model are needed before attributing it to \(E_{\max}\) variability.