Diagnosing Sparse-Data Shrinkage in PK Profiles

Why Sparse PK Profiles Need an Information-Based Diagnosis

Two individuals can contribute the same number of concentration measurements yet have very different amounts of information about their individual pharmacokinetic parameters. Measurements taken where a predicted concentration changes little as a parameter changes may tell us little about that parameter. Measurements near a sensitive part of the concentration-time profile can be more informative, provided their residual errors are not too large.

This matters when interpreting empirical Bayes estimates (EBEs) and eta shrinkage. A profile with few informative observations will generally yield EBEs pulled toward the population typical value. A profile can also have many observations but little individual information if their timing, concentration range, or precision is unhelpful. Thus, sample count is not a substitute for examining what the observations can resolve.

The calculations below use a one-random-effect local approximation to make this comparison explicit. They are diagnostic approximations, not a replacement for fitting or checking the full nonlinear mixed-effects model.

Local Information in a PK Profile

Here, \(g_{ij}\) measures how much the predicted concentration changes per unit change in \(\eta_i\). Its units are concentration per eta unit. Each term \(g_{ij}^2/\sigma_{ij}^2\) is information about the dimensionless eta, in inverse squared eta units. A measurement contributes more information when the prediction is sensitive to eta and its residual standard deviation is relatively small.

For example, consider a hypothetical one-compartment intravenous bolus profile with fixed volume \(V\), population clearance \(CL_{\mathrm{pop}}\), and a log-clearance random effect:

Writing \(k=CL_{\mathrm{pop}}/V\), differentiation gives \(g(t)=-ktC(t,0)\). At time zero, this sensitivity is zero: the bolus concentration in this model does not depend on clearance at that instant. At later times, clearance affects the predicted concentration, but the sensitivity does not necessarily keep increasing indefinitely; it depends on both time and the concentration remaining. Other PK models and other random effects have different sensitivity patterns.

The local approximation has specific assumptions. It treats population parameters and residual variances as known, considers one eta at a time, linearizes the prediction around \(\eta_i=0\), and assumes independent normal residuals with the stated variances. The approximation is most useful when this local linearization is reasonable and the model and residual-error assumptions are adequate. Strong nonlinearity, influential observations, or substantial uncertainty in population parameters can make its numerical results misleading.

From Local Information to an EBE

Under the local approximation, write the observations after subtracting the prediction at \(\eta_i=0\) as \(r_{ij}\approx g_{ij}\eta_i+\epsilon_{ij}\). Combining this linear observation model with the normal prior gives a normal conditional distribution for \(\eta_i\). Its variance \(v_i\) and mean \(m_i\) are:

The conditional mean is the EBE in this normal approximation; for a normal conditional distribution, its mode is the same. The formula shows why low-information profiles tend to have EBEs near zero: when \(J_i\) is small, the prior contributes a large share of the precision, \(1/v_i\). It also shows that information about eta depends on the sensitivity and error variance at each observation, not merely on how many rows appear in an individual’s record.

A useful connection to conventional eta shrinkage follows when all individuals have the same design and the same \(J_i\). The conditional variance is then \(v\) for everyone. Under the model, the variance across individuals of their conditional means is \(\omega^2-v\). To see this, use the variance decomposition \(\mathrm{Var}(\eta_i)=\mathrm{Var}\{\mathrm{E}(\eta_i\mid y_i)\}+\mathrm{E}\{\mathrm{Var}(\eta_i\mid y_i)\}\); the second term is \(v\), so the first is \(\omega^2-v\). Therefore, the expected conventional shrinkage, defined as one minus the standard deviation of the EBEs divided by \(\omega\), is:

This is an expected value under the stated model and equal-design assumptions, not a promise that the sample EBE standard deviation will give that exact number. With different individual designs, one common \(J_i\) no longer describes everyone, and a single design-based value should not be assigned to all profiles.

Concentration Coverage and Residual Error

Concentration coverage matters because it affects both sensitivity and the relevant residual variance. In a PK profile, observations concentrated in a region where predictions barely respond to the parameter of interest can add little information. Observations spanning more informative parts of the curve can help distinguish parameter values, but only if the model predicts meaningful changes there and residual uncertainty does not overwhelm those changes.

Residual error may depend on concentration. For instance, a model may include both additive and proportional error, so that \(\sigma_{ij}^2=\sigma_a^2+(\mathrm{CV}\,f_{ij})^2\), where \(\sigma_a\) is an additive-error scale and \(\mathrm{CV}\) is a proportional-error coefficient. When concentrations are low, the additive component can make the residual standard deviation large relative to the signal. With purely proportional error, the ratio of sensitivity to residual standard deviation can behave differently. Consequently, “late samples are uninformative” is not a universal rule; it depends on the parameter, model, sampling time, concentration, and error model.

Practical Diagnostics and Alternatives

When evaluating sparse-data shrinkage, begin with the actual concentration-time coverage and the parameter being assessed. A profile may support an individual estimate of one parameter better than another: sensitivity to clearance, for example, need not match sensitivity to volume or absorption. The local-information calculation should therefore use the derivative for the specific eta, not a generic label such as “PK information.”

A useful diagnostic sequence is to:

The information calculation is more informative than simply counting samples, but it is still only a local, model-based summary. An alternative is to inspect the nonlinear conditional likelihood or posterior numerically for each individual; this can reveal asymmetry or multiple plausible parameter values that a local normal approximation misses. Design simulations under the full model can assess expected individual precision across sampling schedules and realistic residual error. These approaches require more computation, but they avoid treating a single local curvature calculation as a complete description.

Several limitations deserve emphasis. The calculations above condition on known population parameters, while actual fits estimate them. They consider a single random effect and do not represent dependence among random effects. They also assume the model and residual variances are correct. Sparse profiles can make these assumptions especially consequential: apparent low individual information may reflect poor design, high residual error, model misspecification, or a combination. A large calculated \(J_i\) is not proof that the model is correct, and a low shrinkage estimate is not by itself proof that every individual parameter is accurately recovered.

Common Pitfall