Why Joint Estimation Can Improve Efficiency

Why Information Can Flow From PD Data to PK Parameters

A PK model predicts concentrations, and a PD model relates those concentrations to response. When a PK parameter changes the predicted concentrations, it can also change the PD predictions. Consequently, PD observations may contain information about that PK parameter as well as about the PD parameters. Joint estimation uses this concentration-response dependence in the shared likelihood.

The potential efficiency gain needs a careful comparison. If PK parameters were known exactly, PD estimates would generally be more precise than when those parameters are uncertain. Joint estimation does not make uncertain PK parameters known. Its possible advantage is relative to a staged analysis that estimates PK parameters from concentration data alone and then carries their uncertainty into the PD fit: the PD data can also help estimate the PK parameters in the joint analysis.

This representation assumes a correctly specified likelihood, appropriate conditional independence between PK and PD observations given the model parameters, and parameters that are locally identifiable. The information matrix must be positive definite for the usual inverse-information covariance approximation. In finite samples, the inverse observed Hessian is often used as an approximation; the expected Fisher information gives the corresponding large-sample result. These are local approximations, not guarantees about every dataset or estimator.

Deriving the Joint PD Uncertainty

Under regularity conditions, the covariance matrix of the joint maximum-likelihood estimator is approximately \(\mathcal{I}^{-1}\). The PD block of that inverse is $$ \operatorname{Cov}(\hat{\theta}_{\mathrm{joint}}) \approx \left[D-B^{\mathsf T}(K+A)^{-1}B\right]^{-1}. $$ The matrix inside the inverse is the Schur complement of the PK block \(K+A\). It is the information about \(\theta\) remaining after accounting for the fact that \(\phi\) is estimated rather than known.

To see why the cross block matters, write a local quadratic approximation to the negative log likelihood in increments \(u\) for \(\phi\) and \(v\) for \(\theta\). Its curvature-dependent part is $$ \frac{1}{2}u^{\mathsf T}(K+A)u + u^{\mathsf T}Bv + \frac{1}{2}v^{\mathsf T}Dv. $$ For a fixed \(v\), the minimizing increment in \(u\) is \(u=-(K+A)^{-1}Bv\), found by setting the derivative with respect to \(u\) equal to zero. Substituting that value into the quadratic leaves $$ \frac{1}{2}v^{\mathsf T}\left[D-B^{\mathsf T}(K+A)^{-1}B\right]v. $$ Thus uncertainty in the estimated PK parameters reduces the effective information about PD parameters compared with the hypothetical case in which the PK parameters are known, for which the PD covariance is approximately \(D^{-1}\).

The staged comparison is different. In a local quadratic model, a two-stage fit first estimates \(\phi\) using PK data alone and then fits \(\theta\) conditional on that estimate. Its conditional PD covariance is approximately \(D^{-1}\), and the sensitivity of the conditional estimate to \(\hat{\phi}\) is \(-D^{-1}B^{\mathsf T}\). If the PK and PD score contributions are independent, a first-order uncertainty propagation gives $$ \operatorname{Cov}(\hat{\theta}_{\mathrm{staged}}) \approx D^{-1}+D^{-1}B^{\mathsf T}K^{-1}BD^{-1}. $$ The second term represents the uncertainty transferred from the PK estimate. Unlike a fit that treats PK estimates as fixed, this approximation does not discard that uncertainty. Joint estimation can improve on this staged result when the PD data provide useful information about \(\phi\) and the cross-information links \(\phi\) to \(\theta\). The amount of improvement depends on the information blocks and model; it is not automatic.

What Creates or Removes the Gain?

The cross block \(B\) is central: if \(B=0\), the local likelihood has no PK–PD parameter coupling through this mechanism. The PD observations then do not locally inform the PK parameters in a way that changes the PD covariance. A nonzero \(B\) alone is not enough to promise a practically meaningful gain. The PD likelihood must also contain usable information about \(\phi\), represented by \(A\), and the PK likelihood must be informative enough to support stable estimation.

A second comparison shows how additional PD information about PK can strengthen the gain while keeping the cross-information fixed. In the first example, \(A=6\), \(B=3\), \(D=9\), and \(K=10\). Suppose instead that a different hypothetical design yields \(A=15\), \(B=3\), \(D=9\), and the same PK-only information \(K=10\). The PD information block remains valid because \(15(9)-3^2=126>0\).

Assumptions, Diagnostics, and Alternatives

Information-based precision comparisons are most persuasive when the model is identifiable, the likelihood is well specified, and the local quadratic approximation is adequate. Concentration-response dependence must be supported by the design and data, not merely by a parameterization that mathematically links the predictions. If PK sampling is sparse, the exposure range is narrow, or the PD response is insensitive to concentration in the observed range, the relevant information may be weak. Strong parameter correlations, flat likelihood profiles, or unstable covariance estimates can indicate that the inverse information matrix is not a reliable summary.

Useful checks include inspecting standard errors and parameter correlations from the joint covariance matrix, assessing whether estimates change substantially with starting values, and examining whether the likelihood has flat or asymmetric regions. Compare joint and staged estimates and uncertainty estimates on the same parameter scale. A very large standard error, near-singular information matrix, or implausibly strong apparent precision gain warrants scrutiny of identifiability, model fit, and numerical convergence. Such diagnostics do not by themselves establish which model is correct.

The principal alternatives answer different questions. A staged fit with PK uncertainty propagated by a delta method retains first-order uncertainty from the PK stage, but does not allow PD data to update the PK estimate. A staged fit that treats estimated concentrations as exact omits that uncertainty and can report overly narrow uncertainty. A joint likelihood uses both likelihood contributions together, allowing PD observations to inform PK parameters through their predicted concentrations. Comparing these approaches is meaningful only when they target the same parameters and estimands and use compatible likelihood assumptions.

The inverse-information argument is local and commonly relies on large-sample approximations. It does not guarantee lower finite-sample error, lower bias, or better predictive performance. Model misspecification, weak identification, parameter boundaries, dependence not represented in the likelihood, or numerical problems can undermine the comparison. Joint estimation is valuable when the shared concentration-response structure is credible and informative; complexity alone is not evidence of efficiency.

Common Mistake

A common mistake is to say, “Joint estimation reduces PD uncertainty because it treats PK estimates as known.” That reverses the logic. Joint estimation still estimates PK parameters, so its PD uncertainty is generally greater than the uncertainty under the hypothetical assumption that PK parameters are known. Its possible advantage is instead that PD observations can add information about PK parameters, which may reduce PD uncertainty relative to a staged fit that uses PK data alone to estimate those parameters. A second mistake is to compare a joint standard error with a staged standard error computed as though plug-in PK estimates were exact; that comparison hides the uncertainty being transferred from PK to PD.