Balancing PK and PD Information in the Joint Likelihood

Why Balance Matters

A simultaneous PK/PD analysis does not give concentration data and response data equal influence merely because both are present in the model. Their influence depends on how strongly each stream changes as a parameter changes, on measurement variability, on dependence among observations, and on the other parameters being estimated. A large number of relatively uninformative records can contribute less precision than a small number of precise, sensitive observations.

The useful question is therefore not “Which stream has more observations?” or “Which stream has the larger likelihood value?” It is “How much does each stream constrain the parameter or parameter combination of interest?” Likelihood curvature gives a local answer. Because the PD model may depend on PK parameters through concentration or exposure, response data can also contribute information about PK parameters.

Joint Likelihood and Information Contributions

Let \(C\) denote the concentration data, \(R\) the response data, \(\theta\) the PK parameters shared with the PD model, and \(\beta\) PD-specific parameters. A common factorization, when the response model is conditional on the concentration information used in the analysis, is

L(\theta,\beta;C,R) = L_C(\theta;C)\,L_R(\theta,\beta;R\mid C), \qquad \ell(\theta,\beta) = \ell_C(\theta)+\ell_R(\theta,\beta).

Here \(L\) is the likelihood and \(\ell=\log L\) is the log-likelihood. The factorization does not assert that every concentration and response record is mutually independent. The conditional response likelihood must reflect the model’s dependence structure, including any correlation or latent-variable integration that the analysis specifies.

Definition: The observed information matrix is the negative Hessian of the log-likelihood at a parameter value: \(J(\theta,\beta)=-\partial^2\ell/\partial(\theta,\beta)\partial(\theta,\beta)^\mathsf{T}\). Under the stated factorization, the observed information is the sum of the concentration and conditional-response contributions. Fisher information is the expected observed information under the model. Both describe local curvature; neither is simply a count of observations.

For a scalar parameter, greater positive curvature near the maximum generally means a smaller local standard error, approximately the inverse square root of the information. For multiple parameters, the inverse information matrix describes local covariance, and the information about one parameter depends on how it can trade off against the others. These are local, model-based approximations; they do not by themselves establish that the model is adequate or that a parameter is globally identifiable.

A Normal-Model Derivation

Suppose independent observations \(Y_i\) have normal distributions with known variance \(\sigma^2\) and mean \(\mu_i(\theta)\). Their log-likelihood, omitting constants that do not depend on \(\theta\), is

\ell(\theta) = -\frac{1}{2\sigma^2} \sum_i \{Y_i-\mu_i(\theta)\}^2.

Differentiating twice and taking the expectation under the model gives the Fisher information

I(\theta) = \frac{1}{\sigma^2} \sum_i \left\{\frac{\partial\mu_i(\theta)}{\partial\theta}\right\}^2.

The residual-dependent term in the second derivative has expectation zero because \(E\{Y_i-\mu_i(\theta)\}=0\). Thus each observation contributes information in proportion to squared sensitivity divided by variance. If a stream has a small measurement variance or a mean that changes sharply with \(\theta\), it can be informative even with few observations. If its mean is nearly insensitive to \(\theta\), many observations may still provide little information about that parameter.

Worked Example: Comparing Concentration and Response Information

Consider a hypothetical local analysis of a scalar PK parameter \(\theta\), with the PD-specific parameters treated as fixed for this calculation. Four independent concentration observations have normal errors with standard deviation \(1\) mg/L, and each predicted concentration has local sensitivity \(2\) mg/L per unit of \(\theta\). Six independent response observations have standard deviation \(1.5\) response units, and each predicted response has local sensitivity \(0.5\) response units per unit of \(\theta\). Assume these variances are known and the two streams are conditionally independent under the model.

For the concentration stream, the information is

I_C = 4\frac{(2\ \mathrm{mg/L\ per\ unit})^2}{(1\ \mathrm{mg/L})^2} =16\ \mathrm{per\ unit}^2.

For the response stream, it is

I_R = 6\frac{(0.5\ \mathrm{response\ units\ per\ unit})^2} {(1.5\ \mathrm{response\ units})^2} = 6\frac{0.25}{2.25} =0.6667\ \mathrm{per\ unit}^2.

The total information is \(I=16+0.6667=16.6667\ \mathrm{per\ unit}^2\). The response data contribute \(0.6667/16.6667=0.0400\), or 4.00%, of this scalar local information; the concentration data dominate this particular calculation. The corresponding approximate standard error is \(1/\sqrt{16.6667}=0.2449\) parameter units. This conclusion is limited to the stated parameter, local sensitivities, variances, and independence assumptions. It does not mean that response data are generally unimportant or that they cannot inform other parameters.

Shared Parameters, Correlation, and Nuisance Parameters

Information contributions must be evaluated for the parameter direction of interest. In a joint model, a response likelihood can depend on a PK parameter through predicted concentrations, while also depending on PD parameters such as potency or maximal effect. The response contribution can therefore include curvature in PK directions and cross-curvature between PK and PD parameters.

For a parameter of interest \(\theta\) and a nuisance parameter \(\beta\), partition the local information matrix as

I= \begin{pmatrix} I_{\theta\theta} & I_{\theta\beta}\\ I_{\beta\theta} & I_{\beta\beta} \end{pmatrix}.

When \(I_{\beta\beta}\) is invertible, the local information about \(\theta\) after allowing \(\beta\) to be estimated is the Schur complement \(I_{\theta\theta}-I_{\theta\beta}I_{\beta\beta}^{-1}I_{\beta\theta}\). The subtracted term is nonnegative in the scalar-\(\theta\) case for a positive-definite information matrix. It represents the loss of precision because changes in \(\theta\) can be partly offset by changes in \(\beta\). Thus the diagonal entry \(I_{\theta\theta}\) alone can overstate the information available for estimating \(\theta\).

Worked Example: Response Information With a PD Nuisance Parameter

Suppose a hypothetical joint fit has parameter vector \((\theta,\beta)\), where \(\theta\) is a shared PK parameter and \(\beta\) is a PD-specific parameter. At the fitted values, let the concentration contribution be \(\begin{pmatrix}10&0\\0&0\end{pmatrix}\) and the conditional-response contribution be \(\begin{pmatrix}2&3\\3&8\end{pmatrix}\), in common local information units. Both contributions are positive semidefinite; for the response matrix, its determinant is \(2(8)-3^2=7>0\) and its diagonal entries are positive.

The joint information matrix is

I= \begin{pmatrix} 10&0\\0&0 \end{pmatrix} + \begin{pmatrix} 2&3\\3&8 \end{pmatrix} = \begin{pmatrix} 12&3\\3&8 \end{pmatrix}.

The response stream contributes 2 units of direct curvature in the \(\theta\) direction, but its cross-information with \(\beta\) is 3. After accounting for estimation of \(\beta\), the effective information for \(\theta\) is

I_{\theta\mid\beta} = 12-\frac{3^2}{8} = 10.875.

The local standard error is approximately \(1/\sqrt{10.875}=0.3032\) in the parameter’s units. If \(\beta\) were known and fixed, the relevant curvature would instead be 12 and the corresponding approximation would be \(1/\sqrt{12}=0.2887\). The difference illustrates why a response stream’s apparent contribution to one parameter cannot be assessed from its diagonal curvature alone: nuisance-parameter trade-offs matter.

Correlation also changes information. For a likelihood that correctly models dependence, the data stream’s information comes from the full covariance structure, not from treating every recorded value as an independent replicate. Correlation can make repeated observations carry substantially less information than their raw count suggests. The following calculation treats each response block’s correctly modeled likelihood contribution as a scalar information value; it is not a replacement for specifying the joint distribution within a block.

Worked Example: A Correlated Response Stream

In a hypothetical local calculation for a shared scalar parameter, concentration data contribute 4 information units. There are four independent response blocks. After accounting for dependence within each block, the likelihood curvature from one block is \(2.5/2.05\) units. The blocks are independent of one another and of the concentration contribution, conditional on the model parameters.

The total information is

I_{\mathrm{total}} = 4+4\left(\frac{2.5}{2.05}\right) = 4+4.87804878 = 8.87804878.

The local standard error is therefore

SE \approx \frac{1}{\sqrt{4+4(2.5/2.05)}} = 0.3356

to four decimal places, in the parameter’s units. If one instead used a naïve independent-record calculation with total information 44, the resulting standard error would be \(1/\sqrt{44}=0.1508\), substantially smaller. That naïve value is not justified when the within-block dependence reduces the response curvature as specified. The calculation shows why the dependence-adjusted likelihood contribution, rather than the number of measurements, is the relevant quantity.

Assessing Which Stream Dominates

A defensible dominance assessment is specific: name the parameter or estimable direction, state the fitted model and assumptions, then compare the likelihood curvature contributed by each stream in that direction. In a multivariate model, inspect the contribution matrices and their cross-parameter terms. For a scalar target with nuisance parameters, compare efficient information after nuisance adjustment rather than unadjusted diagonal entries.

Useful checks include:

Check the likelihood factorization. Confirm that conditional dependence, repeated measures, and any latent exposure uncertainty are handled in the likelihood rather than silently treated as independence.
Inspect local curvature. Compare observed information contributions at the fitted values, and check whether small changes in the fit produce large changes in the comparison.
Check parameter scaling. Curvature changes when parameters are re-expressed in different units. Compare information only on a common, clearly stated parameter scale, or use a meaningful standardized parameterization.
Look for weak directions. A nearly singular joint information matrix or strong parameter trade-offs can make a large diagonal contribution misleading. Examine eigen-directions or profile behavior when local curvature is unstable.
Separate precision from fit. The value of a log-likelihood contribution includes data- and unit-dependent constants. Its raw magnitude is not an information measure. Curvature with respect to parameters is the relevant local comparison.

Observed information is evaluated at the fitted data and can be noisy, especially with sparse observations or a strongly nonlinear model. Expected information averages curvature under the model and may be easier to interpret, but depends on the assumed model being an adequate representation. Neither quantity detects every form of model failure. Residual and predictive checks remain necessary, and a local information comparison does not establish that an estimate is unbiased or that the likelihood has only one maximum.

Alternatives and Limits of the Comparison

Sequential estimation, in which PK parameters are estimated first and then used in a PD analysis, answers a different computational question from simultaneous likelihood estimation. It can be useful when its assumptions and uncertainty handling are appropriate, but it does not reveal the full joint likelihood’s response contribution to shared PK parameters. A joint fit retains both likelihood contributions and their shared-parameter dependence; inspecting their curvature helps explain where its local precision comes from.

A likelihood profile is a useful alternative when a quadratic, local curvature summary is unreliable: it examines how the optimized likelihood changes as a target parameter is fixed over a range. It is not a substitute for checking the model or for identifying which data stream contributes the curvature. The information matrix is compact and interpretable near a regular optimum, but can be misleading under weak identification, boundary estimates, or substantial nonlinearity.

Key takeaway: A stream’s contribution to estimation is determined by its correctly specified likelihood curvature in the parameter direction of interest, not by its record count or raw log-likelihood value. Shared parameters allow response data to inform PK estimates, but correlation and nuisance-parameter trade-offs can reduce the effective information.

Common Pitfall

Wrong reasoning: “There are ten times as many response records as concentration records, so the response data must dominate the PK parameter.” Record counts ignore measurement variance, sensitivity to the parameter, within-stream dependence, and trade-offs with PD parameters. Correction: calculate or inspect the concentration and response information contributions on the same parameter scale, using the full likelihood structure, and account for nuisance parameters when the target is a component of a joint model. Also avoid comparing raw log-likelihood values: constants and measurement units can change those values without changing the parameter information.

Check Your Understanding

Answer each question, then compare with the brief solutions.

For independent normal observations with known standard deviation, what two features determine an observation’s information about a scalar parameter?
If the concentration and conditional-response information for a scalar parameter are 9 and 3 units, respectively, what fraction of their combined local information comes from the response stream?
Why can the diagonal information for a PK parameter overstate its effective information when a PD parameter is estimated jointly?
Why is the number of response measurements insufficient to decide whether the response stream dominates?

Solutions. (1) Squared sensitivity of the predicted mean to the parameter and the inverse error variance: \((\partial\mu/\partial\theta)^2/\sigma^2\). (2) The total is \(9+3=12\), so the response fraction is \(3/12=0.25\), or 25%. (3) Because cross-information permits changes in the PK and PD parameters to offset one another; the nuisance-adjusted information subtracts the Schur-complement penalty. (4) Record count ignores sensitivity, variability, dependence among observations, and parameter trade-offs; dominance requires comparing correctly modeled likelihood curvature for a specified parameter direction.