Correlated Random Effects In Population PD Models

Why Correlate PD Random Effects?

People can differ jointly in several aspects of a drug response. For example, a person with a higher-than-typical baseline response might also have a larger maximum drug effect; a larger maximum effect might occur alongside a higher concentration needed to produce that effect. A model that assigns each person a separate random effect for each PD parameter can represent this joint variation, but only if it specifies how those effects vary together.

This tutorial focuses on the variance-covariance matrix for random effects on baseline, potency, and maximum effect. Its correlations describe relationships among individual deviations from typical parameter values on the model's random-effect scales. They are not, by themselves, correlations among raw observations, evidence of a causal mechanism, or proof that a particular biological relationship exists.

A PD Model and Its Random-Effects Matrix

Consider a stimulatory Emax model for subject \(i\) at observation \(j\). Let \(C_{ij}\) be the relevant drug concentration, \(B_i\) the subject-specific baseline, \(E_{\max,i}\) the subject-specific maximum drug effect, and \(EC_{50,i}\) the subject-specific concentration producing half of that maximum effect above baseline. A simple observation model is:

For this example, take \(B_i=B+\eta_{B,i}\), \(\log(EC_{50,i})=\log(EC_{50})+\eta_{P,i}\), and \(\log(E_{\max,i})=\log(E_{\max})+\eta_{M,i}\). The log transformations keep potency and maximum effect positive. The random effects have mean zero, so the untransformed baseline \(B\) and the exponentiated typical values are reference values at zero random effect. Assume the residual errors \(\epsilon_{ij}\) have mean zero and are independent of the random effects; their variance model is specified separately.

In this model, \(\eta_{B,i}\) is measured in the response units of \(B_i\), while \(\eta_{P,i}\) and \(\eta_{M,i}\) are deviations on log scales and are dimensionless. A correlation is dimensionless in all three cases. A positive \(\eta_{P,i}\) means a higher \(EC_{50,i}\), and therefore lower potency; a positive \(\eta_{M,i}\) means a higher maximum effect. Thus a positive potency–maximum-effect correlation means that subjects with higher \(EC_{50}\) deviations tend also to have higher maximum-effect deviations. It does not mean that higher potency accompanies higher maximum effect.

A diagonal \(\Omega\) sets all three random-effect correlations to zero. A full \(3\times3\) matrix estimates three variances and three correlations. The matrix must be symmetric and positive definite (or, in boundary cases, positive semidefinite) to represent a valid covariance structure. With a multivariate normal distribution, zero correlation between two random effects implies their independence; without that distributional assumption, zero correlation alone does not imply independence.

How Correlation Changes Response Uncertainty

A random-effects correlation matters because a predicted response depends on several parameters at once. Even if the marginal variability of each parameter stays fixed, the joint pattern of deviations can change the variability of a response at a particular concentration. A useful local approximation is the delta method: for a differentiable quantity \(g(\boldsymbol{\eta})\), approximate it near the zero random-effect vector by \(g(\boldsymbol{0})+\nabla g(\boldsymbol{0})^\mathsf{T}\boldsymbol{\eta}\). The variance of this linear approximation is \(\nabla g(\boldsymbol{0})^\mathsf{T}\Omega\nabla g(\boldsymbol{0})\). This follows by taking the variance of the linear term and applying \(\operatorname{Var}(\boldsymbol{a}^\mathsf{T}\boldsymbol{\eta})=\boldsymbol{a}^\mathsf{T}\Omega\boldsymbol{a}\).

For the Emax drug-effect component \(R=E_{\max}C/(EC_{50}+C)\), consider variation in \(\eta_P\) and \(\eta_M\), holding baseline fixed. Write \(q=C/(EC_{50}+C)\), evaluated at the typical parameters. Since \(EC_{50,i}=EC_{50}\exp(\eta_P)\) and \(E_{\max,i}=E_{\max}\exp(\eta_M)\), the derivatives at zero random effects are \(\partial R/\partial\eta_P=-R(1-q)\) and \(\partial R/\partial\eta_M=R\). Consequently, the local variance is:

The minus sign is important: at a fixed concentration, a higher \(EC_{50}\) reduces the response, whereas a higher \(E_{\max}\) increases it. Positive correlation between those deviations can therefore offset some response variation at concentrations where both parameters influence the response. At other concentrations, the size of the effect differs because \(q\) differs.

Estimating Correlations: Information and Complexity

Estimating correlations requires information about how subjects' random effects co-vary, not merely enough observations to estimate each marginal variance. Repeated PD observations across informative concentration ranges can help distinguish potency from maximum effect: observations only at concentrations far below or far above \(EC_{50}\) may provide limited information about one or both. A correlation estimate can consequently be unstable even when the model converges. The usual model assumptions still matter: subjects are independent draws from the stated random-effects distribution, the parameter transformations and structural PD model are appropriate, and the residual model describes remaining observation-level variation.

A full covariance matrix is more flexible than a diagonal one, but flexibility costs parameters and can make estimation less stable. If all three random effects are included, the diagonal structure estimates three variances; the full structure estimates those same three variances plus three correlations. A structured intermediate choice may estimate selected correlations while fixing others to zero, but that structure should have a defensible rationale and adequate data support. Correlations should not be added just because the software can estimate them.

Diagnostics, Alternatives, and Limitations

A correlation estimate is more credible when the likelihood and estimates are stable, the covariance matrix remains positive definite, uncertainty is not excessive, and the fitted model reproduces relevant features of the data. Warning signs include correlations close to \(-1\) or \(1\), large standard errors, substantial changes from small modeling adjustments, failure to converge, or a covariance matrix near singularity. Such patterns can indicate weak information, excessive model complexity, or confounding among the random effects. A successful optimizer termination is not sufficient evidence that a correlation is well estimated.

Useful checks include inspecting the estimated standard deviations and correlations together, verifying positive definiteness, and examining whether uncertainty or predictions change substantially across plausible covariance structures. Subject-level residual patterns and simulated or otherwise model-based checks can reveal systematic discrepancies, although a good predictive fit does not by itself validate the interpretation of a latent correlation. When uncertainty is material, likelihood profiles or bootstrap analyses can help assess stability; their results should be interpreted in light of the information available in the data.

The diagonal model is not automatically a neutral default: it makes a substantive assumption of zero random-effect correlations under the specified distribution. Conversely, a full covariance estimate is not automatically superior because it captures a sample pattern. Correlations are conditional on the parameter scales, model structure, and included effects; changing transformations or the structural PD model can change their interpretation and estimates. Nonlinear transformations also mean that a correlation between random effects need not equal the correlation between the corresponding natural-scale parameters.

Common Pitfall

Wrong reasoning: “The observed baseline and maximum response are correlated, so I should set the baseline–maximum random-effect correlation to that same value.” The observed measurements include concentration, residual error, repeated-measurement structure, and other model components, so their correlation is not generally the latent random-effect correlation. Correction: Estimate or justify the correlation on the defined random-effect scales within the specified model, then assess whether the data support it. Treat it as a description of joint between-subject variation under that model, not as a causal claim.