Lognormal and Correlated Random-Effects Priors

Why the Prior Scale Matters

Individual pharmacokinetic parameters such as clearance and volume must be positive. A normal distribution on the original parameter scale can assign probability to impossible negative values, while a normal distribution on the logarithm of the parameter naturally gives a positive-valued distribution. In a population model, this construction also gives a useful interpretation to variability: individuals differ multiplicatively around a typical value.

When several parameters vary together, specifying a separate prior for each does not describe their joint distribution. A multivariate prior on transformed random effects also records whether individuals with relatively high values of one parameter tend to have relatively high or low values of another. The details of this construction matter: log-scale means are not generally arithmetic means on the parameter scale, and log-scale correlations are not identical to correlations between the original parameters.

Lognormal Priors for Positive Parameters

In a typical population PK parameterization, an individual parameter is written \(\theta_i=\theta_{\mathrm{typ}}\exp(\eta_i)\), where \(\theta_{\mathrm{typ}}>0\) is the typical value and \(\eta_i\sim N(0,\omega^2)\). This makes \(\theta_{\mathrm{typ}}\) the population median, not the arithmetic mean: \(\operatorname{median}(\theta_i)=\theta_{\mathrm{typ}}\), whereas \(E(\theta_i)=\theta_{\mathrm{typ}}\exp(\omega^2/2)\). The distinction is important whenever a reported population summary is meant to be an arithmetic average.

A convenient way to specify the spread is the coefficient of variation (CV), defined as the standard deviation divided by the arithmetic mean. For a lognormal distribution, the relation is \(\mathrm{CV}=\sqrt{\exp(\sigma^2)-1}\); thus, \(\sigma^2=\log(1+\mathrm{CV}^2)\). This CV describes between-individual variability in the prior, not residual measurement error.

Choosing Whether to Center on the Median or Mean

The random-effects form \(\theta_i=\theta_{\mathrm{typ}}\exp(\eta_i)\) with \(E(\eta_i)=0\) centers the log parameter at \(\log(\theta_{\mathrm{typ}})\). That is often natural when \(\theta_{\mathrm{typ}}\) means the typical, or median, individual. If the information to encode instead specifies an arithmetic mean \(m\), the log-scale location must be shifted down to account for the positive skew.

Correlated Random Effects

For \(q\) positive individual parameters, collect their transformed random effects in \(\boldsymbol{\eta}_i=(\eta_{i1},\ldots,\eta_{iq})^\mathsf{T}\). The standard multivariate construction is \(\boldsymbol{\eta}_i\sim N(\mathbf{0},\boldsymbol{\Omega})\), with individual parameters \(\theta_{ij}=\theta_{\mathrm{typ},j}\exp(\eta_{ij})\). The diagonal entries of \(\boldsymbol{\Omega}\) are log-scale variances; off-diagonal entries are log-scale covariances. For parameters \(j\) and \(k\), the log-scale correlation is \(\rho_{jk}=\omega_{jk}/\sqrt{\omega_{jj}\omega_{kk}}\).

The covariance matrix must be symmetric and positive semidefinite; a positive-definite matrix is needed for a nonsingular multivariate normal density. In the two-parameter case, this requires nonnegative variances and \(|\rho_{12}|\leq 1\). In higher dimensions, checking each pairwise correlation alone is not sufficient: the complete matrix must be positive semidefinite.

Log-scale correlation has a precise, but scale-specific, meaning. Positive \(\rho_{jk}\) means that above-typical values of one transformed effect tend to occur with above-typical values of the other. It does not mean the raw parameter correlation has exactly the same numerical value. For the lognormal construction above, the correlation on the original scale is

This follows because multiplying by fixed typical values does not change correlation, while the moments of jointly normal variables give \(\operatorname{Cov}(\exp(\eta_{ij}),\exp(\eta_{ik}))=\exp((\omega_{jj}+\omega_{kk})/2)(\exp(\omega_{jk})-1)\). Dividing that covariance by the two marginal standard deviations yields the displayed expression. Consequently, the original-scale correlation depends on both the covariance and the marginal variances.

Construction Checks and Alternatives

A prior is only as interpretable as its parameterization. Before using a lognormal random-effects prior, state whether each typical value is a median or an arithmetic mean, identify whether variability is supplied as a CV or a log-scale standard deviation, and attach the original parameter’s units to summaries on its natural scale. Logarithms apply to dimensionless ratios in principle; writing a logarithm of a parameter implicitly means that the parameter has first been expressed relative to a fixed unit.

Useful checks include verifying that simulated parameters are positive, that their medians and CVs match the intended specifications, and that joint simulated values reproduce the intended direction and approximate strength of dependence. Inspect simulated quantiles as well as means: a small log-scale standard deviation can still produce a long upper tail, and extreme draws may matter for a PK model. For a multivariate prior, inspect symmetry and eigenvalues of \(\boldsymbol{\Omega}\), or use a Cholesky factorization when generating draws. A negative eigenvalue signals an invalid covariance specification; rounding a matrix can also make a near-singular specification invalid.

Independent lognormal priors are a simpler alternative when dependence is intentionally absent, but setting all off-diagonal covariances to zero is a substantive assumption. A normal prior on the original parameter scale can be reasonable for parameters whose support is not restricted to positive values, but for positive PK parameters it may permit impossible values. Other positive distributions can represent different tail behavior, but changing the family changes the implied prior summaries and dependence structure. The lognormal construction is not automatically correct simply because parameters are positive: its multiplicative spread and tail behavior must be scientifically plausible.

Covariate adjustment changes the typical value used for an individual, not the basic random-effects construction. For a covariate-specific typical value \(\theta_{\mathrm{typ},j}(x_i)\), the parameter can be written \(\theta_{ij}=\theta_{\mathrm{typ},j}(x_i)\exp(\eta_{ij})\); the random-effects covariance describes remaining between-individual variation conditional on that typical-value model. This prior specifies population-level variation to be used in individualization; it does not by itself say how concentration observations update an individual’s parameters.

Common Mistakes and Limitations

• Treating the typical value as an arithmetic mean. A zero-mean log random effect makes the typical value the median. If the arithmetic mean is intended, subtract half the log-scale variance from the log-scale location.
• Putting a CV directly into the covariance matrix. A CV is not a log-scale standard deviation. Convert using \(\sigma^2=\log(1+\mathrm{CV}^2)\).
• Interpreting log-scale correlation as raw-scale correlation. The relationship is nonlinear and depends on both marginal variances; use the transformed-correlation formula when the raw-scale value is needed.
• Assuming pairwise-valid correlations guarantee a valid joint prior. For three or more effects, validate the entire covariance matrix, not only each correlation separately.
• Ignoring implausible tails. A mathematically valid lognormal prior can still generate clinically implausible parameter values. Examine prior simulations and revisit the variability assumptions when necessary.

These checks do not establish that a prior is scientifically appropriate; they establish that its mathematical specification matches the intended summaries and dependence assumptions. Sparse information about the population, strong covariate extrapolation, or a poorly supported correlation can leave substantial uncertainty in those assumptions.