Random Effects On Derived Quantities Such As Half-Life

Why a Derived Quantity Has Its Own Variability

In a one-compartment model with first-order elimination, the elimination half-life is determined by volume of distribution and clearance. It is not an independent parameter: a subject’s half-life is the ratio of that subject’s volume to that subject’s clearance, multiplied by \(\ln 2\). Consequently, its population distribution depends on the variability in both parameters and on how their random effects covary.

This distinction matters in population pharmacokinetics because neither the clearance variance nor the volume variance alone describes how widely half-lives vary. For example, when clearance and volume random effects move together, some of their variation can cancel in the ratio. When they move in opposite directions, variation in the ratio can increase. The derivation below makes that relationship explicit.

Random Effects on Clearance and Volume

Assume clearance and volume follow correlated lognormal random-effects models:

Here, \(CL_{\mathrm{pop}}\) and \(V_{\mathrm{pop}}\) are positive typical values, and the subject-level random effects have a joint bivariate normal distribution with mean zero:

Thus, \(\omega_{CL}^{2}\) and \(\omega_{V}^{2}\) are the variances of the log-scale effects, and \(\omega_{CL,V}\) is their covariance. The covariance matrix must be positive semidefinite; equivalently, \(|\omega_{CL,V}|\leq \omega_{CL}\omega_V\). The correlation is \(\rho_{CL,V}=\omega_{CL,V}/(\omega_{CL}\omega_V)\) when both variances are positive.

Substituting the subject-specific clearance and volume into the definition of half-life gives

Define \(t_{1/2,\mathrm{typ}}=\ln(2)V_{\mathrm{pop}}/CL_{\mathrm{pop}}\), the half-life calculated from the typical clearance and volume. The log of the individual half-life is therefore the log typical half-life plus a contrast of two random effects:

Deriving the Distribution and Summaries

A linear combination of jointly normal random variables is normal. Its mean is the same linear combination of their means, and its variance follows from the variance rule for a difference:

Write this variance as \(\sigma_{h}^{2}\). Then \(\log(t_{1/2})\) is normal with mean \(\log(t_{1/2,\mathrm{typ}})\) and variance \(\sigma_h^2\). The corresponding half-life summaries are:

The median equals the typical half-life because the random-effects contrast has mean zero on the log scale. The arithmetic mean is larger when there is nonzero variability; this is the usual right-skew of a lognormal distribution. The coefficient of variation (CV) is dimensionless, while the mean, median, and standard deviation of half-life have units of time. For a lognormal variable, its standard deviation is its mean multiplied by its CV.

Dependence Between Half-Life and Its Components

Half-life is mathematically constructed from clearance and volume, so it is generally dependent on both. The direction and strength of that dependence are not determined merely by the ratio formula: they also depend on the joint random-effects structure. Under the assumptions above, let \(d=\eta_V-\eta_{CL}\). For a normally distributed pair of effects, the covariance between exponentiated linear combinations can be calculated using the lognormal covariance identity.

For example, the half-life–clearance covariance is negative when \(\omega_{CL,V}<\omega_{CL}^{2}\), zero when the two are equal, and positive when \(\omega_{CL,V}>\omega_{CL}^{2}\). A negative association is intuitive if volume is held fixed, but it is not guaranteed when volume and clearance vary jointly. Similarly, half-life can be positively or negatively associated with volume depending on \(\omega_V^{2}-\omega_{CL,V}\). These covariance results describe the population model; they do not establish a causal effect of changing one parameter while holding everything else fixed.

How the Clearance–Volume Correlation Changes Half-Life

The covariance contribution is especially clear when the marginal variances are held fixed. Since \(\omega_{CL,V}=\rho_{CL,V}\omega_{CL}\omega_V\), increasing the clearance–volume correlation decreases the variance of their difference \(\eta_V-\eta_{CL}\). Positive correlation tends to make the two effects move together, reducing their contrast; negative correlation tends to make them move in opposite directions, increasing it.

Assumptions, Diagnostics, and Alternatives

The lognormal result requires a specific set of assumptions: half-life is defined by the one-compartment, first-order elimination relationship; clearance and volume are positive and have the stated exponential random-effects forms; and their random effects are jointly normal with finite covariance. The covariance matrix must be valid. Under those conditions, the distribution and moment formulas follow directly. If the random effects are not jointly normal, the identity \(\log(t_{1/2})=\log(t_{1/2,\mathrm{typ}})+\eta_V-\eta_{CL}\) still holds, but the contrast need not be normal, and the lognormal summaries need not apply.

Useful checks include verifying that the estimated covariance matrix is positive semidefinite, examining the estimated random-effects distributions, and computing the half-life distribution from simulated paired clearance and volume values. Pairing matters: independently simulating clearance and volume would erase their fitted covariance and could misstate the derived variability. For a fitted model, compare the distribution of simulated half-lives with the algebraic result when the joint-normal assumptions hold. Large disagreement can indicate an implementation or calculation error, or that the simulation includes features beyond the assumptions used in the formula.

The direct algebraic calculation is exact under the stated model and is preferable when its assumptions hold. Simulation is a flexible alternative for describing derived quantities under more complicated models or non-normal random effects, but its accuracy depends on the number of draws and on preserving the dependence structure. A first-order approximation such as the delta method can be useful for other derived functions, especially when an exact distribution is unavailable; for this ratio of lognormal parameters, it is unnecessary under the assumptions above.

Another possible model is to assign a random effect directly to half-life. That approach is not automatically equivalent to assigning random effects to clearance and volume. To preserve the implied half-life distribution, its log-scale variance must equal \(\omega_V^2+\omega_{CL}^2-2\omega_{CL,V}\), and the joint relationships with clearance and volume must also be compatible. Treating half-life as an unrelated random effect can therefore discard important dependence or create an incoherent parameterization.