Priors for Population Parameters

Why Priors for Population Parameters Need Care

A prior for a typical pharmacokinetic parameter encodes uncertainty about a population-level quantity before the current study data are used. For example, it may describe plausible values for typical clearance, volume of distribution, or absorption rate. Such priors matter because these quantities determine the model’s predictions, and their scales and supports differ: a rate or volume must be positive, while a prior written directly on an unconstrained scale may assign probability to impossible values.

Building on the earlier material on conditional and marginal Bayesian NLME, the distinction between population parameters and individual random effects is essential. A prior on typical clearance, \(\theta_{CL}\), describes uncertainty about that population parameter. It is not the distribution of individual clearances, and its spread is not the random-effect standard deviation. Those are separate parts of the hierarchical model.

Definition: A prior distribution for a population parameter is a probability distribution representing uncertainty about that parameter before conditioning on the current study data. A log-scale prior for a positive parameter \(\theta\) specifies a distribution for its logarithm. For example, \(\log(\theta/u)\sim N(m,s^2)\), where \(u\) is a stated unit, means that the logarithm of the parameter expressed in units of \(u\) is normally distributed with mean \(m\) and variance \(s^2\).

The unit \(u\) makes the logarithm’s argument dimensionless. For clearance one might use \(u=1\ \mathrm{L/h}\), for volume \(u=1\ \mathrm{L}\), and for an absorption rate \(u=1\ \mathrm{h}^{-1}\). The stated unit is part of the specification: changing units changes the numerical value of the log-scale location, even though the intended physical distribution can remain the same.

Log-Scale Priors and Their Natural-Scale Meaning

Suppose \(Y=\log(\theta/u)\sim N(m,s^2)\), with \(s>0\). Then \(\theta\) has a lognormal distribution. The normal distribution for \(Y\) guarantees that \(\theta\) is positive, and its asymmetry on the natural scale can represent uncertainty that extends farther upward than downward.

\operatorname{median}(\theta)=u\exp(m), \qquad E(\theta)=u\exp\left(m+\frac{s^2}{2}\right), \qquad \operatorname{GSD}(\theta)=\exp(s).

Here \(\operatorname{GSD}\) is the geometric standard deviation: multiplying or dividing the median by this factor gives values one log-scale standard deviation above or below the median. The median is not generally the mean. In fact, the mean exceeds the median when \(s>0\); as the log-scale spread grows, this difference grows too. The \(q\) quantile, for \(0<q<1\), is \(u\exp(m+s z_q)\), where \(z_q\) is the \(q\) quantile of the standard normal distribution.

The natural-scale density follows by transformation. Since \(y=\log(\theta/u)\), the derivative is \(dy/d\theta=1/\theta\). Multiplying the normal density of \(y\) by this absolute derivative gives:

f(\theta)= \frac{1}{\theta s\sqrt{2\pi}} \exp\left\{-\frac{[\log(\theta/u)-m]^2}{2s^2}\right\}, \qquad \theta>0.

The factor \(1/\theta\) is the Jacobian. It is required when expressing the prior density with respect to the natural parameter \(\theta\). If computation instead uses \(y=\log(\theta/u)\) as the parameter and represents the density with respect to \(y\), the prior is simply the normal density for \(y\); a second Jacobian should not be added to that density. The relevant density is always defined relative to the variable with respect to which it is written.

Derivation: For \(Y\sim N(m,s^2)\), \(E(\theta)=uE[\exp(Y)]\). The normal moment-generating function is \(E[\exp(tY)]=\exp(tm+t^2s^2/2)\). Setting \(t=1\) gives \(E(\theta)=u\exp(m+s^2/2)\). The median follows because the normal median of \(Y\) is \(m\), and the exponential function is increasing. The quantile formula follows by applying that same increasing transformation to the normal quantile \(m+s z_q\).

Worked Example: A Lognormal Prior for Typical Clearance

Suppose, for illustration only, an analyst specifies \(\log[\theta_{CL}/(1\ \mathrm{L/h})]\sim N(\log 4,(\log 2)^2)\). This prior is on typical clearance, not on an individual’s clearance.

The median is \(1\ \mathrm{L/h}\times\exp(\log 4)=4.00\ \mathrm{L/h}\). The geometric standard deviation is \(\exp(\log 2)=2.00\). Thus, values one log-scale standard deviation below and above the median are \(4/2=2.00\) and \(4\times2=8.00\ \mathrm{L/h}\).

The natural-scale mean is \[ 1\ \mathrm{L/h}\times \exp\left(\log 4+\frac{(\log 2)^2}{2}\right) =4\exp(0.24023)\ \mathrm{L/h} \approx 5.086\ \mathrm{L/h}. \] This exceeds the median because the lognormal distribution is right-skewed.

For a central 95% prior interval, use \(z_{0.975}\approx1.96\). The multiplicative factor is \(\exp(1.96\log 2)\approx3.891\), so the interval is \[ [4/3.891,\;4\times3.891]\ \mathrm{L/h} \approx[1.028,\;15.56]\ \mathrm{L/h}. \] This is a range containing 95% of the prior probability under the stated specification; it is not an interval calculated from study data.

Choosing and Scaling Priors for PK Parameters

A practical specification begins with the parameter’s meaning, support, units, and plausible scale. For positive parameters such as typical clearance \(\theta_{CL}\), typical volume \(\theta_V\), and typical absorption rate \(\theta_{ka}\), a lognormal prior is one common option. Its location and spread should be chosen so that the resulting natural-scale distribution expresses the intended uncertainty, rather than selected by copying numerical values from a different parameter or unit.

State the parameter and its units.
Write, for example, typical volume in litres or typical absorption rate in inverse hours. Define the quantity as a population parameter, not an individual-level random effect.

Choose a scale and distribution.
For a positive parameter, decide whether a lognormal prior on the natural scale or another positive-support distribution expresses the intended knowledge.

Translate and check.
For a lognormal prior, calculate the natural-scale median, mean, and useful quantiles. Confirm that these summaries and the tails are plausible for the model and unit.

Independence is another modeling choice, not an automatic property of separate parameters. If the prior is specified as a product of separate priors for clearance, volume, and absorption rate, it asserts prior independence among those population parameters. When prior knowledge implies dependence, a joint prior may be more appropriate. The choice concerns the prior distribution for the population parameters; it does not replace the model’s separate specification of random effects.

Worked Example: Scaling a Prior for Typical Volume

Suppose a hypothetical model uses \(\log[\theta_V/(1\ \mathrm{L})]\sim N(\log 20,(\log 1.5)^2)\). The median typical volume is \(20\ \mathrm{L}\), and the geometric standard deviation is \(1.5\). The log-scale standard deviation is \(\log 1.5\approx0.4055\).

The central 95% interval has multiplicative factor \(\exp[1.96\log(1.5)]\approx2.214\). Its endpoints are \(20/2.214\approx9.03\ \mathrm{L}\) and \(20\times2.214\approx44.28\ \mathrm{L}\).

The prior probability that typical volume exceeds \(30\ \mathrm{L}\) is computed on the log scale: \[ P(\theta_V>30\ \mathrm{L}) =P\left(Z>\frac{\log(30/20)}{\log(1.5)}\right) =P(Z>1)\approx0.1587, \] where \(Z\sim N(0,1)\). The ratio \(30/20=1.5\) makes the standardized threshold exactly one. This probability is a property of the specified prior, not a probability derived from patient data.

This calculation also shows why reporting only the log-scale standard deviation is not enough for interpretation. The natural-scale interval and tail probability make the prior’s implications easier to assess in the units of the parameter.

Worked Example: An Absorption-Rate Prior and Its Implications

For illustration, suppose \(\log[\theta_{ka}/(1\ \mathrm{h}^{-1})]\sim N(\log 1.2,(\log 2)^2)\). The median typical absorption rate is \(1.20\ \mathrm{h}^{-1}\), and its geometric standard deviation is \(2.00\). The mean is \[ 1\ \mathrm{h}^{-1}\times1.2\exp\left(\frac{(\log 2)^2}{2}\right) \approx1.526\ \mathrm{h}^{-1}. \]

The probability that the rate is below \(0.60\ \mathrm{h}^{-1}\) is \[ P(\theta_{ka}<0.60\ \mathrm{h}^{-1}) =\Phi\left(\frac{\log(0.60/1.20)}{\log 2}\right) =\Phi(-1)\approx0.1587, \] where \(\Phi\) is the standard normal cumulative distribution function. The threshold is half the median, or one geometric standard deviation below it.

This prior assigns probability only to positive rates. It does not, by itself, establish that its median or spread is scientifically suitable. Those choices must be checked against the intended model, available knowledge, and the implications for predicted concentration-time behavior.

Alternatives, Checks, and Limitations

A lognormal prior is not the only possible prior for a positive parameter. A gamma distribution is also supported on positive values and can be parameterized in different ways to express a range of shapes. A normal prior placed directly on the natural scale is symmetric, but it can assign probability to negative values; truncating it at zero avoids that support problem but changes its distribution and summaries. No distribution is automatically appropriate simply because its support is convenient.

Before using a prior, check that its location, spread, and units match the quantity being modeled. For a lognormal specification, inspect natural-scale summaries as well as the log-scale values. A prior can have a seemingly reasonable median but an implausibly large upper tail, or a mean substantially above its median. Also check that calculations or software use the intended parameterization and density scale.

Support check: Does the prior assign probability to values the parameter cannot take?
Unit check: Is the log applied to a dimensionless ratio, and are the prior summaries reported in the correct units?
Scale check: Are the median, mean, and relevant quantiles plausible for the population parameter?
Joint-prior check: Does an independence assumption among population parameters reflect the intended prior knowledge?

A common mistake is to interpret \(\log(\theta/u)\sim N(m,s^2)\) as saying that the natural-scale mean is \(u\exp(m)\). That quantity is the median, not the mean: the mean is \(u\exp(m+s^2/2)\). Another mistake is to use the log-scale spread as though it were the standard deviation of the individual random effects. The correction is to identify exactly which parameter the prior describes and to transform its summaries using the stated distribution.

Key takeaway: Specify priors for typical population parameters with explicit units and support. For lognormal priors, interpret the distribution on the natural scale: the median is \(u\exp(m)\), the mean is \(u\exp(m+s^2/2)\), and the geometric standard deviation is \(\exp(s)\). Keep these priors distinct from the distributions of individual random effects.

Check Your Understanding

Try each question, then compare your reasoning with the solutions below.

If \(\log[\theta/(1\ \mathrm{L/h})]\sim N(\log 4,(\log 2)^2)\), what are the natural-scale median and geometric standard deviation?
For the prior in question 1, calculate the natural-scale mean to three significant figures.
Why can a normal prior directly on a positive clearance parameter be problematic?
When a lognormal prior is written as a density with respect to \(\theta\), what transformation factor appears, and why?

Solutions. (1) The median is \(1\ \mathrm{L/h}\times\exp(\log4)=4\ \mathrm{L/h}\), and the geometric standard deviation is \(\exp(\log2)=2\). (2) The mean is \(4\exp[(\log2)^2/2]\ \mathrm{L/h}=4\exp(0.24023)\ \mathrm{L/h}\approx5.09\ \mathrm{L/h}\). (3) A normal distribution on the natural scale assigns some probability to negative clearance, which is outside the parameter’s positive support; a positive-support distribution or a suitable transformation avoids that problem. (4) The density includes the Jacobian \(1/\theta\), because \(d\log(\theta/u)/d\theta=1/\theta\). It is included when the density is with respect to \(\theta\), not added again to a density already expressed with respect to the log parameter.

Priors for Population Parameters

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