Priors for Design Parameters

Why Specify a Distribution for Design Parameters?

A local design calculation evaluates a design at a particular parameter value. When the parameter values are uncertain, a single estimate may not describe the range of plausible conditions under which the design will be used. A design-stage parameter distribution makes that uncertainty explicit: it assigns relative probability to alternative values of the unknown model parameters before the prospective study supplies new data.

For example, a prior distribution for typical clearance describes uncertainty about the population-level clearance parameter. It is not the distribution of clearances across individual patients. That distinction matters in pharmacometrics: the model may have a typical clearance and a random effect describing individual deviations from it, while the design distribution describes uncertainty about the typical clearance itself. These are different sources of variation, with different roles in a design calculation.

Definition: A design-stage parameter distribution is a probability distribution over unknown model parameters used to represent uncertainty about their values when evaluating or choosing a study design. Its random variables represent uncertain parameters of the model, not individual-level random effects or residual observations.

A distribution can be informed by estimates and standard errors from previous data, by correlations among parameter estimates, by external knowledge, or by plausible parameter ranges. Those inputs do not determine a unique distribution without assumptions. The distributional form and parameterization must therefore be stated, and its implications checked against the scientific problem.

From Estimates and Standard Errors to a Distribution

A common approximation is to use an estimate as the center of a normal design distribution and its standard error as the distribution's standard deviation. If \(\hat{\theta}\) estimates an unknown parameter \(\theta\) and \(SE(\hat{\theta})\) is available, this approximation is

\theta \sim N\left(\hat{\theta}, SE(\hat{\theta})^2\right).

This is a practical representation of uncertainty, not an automatic consequence of reporting an estimate and standard error. It is most defensible when the estimate is approximately normal on the chosen scale, its standard error is a reasonable measure of uncertainty, and the estimate pertains to the same parameter definition and population as the proposed design. An estimate and standard error alone do not establish a Bayesian prior distribution. When they are used this way, the approximation and its source should be documented.

The scale is especially important for positive parameters such as clearance, volume, and rate constants. A normal distribution on the original scale assigns some probability to negative values. If that probability is material or negative values are scientifically impossible, it is generally preferable to model the log parameter as normal. The resulting parameter has a lognormal distribution and remains positive.

Worked Example: A Positive Clearance Parameter

Suppose previous information gives an estimated typical clearance of \(5.00\) L/h with standard error \(0.20\) on the natural-log scale. Let \(\phi=\log(CL)\). For this hypothetical example, use \(\hat{\phi}=1.6094\) and \(SE(\hat{\phi})=0.20\). Specify the design distribution and calculate its central 95% interval for clearance.

Distribution on the modeled scale. Approximate uncertainty in \(\phi\) by \( \phi \sim N(1.6094,0.20^2) \). Thus \(CL=\exp(\phi)\) is lognormal and is positive for every draw.

Central interval. For a normal distribution, the central 95% interval is approximately the mean plus or minus 1.96 standard deviations. On the log scale this is \(1.6094 \pm 1.96(0.20)=1.6094\pm0.392\), or \([1.2174,2.0014]\). Exponentiating both endpoints gives \( [\exp(1.2174),\exp(2.0014)] \approx [3.38,7.40]\) L/h. Under this specified design distribution, 95% of its probability lies between approximately 3.38 and 7.40 L/h. This is a statement about the chosen distribution, not a guarantee about the unknown clearance.

Centering detail. The median clearance is \(\exp(1.6094)\), approximately \(5.00\) L/h. The mean is larger because the lognormal distribution is right-skewed. For \(\phi\sim N(m,s^2)\), completing the square in the normal-density integral gives

E[\exp(\phi)] =\int_{-\infty}^{\infty}\exp(\phi) \frac{1}{s\sqrt{2\pi}}\exp\left(-\frac{(\phi-m)^2}{2s^2}\right)d\phi =\exp\left(m+\frac{s^2}{2}\right).

Here the mean is \(\exp(1.6094+0.20^2/2)\), approximately \(5.10\) L/h. Reporting the distribution as lognormal therefore requires being clear whether the supplied estimate is a median (or geometric mean) or an arithmetic mean.

When estimates are provided on the original scale, do not automatically treat their standard errors as log-scale standard deviations. A transformation or approximation is needed, and it should match the meaning of the estimate. For a parameter estimated with a small relative standard error, a delta-method approximation is \(SE(\log\hat{\theta})\approx SE(\hat{\theta})/\hat{\theta}\). This approximation relies on uncertainty being small relative to the estimate; it can be poor for highly uncertain or boundary-near parameters.

Representing Correlated Parameter Uncertainty

Parameters in a model are often estimated jointly, so their uncertainties can be correlated. Let \(\boldsymbol{\theta}\) be a vector of uncertain parameters and let \(\hat{\boldsymbol{\theta}}\) be their estimates. A multivariate normal design distribution is specified by a mean vector and covariance matrix. The covariance between parameters \(i\) and \(j\) is their correlation multiplied by their standard deviations:

\Sigma_{ij}=\rho_{ij}SE_iSE_j,\qquad \boldsymbol{\theta}\sim N(\hat{\boldsymbol{\theta}},\Sigma).

The diagonal entries of \(\Sigma\) are variances, not standard errors. The off-diagonal entries retain the direction and magnitude of dependence. Omitting a meaningful correlation changes the joint distribution, even if each parameter's marginal distribution remains unchanged.

Worked Example: Constructing a Two-Parameter Distribution

For a hypothetical PK model, suppose typical clearance is estimated as \(5.0\) L/h with standard error \(1.0\) L/h, and typical volume is estimated as \(50\) L with standard error \(10\) L. Suppose their estimate correlation is \(-0.60\). Construct the approximate joint design distribution and check its consistency.

The covariance is \((-0.60)(1.0)(10)=-6.0\) L\(^2\)/h. The variances are \(1.0^2=1.0\) (L/h)\(^2\) and \(10^2=100\) L\(^2\). Therefore,

\begin{pmatrix}CL\\V\end{pmatrix} \sim N\left[ \begin{pmatrix}5.0\\50\end{pmatrix}, \begin{pmatrix}1.0&-6.0\\-6.0&100\end{pmatrix} \right].

One way to verify the covariance is to let \(Z_1\) and \(Z_2\) be independent standard normal variables and write \(CL=5+Z_1\) and \(V=50+10(-0.60Z_1+0.80Z_2)\). Then \(Var(CL)=1\), while \(Var(V)=100(0.60^2+0.80^2)=100\). Also, \(Cov(CL,V)=10(-0.60)=-6\), giving correlation \(-6/(1\cdot10)=-0.60\), as specified.

For a two-parameter covariance matrix, positive definiteness requires positive diagonal entries and a positive determinant. Here the determinant is \(1(100)-(-6)^2=64>0\), so the matrix is positive definite and defines a valid non-degenerate joint normal distribution. The normal approximation is plausible for this example's positive parameters because their means are five standard errors above zero, but positivity should still be checked when drawing values. If substantial probability falls on impossible values, a distribution on transformed parameters is more suitable.

For more than two parameters, a collection of pairwise correlations is not automatically a valid joint distribution. The resulting correlation matrix must be positive semidefinite (and positive definite when a non-degenerate multivariate normal is required). In practice, use a joint covariance or correlation estimate when available, and check that it is mathematically valid before using it to generate parameter values.

Using Plausible Ranges

Sometimes the available information is a plausible lower and upper bound rather than an estimate with standard error. Bounds constrain support but do not specify how probability is distributed within that support. A uniform distribution assigns equal density throughout the interval; a triangular distribution concentrates density around a stated mode; a truncated normal retains a normal shape only within the specified bounds and must be renormalized. These choices encode different judgments.

Worked Example: Turning a Plausible Range into a Distribution

Suppose a hypothetical analysis regards typical clearance values from 2 to 8 L/h as plausible, with no defensible basis for favoring one value within that interval. As an explicit simplifying choice, assign a uniform distribution on \([2,8]\) L/h. Find its mean, standard deviation, and central 95% interval.

For \(CL\sim Uniform(a,b)\), the mean is \((a+b)/2\), the variance is \((b-a)^2/12\), and the density is constant on the interval. Substituting \(a=2\) and \(b=8\) gives \(E[CL]=(2+8)/2=5.00\) L/h and \(SD(CL)=\sqrt{(8-2)^2/12}=\sqrt{3}=1.732\) L/h.

For a uniform distribution, the central 95% interval excludes 2.5% of probability at each end. Each excluded tail has width \(0.025(8-2)=0.15\) L/h. The interval is therefore \([2+0.15,8-0.15]=[2.15,7.85]\) L/h. The full support remains \([2,8]\) L/h; it is not the same as the central 95% interval.

The uniform choice is justified here only as a declared assumption in the absence of a preferred value. If the midpoint is particularly plausible, a triangular distribution might better represent that belief; if the range is intended as a high-probability interval, a bounded or truncated distribution could be more appropriate. The range alone cannot decide among them.

A plausible range should also be distinguished from a confidence interval, a credible interval, and a hard physical bound. A confidence interval summarizes a procedure's repeated-sampling behavior; a credible interval summarizes probability under a specified Bayesian distribution; a hard bound rules out values outside the interval by assumption. Treating a reported interval as a hard support boundary, or treating a broad plausible range as a 95% interval, can materially distort the parameter distribution.

Checks, Alternatives, and Limitations

Before a design-stage distribution is used, check that it is coherent with the parameterization and with the model. A practical review includes:

Scale: confirm whether estimates and standard errors are on the original, logarithmic, or another transformed scale.
Units: confirm that each standard deviation and covariance uses compatible units.
Support: check for negative values of positive parameters, probabilities outside \([0,1]\), or other impossible draws.
Dependence: check that correlations refer to the same parameterization and that the joint covariance matrix is valid.
Range meaning: record whether bounds are hard constraints, plausible limits, or an interval with a stated probability interpretation.
Source and target: check whether the estimates describe the same population, model, and covariate conditions as the intended design.

If a normal approximation creates implausible values, alternatives include normal distributions on transformed scales, truncated distributions, or distributions chosen to match an explicitly elicited range and shape. A multivariate normal is convenient and often useful, but it may not represent skewness, multiple plausible parameter regions, or nonlinear dependence. Conversely, independent marginal distributions are simpler but discard parameter correlations; they are not a neutral substitute when dependence is supported by evidence.

These distributions are approximations, conditional on a model and on available information. They cannot compensate for a misspecified structural model, unrepresented populations, or poorly understood covariates. Their purpose here is to state uncertainty in design parameters transparently, not to claim that uncertainty has been eliminated. The later analysis of how uncertainty propagates through design criteria requires the parameter distribution as an input; constructing that input carefully is a separate step.

Common Pitfall

Common mistake: “The between-subject standard deviation of clearance is 1 L/h, so I will use \(N(5,1^2)\) as the design prior for typical clearance.” This confuses variability among individual clearances with uncertainty about the population's typical clearance. The correction is to identify the quantity being distributed: use uncertainty in the estimated typical clearance to specify a distribution over that population parameter, and use the between-subject distribution separately when representing individual variability in the model.

A related mistake is to copy standard errors into a multivariate distribution while setting all correlations to zero. That is justified only when independence is appropriate for the uncertain parameters on the chosen scale. Otherwise, retain the covariance information or state why it is unavailable and what approximation is being made.

Check Your Understanding

Answer each question, then compare with the concise solutions below.

A positive parameter has a normal design distribution on its natural-log scale. Why does that imply a lognormal distribution on the original scale?
Two parameter estimates have standard errors 2 and 3, and correlation \(0.50\). What is their covariance?
A uniform distribution is assigned to a plausible interval. Does the interval alone establish that uniformity is appropriate?
Explain the difference between a distribution for uncertain typical clearance and a distribution for individual clearances.
A proposed two-parameter covariance matrix is \(\begin{pmatrix}1&2\\2&1\end{pmatrix}\). Is it positive definite?

Solutions. (1) Exponentiating a normally distributed log parameter yields a lognormal variable, which is positive. (2) The covariance is \((0.50)(2)(3)=3\), in the product of the parameters' units. (3) No. The interval gives bounds but not the probability shape; uniformity is an additional assumption of equal density across the interval. (4) The first represents uncertainty about a population-level model parameter; the second represents between-subject variability around that parameter. (5) No. Its determinant is \(1(1)-2^2=-3<0\), so it is not positive definite and cannot be the covariance matrix of a non-degenerate multivariate normal distribution.

What You'll Learn