Choosing Random-Effects Distributions For PD Parameters

Why the Random-Effects Distribution Matters

In a population PD model, a random effect describes how an individual’s parameter differs from the population value. Its distribution is not just a computational detail: it determines which individual parameter values are possible and how probability is assigned across the population. A distribution that permits negative values for a parameter that must be positive can imply biologically impossible individuals, even when the population estimate looks reasonable.

The earlier material on hierarchical population PD models and additive and proportional random effects established how individual parameters can vary around population values. Here the focus is the shape and support of that variation. The examples use one random effect at a time; a marginal distribution for one parameter does not, by itself, specify how that parameter varies jointly with other PD parameters.

Write \(\theta_i\) for an individual PD parameter and \(\theta_{\mathrm{pop}}\) for its population-typical value. A generic transformed model is

Here \(g\) is a chosen transformation, and \(\omega\) is the standard deviation of the random effect on the transformed scale. The normal assumption applies to \(\eta_i\), not necessarily to \(\theta_i\). The transformation determines the support and shape of the parameter distribution.

Positive Parameters: Normal Effects on the Log Scale

For a parameter that must be positive, a common choice is \(g(\theta)=\log(\theta)\). The model becomes

or, equivalently, \(\theta_i=\theta_{\mathrm{pop}}\exp(\eta_i)\). This is a lognormal distribution: every individual value is positive, while the distribution is generally right-skewed. The population-typical value \(\theta_{\mathrm{pop}}\) is the median, not the arithmetic mean. In fact,

The median follows because \(\eta_i\) has median zero and exponentiation preserves order. The mean follows from the normal moment-generating function. For \(t=1\), \(E[\exp(t\eta_i)]=\exp(t^2\omega^2/2)\), so \(E(\theta_i)=\theta_{\mathrm{pop}}E[\exp(\eta_i)]=\theta_{\mathrm{pop}}\exp(\omega^2/2)\). Thus, increasing variability raises the arithmetic mean while leaving the median at \(\theta_{\mathrm{pop}}\).

The coefficient of variation (standard deviation divided by arithmetic mean) is \(\sqrt{\exp(\omega^2)-1}\). This can help express log-scale variability in relative terms, but it does not remove the need to inspect the implied range of individual values.

When an Additive Normal Distribution Is Plausible

An untransformed additive model has the form \(\theta_i=\theta_{\mathrm{pop}}+\eta_i\), where \(\eta_i\sim N(0,\omega^2)\). It gives a symmetric normal distribution centered at \(\theta_{\mathrm{pop}}\). This can be suitable for a parameter that is allowed to take positive and negative values, provided a symmetric distribution is credible. For a parameter constrained to be positive, however, the model assigns some probability to invalid negative values unless that probability is negligible for the intended application.

Truncating a normal distribution at zero is another possible construction, but it is not identical to an ordinary additive-normal model with negative values simply discarded. A truncated distribution must be renormalized over its permitted range, changing its mean and other distributional properties. The resulting model should be specified and interpreted explicitly.

Bounded Parameters and Other Alternatives

Some PD parameters are restricted to a finite interval. For a fraction \(\phi_i\) between zero and one, a logit transformation gives

This induces a logistic-normal distribution: all individual values lie strictly between zero and one, and the distribution need not be symmetric on the original scale. As with the lognormal model, \(\phi_{\mathrm{pop}}\) is the median because the random effect is centered and the inverse-logit is increasing. It is not generally the arithmetic mean.

A beta distribution is another option for a parameter strictly between zero and one. It can represent different shapes, including distributions concentrated near a boundary, but its parameters and population interpretation differ from those of a logistic-normal model. For positive parameters, gamma distributions are another possible alternative. Such alternatives should be considered when their shape has a scientific or data-supported rationale, not merely because they offer extra flexibility.

Diagnostics, Assumptions, and Failure Modes

A distributional choice assumes that the random effects have the stated distribution across the target population, conditional on the model’s population structure. For the examples above, the normality assumption is on the transformed scale. It also assumes that the parameter transformation is appropriate and that the model has enough information to estimate the relevant variability. A fitted distribution cannot be judged solely by whether the estimation algorithm converges.

A practical diagnostic is to simulate individual parameters from the fitted population model, transform them to the parameter scale, and inspect quantiles and the fraction near implausible regions. Compare those implications with biological limits and with the range of values that could reasonably produce the observed PD data. For positive parameters, inspect both lower and upper tails; for bounded parameters, inspect how much probability accumulates near zero or one. A model can meet a hard support constraint yet still imply values that are scientifically implausible.

Individual empirical Bayes estimates can be useful for visual checks, but they are not direct observations of the population random effects. They are estimated using each subject’s data and are typically pulled toward the population value, particularly when an individual has sparse or noisy observations. A compressed spread of those estimates therefore does not, by itself, demonstrate that the population random-effects distribution is too narrow or nearly normal.

When comparing candidate distributions, keep the PD structure and residual-error assumptions consistent so the comparison targets the distributional choice. Assess fit, stability, and predictive implications, not just a single summary statistic. More flexible distributions can be difficult to estimate when the data contain little information about the tails. Warning signs include unstable variance estimates, sensitivity to starting values, numerical integration problems, or large changes in tail predictions from small changes in the fitted model.

Common Mistake

Wrong reasoning: “EC₅₀ has a positive population estimate, so an additive-normal random effect is acceptable.” A positive center does not prevent the model from assigning probability to negative individual EC₅₀ values.

Correction: Calculate the probability outside the scientifically permitted range, and inspect the distribution’s relevant quantiles. If that probability is not negligible, consider a model whose support matches the parameter, such as a log-scale model for a positive parameter. Then check whether its induced skewness and range are plausible; support alone is not a complete diagnostic.