Why Transform Parameters?
Uncertainty distributions for PBPK inputs must respect the values a parameter can physically take. A clearance, volume, or rate constant is strictly positive; a fraction or probability lies within a finite interval. A normal distribution placed directly on either kind of parameter can generate impossible values. Transformations provide a way to express uncertainty on an unrestricted scale and map it back to the parameter scale without crossing those limits.
The choice of transformation is part of specifying the uncertainty model, not merely a computational adjustment. A normal distribution on a transformed scale generally induces a non-normal distribution on the original scale. Its skewness, mean, median, and interval widths depend on both the transformation and the uncertainty assigned on the transformed scale. The earlier material on selecting marginal distributions for PBPK inputs provides the broader context for choosing an appropriate marginal model.
Positive Parameters: The Log Transformation
For a parameter that must be strictly positive, let \(\theta_{\mathrm{ref}}>0\) be a fixed reference value with the same units as \(\theta\). Define the dimensionless log-scale quantity \(\eta=\log(\theta/\theta_{\mathrm{ref}})\). The inverse transformation is \(\theta=\theta_{\mathrm{ref}}\exp(\eta)\), which is positive for every real \(\eta\). Using a reference value avoids taking the logarithm of a dimensional quantity.
A common specification is \(\eta\sim N(m,s_{\eta}^{2})\). It follows that \(\theta\) has a lognormal distribution with median \(\theta_{\mathrm{ref}}\exp(m)\), and its \(p\)-quantile is \(\theta_{\mathrm{ref}}\exp(m+z_p s_{\eta})\), where \(z_p\) is the \(p\)-quantile of a standard normal distribution. The original-scale distribution is right-skewed when \(s_{\eta}>0\). Its mean is \(\theta_{\mathrm{ref}}\exp(m+s_{\eta}^{2}/2)\), which is greater than its median unless there is no uncertainty.
Worked Example: A Positive Clearance with a Specified Median
Suppose, hypothetically, that clearance \(CL\) is modeled as lognormal with median \(12\ \mathrm{L/h}\) and geometric standard deviation \(1.30\). A lognormal model expresses uncertainty in the logarithm of the parameter relative to a reference value. Choose \(\theta_{\mathrm{ref}}=1\ \mathrm{L/h}\), so that \(\eta=\log(CL/\theta_{\mathrm{ref}})\). The log-scale median is \(m=\log(12)=2.4849\), and the log-scale standard deviation is \(s_{\eta}=\log(1.30)=0.2624\).
The central 95% interval is obtained from the 2.5th and 97.5th percentiles of the normal log-scale distribution:
Substituting the values gives \(12\exp(\pm 1.96\times 0.2624)\ \mathrm{L/h}\). Since \(1.96\times0.2624=0.5143\), the interval is approximately \((12\exp(-0.5143),12\exp(0.5143))=(7.18,20.1)\ \mathrm{L/h}\). Its endpoints are not equally distant from 12 because the transformation is nonlinear.
The median is \(12\ \mathrm{L/h}\), but the mean is \(12\exp(0.2624^2/2)=12.42\ \mathrm{L/h}\). Thus, describing 12 as the mean would be incorrect under this specification. The interval is a central 95% interval under the stated lognormal model, not a claim that a fixed parameter has a 95% probability of lying in this particular interval.
Parameters Between Two Bounds: Logit and Probit
Suppose a parameter must lie strictly between fixed bounds \(a\) and \(b\), where \(a<b\). Rescale it to \(u=(\theta-a)/(b-a)\), so \(0<u<1\). The logit transformation maps this proportion to the real line: \(\eta=\log\{u/(1-u)\}\). Its inverse is
If \(\eta\sim N(m,s_{\eta}^{2})\), the original parameter is bounded but generally asymmetric. The median is the inverse-logit of \(m\), rescaled to \((a,b)\). Its quantiles are found by applying the inverse transformation to the corresponding normal quantiles. A logit-normal model is not the same as a beta distribution: each can describe bounded uncertainty, but their shapes and parameterizations differ.
The probit transformation uses the standard normal cumulative distribution function \(\Phi\): \(\eta=\Phi^{-1}(u)\), with inverse \(u=\Phi(\eta)\). Thus \(\theta=a+(b-a)\Phi(\eta)\). When \(\eta\sim N(m,s_{\eta}^{2})\), its transformed quantiles are \(a+(b-a)\Phi(m+s_{\eta}z_p)\). Logit and probit both map the real line into the open unit interval, but they use different functions and do not generally yield the same distribution for a given normal model on \(\eta\).
Worked Example: A Bounded Fraction Using a Logit-Normal Model
Suppose, hypothetically, that a tissue fraction \(f\) lies strictly between 0 and 1 and is modeled using \(\eta=\log\{f/(1-f)\}\), with \(\eta\sim N(\log(0.20/0.80),0.70^2)\). The transformed-scale mean is \(\log(0.25)=-1.3863\), so the median fraction is \(1/(1+\exp(1.3863))=0.200\).
The central 95% interval on the transformed scale is \(-1.3863\pm1.96(0.70)\), or approximately \((-2.758,\,-0.0143)\). Transform each endpoint back:
The resulting central 95% interval is approximately \((0.0596,0.496)\). Both values are fractions on the original scale. The upper endpoint is much farther from the median than the lower endpoint, illustrating the asymmetry that can arise even when transformed-scale uncertainty is symmetric.
Worked Example: A Bounded Parameter Using a Probit Transformation
Suppose, hypothetically, that a normalized parameter \(\theta\) is known to lie strictly between \(0.10\) and \(0.90\). Model it as \(\theta=0.10+0.80\Phi(\eta)\), where \(\eta\sim N(0,0.60^2)\). The median is \(0.10+0.80\Phi(0)=0.50\), because \(\Phi(0)=0.5\).
For a central 95% interval, the latent-scale endpoints are \(\pm1.96(0.60)=\pm1.176\). Using \(\Phi(1.176)\approx0.880\) and \(\Phi(-1.176)\approx0.120\), the original-scale endpoints are \(0.10+0.80(0.120)=0.196\) and \(0.10+0.80(0.880)=0.804\). The interval is therefore approximately \((0.196,0.804)\), within the specified bounds. This example illustrates the probit mapping; it does not imply that probit is preferable to logit for every bounded PBPK parameter.
How Densities and Uncertainty Change Scale
A transformation of random variables changes probability density as well as values. For a differentiable, one-to-one transformation \(\theta=g(\eta)\), the density on the original scale is the transformed-scale density evaluated at the inverse transformation, multiplied by the absolute derivative of that inverse. This Jacobian factor ensures that probabilities are preserved.
For the positive log transformation, \(\eta=\log(\theta/\theta_{\mathrm{ref}})\), so \(d\eta/d\theta=1/\theta\). Consequently, \(f_{\theta}(\theta)=f_{\eta}(\log(\theta/\theta_{\mathrm{ref}}))/\theta\) for \(\theta>0\). For the bounded logit transformation, write \(u=(\theta-a)/(b-a)\). The derivative of the inverse is \(d\eta/d\theta=1/\{(b-a)u(1-u)\}\), so the density includes that factor. For the probit transformation, \(d\eta/d\theta=1/\{(b-a)\phi(\eta)\}\), where \(\phi\) is the standard normal density. These factors explain why a normal density on the transformed scale does not remain normal after transformation back.
For a strictly increasing transformation, quantiles map directly: \(Q_{\theta}(p)=g(Q_{\eta}(p))\). This is often the clearest way to report uncertainty intervals. Means and variances do not generally map by simply applying \(g\) to a transformed-scale mean or variance. When uncertainty is small, a local approximation can be useful: if \(\eta\) has mean \(\mu_{\eta}\) and variance \(\sigma_{\eta}^{2}\), then \(\operatorname{Var}(\theta)\) is approximately \(\{g'(\mu_{\eta})\}^{2}\sigma_{\eta}^{2}\). This is a first-order approximation, not an exact result, and may be poor when uncertainty is wide or the transformation is strongly curved over the likely range.
Decide whether the parameter is strictly positive or constrained to an interval, and state its units and any fixed bounds.
Use a log scale for strictly positive values, or a logit or probit map for values strictly between two bounds. The choice also specifies a distributional shape.
State the distribution and its parameters, and relate them to interpretable summaries such as an original-scale median or quantiles.
Transform quantiles back, inspect skewness and interval widths, and verify that the implied values are plausible for the PBPK application.
Assumptions, Diagnostics, and Alternatives
These transformations rely on correctly specified support and bounds. The log transformation assumes that zero and negative values are impossible. Logit and probit models assume strict interior values: they approach, but never attain, \(a\) or \(b\). A numerical implementation may need stable functions for very large positive or negative latent values, since direct exponentiation can overflow even though the mathematical inverse-logit remains bounded.
Check transformed-scale specifications by mapping their quantiles back to the original scale and comparing them with scientific knowledge. A positive distribution may have an implausibly long upper tail; a bounded model may place nearly all uncertainty near a boundary or may be too concentrated around its median. Also check whether the parameter can truly attain a boundary. If zero is a possible value with positive probability, a continuous lognormal model is inadequate. If a fraction can be exactly zero or one, a logit-normal or probit-normal distribution alone is likewise inadequate. A model with explicit probability at the boundary, or another scientifically appropriate distribution, may be needed.
Alternatives include specifying a distribution directly on the original scale, such as a suitable bounded distribution, or using a truncated distribution. Direct specification can be more interpretable when its parameters match available evidence. Truncation restricts values to a range, but it changes the distribution by conditioning on being within that range; it is not generally equivalent to applying a logit or probit transformation. Repeatedly discarding invalid draws also changes the effective distribution if it is used as an informal fix. The relevant comparison is therefore between the full implied distributions, not only whether they obey the bounds.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A positive parameter has \(\log(\theta/\theta_{\mathrm{ref}})\sim N(0,0.20^2)\). What are its median and approximate central 95% interval in units of \(\theta_{\mathrm{ref}}\)?
- Why can a logit-normal model not assign positive probability to an exact fraction of zero?
- For \(\theta=a+(b-a)\Phi(\eta)\), with \(\eta\sim N(0,s_{\eta}^{2})\), what is the median of \(\theta\)?
- Does applying a transformation to a normal variable generally produce a normal parameter? Explain.
Solutions. (1) The median is \(\theta_{\mathrm{ref}}\exp(0)=\theta_{\mathrm{ref}}\). The interval is \(\theta_{\mathrm{ref}}\exp(\pm1.96(0.20))=\theta_{\mathrm{ref}}\exp(\pm0.392)\), or approximately \((0.676,1.480)\theta_{\mathrm{ref}}\). (2) The inverse logit of any finite real number lies strictly between zero and one; reaching zero requires a limiting latent value of negative infinity. (3) The median latent value is zero, so the median parameter is \(a+(b-a)\Phi(0)=(a+b)/2\). (4) No. A nonlinear transformation generally changes the distributional shape; the resulting parameter has the distribution induced by that transformation, not a normal distribution simply because the latent variable was normal.