Parameterizing Fast and Slow Clearance Classes

Why Parameterize Clearance by Class?

A two-class clearance model needs to represent two related features: the typical clearance in each class and the individual variability within each class. Estimating two unrelated class-specific typical values is possible, but it can make their relationship harder to describe. A shared value and a fast-to-slow ratio give a direct interpretation of the separation between classes while preserving the usual random effect for individual clearance.

Here, “fast” and “slow” refer to the ordering of the class-specific typical clearances. They do not imply that every individual in the fast class must have higher clearance than every individual in the slow class. Within-class random effects can make the two population distributions overlap.

Model and Assumptions

For individual \(i\), let \(C_i\in\{S,F\}\) denote the individual’s class. A multiplicative random effect gives the individual clearance:

Clearance and \(CL_{\mathrm{typ}}\) have the same units, such as L/h; \(R_{CL}\) and \(\exp(\eta_i)\) are dimensionless. The common within-class random-effect variance is \(\omega_{CL}^{2}\), on the log-clearance scale. The model assumes that, conditional on class, log clearance is normally distributed with mean set by that class’s typical clearance. For the marginal calculations below, it also assumes that \(\eta_i\) has the same distribution in both classes and is independent of class membership.

Taking logarithms makes the structure explicit:

Because the log-scale random effect has mean zero, setting \(\eta_i=0\) gives the class-specific typical clearance. With the stated normal assumption, that value is also the conditional median of clearance. It is not the conditional arithmetic mean unless \(\omega_{CL}^{2}=0\). This distinction matters whenever “typical” is used to describe a log-normal parameter.

Conditional and Overall Population Distributions

Write \(m_c=CL_{\mathrm{typ},c}\) for the typical clearance in class \(c\). If \(CL\mid C=c=m_c\exp(\eta)\) and \(\eta\sim N(0,\omega_{CL}^{2})\), the conditional mean and variance are

These are the standard mean and variance of a log-normal random variable. The variance includes the within-class random effects; setting it to zero would remove individual clearance variability. A central 95% conditional model interval, when the parameters are treated as fixed, is

This interval describes the modeled spread of individual clearances within a class, not uncertainty in the estimated parameters. It does not guarantee that 95% of a finite sample will fall inside it.

To obtain an overall population distribution, let \(\pi_F=P(C=F)\) be the fast-class proportion and \(\pi_S=1-\pi_F\) the slow-class proportion. The marginal distribution is a weighted mixture of the two conditional distributions. Its mean is the weighted conditional mean, while its variance contains both within-class and between-class contributions:

where \(\mu_c=E(CL\mid C=c)\) and \(\sigma_c^2=\operatorname{Var}(CL\mid C=c)\). The conditional log-normal distributions do not generally combine into one log-normal distribution. The mixture variance formula is the law of total variance: it adds the average within-class variance to the variance of the class-specific means. The class proportions here are inputs for describing a marginal distribution; this parameterization alone does not determine or estimate them.

Choosing and Checking the Parameterization

The centered parameterization is one of several ways to express two class-specific typical values. Its main advantage is that \(CL_{\mathrm{typ}}\) summarizes their geometric center, while \(R_{CL}\) describes their multiplicative separation. The table contrasts it with common alternatives.

A useful check is to reconstruct both class typicals from the fitted shared value and ratio, then verify that their quotient equals \(R_{CL}\). Also check units, the sign and scale of the log random effect, and whether the reported spread is conditional on class or marginal across classes. Extremely large ratios or random-effect variances may imply implausibly broad distributions, but their plausibility must be assessed in the context of the model and data.

Common Mistakes and Limitations

• Using the ratio on the wrong scale. \(R_{CL}\) is a ratio of clearances, not a difference to add. The corresponding fast-versus-slow displacement on the log scale is \(\log(R_{CL})\).
• Calling the shared value the arithmetic population mean. \(CL_{\mathrm{typ}}\) is the geometric center of the two class typicals. Conditional means also depend on \(\omega_{CL}^{2}\), and the marginal mean additionally depends on class proportions.
• Dropping the random effect. Setting \(\eta_i=0\) gives a class-specific typical value; it does not describe the population distribution unless within-class variability is zero.
• Adding conditional variances and stopping. Overall variance also includes differences between class means. Omitting those between-class terms underestimates marginal variability when the class means differ.
• Assuming class ordering applies to every individual. \(R_{CL}>1\) orders typical clearances only. Overlap of the conditional distributions is possible and expected for many parameter values.

The parameterization is a way to describe class-specific clearance, not a guarantee that the data can distinguish the classes or estimate every parameter precisely. It also relies here on a common log-scale random-effect variance and independence between class membership and the random effect. If those assumptions do not fit the scientific model, the resulting conditional and marginal distributions will need a different specification.