Parametric Sampling Of Continuous Covariates

Why Parametric Covariate Sampling?

A virtual population needs covariate values that represent a specified target population. When individual records are unavailable, or when a smooth distribution is the intended representation, a parametric distribution offers a way to generate those values. The choice is consequential: a distribution determines not only typical values but also the frequency of unusual values and, for bounded distributions, which values are possible at all.

This tutorial treats one continuous covariate at a time. It focuses on specifying a marginal distribution, generating values from it, and translating them into useful units. A simulated draw is not automatically a clinically plausible value merely because it came from a well-known distribution. The distribution, its parameterization, and the units must all match the intended target.

The distribution describes variation between individuals in the target population; it is not a model for uncertainty in a pharmacometric parameter estimate. If the target population is described by a mean and standard deviation, those values are population-distribution parameters, not automatically the sample mean \(\bar{x}\) and sample standard deviation \(s\) of a finite simulated population. The earlier material on defining target covariate populations is relevant here: a distribution should represent the population the simulation is meant to address.

Specify the Distribution Before Drawing

A continuous distribution is specified by its family, parameters, and measurement scale. For a normal model, \(X\sim N(\mu,\sigma^2)\), \(\mu\) is the mean and \(\sigma\) is the standard deviation; the second argument is the variance. The model has support over all real numbers. That can be suitable for some measurements, but it can generate negative or otherwise implausible values for covariates that are necessarily positive.

A lognormal model is appropriate for a strictly positive quantity when its logarithm is modeled as normal. If \(\log(X)\sim N(\mu_{\log},\sigma_{\log}^2)\), then \(X\) is lognormal and cannot be negative. Its parameters are on the log scale, which makes it essential to distinguish them from the arithmetic mean and standard deviation on the original scale. For this model, the arithmetic mean is \(\exp(\mu_{\log}+\sigma_{\log}^2/2)\), and the variance is \((\exp(\sigma_{\log}^2)-1)\exp(2\mu_{\log}+\sigma_{\log}^2)\).

For a variable bounded between known limits, a beta distribution can be rescaled to that interval. A beta variable \(Y\) lies between 0 and 1. If the clinically meaningful covariate lies between \(L\) and \(U\), set \(X=L+(U-L)Y\). The beta distribution can represent different shapes within the bounds, but its parameters must be chosen to match plausible target moments or other available information.

Generating Draws and Transforming Scales

Many simulation tools generate standard normal draws \(Z\sim N(0,1)\) or uniform draws \(U\) between 0 and 1. A normal draw is obtained as \(X=\mu+\sigma Z\). More generally, if \(F\) is a continuous cumulative distribution function, an inverse-transform draw is \(X=F^{-1}(U)\). Because \(P(U\leq u)=u\) for \(0\leq u\leq1\), this transformation gives \(P(F^{-1}(U)\leq x)=F(x)\), so the resulting draw has cumulative distribution function \(F\).

For a lognormal variable, draw \(Z\sim N(0,1)\), form \(\log(X)=\mu_{\log}+\sigma_{\log}Z\), and exponentiate. If the target information is reported as an arithmetic mean \(m\) and coefficient of variation \(CV\), the log-scale parameters follow from the lognormal moment formulas:

Here \(CV\) is the standard deviation divided by the arithmetic mean on the original scale. These formulas are specific to the lognormal model; using \(m\) itself as \(\mu_{\log}\), for example, would mix scales and produce an incorrect distribution.

For a beta variable \(Y\sim\operatorname{Beta}(a,b)\), its mean is \(a/(a+b)\) and variance is \(ab/[(a+b)^2(a+b+1)]\). If the desired mean and variance on the unit interval are \(m_Y\) and \(v_Y\), define the concentration \(k=a+b\). Solving the mean and variance equations gives:

This parameterization requires \(0<m_Y<1\) and a variance compatible with a beta distribution, in particular \(0<v_Y<m_Y(1-m_Y)\). If the interval is scaled to \([L,U]\), first convert the target moments to the unit interval: \(m_Y=(m_X-L)/(U-L)\) and \(v_Y=v_X/(U-L)^2\).

Assumptions, Diagnostics, and Alternatives

Parametric sampling is most defensible when the chosen distribution is a reasonable representation of the target covariate’s marginal shape and support. The target parameters should refer to a clearly defined population and measurement scale. Independently generating a covariate for each virtual individual assumes that individual draws follow the specified distribution without additional dependence or clustering being represented. If the target includes clinically important subgroups with different distributions, one overall distribution may not represent it adequately.

Before using simulated values in predictions, inspect whether the generated sample is consistent with the model and whether its values make clinical sense. Useful basic checks include the minimum and maximum, mean and standard deviation on the intended scale, selected quantiles, and the number of values outside any scientifically justified bounds. A histogram can reveal substantial skew, multiple modes, or tail behavior that a chosen family does not capture. These checks do not prove that the distribution represents the target population; that judgment depends on the evidence used to specify the model.

The main alternatives differ in what they assume. Empirical resampling uses observed covariate records and preserves their observed marginal shape, but is limited by the available records and cannot generate values not represented there. Parametric sampling provides a compact model and can generate values beyond observed records, but depends on the selected family and parameters. A normal model is simple and symmetric but has unbounded support. A lognormal model ensures positivity and allows right skew, but may still put too much or too little probability in the upper tail. A scaled beta model respects specified finite bounds, but can impose an unsuitable shape or artificial boundary behavior.

Transformations also require attention to units. A normal draw in kilograms cannot be used as if it were in pounds, and a log-scale parameter is not a clinical value on the original scale. For a deterministic unit conversion such as \(X_{\text{new}}=cX_{\text{old}}\), both the mean and standard deviation are multiplied by \(c\) when \(c>0\); the variance is multiplied by \(c^2\). Record the scale and units alongside the sampled covariate so that downstream model inputs are interpretable.

Common Mistakes and Limitations

• Using the wrong scale. A lognormal \(\mu_{\log}\) is the mean of \(\log(X)\), not the arithmetic mean of \(X\). Convert parameters using the lognormal moment equations.
• Ignoring support. A normal distribution can generate negative values for a positive covariate. Consider whether another family is more defensible rather than silently treating every impossible draw as acceptable.
• Confusing a finite sample with its target. A finite simulated sample will not usually reproduce the specified moments exactly. Differences alone do not show that the sampling code is wrong.
• Assuming a good fit from matching two moments. Many distributions share a mean and variance but have different tails and shapes. Moment matching does not establish that the selected family is clinically appropriate.
• Applying a transformation without tracking units. Exponentiating a log-scale draw returns a value on the original scale, but the clinical unit must still be the one used to define the model.

Parametric sampling is limited by the information used to choose the distribution. A simple family can smooth over multimodality, subgroup differences, or unusual tails. Conversely, adding complexity without adequate evidence can make the simulation less transparent. State the assumptions explicitly and interpret results as conditional on them.