Deriving Parameter Distributions from Assay Summary Statistics

From Assay Summaries to Parameter Uncertainty

PBPK inputs are often reported as a mean with a standard deviation (SD), a standard error (SE), and a sample size. Those summaries can support a quantitative description of uncertainty, but only after identifying what quantity was measured, on which scale, and what the reported spread describes. A standard deviation across individuals, for example, is not interchangeable with the standard error of an estimated mean.

This tutorial uses “parameter distribution” carefully. An assay summary can be used to describe the sampling uncertainty in an estimated parameter, such as the assay population mean. It does not, by itself, establish a probability distribution for the unknown parameter in a Bayesian sense, nor does it determine the distribution of values among individuals. The earlier material on epistemic uncertainty versus interindividual variability distinguishes those targets; here the focus is converting reported assay summaries into a usable uncertainty representation.

State the Target and Assumptions First

Before calculating, state whether the PBPK input of interest is an individual value, a population mean, or another assay-derived quantity. If the aim is to characterize the spread of measurements among comparable individuals, \(s\) may be relevant. If the aim is uncertainty in the estimated mean used as a fixed model input, \(SE\) is generally the relevant summary. Using \(s\) as though it were the uncertainty in \(\bar{x}\) can greatly overstate uncertainty when \(n\) is large; using \(SE\) to represent individual variation can greatly understate it.

The usual calculation assumes that the \(n\) observations are independent and represent comparable measurements of the stated target. The exact small-sample \(t\) interval additionally assumes observations from a normal population. With a sufficiently large sample, a normal approximation for the sampling distribution of the mean may be reasonable under weaker conditions, but severe skewness, influential outliers, clustering, or a small sample can undermine that approximation. Technical replicates from one biological specimen are not automatically independent biological observations: the effective sample size depends on the study design.

For independent observations with population variance \(\sigma^2\), the variance of their mean is \(\sigma^2/n\). Replacing the unknown \(\sigma\) by the observed \(s\) gives the estimated standard error. When the normal-population assumptions hold, the standardized difference between \(\bar{x}\) and \(\theta\), divided by \(s/\sqrt{n}\), follows a \(t\) distribution with \(n-1\) degrees of freedom. This gives the confidence interval:

Here \(t_{1-\alpha/2,\;n-1}\) is the corresponding quantile of the \(t\) distribution. A 95% confidence interval is a procedure that, under its assumptions and repeated sampling, covers the fixed population mean in 95% of repetitions. It is not a statement that the fixed mean has a 95% probability of lying in this particular interval.

A common approximate sampling representation for the estimated mean is \(\bar{x}\) centered at the reported estimate with standard deviation \(SE\). This can be written as \(N(\bar{x},SE^2)\) as a normal approximation to the sampling distribution of the estimator. It is not automatically a probability distribution for the unknown \(\theta\). Treating it as a practical uncertainty distribution for a PBPK input is an explicit modeling choice, not a consequence of the summaries alone.

Recovering the Missing Summary Statistic

Reports do not always provide both \(s\) and \(SE\). If a report gives \(SE\) and \(n\), the corresponding sample SD is \(s=SE\sqrt{n}\), provided the reported SE was calculated as the ordinary sample SD divided by the square root of the sample size. If the report gives \(s\) and \(n\), calculate \(SE=s/\sqrt{n}\). Check the report’s methods: an adjusted, model-based, or clustered SE may not obey this simple relationship.

Respect the Scale Used in the Report

The numerical meaning of a mean and SD depends on the measurement scale. A mean and SD reported for the raw assay values describe arithmetic summaries on that original scale. A mean and SD reported after taking logarithms describe summaries of the log-transformed values instead. The latter do not directly give the arithmetic mean and SD on the original scale.

For a logarithmic report, define \(Y_i=\log_{10}(X_i)\), where \(X_i\) is the positive assay measurement. If the reported mean and SD are for \(Y\), first calculate the SE and interval on the \(Y\) scale. Exponentiating the estimate gives the antilog of the mean log value, commonly interpreted as a geometric mean. Exponentiating the interval endpoints gives an interval on the \(X\) scale. It does not turn the geometric mean into the arithmetic mean. The interpretation also depends on the base of the logarithm: a value reported in \(\log_{10}\) units cannot be treated as a natural-log value without conversion.

Choosing and Checking an Uncertainty Representation

When the purpose is to carry sampling uncertainty in an estimated assay mean into a PBPK analysis, the estimate and its SE can motivate an approximate normal representation on the same scale. That representation is most defensible when the sampling distribution is roughly symmetric and the estimate is not constrained near a physical boundary. A normal distribution can assign probability to impossible values—for example, negative values for a positive parameter or values outside the feasible range for a fraction. This is a diagnostic warning, not a reason to silently change scales or truncate values without justification.

The main alternatives answer different questions. A summary of individual-level dispersion uses the SD; uncertainty in a mean uses its SE. A normal approximation is simple and may be reasonable with adequate data; a \(t\) interval accounts for estimating the SD and is preferable for an exact small-sample interval under normality. Working on a logarithmic scale can suit multiplicative measurement variation, but the interpretation must remain on that scale until a stated transformation is applied. Distribution choice for positive or bounded parameters requires additional considerations beyond these conversions.

Useful diagnostics include checking that the sample size counts independent units rather than technical repeats, verifying the units and transformation stated in the methods, and comparing the implied interval with assay limits and biological constraints. A very small SE despite substantial biological dispersion may simply reflect a large \(n\); it is not evidence that individual values are nearly identical. Conversely, a narrow reported range does not establish a precise estimate if the sample is small or the range is not a confidence interval.

Common Mistake and Practical Limits

A frequent error is to take a reported mean and SD and use the SD as the standard deviation of uncertainty in the estimated mean. For \(n>1\), that confuses the dispersion of observations with the sampling uncertainty of their mean. The correction is to establish the intended target and, when appropriate, calculate \(SE=s/\sqrt{n}\). The reverse error—using SE as the spread of individual biological values—confuses those targets in the other direction.

Summary statistics cannot reveal every feature needed to choose a distribution. They may conceal skewness, outliers, censoring below a quantification limit, batch effects, or dependence among observations. Nor do they correct assay bias: a precise estimate can still be systematically inaccurate. If these issues matter, the raw data, study design, and assay methods are needed. For a PBPK parameter, any derived uncertainty representation should therefore be documented with its target, scale, assumptions, and limitations.