Common Errors In Reporting Parameter Precision

Why Precision Summaries Need Careful Reporting

A population PK report may place an estimate, its relative standard error (RSE), a confidence interval, and a condition number in the same table. These summaries are related, but they do not answer the same question. An RSE summarizes an estimate’s standard error relative to its magnitude; a confidence interval gives a range of parameter values compatible with an estimation procedure and its assumptions; a condition number describes the sensitivity of a matrix calculation to perturbations. None, by itself, establishes that a parameter is accurate, biologically important, or well identified in every relevant sense.

The introductory material on interpreting RSE in context and the earlier material on confidence intervals from RSE, precision of variance components, and condition numbers provide the background. The reporting task here is to state clearly what each quantity refers to, preserve the scale on which it was calculated, and avoid conclusions that the summary cannot support.

RSE Is Not a Percent Error

If a clearance estimate is 5.0 L/h with an RSE of 20%, its estimated standard error is \(0.20\times5.0=1.0\) L/h. This does not mean the estimate is “20% wrong,” nor does it establish that the true clearance is within 20% of 5.0 L/h. The RSE describes the estimated uncertainty of the estimator relative to the estimate, conditional on the model, data, and approximations used to obtain the standard error.

The denominator matters. An RSE can be large because the standard error is large, because the estimate is close to zero, or both. For a parameter whose estimate may be zero or negative, a percentage RSE can be unstable or uninformative; an absolute standard error and an interval on a meaningful scale may be more useful. Reports should not present a low RSE as proof of lack of bias or of adequate model specification.

Confidence Intervals: Name the Scale and the Approximation

An interval should be reported with its confidence level, units, and method or scale when those details affect interpretation. A symmetric Wald interval uses an estimate plus or minus a critical value times its standard error. It relies on an approximately normal sampling distribution and can produce impossible values for a parameter constrained to be positive. A transformed interval can be more suitable when uncertainty is approximately symmetric on a transformed scale.

For a positive parameter \(\theta=\exp(\eta)\), an approximate interval formed on the log scale is obtained by exponentiating the endpoints \(\hat{\eta}\pm1.96SE(\hat{\eta})\). It is asymmetric on the original scale and remains positive. By the delta method, \(SE(\hat{\theta})\) is approximately \(\hat{\theta}SE(\hat{\eta})\), so the original-scale RSE is approximately \(100SE(\hat{\eta})\). These are first-order approximations, not identities for every finite-sample estimator.

A report should not silently mix an RSE calculated on one scale with an interval calculated on another. If the interval is asymmetric, state the endpoints rather than trying to summarize it as though it were a symmetric “estimate plus or minus” range. Also distinguish an interval for a typical parameter from one for an individual’s random effect or predicted concentration: these are different quantities with different sources of uncertainty.

Variability Estimates Are Not Their RSEs

In a population model, a variability parameter describes variation across individuals under the model. Its estimate’s RSE describes uncertainty in estimating that population quantity. Confusing these two concepts can reverse the meaning of a result. For example, a 25% RSE for an estimated random-effect variance does not mean that individuals vary by 25%.

The scale of the reported quantity must also be explicit. For a normally distributed random effect with variance \(\omega^2\), the standard deviation is \(\omega=\sqrt{\omega^2}\). For a lognormal random effect, a coefficient of variation may be derived from the variance on the log scale as \(\sqrt{\exp(\omega^2)-1}\). Standard errors and RSEs do not stay numerically unchanged when a variance is transformed into a standard deviation or a coefficient of variation. If uncertainty for a transformed quantity is reported, the transformation and uncertainty method should be identified.

Condition Numbers: Numerical Warning, Not Precision Percentage

A condition number summarizes how sensitive a matrix-based calculation may be to small perturbations. For a symmetric positive-definite information matrix, a common convention is the ratio of its largest to smallest eigenvalues, \(\kappa=\lambda_{\max}/\lambda_{\min}\). Large values can signal near-singularity and numerical sensitivity. The condition number is not an RSE, a confidence interval, or a direct measure of how accurately each PK parameter has been estimated.

A further reporting pitfall is that a condition number depends on parameter scaling. Changing parameter units or rescaling coordinates can change the condition number even when the underlying scientific model and data are unchanged. Reports should state which matrix and scaling were used, and should avoid presenting a universal pass/fail threshold without justification. A high value is a reason to investigate scaling, numerical behavior, and weakly estimated parameter combinations; it does not by itself identify which parameter is responsible or prove that the model is unusable.

Reporting Checks and Common Mistakes

Before publishing a precision summary, check that the estimate, standard error, RSE, interval, and parameter label refer to the same quantity and scale. In particular, a standard error for a variance should not be attached to a standard deviation label, and a log-scale interval should not be described as symmetric on the original scale. When a parameter is near a boundary or an estimated standard error is unstable, acknowledge that an approximate RSE or Wald interval may be unreliable.

Precision summaries also need to be read alongside estimation diagnostics. Unexpectedly large RSEs, extreme or unstable intervals, non-positive-definite covariance estimates, or sensitivity to parameter scaling can indicate problems requiring investigation. A small RSE alone does not rule out model misspecification, bias, or a poorly represented parameter combination. A condition number alone does not replace examination of parameter-specific uncertainty.

• Wrong: “The RSE is 20%, so the parameter estimate is 20% inaccurate.” Correction: It is an estimated standard error relative to the estimate, not an observed error from the unknown truth.
• Wrong: “There is a 95% probability that the parameter lies in the calculated 95% interval.” Correction: In the frequentist interpretation, the procedure has approximately 95% coverage over repeated samples under its assumptions.
• Wrong: “A 25% RSE means 25% interindividual variability.” Correction: RSE measures uncertainty in an estimate; the variability quantity itself must be identified and interpreted on its stated scale.
• Wrong: “A condition number of 100 means every parameter has 100% uncertainty.” Correction: A condition number describes matrix sensitivity and depends on scaling; it is not a parameter-specific uncertainty percentage.