Two Summaries, Different Questions
An estimation summary may report both relative standard errors (RSEs) and a condition number. They can appear to describe the same issue because both concern precision, but they answer different questions. An RSE is tied to one parameter estimate and its standard error. A condition number summarizes how unevenly information is distributed across directions in the chosen parameter coordinates. Neither number, by itself, describes all aspects of uncertainty.
This distinction matters in population pharmacokinetics, where model parameters can have different units and scales, and where estimates may be correlated. A small RSE for each parameter does not rule out strong joint dependence. A large condition number does not, on its own, tell which parameter has a large RSE or whether absolute uncertainty is large. Interpreting the two requires knowing which matrix was used, how parameters were scaled, and how the parameter-specific standard errors were obtained.
The condition number is global: it has no parameter index. The RSE is parameter-specific: it compares uncertainty in one estimate to that estimate’s magnitude. If an estimated covariance matrix \(C\) is used, the standard error for parameter \(j\) is \(SE(\hat{\theta}_j)=\sqrt{C_{jj}}\). Under a local quadratic approximation to the likelihood, \(C\) is often approximated by \(I^{-1}\). That approximation and the definition of \(I\) need to be stated before interpreting the resulting summaries.
What the Condition Number Does—and Does Not—Measure
For a positive-definite information matrix, all eigenvalues are positive, so \(\kappa_2(I)\ge 1\). A value of 1 means the eigenvalues are equal: information is equally strong in all directions under the coordinates used. A larger value means the least-informed direction has much less information than the most-informed direction. It is a relative comparison between directions, not a direct measure of the overall amount of information.
When \(C=I^{-1}\), the largest and smallest eigenvalues of \(C\) are the reciprocals of the smallest and largest eigenvalues of \(I\). Therefore, \(\kappa_2(C)=\kappa_2(I)\). This equality does not make the condition number interchangeable with the diagonal standard errors: eigenvalues describe joint directions, while diagonal entries of \(C\) determine marginal standard errors for the named parameters.
Assume the likelihood is adequately approximated near the estimate by a quadratic form and that the information matrix \(I\) is symmetric and positive definite. Then the approximate covariance is \(C=I^{-1}\).
If \(I\) has eigenvalues \(\lambda_1,\ldots,\lambda_p>0\), then \(C\) has eigenvalues \(1/\lambda_1,\ldots,1/\lambda_p\). Thus the largest-to-smallest eigenvalue ratio is the same for \(I\) and \(C\).
The condition number uses the extreme eigenvalues of the whole matrix. The RSE for parameter \(j\) uses only \(\sqrt{C_{jj}}\) and \(|\hat{\theta}_j|\). A matrix’s eigenvalues and its diagonal entries encode different features of its uncertainty.
These statements rely on a usable local covariance approximation. If the information matrix is singular, the condition number is not finite in the positive-definite sense and \(I^{-1}\) does not exist. If the matrix is nearly singular, the inverse and resulting standard errors may be sensitive to small changes. If estimates are at boundaries, the likelihood is strongly non-quadratic, or the uncertainty approximation is otherwise poor, reported standard errors and RSEs may not faithfully describe the sampling uncertainty. The condition number does not repair those limitations.
Worked Comparisons
Worked Example: Changing Scale Changes the Condition Number
Consider a hypothetical two-parameter fit with estimates \(\hat{\theta}_1=10\) and \(\hat{\theta}_2=2\). Suppose its information matrix and local covariance approximation are
The condition number is \(\kappa_2(I_\theta)=100/1=100\). The standard errors are \(SE(\hat{\theta}_1)=\sqrt{0.01}=0.1\) and \(SE(\hat{\theta}_2)=\sqrt{1}=1\). Thus,
Now change coordinates by defining \(\theta=A\phi\), where \(A=\operatorname{diag}(0.1,1)\). Information transforms to \(I_\phi=A^\mathsf{T}I_\theta A\), giving \(I_\phi=\operatorname{diag}(1,1)\) and \(\kappa_2(I_\phi)=1\). The estimates on the new scale are \(\hat{\phi}_1=100\) and \(\hat{\phi}_2=2\); their standard errors are 1 and 1. Their RSEs remain \(100(1/100)=1.00\%\) and \(100(1/2)=50.0\%\). This coordinate change alters the condition number but not the relative uncertainty of either corresponding estimate. The condition number must therefore be reported with the parameter scaling or coordinates that produced it.
Worked Example: Equal RSEs, Different Joint Uncertainty
Suppose two hypothetical parameters both have estimates of 10 and standard errors of 1. In one fit their covariance matrix is the identity matrix, while in another it has correlation 0.99:
In both fits each diagonal variance is 1, so both parameters have RSE \(100(1/10)=10.0\%\). For \(C_A\), both eigenvalues are 1, and the condition number is 1. For \(C_B\), the eigenvalues are \(1+0.99=1.99\) and \(1-0.99=0.01\). Its condition number, the same for the inverse information matrix, is
The marginal RSEs alone do not reveal the strong correlation in the second fit or the very different uncertainty along joint parameter directions. Conversely, the condition number alone does not identify which parameter has which standard error. The full covariance or correlation information answers a different question from either summary.
Worked Example: The Same Condition Number, Different RSEs
Consider two hypothetical information matrices, with the same estimates \(\hat{\theta}_1=10\) and \(\hat{\theta}_2=2\):
Both have condition number 100: \(100/1=100\) for \(I_A\), and \(1/0.01=100\) for \(I_B\). But their approximate covariance matrices are \(C_A=\operatorname{diag}(0.01,1)\) and \(C_B=\operatorname{diag}(1,100)\). For \(I_A\), the standard errors are 0.1 and 1, giving RSEs of 1.00% and 50.0%. For \(I_B\), they are 1 and 10, giving
The condition numbers match because each matrix has the same ratio of directional information. The second matrix has less information overall, so its standard errors are larger. A condition number is not an absolute precision scale.
Assumptions, Checks, and Comparisons
Before comparing either summary across fits, verify that they refer to comparable quantities. An RSE requires an estimate that is not zero and a standard error on the same parameter scale. For a parameter that can be negative, the absolute value in the denominator makes the RSE nonnegative; an estimate close to zero can make the ratio extremely large or uninformative. For parameters constrained to be positive, interpret the percentage relative to the positive estimate and retain the parameter’s units when reporting the estimate and standard error.
A condition number requires a specified matrix and coordinates. State whether it is calculated from observed or expected information, or from an estimated covariance matrix, and whether the matrix has been scaled or normalized. Do not compare condition numbers calculated from differently scaled parameterizations as though they were on a common absolute scale. Check that the matrix is symmetric and positive definite for the eigenvalue-ratio definition used here. Nonpositive eigenvalues or a failed inverse mean that the stated positive-definite covariance interpretation is unavailable.
In practice, read the summaries together with the parameter estimates and the covariance or correlation matrix. Ask whether the matrix approximation is credible near the estimates; whether any RSE is large or inflated by an estimate near zero; whether strong covariance changes uncertainty in combinations of parameters; and whether the condition number changes substantially under a reasonable change of units. These checks help distinguish what each summary establishes from what it leaves unanswered.
| Summary | What it describes | What it does not establish by itself |
|---|---|---|
| Parameter-specific RSE | Standard error relative to the magnitude of one estimate | Joint dependence among parameters or the overall shape of uncertainty |
| Condition number | Ratio of information in the strongest and weakest matrix directions, in the chosen coordinates | Absolute precision, the RSE of a named parameter, or which direction is problematic |
| Covariance or correlation matrix | Marginal variances and pairwise dependence in the stated parameterization | Whether the local approximation is adequate or the model is scientifically appropriate |
The main alternative to relying on these two scalar summaries is to examine the covariance structure directly and, where needed, assess uncertainty using methods that do not rely solely on a local quadratic approximation. Such approaches can add information, but they do not make the condition number or RSE meaningless: the summaries remain compact diagnostics when their assumptions and scales are clear. Their limitation is compression. Many distinct covariance structures can share a condition number, and RSEs discard off-diagonal covariance.
Common Mistake
Wrong reasoning: “The condition number is high, so every parameter must have a high RSE,” or “all RSEs are small, so there cannot be a poorly informed parameter combination.” Neither conclusion follows. The first treats a global eigenvalue ratio as a parameter-wise error measure; the second ignores covariance and joint directions. The worked examples show both failures: equal RSEs can accompany very different condition numbers, and identical condition numbers can accompany very different RSEs.
Correction: Use RSEs to describe uncertainty relative to individual estimates, and use the condition number to describe relative information across directions in a specified parameterization. Inspect the covariance structure and the estimation context before drawing a broader conclusion about precision.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A parameter estimate is 8.0 units with standard error 1.2 units. Calculate its RSE.
- An information matrix has eigenvalues 12 and 3. What is its condition number, and what does that number compare?
- Two fits have identical parameter estimates and diagonal covariance entries, but different off-diagonal entries. Can their RSEs match while their condition numbers differ?
- Why does multiplying every entry of a positive-definite information matrix by the same positive constant leave its condition number unchanged, while changing its absolute information?
Solutions. (1) \(RSE=100(1.2/8.0)=15.0\%\). Both quantities are in the same units, so their ratio is unitless. (2) \(\kappa_2=12/3=4\); it compares the largest and smallest eigenvalues, hence the most- and least-informed directions in the specified coordinates. (3) Yes. The diagonal entries determine marginal standard errors and thus RSEs for fixed estimates, while off-diagonal entries can change the eigenvalues and condition number. (4) If \(I\) is multiplied by \(c>0\), both extreme eigenvalues are multiplied by \(c\), so \((c\lambda_{\max})/(c\lambda_{\min})=\kappa_2(I)\). The inverse covariance becomes \(I^{-1}/c\), so its variances change even though the ratio does not.