Why Parameter Units Matter
A condition number can flag numerical sensitivity in an information matrix, but its value is affected by how parameter coordinates are expressed. This matters in population pharmacokinetics because clearance and volume may have different units and very different numerical magnitudes. A large condition number calculated from an unscaled matrix can therefore reflect the choice of units rather than poor information about the parameters.
The key distinction is between numerical scaling and statistical information. Changing a parameter from liters to milliliters changes its numerical value and the corresponding entries of the information matrix. It does not, by itself, change the observations or add information to the model. Scaling can make numerical calculations better behaved, but the resulting condition number must be interpreted in light of the scales used.
How Scaling Transforms Information
Suppose the parameter vector is \(\theta=(CL,V)^\mathsf{T}\), where \(CL\) is clearance and \(V\) is volume. Choose positive reference scales \(s_{CL}\) and \(s_V\), expressed in the same units as their respective parameters, and define dimensionless coordinates \(z\) by \(\theta=Sz\), where \(S=\operatorname{diag}(s_{CL},s_V)\). For example, \(s_{CL}=5\ \mathrm{L/h}\) and \(s_V=50\ \mathrm{L}\) make \(z=(1,1)^\mathsf{T}\) at those parameter values.
For a linear change of coordinates, the information matrix in the new coordinates is obtained by applying the chain rule to the likelihood curvature. This is an exact transformation for linear scaling; it does not change the fitted likelihood or the predictions associated with a given physical parameter vector.
Because \(S\) is diagonal, the \(j,k\) entry becomes \(s_j s_k(I_\theta)_{jk}\). Thus its numerical value depends on the chosen units and scales. The inverse matrix transforms consistently: if \(I_\theta\) is invertible, then \(I_z^{-1}=S^{-1}I_\theta^{-1}S^{-\mathsf{T}}\). The approximate covariance represented in scaled coordinates is therefore expressed on the scales of \(z\), not in the original units of clearance and volume.
The scaling factors should be chosen before interpreting the resulting condition number. Useful choices include typical parameter magnitudes or meaningful changes for the application. A condition number after scaling is still conditional on that choice; it is not a scale-free certificate that a model is well informed.
Worked Example: Equal Relative Precision, Unequal Raw Scales
Consider hypothetical estimates \(CL=5\ \mathrm{L/h}\) and \(V=50\ \mathrm{L}\). Suppose their estimated standard errors are \(0.5\ \mathrm{L/h}\) and \(5\ \mathrm{L}\), respectively, and their estimated covariance is zero. The information matrix is the inverse of this diagonal covariance matrix.
The variances are \(0.5^2=0.25\ (\mathrm{L/h})^2\) and \(5^2=25\ \mathrm{L}^2\). Consequently,
Its eigenvalues are 4 and 0.04, so \(\kappa_2(I_\theta)=4/0.04=100\). Yet each estimate has a standard error equal to 10% of its estimate: \(0.5/5=0.10\) and \(5/50=0.10\). The raw condition number is large because the parameters and their information entries are on different numerical scales, not because one estimate has a larger relative standard error.
Now take \(S=\operatorname{diag}(5,50)\), using the estimates as reference scales. Then
The scaled condition number is 1. The rescaling has made the curvature comparable on the chosen reference scales; it has not reduced the standard errors or increased the information in the hypothetical data.
What a Condition Number Can and Cannot Tell You
For a positive definite matrix, a large condition number means that numerical perturbations can have a substantial effect on calculations involving that matrix. One justification comes from solving a linear system \(Ax=b\), as often occurs in calculations using an information matrix. Suppose the matrix is perturbed by \(E\) and the solution changes from \(x\) to \(x+\delta x\). Subtracting \(Ax=b\) from \((A+E)(x+\delta x)=b\) gives \(A\delta x=-E(x+\delta x)\). Hence
Let \(q=\|A^{-1}\|\|E\|\). If \(q<1\), rearranging the inequality gives \(\|\delta x\|/\|x\|\leq q/(1-q)\). Since \(q\leq\kappa_2(A)\|E\|/\|A\|\), a matrix with a large condition number can be more sensitive to a given relative perturbation. This is a statement about numerical sensitivity of the linear system, not a direct statement about the probability that an estimate is correct or about its absolute uncertainty.
When the parameters have different units, the raw matrix combines entries with different unit combinations. Its eigenvalue ratio can be dominated by that mismatch. Scaling to dimensionless coordinates makes comparisons more interpretable, but a large condition number can remain after sensible scaling. That remaining value suggests uneven curvature on the chosen scales; it does not identify, by itself, which parameter combination is responsible. The condition number of the information matrix is not a substitute for checking estimates, uncertainty summaries, correlations, or model diagnostics.
Worked Example: Separating Scale Imbalance From Remaining Dependence
For a hypothetical clearance and volume model, let the dimensionless information matrix on reference scales \(5\ \mathrm{L/h}\) and \(50\ \mathrm{L}\) be
Its eigenvalues are \(100+80=180\) and \(100-80=20\), so \(\kappa_2(I_z)=180/20=9\). This matrix is positive definite because both eigenvalues are positive. The condition number is not 1 after scaling: the two scaled directions do not have equal curvature.
To see the effect of expressing the parameters on the original scales, use \(I_\theta=S^{-1}I_zS^{-1}\), where \(S=\operatorname{diag}(5,50)\). The entries are
For a symmetric two-by-two matrix, the eigenvalues are \((\operatorname{tr}(I_\theta)\pm\sqrt{\operatorname{tr}(I_\theta)^2-4\det(I_\theta)})/2\). Here the trace is \(4.04\), the determinant is \(4(0.04)-0.32^2=0.0576\), and the eigenvalues are approximately \(4.0257\) and \(0.01431\). Thus the raw condition number is about \(4.0257/0.01431=281.3\), compared with 9 on the stated reference scales.
The difference shows that units and scales strongly affect the numerical condition number. The value 9 after scaling also shows that rescaling has not made the information equally strong in every direction. It would be an error to conclude that the raw value 281.3 alone demonstrates poor statistical information, or that the scaled value 9 proves that the estimates are precise.
Unit Changes and Practical Diagnostics
A unit conversion can change a condition number dramatically while leaving the scientific problem unchanged. For example, the numerical value of clearance in \(\mathrm{mL/min}\) is \(1000/60\) times its value in \(\mathrm{L/h}\); the numerical value of volume in milliliters is 1000 times its value in liters. These are coordinate changes, not changes in the physical clearance or volume.
Worked Example: A Unit Conversion Changes the Raw Condition Number
Continue the hypothetical independent-estimate case with \(I_\theta=\operatorname{diag}(4,0.04)\) when clearance is in \(\mathrm{L/h}\) and volume is in \(\mathrm{L}\). Let the new numerical coordinates be clearance in \(\mathrm{mL/min}\) and volume in \(\mathrm{mL}\). The old numerical coordinates equal \(1/16.6667\) times the new clearance coordinate and \(1/1000\) times the new volume coordinate. Therefore \(I_{\mathrm{new}}=D^\mathsf{T}I_\theta D\), where \(D=\operatorname{diag}(1/16.6667,1/1000)\).
The resulting matrix is approximately \(\operatorname{diag}(4/16.6667^2,\ 0.04/1000^2)=\operatorname{diag}(0.0144,\ 0.00000004)\). Its condition number is \(0.0144/0.00000004=360{,}000\), compared with 100 in the original units. Nothing about the hypothetical estimates’ relative standard errors or the underlying information has changed.
If the reference scales are converted as well, they are \(83.3333\ \mathrm{mL/min}\) for clearance and \(50{,}000\ \mathrm{mL}\) for volume. Scaling by those values again gives diagonal information entries of 100 and 100, and a condition number of 1. This confirms that the raw value was sensitive to units.
In practice, diagnose a large condition number by checking the parameterization and scales used to construct the information matrix. Confirm that the matrix corresponds to the same free parameters and the same fitted model across calculations, and check that it is symmetric and positive definite when an inverse is expected. A singular or nearly singular matrix requires care: rescaling cannot recover information that is absent, and an inverse may be unstable or unavailable.
Useful comparisons include the raw condition number, a condition number after scaling by scientifically reasonable reference values, and the parameter-specific precision summaries used elsewhere in the analysis. If the raw value is large but the scaled value is modest, unit imbalance is a plausible explanation for much of the raw value. If a large value persists across defensible scales, numerical sensitivity remains a concern and may warrant further investigation. Neither finding alone establishes why precision is limited.
Common Pitfalls and Alternatives
- Treating every large raw condition number as weak identification. Different parameter units can inflate the value. Recalculate on stated, meaningful dimensionless scales before drawing a statistical conclusion.
- Choosing scales solely to obtain a small number. Arbitrary scales can make a condition number look favorable without making the model more informative. State and justify the scales used.
- Assuming scaling improves statistical precision. Scaling changes coordinates and numerical behavior; it does not add observations, reduce uncertainty, or repair a misspecified model.
- Using a condition number as the only diagnostic. It does not replace checking parameter estimates, standard errors, correlations, boundary behavior, or whether the fitted model is appropriate.
A correlation matrix or standardized information matrix can help compare relative curvature without the original unit magnitudes, but it does not convey all aspects of absolute precision. Parameter-specific standard errors and relative standard errors answer different questions from a matrix condition number. In difficult cases, numerical scaling is one diagnostic and computational aid; further assessment may require examining the model fit and the uncertainty summaries relevant to the scientific question.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- An information matrix is diagonal with entries 9 and 0.09. What is its condition number? If the parameters are scaled by \(S=\operatorname{diag}(3,10)\), what is the scaled matrix and its condition number?
- Why can changing volume from liters to milliliters change the information-matrix condition number without changing the physical model?
- A raw condition number is very large, but it becomes modest after scaling clearance and volume by defensible reference values. What conclusion is justified, and what conclusion is not?
- After reasonable scaling, the condition number remains large. Does this alone prove that one parameter is poorly estimated? Explain.
Solutions. (1) The raw condition number is \(9/0.09=100\). The scaled matrix is \(SI S=\operatorname{diag}(3,10)\operatorname{diag}(9,0.09)\operatorname{diag}(3,10)=\operatorname{diag}(81,9)\), with condition number \(81/9=9\). (2) The numerical coordinate and the corresponding information entries transform with the unit conversion; the data and physical parameter values do not. (3) The raw value is sensitive to scaling, so unit imbalance plausibly explains part of it. This does not prove that the model is well informed or that estimates are precise. (4) No. It indicates remaining numerical sensitivity or uneven curvature on the chosen scales, but does not by itself identify a poorly estimated parameter or explain the cause. Examine parameter-specific uncertainty and other diagnostics as well.