Why Parameter Coordinates Matter
A population pharmacokinetic model can describe the same predictions using different parameter coordinates. For example, a regression intercept may refer to a covariate value of zero or to its mean; time may be measured in hours or in minutes; and a one-compartment concentration curve may be expressed using clearance and volume or using an initial concentration and an elimination rate constant. These choices do not necessarily change the model’s scientific content, but they can change how difficult it is for an optimizer to calculate parameter estimates.
The central distinction is between numerical conditioning and information from the data. A coordinate change can make calculations more stable or efficient. If it is one-to-one and the likelihood is transformed correctly, it does not create new observations or distinguish parameter values that the original model could not distinguish.
In parameter estimation, a condition number often describes the local curvature or information matrix. It depends on the units and coordinates used for the parameters. Earlier material on condition numbers, parameter scaling, and weakly estimated parameter combinations provides the background for interpreting this diagnostic. Here the emphasis is how an intentional coordinate change affects computation and what that change does—and does not—mean.
What a Reparameterization Preserves
Let \(\theta\) denote the original parameter vector and let \(\phi\) denote a new vector related by a smooth, one-to-one transformation \(\theta=g(\phi)\). The likelihood in the new coordinates is the original likelihood evaluated at the corresponding parameter value:
Every candidate value in one coordinate system corresponds to exactly one candidate value in the other. Consequently, the maximum likelihood fit has the same likelihood and gives the same model predictions after translating the estimate back to the original parameters. The likelihood has not gained data or become more informative just because its coordinates have changed.
The information matrix, however, does not generally keep the same numerical entries or condition number. Write \(J=\partial\theta/\partial\phi\) for the Jacobian matrix of the transformation. Under the usual regularity conditions for Fisher information, the information in the new coordinates is
This is a change of coordinates, not an equality of matrices. A nonsingular \(J\) preserves rank, so it cannot turn a truly unidentified direction into an identified one. But it can change eigenvalues and their ratio, sometimes substantially. For the observed negative Hessian (the local curvature of the negative log-likelihood), the same congruence relation holds at a stationary point; away from a stationary point, additional terms involving the gradient and second derivatives of the transformation may appear.
A useful justification follows directly from the score. By the chain rule, the score with respect to \(\phi\) is \(J^T\) times the score with respect to \(\theta\). Taking the expected outer product of these scores gives \(J^T I_{\theta}J\), provided the standard regularity conditions make that expected outer product equal to the Fisher information. This derivation shows why a nonsingular coordinate change preserves rank but need not preserve conditioning.
For a linear transformation, the likelihood surface is represented by a linear change of axes. For a nonlinear transformation, the new axes can also vary across parameter space. In either case, a better-conditioned numerical representation can help optimization, but it should not be mistaken for evidence that the observations constrain the underlying scientific quantities more strongly.
Centering a Covariate
Consider the linear predictor \(\eta=\beta_0+\beta_1x\). Define a centered covariate \(z=x-c\), where \(c\) is a fixed reference value such as the sample mean. Then
This transformation is one-to-one: \(\beta_0=\gamma_0-c\gamma_1\) and \(\beta_1=\gamma_1\). Fitted values are unchanged. The intercept now describes the predictor at \(x=c\), which may also be more meaningful than the predictor at \(x=0\).
Worked Example: Centering a Covariate With a Large Baseline
Suppose a hypothetical fixed-effects model has an intercept and a slope for a covariate observed at \(x=100,101,102,103\). Compare the curvature matrices for the original and centered design. For this calculation, assume independent Gaussian residuals with mean zero and common variance \(\sigma^2\), with the covariate values treated as fixed.
Under these assumptions, the log-likelihood for the regression coefficients has curvature matrix \(X^TX/\sigma^2\). The design matrix has columns for the intercept and \(x\), so
Its trace is \(41218\), and its determinant is \(4(41214)-406^2=20\). The eigenvalues are approximately \(41217.9995\) and \(0.0004852\), giving \(\kappa_2(X^TX)\approx 8.495\times10^7\). Multiplication by the common factor \(1/\sigma^2\) does not change this condition number.
Now take \(c=\bar{x}=101.5\). The centered values are \(-1.5,-0.5,0.5,1.5\); they sum to zero and their squared values sum to \(5\). Thus,
The calculation demonstrates a large improvement in conditioning for these coordinates. The model has not acquired observations: the centered intercept is \(\gamma_0=\beta_0+101.5\beta_1\), and the slope is unchanged. The same fitted line can be translated exactly back to the original intercept and slope.
Rescaling Parameters and Predictors
Centering removes a large offset; rescaling addresses differences in magnitude. Suppose \(z\) is replaced by \(r=az\), where \(a\ne0\). In \(\eta=\alpha+bz\), write \(\eta=\alpha+cr\), where \(c=b/a\). This is an exact coordinate change. If a column of a design matrix is multiplied by a large factor, its contribution to the matrix’s eigenvalue scale can become much larger than that of other columns. Scaling that column to a comparable numerical range can improve the matrix condition number.
Worked Example: Changing the Units of a Predictor
For a hypothetical linear predictor, suppose three centered covariate values are \(z=-1,0,1\), with an intercept column. The design cross-product is
Now record the same covariate as \(r=1000z\). The transformed design cross-product becomes
The large condition number is a consequence of the changed numerical scale, not a change in the three observations. If the original predictor coefficient is \(b\), then the coefficient for \(r\) is \(c=b/1000\), because \(bz=(b/1000)r\). Translating \(c\) back recovers the original predictor effect exactly.
The same principle applies when an optimizer works with parameters of very different magnitudes. Optimizing a scaled parameter, such as \(\phi=\theta/s\) for a nonzero scale \(s\), can make typical steps more balanced. Positive parameters are often represented on a log scale to enforce positivity and give the optimizer a different geometry. These choices can help computation, but the parameters and their units must be translated back before interpreting results.
A PK Example of Changing Parameterization
For a one-compartment intravenous bolus model with dose \(D>0\), clearance \(CL>0\), and volume \(V>0\), the concentration is
Define \(A=D/V\) and \(k=CL/V\). Then \(C(t)=A\exp(-kt)\). With \(D\) fixed, this change is one-to-one for positive parameters: \(V=D/A\) and \(CL=kD/A\). It changes the coordinates used to fit the model, but every predicted concentration remains the same when the parameters are converted correctly.
Worked Example: Expressing the Same Concentration Curve Two Ways
Take hypothetical values \(D=100\) mg, \(V=20\) L, and \(CL=4\) L/h. The original parameters imply \(A=100/20=5\) mg/L and \(k=4/20=0.2\) h\(^{-1}\). At \(t=5\) h, the concentration is
Using the original parameters gives the same result: \(C(5)=(100/20)\exp[-(4/20)5]\approx1.8397\) mg/L. Conversely, \(V=D/A=100/5=20\) L and \(CL=kD/A=0.2(100)/5=4\) L/h. This transformation has not altered the concentration data or predictions. Depending on the model and optimizer, \((A,k)\) may be more convenient to estimate, while \((V,CL)\) may be more directly useful for interpretation.
A nonlinear transformation can still change local conditioning. For example, log coordinates for \(A\) and \(k\) enforce positivity; the Jacobian then rescales and combines local directions in the curvature matrix. Whether this helps depends on the likelihood, the data, and the numerical algorithm. A parameterization should therefore be assessed by checking convergence and translated estimates, not by assuming that any particular set of parameters is always better conditioned.
Diagnostics, Alternatives, and Limitations
A coordinate change is worth considering when optimization is unstable, takes unusually small or large steps, or produces numerical warnings. Compare estimates and predictions across parameterizations after translating them to common units. Check that the same objective value is attained, that the optimizer reports successful convergence, and that small changes in numerical settings do not cause large unexplained changes in the result. A dramatic change in a condition number can diagnose a computational representation problem, but it does not by itself establish that the data identify every parameter well.
Centering and scaling are computational alternatives to leaving predictors in their original units; they usually preserve the fitted model when the coefficient transformation is one-to-one. A substantive alternative is to collect more informative data, such as observations in a time range that distinguishes competing model behaviors. Unlike a coordinate change, new data can add information. Constraining parameters or adding a penalty can also stabilize estimation, but these alter the estimation problem and may affect the resulting estimates; they are not merely changes of coordinates.
There are important limits. First, a non-one-to-one transformation can discard distinctions between original parameter values. Second, constraints or model changes introduced alongside a transformation may mean the fitted model is no longer equivalent. Third, when a prior distribution is used, it must be transformed with the appropriate density Jacobian to represent the same prior beliefs in new coordinates; simply reusing the same numerical prior form can change the analysis. Finally, better conditioning does not guarantee successful optimization or resolve a data limitation.
- Wrong reasoning: “Centering reduced the condition number, so the data now identify the slope more strongly.”
- Correction: Centering changes the parameter coordinates and can improve numerical calculations. If the transformation is one-to-one and the model is otherwise unchanged, it does not add information to the data.
- Wrong reasoning: “The likelihood is unchanged, so the information matrix must have the same eigenvalues.”
- Correction: Information transforms by a Jacobian congruence, \(I_{\phi}=J^TI_{\theta}J\). Rank is preserved for nonsingular \(J\), but eigenvalues and condition numbers can change.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- If an invertible linear reparameterization reduces a condition number, what has changed, and what has not necessarily changed?
- For \(z=x-10\), rewrite \(\eta=\beta_0+\beta_1x\) using \(z\), and give the inverse coefficient transformation.
- A predictor is multiplied by 100 while its coefficient is divided by 100. Are fitted values changed?
- Why does a nonsingular reparameterization preserve the rank of the information matrix but not necessarily its condition number?
Solutions. (1) The coordinates and numerical conditioning have changed; the observations and, for an equivalent one-to-one model, the likelihood’s information about the underlying model have not necessarily changed. (2) Since \(x=z+10\), \(\eta=(\beta_0+10\beta_1)+\beta_1z\). Thus \(\gamma_0=\beta_0+10\beta_1\), \(\gamma_1=\beta_1\), with \(\beta_0=\gamma_0-10\gamma_1\) and \(\beta_1=\gamma_1\). (3) No. If \(r=100x\), then \(\beta_1x=(\beta_1/100)r\), so the linear predictor is identical. (4) The transformed information is \(J^TIJ\). Since \(J\) is nonsingular, this multiplication preserves rank; its effect on eigenvalue magnitudes and ratios depends on \(J\), so the condition number can change.