Parameter Scaling In Alphabetic Criteria

Why Parameter Units Matter

A Fisher information matrix is expressed in relation to the parameters it informs. If one parameter is measured in milligrams and then expressed in micrograms, its numerical information changes, even though the experiment and its underlying information have not changed. Alphabetic criteria summarize a matrix using determinants, traces, or eigenvalues; those summaries do not all respond to a change of parameter units in the same way.

This distinction is important when comparing designs. A determinant-based comparison has a useful invariance property under a fixed, nonsingular change of coordinates. In contrast, an inverse-information trace or a smallest eigenvalue can depend strongly on the chosen units. Standardizing parameters provides one principled way to make those criteria interpretable, but the scales used for standardization are substantive choices, not cosmetic adjustments.

Information Under a Change of Parameter Scale

Let \(\theta\) be a \(p\)-dimensional parameter vector, and let \(I_\theta(\xi)\) denote the Fisher information matrix for design \(\xi\), evaluated at the parameter value used for the information calculation. Consider an affine reparameterization \(\theta=a+C\eta\), where \(a\) is a fixed vector and \(C\) is a fixed, nonsingular matrix. The new coordinate vector is \(\eta\). For unit changes, \(C\) is commonly a diagonal matrix of scale factors.

The information matrix is defined through derivatives of the log-likelihood (or the corresponding expected information). By the chain rule, the score with respect to \(\eta\) is \(C^T\) times the score with respect to \(\theta\). Consequently, the information in the new coordinates is a congruence transformation:

I_\eta(\xi)=C^T I_\theta(\xi)C.

This relation assumes the same statistical model and design are being expressed in two coordinate systems, with \(C\) nonsingular. In a local information calculation for a nonlinear model, the matrix is evaluated at corresponding parameter values. If the change of coordinates is nonlinear, its Jacobian at that value replaces \(C\); the simple constant-scale interpretation then applies locally.

Definition: Standardized parameters are dimensionless coordinates formed by choosing a reference value \(s_j>0\) for each parameter and writing \(\theta_j=a_j+s_j\eta_j\). In matrix form, \(C=\operatorname{diag}(s_1,\ldots,s_p)\). A unit change in \(\eta_j\) then represents a change of \(s_j\) units in \(\theta_j\).

How the Main Matrix Summaries Transform

The determinant follows directly from the determinant of a product. For nonsingular \(C\),

\det I_\eta =\det(C^T)\det(I_\theta)\det(C) =(\det C)^2\det I_\theta.

Thus the determinant is multiplied by a factor depending on the coordinate scale, but not on the design, provided the same \(C\) is used for every design being compared. If the matrices are positive definite, the corresponding log-determinants differ by the additive constant \(2\log|\det C|\). The D-efficiency ratio between two designs is therefore unchanged under this common reparameterization. If a matrix is singular, its determinant remains defined and equals zero; its log-determinant is not finite.

The inverse transforms as \(I_\eta^{-1}=C^{-1}I_\theta^{-1}C^{-T}\), when the information matrix is positive definite. Therefore,

\operatorname{tr}(I_\eta^{-1}) =\operatorname{tr}\!\left(C^{-1}I_\theta^{-1}C^{-T}\right).

This is generally not the same as \(\operatorname{tr}(I_\theta^{-1})\). Since A-optimality minimizes the trace of the inverse information matrix, a change of units can change both the numerical value and the ranking of designs. The trace of the information matrix itself also generally changes: \(\operatorname{tr}(C^T I_\theta C)\) is not usually a fixed multiple of \(\operatorname{tr}(I_\theta)\).

Eigenvalues are also coordinate-dependent under a general nonsingular change of scale. A congruence transformation preserves rank and positive definiteness, but it does not generally preserve eigenvalues. The E-criterion, which maximizes the smallest eigenvalue of the information matrix, can consequently yield a different ranking after rescaling. A singular symmetric information matrix still has defined, finite eigenvalues. In particular, if it is positive semidefinite and singular, its smallest eigenvalue is zero. Positive definiteness is needed for a finite A-criterion, not for eigenvalues to exist.

Worked Example: D-Comparison After Converting Units

Suppose two hypothetical designs have positive definite information matrices, in the original parameter coordinates, with determinants \(\det I_\theta(\xi_A)=12\) and \(\det I_\theta(\xi_B)=8\). Suppose the first parameter is converted to a coordinate that is \(1000\) times smaller per unit, so that \(\theta=C\eta\) with \(C=\operatorname{diag}(1000,1)\).

The determinant multiplier is \((\det C)^2=1000^2=1{,}000{,}000\). Hence \(\det I_\eta(\xi_A)=12{,}000{,}000\) and \(\det I_\eta(\xi_B)=8{,}000{,}000\). Both determinants change greatly in magnitude, but their ratio does not:

\frac{\det I_\eta(\xi_A)}{\det I_\eta(\xi_B)} =\frac{12{,}000{,}000}{8{,}000{,}000} =\frac{12}{8}=1.5.

For two parameters, the D-efficiency of design \(B\) relative to design \(A\) is \(\left\{\det I(\xi_B)/\det I(\xi_A)\right\}^{1/2}\). Substitution gives \(\sqrt{8/12}=0.8165\), or about 81.65%. The same value results in the converted units. The unit conversion changes the reported determinants, not the relative D-comparison.

Standardization and Criteria That Depend on Scale

Standardization gives a transparent interpretation to scale-sensitive criteria. Choose \(s_j\) to represent a scientifically meaningful change in parameter \(j\), such as a reference magnitude, a plausible range, or a clinically relevant difference. The dimensionless \(\eta_j\) then measures changes in units of that reference scale. A criterion applied to \(I_\eta\) assesses information about those standardized changes.

This does not make A- or E-optimality universally invariant. Instead, it makes the weighting choice explicit. For example, minimizing \(\operatorname{tr}(I_\eta^{-1})\) is equivalent to minimizing a weighted trace in the original coordinates:

\operatorname{tr}(I_\eta^{-1}) =\operatorname{tr}\!\left(C^{-T}C^{-1}I_\theta^{-1}\right).

For diagonal \(C\), the \(j\)th variance contribution is weighted by \(1/s_j^2\). The criterion therefore summarizes uncertainty in units of the chosen reference changes, rather than treating a one-unit change in every original parameter as equally meaningful. Similarly, the smallest eigenvalue of \(I_\eta\) describes the least-informed direction among standardized parameter combinations.

Worked Example: A- and E-Rankings Can Reverse

Consider two hypothetical designs with diagonal information matrices \(I_\theta(\xi_A)=\operatorname{diag}(100,1)\) and \(I_\theta(\xi_B)=\operatorname{diag}(25,4)\). Both are positive definite, so their A-criteria are finite. In the original coordinates, the A-criterion values are

\operatorname{tr}(I_\theta(\xi_A)^{-1}) =\frac{1}{100}+1=1.01,\qquad \operatorname{tr}(I_\theta(\xi_B)^{-1}) =\frac{1}{25}+\frac{1}{4}=0.29.

The smaller value favors design \(B\). The smallest information eigenvalues are \(1\) for \(A\) and \(4\) for \(B\), so the E-criterion also favors \(B\) in these coordinates.

Now use \(\theta=C\eta\), where \(C=\operatorname{diag}(0.1,1)\). The transformed information matrices are \(I_\eta(\xi_A)=\operatorname{diag}(1,1)\) and \(I_\eta(\xi_B)=\operatorname{diag}(0.25,4)\). Their A-criterion values are \(2\) and \(4+0.25=4.25\), respectively, so A-optimality now favors \(A\). Their smallest eigenvalues are \(1\) and \(0.25\), so E-optimality also favors \(A\). The design information itself has not changed; the new coordinates define a different relative scale for the two parameter directions. This reversal is why scale-sensitive comparisons must state their parameter units or standardization.

Worked Example: Interpreting Standardized Information

Suppose a hypothetical two-parameter model has information matrix \(I_\theta=\begin{pmatrix}16&4\\4&4\end{pmatrix}\), and choose reference scales \(s_1=0.25\) and \(s_2=0.5\). Thus \(C=\operatorname{diag}(0.25,0.5)\). The standardized information is

I_\eta=C^T I_\theta C =\begin{pmatrix} 0.25(16)(0.25)&0.25(4)(0.5)\\ 0.5(4)(0.25)&0.5(4)(0.5) \end{pmatrix} =\begin{pmatrix}1&0.5\\0.5&1\end{pmatrix}.

Its determinant is \(1(1)-(0.5)^2=0.75\). Its eigenvalues solve \((1-\lambda)^2-(0.5)^2=0\), giving \(\lambda=1.5\) and \(0.5\). Therefore the E-criterion value is \(0.5\) on this standardized scale. The inverse is \(\frac{1}{0.75}\begin{pmatrix}1&-0.5\\-0.5&1\end{pmatrix}\), so the A-criterion is \(\operatorname{tr}(I_\eta^{-1})=2/0.75=2.667\).

These numbers summarize information for changes of \(0.25\) units in the first original parameter and \(0.5\) units in the second. Different reference scales would produce different A- and E-values. They would not, by themselves, establish that one choice of scales is scientifically preferable.

Choosing and Checking a Scale

For interpretable comparisons, report the parameterization and the scale matrix \(C\), or equivalently the reference scale for each parameter. Choose scales from the question being asked, not to obtain a preferred ranking. If the aim is precision relative to clinically meaningful changes, use those changes as the reference scales. If no defensible scale is available, report scale-sensitive criteria alongside the parameter units and avoid treating their raw magnitudes as universal.

A practical check is to recompute the criterion after a simple unit conversion. A D-efficiency ratio between the same designs should remain unchanged, up to numerical precision, when the same nonsingular coordinate transformation is applied to both. If an A- or E-ranking changes, that can be a genuine consequence of the changed weighting, rather than a computational error. Conversely, a D-ranking that changes under a fixed reparameterization may indicate that different transformations were applied to the designs, that the calculation is inconsistent, or that numerical conditioning has affected the determinant calculation.

Before using the criteria, also check the matrix properties relevant to each one. Information matrices from regular models are symmetric positive semidefinite. A finite A-criterion requires a positive definite matrix, so its inverse exists and has finite entries. D-criteria based on a positive determinant likewise require full rank for a positive determinant. The E-criterion can be evaluated for a singular symmetric information matrix; for a singular positive-semidefinite matrix its smallest eigenvalue is zero, indicating at least one locally uninformed direction. A zero E-value should not be mistaken for an undefined eigenvalue.

Common Pitfalls and Limitations

Comparing raw A-values across incompatible units. The inverse-information trace has squared parameter units and changes with scaling. State the units or compute it on explicitly standardized parameters.
Assuming all criteria are invariant because D is. A fixed reparameterization adds a common log-determinant constant, so D-rankings are preserved. The same conclusion does not hold for A- or E-criteria.
Calling a singular matrix's eigenvalues undefined. A singular symmetric matrix has defined eigenvalues. For positive-semidefinite information, singularity makes the smallest eigenvalue zero; it prevents a finite inverse-based A-criterion.
Choosing scales to favor a design. Since standardization changes the relative weighting of parameter directions, arbitrary scales can make a ranking difficult to defend. Justify scales using the scientific estimand, plausible parameter changes, or a clear reference convention.
Treating local information as a complete uncertainty assessment. These criteria summarize a local information matrix under its model and evaluation assumptions. They do not remove model misspecification, weak identification, or dependence of nonlinear-model information on the parameter value used.

The main alternative is to retain the original parameterization and report the criteria with units and interpretation, rather than standardizing. That approach can be clearest when parameters already share a meaningful scale. Standardization is useful when parameters have very different units, but its conclusions are conditional on the stated reference scales. D-optimality offers a coordinate-invariant ranking under a common nonsingular linear transformation, while A- and E-criteria answer scale-dependent questions about average and weakest-direction information.

Key takeaway: Under \(\theta=a+C\eta\), information becomes \(I_\eta=C^T I_\theta C\). Determinants acquire a common factor \((\det C)^2\), preserving D-rankings across designs under the same transformation; inverse traces and eigenvalues generally change. Standardized parameters make those scale choices explicit.

Check Your Understanding

Use the transformations and assumptions stated above. Solutions follow each question so that the calculations can be checked directly.

For a two-parameter information matrix, let \(C=\operatorname{diag}(2,3)\). By what factor does the determinant change? Solution: The multiplier is \((\det C)^2=(2\cdot3)^2=36\). Thus \(\det I_\eta=36\det I_\theta\).
Two designs have determinants 18 and 8 under the same parameterization. What is the D-efficiency of the second relative to the first for two parameters, and would a common nonsingular reparameterization change it? Solution: The relative efficiency is \(\sqrt{8/18}=\sqrt{4/9}=2/3=0.6667\), or 66.67%. A common nonsingular transformation multiplies both determinants by the same factor, so this ratio is unchanged.
Can the E-criterion be evaluated for a singular positive-semidefinite information matrix? What is its smallest eigenvalue? Solution: Yes. A symmetric matrix, including a singular one, has defined eigenvalues. Positive semidefiniteness makes them nonnegative, and singularity means at least one is zero; therefore the smallest eigenvalue is zero.
Why does a change of parameter units generally change the A-criterion? Solution: When the information is positive definite, \(I_\eta^{-1}=C^{-1}I_\theta^{-1}C^{-T}\). Taking the trace of this transformed matrix generally does not equal \(\operatorname{tr}(I_\theta^{-1})\). For diagonal scaling, the variance contributions receive scale-dependent weights.
A colleague reports that an A-criterion is infinite because an information matrix is singular, but says its smallest eigenvalue is undefined. Which part is correct? Solution: The inverse-based A-criterion is not finite because a singular matrix has no inverse. The smallest eigenvalue is still defined; for a singular positive-semidefinite information matrix it equals zero.