Why Optimize a Determinant?
An NLME design often has to estimate several population parameters together. A schedule that gives precise estimates of one parameter may leave another imprecise, and the estimates may be correlated. The determinant of the information matrix summarizes joint precision: in a regular large-sample approximation, greater information determinant corresponds to a smaller joint confidence region for the parameter vector.
D-optimality is therefore a criterion for choosing a design based on the volume of that region. It is not a guarantee that every parameter will have a small standard error, nor does it directly optimize the precision of an individual prediction. The parameter vector and the model must be specified before candidate designs can be meaningfully compared.
For NLME models, the information used for this comparison is commonly the expected Fisher information from the likelihood after accounting for the random effects. It is not generally the same as treating all observations as independent conditional measurements with no between-subject variation. The marginal likelihood, the design, and the assumed parameter values determine the information matrix.
From Information to Confidence-Region Volume
The volume interpretation follows from the usual large-sample normal approximation. Suppose the estimator \(\hat{\boldsymbol{\theta}}\) is approximately normal with covariance \(I^{-1}\), and \(I\) is positive definite. An approximate joint confidence region has the ellipsoid form below. Here \(\chi^2_{p,1-\alpha}\) is the \(1-\alpha\) quantile of a chi-squared distribution with \(p\) degrees of freedom.
To see how its volume depends on information, let \(I^{1/2}\) be a symmetric square root of \(I\), and set \(\mathbf{z}=I^{1/2}(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta})\). This maps the ellipsoid to a sphere of radius \(r=\sqrt{\chi^2_{p,1-\alpha}}\). The absolute determinant of the map is \(\sqrt{\det I}\), so volumes in the original parameter coordinates are the sphere's volume divided by that factor.
Here \(V_p\) is the volume of the unit ball in \(p\) dimensions. For a fixed confidence level and parameter dimension, the numerator is constant. Maximizing \(\det I\) therefore minimizes the approximate confidence-region volume. This is a statement about the joint ellipsoid under the stated approximation, not a claim that the parameter has a particular probability of lying in a realized interval or region.
If the information matrix is singular, its determinant is zero and it has a direction with zero first-order Fisher information. The usual full-dimensional normal confidence-region approximation is then unavailable. Singularity alone does not establish that the model parameter is unidentifiable: a parameter can be identifiable even when first-order information vanishes at a particular point.
Worked NLME Comparison: Two Sampling Schedules
Worked Example: D-Optimality for a Random-Intercept PK Model
Consider a hypothetical one-compartment concentration model on the log scale: \(\log C_{ij}=\alpha-k t_{ij}+b_i+\varepsilon_{ij}\), where \(i\) indexes subjects and \(j\) indexes samples. The population parameter vector is \(\boldsymbol{\theta}=(\alpha,k)^\mathsf{T}\), with \(k>0\). The subject random intercept satisfies \(b_i\sim N(0,\tau^2)\), and residual errors satisfy \(\varepsilon_{ij}\sim N(0,\sigma^2)\). Assume independent subjects, independent residual errors, and known variance components \(\tau^2=0.09\) and \(\sigma^2=0.04\). These assumptions make the marginal covariance for a subject's two log-concentration measurements explicit.
Compare two schedules, each with two samples per subject: schedule A samples at \(0.5\) and \(2.0\) hours, while schedule B samples at \(0.5\) and \(2.5\) hours. The sampling times are the only design difference. For either subject, the marginal covariance matrix is \(V=0.04I_2+0.09\mathbf{1}\mathbf{1}^\mathsf{T}\). Thus its diagonal entries are \(0.13\), its off-diagonal entries are \(0.09\), and \(\det V=0.13^2-0.09^2=0.0088\).
For a schedule with design matrix \(X\), whose rows are \((1,-t)\), the marginal information for \((\alpha,k)\) from one subject is \(X^\mathsf{T}V^{-1}X\). The inverse covariance is \(V^{-1}=(1/0.0088)\begin{pmatrix}0.13&-0.09\\-0.09&0.13\end{pmatrix}\). For two measurements, the determinant identity \(\det(X^\mathsf{T}V^{-1}X)=\det(X)^2/\det(V)\) gives a convenient check. In this model, \(|\det(X)|\) is the time separation.
For schedule A the time separation is \(2.0-0.5=1.5\) hours. Its per-subject information determinant is \[ \det(I_A)=\frac{1.5^2}{0.0088} =\frac{2.25}{0.0088} =255.7. \] For schedule B the separation is \(2.5-0.5=2.0\) hours, giving \[ \det(I_B)=\frac{2.0^2}{0.0088} =\frac{4.00}{0.0088} =454.5. \] Both determinants are positive, so each two-parameter information matrix is positive definite in this example. Schedule B is D-preferred because \(454.5>255.7\).
For an explicit matrix check, \(V^{-1}\) has diagonal entries \(14.7727\) and off-diagonal entries \(-10.2273\). Under schedule A, \(X_A=\begin{pmatrix}1&-0.5\\1&-2.0\end{pmatrix}\), yielding \(I_A\approx\begin{pmatrix}9.0909&-11.3636\\-11.3636&42.3295\end{pmatrix}\). Its determinant is approximately \(9.0909(42.3295)-(-11.3636)^2=255.68\), consistent with the identity. Under schedule B, \(X_B=\begin{pmatrix}1&-0.5\\1&-2.5\end{pmatrix}\), yielding \(I_B\approx\begin{pmatrix}9.0909&-13.6364\\-13.6364&70.4545\end{pmatrix}\). Its determinant is approximately \(454.55\), again consistent with the identity.
For the same two-parameter confidence level, the approximate region-volume ratio is \(\operatorname{Vol}(C_B)/\operatorname{Vol}(C_A)=\sqrt{\det(I_A)/\det(I_B)} =\sqrt{255.68/454.55}=0.7500\). Thus schedule B's approximate joint confidence region is 25.00% smaller than schedule A's under this model and these assumptions. This comparison does not imply that both marginal standard errors are 25.00% smaller.
How to Use the Criterion and Check Its Meaning
A practical comparison starts by fixing the target parameter vector, nominal parameter values, model, and assumptions used to calculate information. Then calculate or approximate the information matrix for each candidate schedule and compare determinants. For numerical stability, computations often use \(\log\det I\): since the logarithm is increasing, maximizing \(\log\det I\) gives the same ranking as maximizing \(\det I\), while reducing overflow and underflow risks.
Name the population parameters whose joint precision is the design objective. State the nominal values used if the calculation is local.
Use the model's appropriate Fisher information, including the random-effects contribution and marginal covariance structure assumed for the NLME model.
Confirm that each candidate information matrix is finite and positive definite for a full-dimensional volume comparison. Compare \(\det I\) or \(\log\det I\); larger values indicate a smaller approximate joint region.
Worked Example: More Subjects and Region Volume
Suppose a hypothetical design has per-subject information \(I_1=\begin{pmatrix}4&1\\1&3\end{pmatrix}\) for a two-parameter vector. Its determinant is \(4(3)-1^2=11\). If \(n\) independent subjects contribute the same information and all assumptions remain unchanged, total information is \(I_n=nI_1\).
With 30 subjects, \(I_{30}=30I_1\), and \(\det(I_{30})=30^2(11)=9900\), because multiplying a \(p\)-dimensional matrix by \(n\) multiplies its determinant by \(n^p\), here \(p=2\). With 60 subjects, \(\det(I_{60})=60^2(11)=39600\), four times the determinant for 30 subjects. The approximate region-volume ratio is \(\sqrt{9900/39600}=0.5000\). Doubling the subject count halves the approximate joint region volume in this two-parameter normal approximation; it does not halve every standard error.
D-optimality is invariant to a fixed nonsingular linear change of parameter units in its ranking of candidate designs. If \(\boldsymbol{\phi}=A\boldsymbol{\theta}\), then \(I_{\boldsymbol{\phi}}=A^{-\mathsf{T}}I_{\boldsymbol{\theta}}A^{-1}\), so \(\det I_{\boldsymbol{\phi}}=\det(I_{\boldsymbol{\theta}})/\det(A)^2\). The factor is the same for all candidate designs, leaving their ordering unchanged. Nevertheless, reported region volumes depend on the parameter coordinates and units, so state the parameterization when interpreting them.
Worked Example: A Rank Check Before Comparing Designs
A hypothetical two-parameter design has \(I=\begin{pmatrix}5&5\\5&5\end{pmatrix}\). Its determinant is \(5(5)-5(5)=0\), and \(I(1,-1)^\mathsf{T}=(0,0)^\mathsf{T}\). Thus the direction \((1,-1)^\mathsf{T}\) has zero first-order Fisher information, and the standard full-dimensional ellipsoid calculation cannot be used. The correct conclusion is not automatically that the underlying parameters are unidentifiable; the zero information may be a local first-order feature. The model, parameter point, and higher-order behavior would need consideration before making an identifiability claim.
For a nonsingular check, changing the lower-right entry to \(6\) gives \(I^*=\begin{pmatrix}5&5\\5&6\end{pmatrix}\), with determinant \(5(6)-25=5>0\). Its leading diagonal entry is positive and its determinant is positive, so this symmetric matrix is positive definite and supports a full-dimensional local volume calculation.
Interpretation, Alternatives, and Limitations
D-optimality balances information across the target vector through the determinant, which reflects the product of information along the principal directions. Its strength is that it gives a single joint-precision criterion even when parameter estimates are correlated. Its limitation is that a large determinant can coexist with a weakly estimated direction if other directions are very well estimated. It also depends on the target parameter set: changing which parameters are included can change the preferred schedule.
Other design objectives answer different questions. An objective based on average parameter variances emphasizes the sum of marginal variances, whereas one based on the weakest information direction emphasizes protection against the least-informed direction. A design chosen to optimize individual prediction may also differ from one chosen to estimate population parameters precisely. D-optimality should therefore follow the scientific decision question, not replace it.
The local criterion depends on assumed parameter values and variance components. If those values are inaccurate, the calculated ranking may not hold. A schedule can also be impractical or violate constraints on visit times, blood volume, or assay capacity even if its determinant is high. In routine use, inspect the information matrix and parameter-specific precision alongside the determinant; do not treat the scalar objective as a complete diagnostic.
Common Pitfalls
- Calling the determinant a measure of each parameter's precision. It summarizes joint information, not every marginal variance or correlation separately.
- Comparing determinants from different parameter sets. The determinant is meaningful for a stated vector and parameterization; adding or removing parameters changes the question.
- Interpreting zero determinant as proof of nonidentifiability. It indicates a zero first-order information direction and invalidates the regular full-dimensional ellipsoid calculation, but does not alone settle identifiability.
- Ignoring the NLME covariance structure. Repeated observations on a subject are not necessarily independent after marginalizing over random effects. Use information consistent with the specified model.
- Overstating the region-volume result. It is based on an approximate normal confidence region under stated assumptions, not an exact guarantee about finite-sample estimates.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- For \(p=3\), information changes from \(I\) to \(2I\). By what factor does the determinant change, and by what factor does approximate region volume change?
- Why does a larger determinant generally imply a smaller approximate joint confidence region?
- In the worked PK example, what feature of schedule B increased its information determinant?
- Does a singular information matrix, by itself, prove that the model parameters are unidentifiable?
- Give one reason a D-optimal design might not be the best design for a particular study.
Solutions. (1) The determinant is multiplied by \(2^3=8\); volume is multiplied by \(1/\sqrt{8}=0.3536\). (2) For fixed confidence level and dimension, volume is proportional to \(1/\sqrt{\det I}\). (3) Its two sampling times are farther apart, increasing the time separation from 1.5 to 2.0 hours and thus increasing \(\det(X)^2/\det(V)\). (4) No. Singularity shows zero first-order Fisher information in at least one direction; it does not alone establish nonidentifiability. (5) The study may prioritize prediction, a particular parameter, feasibility, or protection of a weak direction rather than joint confidence-region volume.