c-Optimality For A Targeted Parameter Contrast

Why Optimize for a Particular Contrast?

A design that estimates every model parameter reasonably well need not be the best design for the question that matters. In pharmacometrics, the target may be a particular parameter, a difference between parameters, or another linear combination, such as \(c^\top\theta\). The precision of that target depends on the whole information matrix: uncertainty and correlation in parameters that are not named in the contrast can still affect its variance.

c-optimality focuses the design criterion on one specified contrast. It is therefore not a general measure of how much information a design provides about all parameters. It ranks designs according to the variance they are expected to achieve for the target, under a specified model and its information assumptions.

The components of \(c\) specify the target. For example, \(c=(1,0,\ldots,0)^\top\) targets the first parameter, while \(c=(1,-1,0,\ldots,0)^\top\) targets the difference between the first and second parameters. Multiplying all components of \(c\) by the same constant multiplies the criterion by the square of that constant, but does not change which design minimizes it.

Why This Is a Variance Criterion

Suppose the model is locally regular at the parameter value of interest, the observations are independent under the assumed model, and the information matrix is nonsingular. Under these conditions, the estimator \(\hat{\theta}\) has the usual large-sample covariance approximation \(I(\xi)^{-1}\). In a linear model with the corresponding variance assumptions, the covariance expression can be exact. The variance of a linear combination follows directly from the covariance matrix.

To see the first equality, write the contrast error as \(c^\top(\hat{\theta}-\theta)\). The variance of any linear transformation \(A Z\) is \(A\operatorname{Cov}(Z)A^\top\); taking \(A=c^\top\) gives \(c^\top\operatorname{Cov}(\hat{\theta})c\). Substitution of the inverse-information covariance approximation gives the c-criterion. Its square root is the approximate standard error of the estimated contrast.

If a design is represented by proportions of observations at candidate settings, its information matrix is often a per-observation or normalized matrix \(M(\xi)\). With \(n\) independent observations and the same per-observation information structure, \(I(\xi)=nM(\xi)\). Thus the approximate contrast variance is \(c^\top M(\xi)^{-1}c/n\). If a common observation variance \(\sigma^2\) has been factored out of the information matrix, the variance is instead \(\sigma^2 c^\top M(\xi)^{-1}c/n\). Keep this scale factor explicit when comparing numerical criteria with variances in the data's units.

Optimizing an Allocation for a Contrast

A simple allocation problem makes the targeted nature of the criterion explicit. Suppose there are two possible observation types. One provides information only about \(\theta_1\), and the other only about \(\theta_2\), with equal information per observation. Let \(w\) be the fraction allocated to the first type and \(1-w\) the fraction allocated to the second. For a total of \(n\) observations with common variance \(\sigma^2\), the normalized information matrix is \(M(w)=\operatorname{diag}(w,1-w)\). The design is full rank when \(0<w<1\).

The equal allocation follows from this particular information structure and contrast; it is not a general rule. If one parameter were more influential in the target, or if observation types supplied different amounts or patterns of information, the minimizing allocation could differ. The relevant calculation is always based on the information matrix and the specified \(c\).

A Regression Example: The Target Changes the Design

Consider a hypothetical linear model \(Y=\beta_0+\beta_1 x+\varepsilon\), with independent errors of mean zero and common variance \(\sigma^2\). Assume the design points satisfy \(-1\leq x\leq1\). For a design distribution \(\xi\), define \(E_\xi(x)\) and \(E_\xi(x^2)\) as the weighted averages of the design points and their squares. The normalized information matrix, with the common variance factored out, is \(M(\xi)=\begin{pmatrix}1&E_\xi(x)\\E_\xi(x)&E_\xi(x^2)\end{pmatrix}\). It is nonsingular when the design has positive spread in \(x\).

Assumptions, Diagnostics, and Limitations

The criterion is meaningful only relative to a model, a target, and a set of allowable design choices. In model-based design, information may depend on parameter values; the comparison then applies to the nominal values used to calculate it. The covariance interpretation also relies on appropriate regularity and model assumptions. If the model is misspecified, observations are dependent when independence was assumed, or variability is incorrectly represented, the inverse information matrix may not describe the estimator's actual covariance.

Before trusting a calculation, check that the contrast is specified in the parameterization actually used, the information matrix corresponds to the proposed design and observation model, and the matrix is nonsingular for the stated inverse. Verify that the resulting criterion is positive and that its units are squared units of the contrast. In numerical work, inspect whether the matrix is nearly singular: an unstable inverse can make the calculated criterion unreliable. A small value is not automatically evidence of a good practical design if it is driven by unrealistic nominal parameters or assumptions.

The ordinary formula assumes an estimable contrast with nonsingular information. If the full parameter vector is not identifiable, the inverse does not exist, so this formula cannot be applied as written. In some settings a contrast may still be estimable even when all parameters are not; such cases require an explicit estimability argument and an appropriate generalized-inverse treatment. They should not be handled by silently substituting an arbitrary inverse.

c-optimality differs from criteria that summarize uncertainty across all parameters or emphasize the weakest direction. It answers a narrower question: which allowed design minimizes uncertainty for this particular linear target? This focus is valuable when a scientific decision depends on one contrast, but it is also a limitation: the resulting design may perform poorly for other parameters or targets. Prediction-variance criteria address different targets and should not be substituted without checking what quantity is being optimized.