Sensitivity Functions As Optimality Diagnostics

Why Examine Sensitivity?

A design's information matrix summarizes the information contributed by its selected experimental conditions. D-optimality compares designs using the determinant of that matrix, but the determinant alone does not say where a current design could improve. The sensitivity function supplies that local diagnostic: it evaluates the information at each candidate condition against the inverse information already accumulated by the design.

A large value means that adding observations at that condition has potential to increase the log determinant. A value at or below a fixed benchmark everywhere in the design space indicates that no infinitesimal move of design weight to a single candidate point can improve D-optimality. This tutorial focuses on constructing and interpreting that function; it is not a substitute for checking the model, candidate design space, or numerical calculations.

Definition and Conditions

Let $x$ denote a candidate design point, such as a dose, sampling time, or experimental setting. Let $J(x)$ be its per-observation information matrix for a model with $p$ parameters. For an approximate design $\xi$, let $M(\xi)$ be its per-observation design information matrix: the weighted average of per-point information over the design. In a discrete design with points $x_i$ and weights $w_i$, this is $M(\xi)=\sum_i w_iJ(x_i)$, where $w_i\geq 0$ and $\sum_iw_i=1$.

The sensitivity function is defined for any candidate point $x$ when $M(\xi)$ is nonsingular. It measures the per-point information relative to the information matrix of the current design. For a fixed total sample size, multiplying the design information by that sample size does not change the sensitivity function.

Definition: The D-optimal sensitivity function of design $\xi$ at candidate point $x$ is $$ \phi_D(x;\xi)=\operatorname{tr}\!\left[M(\xi)^{-1}J(x)\right], $$ where $J(x)$ is the per-observation information matrix and $M(\xi)$ is the design information matrix. The benchmark is $p$, the number of model parameters.

The usual local D-optimality result assumes that $J(x)$ is symmetric positive semidefinite throughout the candidate design space, that $M(\xi)$ is positive definite, and that the same model parameter value and information convention are used to calculate every matrix. The design is called local when its information is evaluated at a specified parameter value. If $M(\xi)$ is singular, its inverse and hence this sensitivity function are not defined; a design lacking full-rank information cannot be assessed by this formula.

The function is nonnegative because it is the trace of a positive-semidefinite information matrix after expressing it relative to a positive-definite $M(\xi)$. It is dimensionless under a consistent information calculation. It is not itself a probability or a variance.

Why the Benchmark Is the Number of Parameters

Sensitivity is useful because it is the directional rate of change in the D-criterion. To see this, move a small fraction $\epsilon$ of design weight from the current design to a point $x$. The perturbed design is $\xi_\epsilon=(1-\epsilon)\xi+\epsilon\delta_x$, where $\delta_x$ places all its weight at $x$. Its information matrix is $M(\xi_\epsilon)=(1-\epsilon)M(\xi)+\epsilon J(x)$.

For a positive-definite matrix $M$ and a symmetric perturbation $H$, the derivative of $\log\det(M+\epsilon H)$ at $\epsilon=0$ is $\operatorname{tr}(M^{-1}H)$. Here $H=J(x)-M(\xi)$. Therefore:

\left.\frac{d}{d\epsilon}\log\det M(\xi_\epsilon)\right|_{\epsilon=0} =\operatorname{tr}\!\left[M(\xi)^{-1}(J(x)-M(\xi))\right] =\phi_D(x;\xi)-p.

The last equality follows from $\operatorname{tr}(M^{-1}M)=\operatorname{tr}(I_p)=p$. Thus $\phi_D(x;\xi)>p$ identifies a direction in which transferring a sufficiently small amount of weight to $x$ increases the log determinant. If the sensitivity is below $p$, that particular transfer has a negative first-order effect. A value equal to $p$ means the first-order change is zero; it does not, by itself, prove that every finite change has zero effect.

A second useful identity comes from averaging the function under the current design:

\int \phi_D(x;\xi)\,d\xi(x) =\operatorname{tr}\!\left[M(\xi)^{-1}\int J(x)\,d\xi(x)\right] =\operatorname{tr}(I_p)=p.

For a discrete design, the integral is the weight-averaged sensitivity, $\sum_iw_i\phi_D(x_i;\xi)$. The average is always $p$ for a nonsingular design. In particular, sensitivities at positive-weight support points cannot all be strictly below $p$.

Theorem: For a concave design space of approximate designs with a common candidate-point information matrix $J(x)$, a nonsingular design $\xi$ is D-optimal if and only if $\phi_D(x;\xi)\leq p$ at every candidate point. At every support point with positive weight, $\phi_D(x;\xi)=p$. A value above $p$ anywhere rules out D-optimality. These are the sensitivity conditions of the D-optimal equivalence theorem.

The condition is global over the stated candidate design space, not merely over points already used. Checking only support points is therefore insufficient: a candidate outside the support may have sensitivity above $p$. Conversely, a numerical grid check is only as complete as the grid; a narrow peak between grid points can be missed.

Worked Examples

Worked Example: A D-Optimal Linear Regression Design

Suppose the mean response is $\beta_0+\beta_1x$, the observation variance is known to be 1, and the candidate settings are all $x\in[-1,1]$. Assume independent observations and the usual normal linear model. The per-observation information is $J(x)=f(x)f(x)^\mathsf{T}$, where $f(x)=(1,x)^\mathsf{T}$. Consider the design assigning weight $1/2$ to each endpoint.

At $x=-1$, $f=(-1\text{ in the slope component};\ 1\text{ in the intercept component})$, so in intercept-slope order $f=(1,-1)^\mathsf{T}$; at $x=1$, $f=(1,1)^\mathsf{T}$. Averaging the two outer products gives:

M(\xi)=\frac12 \begin{pmatrix}1&-1\\-1&1\end{pmatrix} +\frac12 \begin{pmatrix}1&1\\1&1\end{pmatrix} = \begin{pmatrix}1&0\\0&1\end{pmatrix}, \qquad M(\xi)^{-1}=I_2.

Therefore the sensitivity at any candidate setting is:

\phi_D(x;\xi)=f(x)^\mathsf{T}M(\xi)^{-1}f(x) =(1,x)\begin{pmatrix}1&0\\0&1\end{pmatrix} \begin{pmatrix}1\\x\end{pmatrix} =1+x^2.

On $[-1,1]$, $1+x^2\leq 2$, with equality at $x=-1$ and $x=1$. Here $p=2$, so the maximum sensitivity is $p$, and both positive-weight support points have sensitivity 2. The equivalence condition is satisfied throughout the candidate space: this design is D-optimal under the stated model and assumptions. At the midpoint, the sensitivity is $1+0^2=1$, so moving a small amount of weight there would reduce the log determinant to first order.

Worked Example: A Sensitivity Peak Reveals a Better Setting

Keep the same linear regression model, variance, and candidate interval $[-1,1]$, but use weight $1/2$ at $x=0$ and weight $1/2$ at $x=1$. The information matrix and inverse are:

M(\xi)=\frac12 \begin{pmatrix}1&0\\0&0\end{pmatrix} +\frac12 \begin{pmatrix}1&1\\1&1\end{pmatrix} = \begin{pmatrix}1&1/2\\1/2&1/2\end{pmatrix}, \qquad M(\xi)^{-1}= \begin{pmatrix}2&-2\\-2&4\end{pmatrix}.

The determinant of $M(\xi)$ is $1(1/2)-(1/2)^2=1/4$, so this matrix is nonsingular. For $f(x)=(1,x)^\mathsf{T}$, the sensitivity is:

\phi_D(x;\xi) =(1,x)\begin{pmatrix}2&-2\\-2&4\end{pmatrix} \begin{pmatrix}1\\x\end{pmatrix} =2-4x+4x^2.

The support sensitivities are $\phi_D(0;\xi)=2$ and $\phi_D(1;\xi)=2-4+4=2$, as expected for these positive-weight points. But at the unused candidate $x=-1$, $\phi_D(-1;\xi)=2+4+4=10$, well above $p=2$. The local derivative from transferring weight toward $x=-1$ is $10-2=8$. This is a strong diagnostic signal that the current design is not D-optimal.

As a direct comparison, putting equal weight at $-1$ and $1$ gives $M=I_2$, whose determinant is 1 rather than $1/4$. The alternative is therefore better by this determinant comparison as well. Sensitivity identified a promising direction; a finite design search can then evaluate the actual proposed weights.

Worked Example: Evaluating a Candidate With Matrix Information

Suppose a two-parameter experiment has three candidate settings, A, B, and C, with per-observation information matrices:

J(A)=\begin{pmatrix}4&0\\0&0\end{pmatrix},\qquad J(B)=\begin{pmatrix}0&0\\0&2\end{pmatrix},\qquad J(C)=\begin{pmatrix}2&0\\0&3\end{pmatrix}.

These matrices are symmetric positive semidefinite. Consider a design with equal weight on A and B. Its information is $M(\xi)=\frac12J(A)+\frac12J(B)=\begin{pmatrix}2&0\\0&1\end{pmatrix}$, which is positive definite, and $M(\xi)^{-1}=\begin{pmatrix}1/2&0\\0&1\end{pmatrix}$. There are $p=2$ parameters.

At C, calculate the sensitivity directly:

\phi_D(C;\xi) =\operatorname{tr}\!\left[ \begin{pmatrix}1/2&0\\0&1\end{pmatrix} \begin{pmatrix}2&0\\0&3\end{pmatrix}\right] =\operatorname{tr}\begin{pmatrix}1&0\\0&3\end{pmatrix} =4.

Since $4>2$, the design fails the D-optimality condition. More concretely, shift a fraction $\epsilon$ of its weight from the current A/B mixture to C. The perturbed information matrix is $\operatorname{diag}(2,1+2\epsilon)$, so its determinant is $2+4\epsilon$, greater than the original determinant 2 for every $0<\epsilon\leq1$. The derivative of its log determinant at zero is $4/2=2$, agreeing with $\phi_D(C;\xi)-p=4-2=2$. This confirms both the diagnostic and its direction.

Reading the Shape and Checking the Calculation

In a continuous design space, plotting $\phi_D(x;\xi)$ against the design variable can reveal where a design is underrepresented. A peak above $p$ points to a possible improvement; a broad region above $p$ may indicate that several settings deserve consideration. A value equal to $p$ at a support point is expected for an optimal design, while values below $p$ at unused points are compatible with optimality. The plot does not imply that every high-sensitivity setting should receive equal weight: the matrix contributions are linked, and the best weight allocation must be assessed as a design.

Useful checks include confirming that $M(\xi)$ is symmetric and positive definite, verifying that the weights sum to one, and checking the weighted-average identity $\sum_i w_i\phi_D(x_i;\xi)=p$ for a discrete design. A substantial failure of that identity often indicates a coding, inversion, or weight-normalization error. Small discrepancies can arise from rounding. For a continuous candidate space, examine boundaries and refine the search around apparent maxima instead of relying on a coarse grid.

Sensitivity is local to the information matrices being used. In a nonlinear model, changing the parameter value used to calculate $J(x)$ can change the entire function and its diagnostic. Model misspecification, parameter uncertainty, constraints on allowable designs, and inaccurate numerical optimization can also affect conclusions. The formula itself does not account for these issues; they must be addressed in the design problem's specification and implementation.

Common Pitfalls and Alternatives

Checking only the selected settings. Support-point equality does not rule out a better unused setting. Evaluate the sensitivity over the full candidate design space.
Interpreting a high value as a completed design. A value above $p$ identifies a promising direction, not the optimal weight allocation or final design. Recalculate information and compare proposed designs.
Using a singular information matrix. If $M(\xi)$ is singular, the inverse in the sensitivity definition does not exist. First construct a design with full-rank information or use a suitable constrained design approach.
Comparing with the wrong benchmark. The benchmark is the number $p$ of parameters in the information matrix, not the sample size, the determinant, or the number of design points.
Confusing the diagnostic with other criteria. A-optimality, E-optimality, c-optimality, G-optimality, and I-optimality target different summaries. A D-sensitivity plot diagnoses the D-criterion; it is not automatically a diagnostic for those alternatives.

Unlike comparing a single determinant with a benchmark design, sensitivity evaluates possible local changes across candidate settings. Unlike a direct numerical search, it explains which settings may improve the current design and by what first-order criterion. Its limitation is that the derivative is local: a finite move can behave differently, and a search over a restricted or poorly specified candidate space cannot establish optimality over a larger one.

Check Your Understanding

Answer each question, then compare with the solutions below.

A model has three parameters and a nonsingular design. What is the D-optimal sensitivity benchmark? What does a candidate sensitivity of 3.8 imply?
A discrete design has two support points with weights 0.25 and 0.75. Their sensitivities are 4 and 2, respectively. Check the weighted-average identity and state whether these values alone establish D-optimality for a model with two parameters.
For a two-parameter model, a candidate point has $J(x)=\begin{pmatrix}1&0\\0&4\end{pmatrix}$, while the current design has $M(\xi)=\begin{pmatrix}2&0\\0&2\end{pmatrix}$. Calculate its sensitivity and interpret it.
Why is checking sensitivity only at the support points inadequate for certifying D-optimality?

Solutions. (1) The benchmark is $p=3$. Since $3.8>3$, moving a sufficiently small amount of weight toward that point increases the log determinant to first order, so the design is not D-optimal over a candidate space that includes it. (2) The weighted average is $0.25(4)+0.75(2)=1+1.5=2$, which equals $p$. This consistency check does not establish optimality: an unused point could have sensitivity above 2. (3) The inverse is $M^{-1}=\operatorname{diag}(1/2,1/2)$. Thus $\phi_D=\operatorname{tr}(\operatorname{diag}(1/2,1/2)\operatorname{diag}(1,4))=\operatorname{tr}(\operatorname{diag}(1/2,2))=2.5$. Since $2.5>p=2$, the candidate indicates an improving local direction. (4) The theorem requires sensitivity to be at most $p$ at every allowable candidate point. Support-point checks cannot exclude a peak at a setting the design does not currently use.