Choosing Doses and Perturbations for Estimability

Why Dose and Perturbation Choices Matter

A dose-response experiment can collect many precise measurements and still leave a mechanistic question unresolved. If two parameters change the predicted response in nearly the same way at every tested dose, their effects can compensate for one another. If two proposed mechanisms make nearly identical predictions over the tested dose range, those data may not distinguish them. Changing the dose or selectively perturbing a pathway can alter these patterns of response and make the experiment more informative.

The design question here is local: at specified parameter values, how do candidate doses and perturbations change the sensitivities of model predictions to parameters, and what does that imply for Fisher information? Sensitivity is not a guarantee of successful estimation or mechanism selection. It is a way to assess whether a proposed set of conditions is likely to provide different information about the parameters or mechanisms of interest.

Model and Assumptions for Fisher Information

Definition: Let \(y_i\) be an observation under experimental condition \(c_i\), where a condition specifies such features as dose and pathway perturbation. Suppose \(y_i\mid\theta\sim N(\mu(c_i;\theta),\sigma_i^2)\), independently across observations, with known variances \(\sigma_i^2\) that do not depend on the parameter vector \(\theta\). The Fisher information matrix is \(\mathcal{I}(\theta)=\sum_i \sigma_i^{-2}g_i g_i^T\), where \(g_i=\partial\mu(c_i;\theta)/\partial\theta\) is the column vector of sensitivities evaluated at \(\theta\).

The Gaussian distribution and parameter-independent variances are important assumptions: independence and knowledge of the variances alone do not make the sensitivity formula the Fisher information. If the observation distribution differs from the one stated, its full likelihood must be used to derive Fisher information. A weighted sensitivity matrix can still be a useful approximation in some settings, but it should not be labelled exact Fisher information without the corresponding distributional assumptions.

The information is local because each sensitivity vector is evaluated at a particular parameter vector. It also depends on the observation scale: a response change that is large relative to its standard deviation contributes more information than the same change with much noisier measurement. A condition with a large sensitivity magnitude is not automatically useful for every parameter; the direction of its sensitivity vector matters too.

Why Different Conditions Can Improve Estimability

For the Gaussian model above, the log likelihood, apart from a constant, is \(-\tfrac12\sum_i (y_i-\mu_i)^2/\sigma_i^2\). Differentiating with respect to \(\theta\) gives the score \(\sum_i (y_i-\mu_i)g_i/\sigma_i^2\). The expected negative Hessian equals the information matrix in the definition. To see why, the terms involving second derivatives of \(\mu_i\) are multiplied by \(y_i-\mu_i\), whose expectation is zero under the model; the remaining terms are \(\sum_i g_i g_i^T/\sigma_i^2\). Thus, under the stated assumptions, the formula follows from the likelihood rather than from the mean and variance alone.

Each observation contributes a weighted outer product \(g_i g_i^T/\sigma_i^2\). If every sensitivity vector points in the same direction in parameter space, the sum cannot contain information in directions perpendicular to that shared direction. Observations at different doses or under different perturbations can help when they change the direction of sensitivity, not merely when they increase its magnitude. This is why evaluating a range of conditions can be more revealing than repeating one condition.

Worked Example: Choosing Two Doses to Estimate Potency and Maximum Effect

Consider the hypothetical saturable response model \(\mu(d;E_{\max},K)=E_{\max}d/(K+d)\), where \(d\) is dose in mg/kg, \(E_{\max}\) is the maximum response in normalized response units, and \(K\) is the dose giving half of \(E_{\max}\). Suppose the nominal values are \(E_{\max}=100\) response units and \(K=5\) mg/kg. Observations are independent Gaussian with known standard deviation \(5\) response units at each dose.

The sensitivity vector, ordered as \((E_{\max},K)\), is \(g(d)=(d/(K+d),-E_{\max}d/(K+d)^2)^T\). At \(d=1\) mg/kg it is \((1/6,-25/9)^T\), or approximately \((0.1667,-2.778)^T\). At \(d=10\) mg/kg it is \((2/3,-40/9)^T\), or approximately \((0.6667,-4.444)^T\).

With one observation at each dose, \(\sigma^2=25\) response-units squared, so

\mathcal{I}=\frac{1}{25}\left[ \begin{pmatrix}1/6\\-25/9\end{pmatrix} \begin{pmatrix}1/6&-25/9\end{pmatrix} + \begin{pmatrix}2/3\\-40/9\end{pmatrix} \begin{pmatrix}2/3&-40/9\end{pmatrix} \right] \approx \begin{pmatrix} 0.01889 & -0.1370\\ -0.1370 & 1.099 \end{pmatrix}.

The determinant of the unweighted two-row sensitivity matrix is \((1/6)(-40/9)-(2/3)(-25/9)=10/9\), which is nonzero. Consequently, the two sensitivity directions are linearly independent, and the information matrix has full rank for these two parameters. In contrast, two observations both at \(d=5\) mg/kg have identical sensitivity directions. Their information matrix has rank one: replication improves precision along the direction measured at that dose, but does not create a second sensitivity direction to separate \(E_{\max}\) from \(K\).

Using Dose Levels to Separate Candidate Mechanisms

Estimating parameters within a specified model and distinguishing different models are related but distinct tasks. Fisher information describes local sensitivity to parameters within a model. To examine whether two specified dose-response mechanisms make different predictions, compare their predicted means at candidate doses relative to the expected observation variability. A dose at which their predictions coincide is uninformative for distinguishing those fixed predictions, even if it gives a precise measurement.

Worked Example: Finding a Dose That Separates Saturable and Linear Responses

Suppose two hypothetical mechanisms predict response in normalized response units. The saturable model is \(\mu_S(d)=100d/(5+d)\), and a linear model is \(\mu_L(d)=10d\). These parameter values make both models predict 50 response units at \(d=5\) mg/kg. Assume independent Gaussian observations with standard deviation \(5\) response units at each tested dose.

At \(d=1\) mg/kg, the saturable prediction is \(100/6=16.67\), while the linear prediction is \(10\). Their mean difference is \(6.67\) response units, or \(6.67/5=1.333\) standard deviations. At \(d=20\) mg/kg, the predictions are \(2000/25=80\) and \(200\) response units, respectively: a difference of \(120\) response units, or \(24\) standard deviations. At \(d=5\) mg/kg, the difference is zero.

For these fixed predictions, the sum of squared standardized mean differences at doses 1 and 20 is \((6.67/5)^2+(120/5)^2\approx577.8\). This is a comparison of predicted separation under the assumed error scale, not a formal test statistic or a guarantee of model-selection performance. In practice, parameters would be uncertain and may be estimated from the observations. The example shows why choosing doses only near a point where mechanisms agree can conceal their differences, while a dose range that exposes their different shapes may be more useful.

Using Pathway Perturbations to Reveal Hidden Parameters

A perturbation can change which pathways contribute to an observed response. Its value for estimability depends on how the model represents that change. In the example below, the perturbation is assumed to selectively switch on a feedback contribution. This is a substantive model assumption, not something guaranteed by using a perturbing treatment. If the perturbation also changes other processes, those effects must be represented or the sensitivity calculation may be misleading.

Worked Example: A Selective Perturbation Reveals a Feedback Effect

Consider \(\mu(d,q;A,B)=A\,d/(5+d)+Bq\,d/(5+d)\), where \(A\) and \(B\) are response amplitudes in normalized response units, \(d\) is dose in mg/kg, and \(q=0\) denotes no perturbation while \(q=1\) denotes a selective perturbation that reveals the feedback contribution. Assume the dose scale in both terms is known, and observations are independent Gaussian with known standard deviation \(4\) response units. The unknown parameters are \(A\) and \(B\).

At \(d=5\) mg/kg, the sensitivity vector for \((A,B)\) is \((0.5,0)^T\) when \(q=0\), and \((0.5,0.5)^T\) when \(q=1\). With one observation in each condition, the information matrix is

\mathcal{I}=\frac{1}{16} \left[ \begin{pmatrix}0.5\\0\end{pmatrix} \begin{pmatrix}0.5&0\end{pmatrix} + \begin{pmatrix}0.5\\0.5\end{pmatrix} \begin{pmatrix}0.5&0.5\end{pmatrix} \right] = \begin{pmatrix} 0.03125 & 0.015625\\ 0.015625 & 0.015625 \end{pmatrix}.

Its determinant is \(0.0002441\), which is positive, so this two-parameter information matrix has full rank. Without the perturbation, both observations would have sensitivity \((0.5,0)^T\), giving zero information about \(B\) and a rank-one matrix. Under the selective-perturbation assumption, the perturbed condition adds a sensitivity direction that responds to \(B\). If the perturbation instead changes \(A\), the dose scale, or the observation variance, this calculation would need to include those changes.

Practical Design Checks and Failure Modes

A useful design assessment starts with the model predictions and sensitivities over plausible parameter values, not just at one nominal fit. Check whether candidate conditions produce sensitivity vectors that differ in direction for the parameters of interest. Singular values of a sensitivity or information matrix can reveal directions with little information, while parameter correlations can indicate compensatory changes. These are local diagnostics; they do not replace checking model fit, uncertainty, or the plausibility of the assumed mechanism.

Dose escalation alone is not a universal solution. In the saturable model above, as \(d\) becomes very large, \(d/(K+d)\) approaches one, while the sensitivity to \(K\), \(-E_{\max}d/(K+d)^2\), approaches zero. Very high doses can therefore inform the maximum response while contributing little information about potency. At very low doses, both derivatives may also be small. Intermediate doses and a suitably broad range can be more useful for separating parameters, although the best choices depend on the model, noise, and feasible dose range.

Check units and scaling. A sensitivity to a parameter measured in mg/kg cannot be compared numerically with sensitivity to an amplitude in response units without accounting for parameter scale. Reparameterizing, for example using log-parameters for positive quantities, changes the numerical sensitivity coordinates; state which parameterization is used.
Check the error model. The worked information calculations assume fixed, known Gaussian variances. If variability changes with dose or depends on parameters, the stated formula may not be the full Fisher information.
Check perturbation selectivity. A perturbation intended to affect one pathway may alter several processes. Ignoring those effects can make a design appear more informative than it is.
Check local versus global conclusions. Full rank at one parameter vector is not proof of global identifiability, and a locally informative design may perform poorly at other plausible parameter values.

Compared with adding replicates at an existing condition, adding a different dose or perturbation can create a new sensitivity direction; replicates primarily increase information in directions already represented. Compared with using dose conditions to estimate parameters within one model, contrasting predicted means across candidate mechanisms directly targets their separation. Neither approach removes the need to assess assumptions, measurement error, and parameter uncertainty. A design based only on the largest predicted response can fail if that condition is insensitive to the parameter that distinguishes the mechanisms.

Key takeaway: Choose doses and perturbations for the sensitivity directions and mechanistic predictions they generate, not simply for response magnitude. Fisher-information calculations are exact only under their stated observation model, and their conclusions are local to the parameter values and assumptions used.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

Under what observation assumptions does the sensitivity outer-product formula give Fisher information as written here?
In the saturable response example, why can two observations at the same dose fail to separate \(E_{\max}\) and \(K\), even though both observations are precise?
For the two fixed candidate mechanisms in the dose-separation example, what do they predict at \(d=5\) mg/kg, and what does that imply for separating those predictions at that dose?
In the perturbation example, what change makes the feedback amplitude \(B\) visible in the sensitivities?

Solutions. (1) Observations must be independent Gaussian with known variances that do not depend on the parameters; the full likelihood may give a different information formula otherwise. (2) At a single dose, repeated observations have proportional, identical sensitivity directions, so their information has rank one and does not span both parameter directions. (3) Both predict 50 normalized response units, so their mean difference at that dose is zero; under the stated comparison, that dose provides no separation of these fixed predictions. (4) The selective perturbation changes the sensitivity to \(B\) from zero at \(q=0\) to \(0.5\) at \(q=1\), adding a direction that contains information about the feedback amplitude under the model assumptions.