Identifiability and Design for Exposure-Hazard Models

Why Identifiability Is a Design Question

An exposure-hazard model can fit observed event times well and still fail to determine which parameter values generated them. A high baseline hazard combined with a weak drug effect may predict nearly the same event pattern as a lower baseline hazard combined with a stronger effect. A delay parameter can be equally difficult to estimate if the observed concentrations change little over time. Identifiability asks whether the data-generating pattern can, in principle, distinguish the parameters; design asks how to collect data that make those distinctions informative.

Consider an illustrative delayed log-linear hazard model for one event type:

h(t\mid C)=\exp(a)\exp\{\beta C(t-\tau)\}

Here \(h(t\mid C)\) is the instantaneous event rate at time \(t\), \(a\) is the log baseline hazard, \(\beta\) is the log-hazard change per concentration unit, \(C(u)\) is the concentration history at time \(u\), and \(\tau\) is a nonnegative delay in days. Concentration before dosing is defined as zero when appropriate. This model is a convenient example, not a claim that every exposure effect is log-linear or delayed by a fixed shift.

Definition: A parameter is structurally identifiable for a specified model and design if distinct parameter values imply distinct distributions of the observable data, under the stated assumptions. It is locally structurally identifiable at a parameter value if no distinct parameter value in some neighborhood produces the same observable-data distribution. Practical identifiability concerns whether finite, noisy data can estimate the parameter with useful precision.

For an individual followed from time zero without competing events, the event-time distribution is determined by the hazard through \(S(t)=\exp\{-\int_0^t h(u)\,du\}\), where \(S(t)\) is the probability of remaining event-free beyond \(t\). If two parameter values give the same hazard over the observed time range, they give the same survival distribution there. When censoring or other observation processes are present, identifiability must be assessed for the full observable-data distribution, not just for the hazard formula. The assumptions here include a correctly specified hazard and concentration history, consistent time origins and units, and censoring that is independent conditional on the modeled information. If exposure is uncertain, treating it as known can make the assessment overconfident.

Where Confounding Comes From

The model makes clear why study design matters. If concentration is constant at \(c\) during observation, then \(h=\exp(a+\beta c)\). The data can inform this combined log hazard, but the exposure pattern supplies no contrast that separates \(a\) from \(\beta\). Similarly, a delay is difficult to learn when shifting the concentration curve barely changes the exposure values that enter the hazard. A parameter may be structurally identifiable but practically weakly identified if those changes are small relative to event and censoring variability.

Worked Example: A Baseline–Exposure Confounding

Suppose the observed concentration is \(2\) mg/L throughout follow-up, and the model is \(h=\exp(a+\beta C)\). Compare \((a,\beta)=(-4,0.5)\) with \((-3,0)\). The first gives \(a+\beta C=-4+(0.5)(2)=-3\); the second gives \(-3+(0)(2)=-3\). Thus both give \(h=\exp(-3)=0.04979\) events per day.

The equality is not just similar fit in a small sample: with constant concentration, these parameter pairs give exactly the same hazard and event-time distribution. In fact, for any change \(d\) in \(\beta\), replacing \(a\) by \(a-2d\) preserves \(a+2\beta\). Without additional exposure variation or external information, the baseline and exposure effect are structurally confounded in this design.

Delay and effect size can also trade off. For the model above, the instantaneous log-hazard sensitivities are:

\frac{\partial \log h(t)}{\partial a}=1,\qquad \frac{\partial \log h(t)}{\partial \beta}=C(t-\tau),\qquad \frac{\partial \log h(t)}{\partial \tau}=-\beta C'(t-\tau).

The delay sensitivity depends on the concentration slope, not just the concentration level. A flat concentration profile has zero first-order sensitivity to delay at the observed times. A rising or falling profile can carry delay information, provided the effect is nonzero and the resulting sensitivity is distinguishable from the baseline and effect sensitivities.

Local Information: Useful, but Not a Structural Verdict

A local information calculation summarizes how small parameter changes alter the model near a specified value. Let \(s(t)\) be the column vector of the three sensitivities above. Under exact event times and conditionally independent, noninformative censoring, a typical expected information contribution has the form

\mathcal I(\theta)=\int_0^\infty S(t)G(t)h(t)s(t)s(t)^{\mathsf T}\,dt,\qquad \theta=(a,\beta,\tau),

where \(G(t)\) is the probability of remaining uncensored to \(t\), and \(S(t)\) is event-free survival. The expression follows because, at each time still observed and event-free, the event contribution is weighted by its hazard \(h(t)\); the local change in log hazard is \(s(t)\). Integrating the weighted sensitivity outer products adds information across follow-up. The precise expression changes with the observation scheme, but the sensitivity logic remains useful.

If the sensitivity directions fail to span the parameter space, the local information matrix is rank deficient and signals a first-order identification problem. This does not by itself prove local structural nonidentifiability. First derivatives can vanish even when distinct nearby parameter values produce different observable-data distributions through higher-order changes. To establish structural nonidentifiability, check whether distinct nearby values actually give the same distribution under the design; a singular local approximation alone is not enough.

Worked Example: Zero Delay Sensitivity Does Not Settle Identifiability

Suppose \(\beta\ne 0\), the baseline is fixed, and observations cover \(0\le t\le1\) day. At the true delay \(\tau=0\), let the concentration profile be flat at \(c\) for arguments \(0\le u\le1\), and let it change smoothly just outside this interval. Throughout the observed interval, \(C'(t-\tau)=0\), so \(\partial\log h(t)/\partial\tau=0\). The first-order information for \(\tau\) from these sensitivities is zero.

Now increase \(\tau\) slightly, the only admissible local direction at \(\tau=0\). Part of the shifted interval \(t-\tau\) then extends just below zero. If the concentration differs from \(c\) at those arguments, then, since \(\beta\ne0\), the hazard differs over part of the observed time range, and the event-time distribution can differ as well. Thus zero first-order sensitivity is a warning, not proof that nearby delays are structurally indistinguishable. If instead the concentration remains flat across every shifted argument for a neighborhood of \(\tau\), the hazard is unchanged for those delays and delay is locally structurally unidentifiable in that design.

The practical lesson is to combine derivatives or information-matrix diagnostics with direct comparisons of predictions or likelihoods at nearby parameter values. This is especially important for delay, threshold, and other parameters whose effects may appear only beyond first order.

Using Simulated Profiles to Choose a Design

A design comparison should simulate the exposure histories expected under candidate regimens and evaluate the corresponding hazard changes over the intended observation period. Useful designs often include contrasting dose levels, time-varying concentrations, and enough follow-up to observe the delayed response. Repeated concentration measurements around expected rises and declines help establish the timing of the profile that drives the hazard. Event follow-up must also be long enough to yield informative risk sets; adding PK samples alone cannot compensate for too few events.

Worked Example: A Profile That Separates Three Parameters

Suppose the local parameter values are \(\beta=0.5\) L/mg and an unknown baseline \(a\). A simulated concentration profile supplies three representative exposure states: a pre-exposure point with \(C=0\) mg/L and \(C'=0\), a plateau at \(C=2\) mg/L and \(C'=0\), and a decline at \(C=1\) mg/L with \(C'=-0.5\) mg/L/day. For the parameter order \((a,\beta,\tau)\), the log-hazard sensitivity rows are respectively \((1,0,0)\), \((1,2,0)\), and \((1,1,0.25)\); the final delay component is \(-\beta C'=-0.5(-0.5)=0.25\) per day.

The sensitivity matrix is

J= \begin{pmatrix} 1&0&0\\ 1&2&0\\ 1&1&0.25 \end{pmatrix}, \qquad \det(J)=0.5.

Because its determinant is nonzero, these three local sensitivity directions span the three-parameter space at this point. This is evidence that the design can provide first-order information about baseline, effect, and delay. It is not a guarantee of narrow confidence intervals: event counts, censoring, concentration measurement error, and the validity of the model still matter. By contrast, if every observed concentration point lies on a flat plateau, all delay sensitivities are zero, so this local design cannot provide first-order information about \(\tau\).

The same simulated profiles can be used to compare absolute event predictions. For illustration, take a baseline hazard of \(0.02\) per day and a constant concentration for 10 days after the delay. At \(1\) mg/L, \(h=0.02\exp(0.5)=0.03297\) per day, so the cumulative hazard is \(H=10(0.03297)=0.3297\) and event risk is \(1-\exp(-0.3297)=0.2809\). At \(2\) mg/L, \(h=0.02\exp(1)=0.05437\) per day, \(H=10(0.05437)=0.5437\), and risk is \(1-\exp(-0.5437)=0.4194\). The higher concentration creates a larger predicted risk contrast that can help identify \(\beta\), while the constant plateaus by themselves still provide little or no information about delay. Varying exposure over time is needed to learn timing.

A practical simulation workflow is to generate concentration histories for plausible subjects and regimens, calculate hazards and event probabilities under candidate parameter values, and then inspect both the prediction differences and the sensitivity directions. Candidate schedules should be compared at the same planned follow-up and with plausible event rates. If predicted event risks are nearly zero in every arm, or nearly one before the informative exposure periods, the design may yield little useful timing information even if the mathematical sensitivity matrix has full rank.

Diagnostics, Alternatives, and Limitations

Several checks help reveal an identification problem before interpreting parameter estimates:

Plot simulated concentration histories after applying each candidate delay. Overlapping shifted profiles suggest weak delay information.
Compare predicted hazards and event risks after changing one parameter while re-optimizing the others. Nearly unchanged predictions indicate a confounded direction.
Inspect the rank and conditioning of a scaled information matrix. A very small eigenvalue suggests a weakly informed parameter combination; parameter units should be scaled before comparing eigenvalues.
Use profile likelihood or likelihood contours to see whether the likelihood is flat or has a long ridge. A narrow optimizer standard error is not persuasive if it depends on a boundary, a fixed parameter, or a numerical approximation.
Check whether conclusions change under plausible baseline-hazard specifications, concentration histories, and censoring assumptions. Apparent identification may rely on an overly restrictive model.

These tools answer different questions. Structural analysis asks whether exact observational equivalence exists under the model; local information examines first-order sensitivity; simulation and profile likelihood assess finite-sample precision under realistic designs. Fixing a parameter can stabilize estimation, but it does not create information: the resulting estimate is conditional on that fixed value. External data or a defensible prior can add information, but their source and assumptions should be reported.

Compared with an observational design, randomized dose levels can create more independent exposure contrasts and reduce dose-selection confounding. A richer concentration profile, such as a regimen with identifiable rise and decline phases, can be more useful for delay than repeated observations on a long plateau. These changes are not substitutes for sufficient event follow-up. Nor can any design resolve parameters that are intrinsically equivalent under the chosen model, such as baseline and effect under a single constant concentration without further constraints or contrasts.

Identifiability conclusions are conditional on the model, exposure observation, and time window. A parameter may be identifiable in theory but too imprecise to estimate with the available events. Conversely, a numerical estimate can appear precise because of a boundary, strong model restrictions, or an optimization artifact. Design assessment should therefore be performed before data collection when possible and revisited when the observed exposure and event patterns differ from simulation.

Common pitfall: “The delay sensitivity is zero, so delay is structurally unidentifiable.” Zero first-order sensitivity establishes a local information problem, not necessarily exact equivalence of nearby parameter values. Check the shifted profiles and the resulting observable-data distributions; higher-order changes may distinguish delays. The reverse mistake is also possible: a full-rank local matrix does not guarantee adequate precision or a correct model.

Check Your Understanding

Answer each question, then compare with the solutions.

In a log-linear hazard model, concentration is constant at \(3\) mg/L. Which combination of baseline and exposure-effect parameters is directly represented in the log hazard?
Why does a flat concentration profile give zero first-order sensitivity to a time delay in the shifted-profile model?
Does a rank-deficient local information matrix alone prove structural nonidentifiability? Explain.
A simulated regimen has distinct peak and declining concentration phases, but almost no events occur during follow-up. Is the design necessarily practically informative about delay?

Solutions. (1) The directly represented combination is \(a+3\beta\), since \(\log h=a+\beta C\). Without other exposure contrasts or constraints, \(a\) and \(\beta\) cannot be separated by this constant profile. (2) The delay sensitivity is \(-\beta C'(t-\tau)\); on a flat segment \(C'=0\), so this derivative vanishes. (3) No. Rank deficiency indicates a first-order information problem. Nearby parameter values may still produce different distributions through higher-order changes, so compare the observable-data distributions directly. (4) No. The changing profile may supply delay sensitivity, but few events can leave the parameter practically imprecise. Assess simulated event counts, censoring, and the amount of follow-up during informative exposure phases.