Sensitivity Analysis For Unobserved Visit Triggers

Why Vary an Assumption About Unobserved Health?

A patient may arrange an unscheduled visit because their current symptoms have worsened, even when no recent biomarker measurement records that worsening. The visit then provides information about health that is not fully represented by the observed visit history. A model can represent this dependence, but its conclusions may rely on how strongly unobserved current health is assumed to affect visit timing. Sensitivity analysis makes that reliance visible: fit the model under a range of defensible assumptions, then compare the estimated longitudinal trajectories.

The goal is not to find a single “correct” strength of unobserved triggering from the visit records alone. Rather, the analysis asks: if current health affects the chance of a visit more or less strongly than the primary model assumes, how much do the estimated trajectories change? This is particularly important when visits are sparse, when current health between measurements is uncertain, or when clinical knowledge gives a plausible range but not a precise value.

Specify the Sensitivity Parameter

Let $Y_i(t)$ be the biomarker or health measure for person $i$ at time $t$, and let $m_i(t)$ denote the model’s latent current mean health value. For example, $m_i(t)$ may be the value implied by a longitudinal model at times when no measurement was taken. Suppose larger values indicate worse health. Let $U_i(t)=[m_i(t)-c]/s$, where $c$ is a fixed reference value and $s$ is a fixed scale, such as a clinically meaningful difference or a reference standard deviation. Fixing $c$ and $s$ across sensitivity fits keeps the parameter’s interpretation comparable.

Definition: A sensitivity parameter $\gamma$ quantifies the assumed effect of unobserved current health on the intensity of unscheduled visits. One convenient model is $$ \lambda_i^{U}(t\mid \mathcal{H}_i(t),m_i(t),b_i) = \lambda_{0i}^{U}(t\mid \mathcal{H}_i(t),b_i) \exp\{\gamma U_i(t)\}, $$ where $\lambda_i^{U}(t)$ is the conditional rate of an unscheduled visit, $\mathcal{H}_i(t)$ is the observed history, $b_i$ denotes any shared latent effects, and $\lambda_{0i}^{U}$ is the visit rate when $U_i(t)=0$. The parameter $\gamma$ is a log rate ratio per one-unit increase in standardized current health.

If $\gamma=0$, the visit intensity has no additional dependence on $U_i(t)$, conditional on the history and other terms in the model. If $\gamma>0$, worse current health is associated with a higher visit rate; if $\gamma<0$, it is associated with a lower rate. A one-unit increase in $U_i(t)$ multiplies the conditional visit rate by $\exp(\gamma)$. This is an association in the visit-process model, not a claim that health causally produces a visit.

This specification concerns unscheduled visits. Administrative visits with known appointment times should be represented according to their design, rather than treated as if they arose from the same stochastic intensity. The model should also distinguish current health from observed history: the sensitivity term is for the part of current health not already captured by the recorded covariates and past measurements.

Worked Example: Interpreting a Visit-Trigger Assumption

Suppose the baseline conditional unscheduled-visit rate is $0.20$ visits per month when $U_i(t)=0$. A one-unit increase in $U_i(t)$ represents a clinically meaningful worsening. Under a proposed scenario, $\gamma=0.40$.

The rate ratio for a one-unit worsening is $$ \exp(\gamma)=\exp(0.40)=1.492. $$ At $U_i(t)=1.5$, the assumed rate is $$ 0.20\exp(0.40\times1.5) =0.20\exp(0.60) =0.20(1.822) =0.3644 $$ visits per month.

Thus this scenario assumes that a person at $U_i(t)=1.5$ has a conditional rate about 1.82 times the rate at $U_i(t)=0$, holding the other model terms fixed. The value $0.3644$ is an instantaneous rate, not a probability of having a visit during a month. Converting a rate into an interval probability would require additional assumptions about how the intensity behaves over that interval.

Fit a Grid of Scenarios, Not Just One Model

Choose a set of $\gamma$ values before examining which value produces the most favorable trajectory. The grid should span no current-health triggering, plausible positive triggering, and—if clinically defensible—negative triggering. Use clinical input, prior evidence, or a transparent calibration such as the rate ratios implied by the values. The grid is a set of assumptions for sensitivity analysis; it is not automatically a confidence interval for $\gamma$.

For each fixed grid value, fit the longitudinal and visit-process components jointly, estimating the remaining parameters under that value. In particular, do not change the scale $s$, reference $c$, or other modeling choices from one fit to another without a clear reason. Compare the same estimand at the same times in every fit, such as the population-average latent mean trajectory $E\{m_i(t)\}$ for a specified covariate profile. This is distinct from the mean among observed visits, which can be altered by the visit process.

The visit-time contribution explains why changing $\gamma$ can change a trajectory estimate. Conditional on a latent health path, the likelihood for unscheduled visit times $t_{i1},\ldots,t_{iK_i}$ over follow-up $[0,C_i]$ has the form

L_i^{U}(\gamma\mid m_i,b_i) = \left\{\prod_{j=1}^{K_i}\lambda_i^{U}(t_{ij})\right\} \exp\left\{-\int_0^{C_i}\lambda_i^{U}(t)\,dt\right\}.

The product accounts for the observed visits, while the exponential term accounts for the time without a visit. Both depend on the assumed $\gamma$, because the intensity depends on current health. In the joint fit, this contribution is combined with the longitudinal measurement model and integrated over latent quantities such as random effects or unobserved health paths. Consequently, a different fixed $\gamma$ can change the relative support for latent trajectories, even when the recorded biomarker values are unchanged.

Justification: At a visit time, the intensity contribution is larger for a latent path with greater $U_i(t_{ij})$ when $\gamma>0$. But a path with high intensity also accumulates more integrated hazard, decreasing the no-visit contribution $\exp\{-\int_0^{C_i}\lambda_i^{U}(t)\,dt\}$. The model therefore uses both visit occurrence and time without visits; it does not simply assign more weight to every path that predicts a high rate at observed visits.

A practical sequence is to fit the model at each fixed grid value, check convergence and parameter estimates, calculate the chosen trajectory summaries and uncertainty intervals, and display them together. If standard errors or intervals are reported, they are conditional on the specified sensitivity value unless uncertainty about $\gamma$ is explicitly incorporated. A single fitted scenario should not be presented as if the data had selected $\gamma$ when it was fixed for sensitivity analysis.

Calibrate the scale.
Define what one unit of $U_i(t)$ means, and keep that reference and scale fixed across fits.

Choose defensible values.
Translate each candidate $\gamma$ into a rate ratio $\exp(\gamma)$, and document why the range is plausible.

Refit and compare.
Hold $\gamma$ fixed at each candidate value, refit the remaining model, and summarize the same target trajectory each time.

Describe robustness.
Report which substantive conclusions persist across the grid and where, if anywhere, they change.

Worked Example: Comparing Fitted Trajectories Across a Grid

Consider a hypothetical study measuring a symptom score from 0 to 10, where larger scores indicate worse symptoms. The model uses $U_i(t)$ standardized so that one unit is a one-reference-standard-deviation worsening. The fitted population-average symptom means at month 12 are:

Fixed $\gamma$	Rate ratio $\exp(\gamma)$	Estimated mean at month 12
$-0.50$	0.607	5.40 points
0	1.000	5.00 points
0.50	1.649	4.60 points
1.00	2.718	4.30 points

For instance, $\exp(0.50)=1.649$, so the third scenario assumes a 64.9% higher conditional visit rate per one-unit worsening. Under that assumption, the refitted model estimates a month-12 population mean of 4.60 points, compared with 5.00 points when $\gamma=0$. Across the full grid, the estimated means range from 4.30 to 5.40 points. That range describes sensitivity to the specified assumptions; it is not a sampling confidence interval.

The pattern in this hypothetical fit does not establish that positive triggering is true. It shows that the trajectory estimate is sensitive to the assumed dependence in this direction. A complete report would also show uncertainty intervals for each fit, explain how the reference scale was chosen, and state whether the substantive conclusion changes across the grid.

Assess Robustness and Recognize Failure Modes

Sensitivity analysis is informative when the candidate assumptions are interpretable and the model can be fitted reliably under each one. Check that every fitted scenario converges, that estimates are not at unexplained boundaries, and that implied visit rates are clinically and operationally plausible over follow-up. Large changes in the trajectory, random-effect distribution, or visit-rate parameters can signal that the data provide limited information to separate the visit mechanism from the health trajectory. They can also indicate model misspecification; sensitivity analysis alone cannot tell which explanation is correct.

Compare more than a single endpoint when the scientific claim concerns the shape of a trajectory. For example, a similar month-12 mean can conceal different early and late patterns. Examine fitted means at prespecified times or plot the trajectories and their uncertainty intervals. Do not choose the grid after seeing which values preserve a preferred conclusion. If the result changes only under assumptions that imply implausibly large rate ratios, say so, but justify that judgment using external or clinical information rather than the observed result alone.

Changing the scale between fits. If $s$ changes, a fixed $\gamma$ no longer represents the same rate ratio per health difference. Keep the scale fixed or transform $\gamma$ accordingly.
Confusing a fixed scenario with an estimate. A value set by the analyst is an assumption. Do not report it as an effect estimated by the data.
Ignoring uncertainty. A shift in point estimates may be small relative to their uncertainty, or uncertainty may widen substantially. Present both estimates and intervals, with their conditional interpretation.
Assuming monotonicity. The trajectory need not move in one direction as $\gamma$ increases. The visit likelihood, longitudinal data, and latent heterogeneity jointly determine the fit.

The main alternative is to estimate $\gamma$ directly as part of one joint model. That can be useful when the model and data contain enough information, but a precise-looking estimate may still depend strongly on structural assumptions, especially with sparse observations. A visit-ignorant longitudinal fit is another comparison, but it answers the question under an assumption that the visit process can be ignored; it is not automatically a neutral reference. Inverse-intensity weighting instead reweights observed measurements using an estimated visit intensity and relies on correct intensity specification and adequate probability of observation. These approaches answer related questions but do not remove the need to state assumptions about unobserved health.

A limitation of the grid approach is that it cannot establish which scenario is true. It evaluates consequences conditional on each chosen value and model specification. It may also understate uncertainty if the reported intervals condition on $\gamma$ and the grid omits plausible values. Report the grid, the implied rate ratios, the fixed components of the model, the trajectory estimand, and any change in substantive conclusions. Formal checks of dependence between visits and outcomes require additional diagnostic work; a stable sensitivity plot is not by itself proof that the visit process is adequately modeled.

Worked Example: Finding a Tipping Point

Suppose the target is the difference between treatment groups in mean change in symptom score from baseline to month 12, with positive values indicating a worse change in the treatment group. In a hypothetical sensitivity analysis, that estimated difference is $0.50$ points at $\gamma=0$ and $-0.10$ points at $\gamma=0.75$. For a simple linear interpolation between these two fitted values, the estimated difference at $\gamma$ is $$ \Delta(\gamma) = 0.50+\frac{-0.10-0.50}{0.75-0}\gamma = 0.50-0.80\gamma. $$

The interpolated value where the estimated difference is zero solves $$ 0=0.50-0.80\gamma, \qquad \gamma=\frac{0.50}{0.80}=0.625. $$ The corresponding one-unit rate ratio is $\exp(0.625)=1.868$. Thus, under this interpolation, the estimated direction changes near a scenario that assumes about an 87% higher visit rate per one-unit worsening. This is a descriptive tipping point between two fitted scenarios, not a data-identified threshold and not a hypothesis test. The actual grid should be expanded or refined near that value if the location matters, and uncertainty in the treatment contrast should still be reported.

Common Pitfall

A common mistake is to say, “The model estimated $\gamma=0.5$, so the visit rate is 1.65 times higher for worse health,” when $0.5$ was actually fixed by the analyst as one sensitivity scenario. The calculation $\exp(0.5)=1.649$ is correct, but the conclusion about what the data estimated is not. Instead say: “Under the scenario fixing $\gamma=0.5$, the model assumes a 1.65-fold rate per one-unit worsening, and the resulting trajectory estimate is…” Keep the assumption, the model output, and the scientific interpretation distinct.

Key takeaway: A sensitivity analysis for unobserved visit triggers is a structured comparison of refitted trajectories under explicit, interpretable values of the health-to-visit association. Its strength lies in showing which conclusions survive plausible assumptions, not in selecting a preferred value of the sensitivity parameter.

Check Your Understanding

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

If $\gamma=0.7$, what is the assumed rate ratio for a one-unit increase in standardized current health?
Why should the reference scale $s$ be held fixed across the sensitivity grid?
A model fixes $\gamma=0$ for one fit and $\gamma=0.5$ for another. Is the second value an estimate of the health-trigger effect? Explain.
Across a grid, an estimated trajectory changes direction only at the most extreme $\gamma$. What should be reported before calling the primary conclusion robust?

Solutions. (1) $\exp(0.7)=2.014$, so the scenario assumes about 2.01 times the conditional visit rate per one-unit increase, holding other terms fixed. (2) The scale determines what a one-unit health difference means; changing it changes the interpretation of the same $\gamma$, so comparisons would no longer use a common rate-ratio scale. (3) No. If fixed for the fit, $0.5$ is an assumed sensitivity value, not an effect estimated by the data. (4) Report the full grid and its implied rate ratios, the trajectory estimates and uncertainty intervals, and why the extreme scenario is or is not plausible. A conclusion is robust only relative to the stated range of assumptions; the grid does not prove the extreme value impossible.

Fixed \(\gamma\)	Rate ratio \(\exp(\gamma)\)	Estimated mean at month 12
\(-0.50\)	0.607	5.40 points
0	1.000	5.00 points
0.50	1.649	4.60 points
1.00	2.718	4.30 points