Why Blood-Pressure Trajectories Matter After Stroke
After an index stroke, systolic blood pressure is measured repeatedly, while a recurrent stroke may occur between visits or before a planned measurement. The scientific question is not simply whether a high reading predicts another stroke. Readings fluctuate, are measured with error, and can prompt treatment changes. Patients who remain under observation also differ from those who have an event or leave follow-up early. A joint model links the evolving blood-pressure process to the time-to-event process while representing these features explicitly.
The event considered here is time from the index stroke to the first recurrent stroke. The approach is not, by itself, a model for multiple recurrent strokes per person. Its central association concerns the patient's underlying, or latent, blood-pressure level at a given time—not a single noisy measurement.
A Joint Model for Blood Pressure and Stroke Hazard
A useful starting point is a mixed-effects model for the readings:
Here \(m_i(t)\) is the patient's underlying mean systolic pressure at time \(t\), \(A_i(t)\) indicates whether a specified treatment is active at that time, and \(b_{0i}\) and \(b_{1i}\) are patient-specific random intercept and slope. The measurement errors \(\epsilon_{ij}\) are commonly assumed independent of the random effects and normally distributed with mean zero and variance \(\sigma^2\). The random effects are commonly modeled jointly as normal with mean zero and covariance matrix \(D\). These assumptions provide a model for between-patient variation and within-patient measurement noise; they should be checked as far as the data allow.
Let \(W_i\) denote baseline covariates, such as age or prior vascular risk factors. One event submodel is:
The hazard \(h_i(t\mid b_i)\) is the instantaneous rate of first recurrent stroke at time \(t\), conditional on being event-free just before \(t\), the covariates, and the random effects. The baseline hazard \(h_0(t)\) is left unspecified in a Cox-type specification. The coefficient \(\alpha\) represents the conditional association with a 10 mmHg difference in underlying systolic pressure: the corresponding hazard ratio is \(\exp(\alpha)\). The centering at 130 mmHg affects the reference level, not that contrast. The coefficient \(\eta\) represents the association with the treatment indicator, conditional on the other modeled quantities; it is not automatically a treatment effect.
The model assumes that the chosen trajectory adequately represents pressure over time, that the hazard depends on it through the specified current-value association, and that relevant covariates and treatment history are represented appropriately. It also requires a suitable model for censoring: conditional on modeled information, censoring should not carry additional information about the unobserved event time. The interpretation is conditional association, not a causal effect of changing blood pressure.
Why the Two Submodels Are Estimated Together
Given random effects \(b_i=(b_{0i},b_{1i})\), the longitudinal readings and event time are modeled conditionally. For right-censored data, a patient's event contribution is \(h_i(T_i\mid b_i)^{\delta_i}S_i(T_i\mid b_i)\), where \(S_i(t\mid b_i)\) is the conditional probability of remaining free of recurrent stroke through time \(t\). The shared random effects connect this event contribution to that patient's readings. Because the random effects are unobserved, the subject-specific likelihood integrates over their distribution:
This integration is a key part of the method. The event outcome provides information about which latent trajectories are more compatible with the observed event history, while the repeated readings inform each patient's trajectory. The likelihood combines both sources rather than pretending that the latent pressure is observed without error.
To see why this matters, consider a simplified current-value model with one latent pressure \(m\), a normal prior \(m\sim N(\mu,\tau^2)\), and one reading \(Y\mid m\sim N(m,\sigma^2)\). Completing the square in the product of these normal densities gives:
The latent-pressure estimate is therefore a weighted average of the population mean and the reading, not necessarily the reading itself. With multiple readings and a time-varying trajectory, the same principle applies through the mixed model. In the joint likelihood, the event history also contributes information about the random effects. Replacing \(m_i(t)\) with one observed reading in an ordinary time-dependent Cox model ignores this measurement-error structure.
Worked Example: Estimating the Latent Pressure at a Follow-Up Time
Suppose a fitted longitudinal model for a hypothetical patient gives \(\hat{\beta}_0=142\) mmHg, \(\hat{\beta}_1=-3\) mmHg per year, \(\hat{\beta}_2=-4\) mmHg for active treatment, \(\hat b_{0i}=5\) mmHg, and \(\hat b_{1i}=1\) mmHg per year. At one year, treatment is active. The observed reading at that visit is 150 mmHg.
The model-based underlying pressure is:
The observed 150 mmHg reading is 9 mmHg above the fitted latent mean at that time. In this model, that difference is attributed to residual measurement variation rather than being assumed to be the patient's entire underlying pressure. The estimate depends on the fitted model and available readings; it is not a directly observed value.
Worked Example: Interpreting the Blood-Pressure Association
Suppose the fitted event-model coefficient is \(\hat\alpha=0.262\), with a 95% confidence interval for \(\alpha\) of 0.105 to 0.419. The hazard ratio for a 10 mmHg higher underlying pressure is:
Under the model, among patients with the same modeled covariates and treatment status at a given time, a 10 mmHg higher underlying systolic pressure is associated with a 30.0% higher instantaneous hazard of first recurrent stroke. The confidence interval corresponds to an estimated increase ranging from 11.1% to 52.0%. It is an interval for the model-based association; it does not mean there is a 95% probability that the fixed parameter lies in this particular interval, and it does not show that lowering pressure would cause the hazard to fall.
Treatment Changes and Follow-Up History
Treatment after stroke is often adjusted in response to prior pressure readings, clinical status, or suspected risk. The treatment indicator \(A_i(t)\) can represent a time-varying history, but its timing must be handled carefully: the value used at time \(t\) must be known by that time, not determined by future information. A model that includes treatment can describe the blood-pressure association conditional on the recorded treatment history. It does not, merely by including \(A_i(t)\), remove confounding by indication. For example, patients with persistently high pressure may be more likely to receive intensified treatment and may also have other risk factors that are not fully measured.
Follow-up history matters as well. A patient who experiences a recurrent stroke stops contributing subsequent pre-event pressure readings to this first-event analysis. The joint model uses the observed readings and event status together, under its assumptions about the shared trajectory and event process. It does not automatically account for every source of informative visit timing, dropout, or competing mortality. If visit attendance itself depends on health beyond the modeled information, or if death prevents observing a recurrent stroke, the observation or competing-event process may need explicit modeling for the scientific question.
Worked Example: Comparing Patients With Different Treatment Histories
Consider two hypothetical patients at the same follow-up time, with the same baseline covariates. Patient A has an underlying pressure of 150 mmHg and active treatment; Patient B has 135 mmHg and no active treatment. Suppose \(\hat\alpha=0.262\) per 10 mmHg and \(\hat\eta=-0.223\) for active versus inactive treatment. Their conditional hazard ratio, A relative to B, is:
Under this fitted model, Patient A's instantaneous hazard is estimated to be 1.185 times Patient B's at that time, conditional on the modeled quantities. The pressure contrast contributes a factor of \(\exp(0.393)=1.481\), while the active-treatment term contributes \(\exp(-0.223)=0.800\). This calculation is not a comparison of treatment strategies: treatment was not randomized, and the fitted treatment coefficient may reflect differences that led clinicians to prescribe treatment. Nor does it account for any unmodeled history that differs between the patients.
Diagnostics, Alternatives, and Limits
A joint model is only as credible as its component models. Examine longitudinal residuals for systematic patterns over time, changing variability, and influential readings. Check whether the assumed trajectory is too simple—for example, whether pressure changes nonlinearly or has abrupt shifts not captured by a linear time trend. Examine the random-effects distribution and whether the event association is sensitive to plausible alternatives, such as including a slope or a cumulative exposure rather than only the current value. Event-model checks should assess fit over follow-up and whether proportional-hazards assumptions are reasonable for the specified covariates and association.
Also inspect visit schedules, missingness, treatment timing, and censoring. A joint model with shared random effects can address association between the latent trajectory and event time under its assumptions; it does not automatically model an informative visit process. If treatment changes respond to past blood pressure and measured clinical history, standard adjustment may not support causal conclusions. Stronger causal questions require a design and methods that address time-varying confounding.
| Approach | What it uses | Main limitation for this question |
|---|---|---|
| Baseline-pressure survival model | One pressure measurement at or near the index stroke | Does not represent subsequent trajectory changes. |
| Time-dependent survival model using observed readings | Most recent available pressure as a changing predictor | Can be sensitive to measurement error and irregular measurement timing. |
| Separate longitudinal and survival models | A fitted trajectory and a distinct event analysis | Does not fully propagate their shared uncertainty or link the latent trajectory and event process in a joint likelihood. |
| Joint longitudinal–time-to-event model | Repeated readings, event history, and shared latent effects | Depends on trajectory, association, random-effects, and censoring assumptions; is not automatically causal. |
A common mistake is to interpret \(\exp(\hat\alpha)\) as the effect of lowering blood pressure by 10 mmHg. The coefficient compares modeled hazards associated with different latent pressure levels, conditional on the model. Patients were not necessarily exchangeable with respect to treatment decisions, health status, or unmeasured risk, so the comparison does not identify what would happen under an intervention. Another mistake is to call every censored patient event-free indefinitely. Censoring means the event was not observed before follow-up ended; inference requires appropriate assumptions about why follow-up ended.
Finally, if the scientific question concerns the number and timing of multiple recurrent strokes, rather than time to the first recurrence, a first-event model discards later event information and is not sufficient. Likewise, when death is common and precludes observing a recurrent stroke, treating death as ordinary noninformative censoring may not answer the intended question. The event definition and follow-up process must match the target scientific question.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- If \(\hat\alpha=0.182\) per 10 mmHg, what is the estimated hazard ratio for a 10 mmHg higher underlying pressure?
- Why can a single observed systolic pressure be a poor substitute for \(m_i(t)\) in the event model?
- In a model with a time-varying treatment indicator, why is the treatment coefficient not automatically a causal treatment effect?
- What does \(\delta_i=0\) mean for a patient whose follow-up ends without an observed recurrent stroke?
- Name one situation in which the first-event model described here would not answer the full scientific question.
Solutions. (1) \(\exp(0.182)=1.200\), so the fitted model estimates a 20.0% higher instantaneous hazard per 10 mmHg higher underlying pressure, conditional on modeled quantities. (2) The observed value includes measurement error and may not represent the latent trajectory at the relevant event time. (3) Treatment may be prescribed in response to prior pressure or clinical risk, creating confounding by indication; adjustment alone does not establish a causal contrast. (4) Follow-up was censored before a recurrent stroke was observed; it does not establish that the patient would never have one. (5) If the target is the rate or timing of multiple recurrent strokes, a first-event model omits later events and a recurrent-event approach is needed.