Why Model NT-proBNP as a Trajectory?
NT-proBNP is a circulating biomarker associated with cardiac wall stress and is used in the assessment of heart failure. Its value can change over follow-up, and measurements from the same person are correlated. A single baseline value describes status at one time; it does not summarize whether the marker is rising, falling, or remaining stable. Those patterns may carry different prognostic information for the time to a heart failure hospitalization or death.
A joint model represents the repeated biomarker measurements and the event time together. In this case, the clinical question is whether the event hazard is associated with the person’s underlying NT-proBNP level over time, its rate of change, or both, after accounting for measured covariates. This is an association question: a modeled trajectory–hazard association does not establish that changing NT-proBNP would itself change a person’s event risk.
NT-proBNP measurements are often right-skewed, so a log scale can be appropriate if it improves the longitudinal model and gives scientifically interpretable contrasts. The choice of scale affects coefficient units: a one-unit difference on the natural-log scale corresponds to an \(e\)-fold difference in the original concentration, while a doubling corresponds to a difference of \(\log(2)\). The scale and time units must be stated explicitly.
Joint-Model Specification for This Case
For illustration, suppose the observed outcome is \(Y_{ij}=\log(\text{NT-proBNP}_{ij})\), with NT-proBNP measured in pg/mL. A longitudinal model can describe the mean trajectory using time, baseline characteristics, and person-specific random effects:
Here \(X_i\) contains measured characteristics, \(b_{0i}\) and \(b_{1i}\) allow person-specific departures in level and slope, and \(\epsilon_{ij}\) is measurement error. This linear form is only an example; the time pattern should reflect the scientific question and observed data. A flexible time function may be needed when change is not approximately linear. The event submodel for the first occurrence of heart failure hospitalization or death can be written as:
In this specification, \(h_0(t)\) is the baseline hazard, \(W_i\) contains event-model covariates, \(\alpha_v\) is the current-value association parameter, and \(\alpha_s\) is the current-slope association parameter. A positive \(\alpha_v\) means that, at the same time and conditional on the other terms, a higher underlying log NT-proBNP level is associated with a higher hazard. A positive \(\alpha_s\) means that a more rapidly increasing underlying log level is associated with a higher hazard, conditional on current level. These are conditional associations, not necessarily causal effects.
The model requires a defensible longitudinal mean and error structure, an adequately specified event hazard and association structure, and an appropriate distribution for the random effects. It also assumes that the event and longitudinal processes are conditionally linked as specified, and that censoring is noninformative conditional on the modeled information. The observation process should not contain unmodeled information about impending events: for example, if people are measured more often because clinicians suspect deterioration, the visit process may need attention. A joint model accounts for dependence through the shared latent trajectory, but does not automatically correct every form of informative measurement scheduling.
Worked Example: Converting Concentrations to the Model Scale
Suppose a hypothetical participant has NT-proBNP measurements of 900 pg/mL and 1,800 pg/mL at two visits. On the natural-log scale, the measurements are:
The log-scale change is \(7.4955-6.8024=0.6931\). This equals \(\log(2)\), consistent with the original concentration doubling from 900 to 1,800 pg/mL. If the visits are one year apart, the observed two-point change is \(0.6931\) log units per year. It is not automatically the model-based slope: the fitted slope estimates the underlying trajectory using all available measurements and the longitudinal model, rather than treating two noisy observations as the true trajectory.
This transformation gives the model a scale on which multiplicative changes in concentration become additive differences. Its adequacy still needs to be checked against the observed data and model residuals.
How a Trajectory Enters the Hazard
A joint-model association can be understood by comparing two people at the same follow-up time, with the same event-model covariates and the same baseline hazard. Their hazard ratio is obtained by dividing their hazards. The common factors cancel, leaving a contrast in their underlying trajectories:
This comparison is conditional on the stated model and covariates. It is a relative instantaneous hazard at time \(t\), not a ratio of cumulative probabilities of experiencing the event. The current-value contribution depends on the difference in log NT-proBNP at that time; the slope contribution depends on the difference in log-scale rates of change. The following calculation shows why a baseline value alone can give a different summary from the evolving trajectory.
Worked Example: Comparing a Changing Trajectory With a Baseline-Only Contrast
Consider hypothetical fitted association parameters \(\alpha_v=0.70\) per log unit and \(\alpha_s=0.30\) per log unit per year. At baseline, person A’s underlying NT-proBNP is twice person B’s, so their initial log-scale difference is \(\log(2)=0.6931\). Over the next year, suppose their underlying slopes are \(0.05\) and \(0.40\) log units per year, respectively. At year 1, the current-value difference is:
Their slope difference is \(0.05-0.40=-0.35\) log units per year. Substituting both differences into the joint-model hazard contrast gives:
Under this hypothetical fitted model, at year 1 person A’s instantaneous hazard is about 1.145 times person B’s, conditional on the model covariates and at the same time. The value and slope components partly offset: A’s current log NT-proBNP remains higher, but A’s slope is less steep. A baseline-only calculation using the value coefficient and the initial difference would instead be \(\exp\{0.70(0.6931)\}=\exp(0.4852)\approx1.625\). That calculation ignores the subsequent change and the slope association; it is not the year-1 contrast from the joint model. The two figures answer different model-based questions.
Worked Interpretation of Association Parameters
Worked Example: Interpreting a Log-Scale Association
Suppose a fitted model estimates \(\alpha_v=0.70\), with the longitudinal marker modeled as the natural logarithm of NT-proBNP in pg/mL. Compare two otherwise identical people at the same time whose underlying concentrations differ by a factor of two, and whose slopes are equal. The log-scale difference is \(\log(2)=0.6931\); because the slopes match, the slope term cancels. The hazard ratio is:
Thus, under this model, a doubling in the underlying NT-proBNP concentration is associated with an instantaneous hazard about 1.625 times as high, conditional on the same slope and event-model covariates. This is not a claim that doubling the concentration causes a 62.5% increase in risk. If \(\alpha_s=0.30\), a slope difference of \(0.10\) log units per year contributes \(\exp\{0.30(0.10)\}=\exp(0.03)\approx1.030\) to the hazard ratio, holding the current value fixed. The slope effect is expressed per log unit per year, so omitting the time unit would make its interpretation incomplete.
Assumptions, Diagnostics, and Clinical Limits
The clinical endpoint must be defined before fitting. If the event is the first heart failure hospitalization or death, then death is part of the composite endpoint, not a competing event that is censored. The estimated association concerns time to that first composite event. It does not separately identify the association with hospitalization and the association with death; those require distinct endpoint analyses and assumptions.
Check whether the chosen time scale and longitudinal trajectory form can represent the observed NT-proBNP patterns. Plot observed values and fitted trajectories against time, inspect residual patterns and unusually influential participants, and assess whether the assumed random-effects and residual distributions are plausible. For the event component, inspect the adequacy of the baseline hazard and proportional-hazards structure, and compare observed and model-based event behavior where suitable. Numerical warnings, unstable estimates, or markedly different results under reasonable trajectory specifications are reasons to investigate, not merely presentation details.
Also examine whether the estimated current-value and slope effects are too strongly correlated to distinguish reliably. A model including both can be clinically attractive, but if the data contain few measurements per person or little variation in slopes, estimates may be imprecise. NT-proBNP also reflects factors beyond cardiac status, including renal function and other patient characteristics. Measured confounders should be considered on clinical grounds; adjustment does not guarantee that all confounding has been removed.
A baseline-only survival model is simpler and can be useful when only baseline information is available, but it cannot represent evolving marker status. A conventional time-updated covariate model uses observed values as predictors, potentially treating noisy measurements as if they were measured without error and requiring careful handling of measurement timing. Separate longitudinal and survival fits do not, by themselves, propagate the shared trajectory uncertainty into the event analysis. A joint model links the two components through the latent trajectory and estimates them together, but relies on its modeling assumptions and may be computationally demanding. No approach is universally preferable: the target question, measurement density, missingness, and model fit matter.
The estimated hazard association is not an individual’s absolute event probability. Absolute risk prediction additionally depends on the baseline hazard, covariate values, the person’s longitudinal information, and the prediction time horizon. Further, an association with a biomarker trajectory does not show that a biomarker-guided intervention improves outcomes. Clinical prediction and causal treatment effects are different targets.
Common Mistake
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- In the specified model, what does a positive \(\alpha_s\) mean when the current value is held fixed?
- If two people’s underlying log NT-proBNP values differ by \(0.20\) at a given time and their slopes are equal, with \(\alpha_v=0.70\), calculate their hazard ratio.
- Why does including death in a first-event composite change how it should be treated in the event analysis?
- Name one limitation of using a single baseline NT-proBNP value instead of a serial trajectory.
- Does a positive trajectory–hazard association prove that lowering NT-proBNP will reduce event risk? Why or why not?
Solutions: (1) A more rapidly increasing underlying log NT-proBNP is associated with a higher instantaneous hazard, conditional on current value and other modeled terms. (2) \(\exp(0.70\times0.20)=\exp(0.14)\approx1.150\); the hazard is about 1.150 times as high under the model. (3) Death is itself an event in the composite, so it is not censored as though the composite event had not occurred. (4) A baseline value cannot represent subsequent rises, falls, or changing slopes. (5) No. The fitted association is not, by itself, a causal treatment effect; confounding and other causal assumptions would need to be addressed.