Why Accumulate Deviation From Baseline?
A cumulative marker association allows the event hazard to depend on a subject’s marker history, rather than only on the marker value at the current time. But the way that history is accumulated matters. Integrating the marker itself counts the subject’s starting level throughout follow-up. Integrating the marker’s deviation from its baseline level instead asks how much marker exposure has accumulated above or below that subject’s own starting point.
This distinction is useful when the scientific question concerns change from baseline. It also changes the units and interpretation of the association. Baseline centering is a definition of the cumulative exposure; it does not, by itself, remove confounding, correct measurement error, or guarantee that the model has controlled for baseline marker level.
The integrand can be positive or negative. A trajectory persistently above baseline produces positive accumulation; one persistently below baseline produces negative accumulation. The accumulated value can also return to zero if positive and negative deviations cancel. Thus \(A_i(t)\) is not generally the same as the total amount of marker present over follow-up.
Connection to Unadjusted Cumulative Exposure
The unadjusted cumulative marker exposure is \(C_i(t)=\int_0^t m_i(s)\,ds\). Since \(m_i(0)\) is constant with respect to the integration variable \(s\), its integral from 0 to \(t\) is \(t\,m_i(0)\). Therefore,
This identity shows exactly what baseline adjustment subtracts: the exposure that would have accumulated if the marker had remained at its baseline value throughout the interval. At a fixed time, subjects with different baseline levels can therefore have identical baseline-adjusted cumulative exposure but different unadjusted cumulative exposure.
Worked Example: Same Change, Different Starting Levels
Suppose two hypothetical subjects have linear latent marker trajectories, measured in marker-units, with time \(t\) in years: $$ m_1(t)=11+0.6t,\qquad m_2(t)=9+0.6t. $$ Calculate both cumulative exposures at \(t=4\) years and compare them.
For a linear trajectory \(m_i(t)=b_{0i}+b_{1i}t\), the deviation from baseline is \(b_{1i}t\). Thus, $$ A_i(4)=\int_0^4 b_{1i}s\,ds=\frac{b_{1i}(4)^2}{2}. $$ Substituting the common slope \(0.6\) gives \(A_1(4)=A_2(4)=0.6(16)/2=4.8\) marker-unit-years.
For subject 1, the unadjusted cumulative exposure is $$ C_1(4)=\int_0^4(11+0.6s)\,ds=11(4)+0.6(16)/2=44+4.8=48.8 $$ marker-unit-years. For subject 2, $$ C_2(4)=\int_0^4(9+0.6s)\,ds=9(4)+4.8=36+4.8=40.8 $$ marker-unit-years. Their unadjusted cumulative exposures differ by \(48.8-40.8=8.0\) marker-unit-years, exactly \(4(11-9)=8.0\). Their baseline-adjusted exposures are equal because they have the same increase from their respective baselines.
Association With the Event Hazard
A common joint-model specification links the cumulative exposure to the event hazard conditional on the subject’s latent trajectory and baseline covariates. For example, let \(w_i\) be a vector of baseline covariates, \(h_0(t)\) the baseline hazard, and \(\gamma\) and \(\alpha\) regression parameters. The model is
This is a conditional hazard model: at time \(t\), it describes the instantaneous event rate among subjects still event-free just before \(t\), given their latent marker trajectory and covariates. The cumulative exposure is evaluated through the current time, not through future times. The model assumes that the log hazard is linear in \(A_i(t)\), that the association parameter \(\alpha\) is constant over time, and that the remaining hazard structure is adequately represented by \(h_0(t)\) and the included covariates.
For two subjects compared at the same time, with the same \(w_i\) and otherwise the same model terms, a difference \(\Delta A\) in baseline-adjusted cumulative exposure gives the conditional hazard ratio
If \(A_i(t)\) is in marker-unit-years, then \(\alpha\) is measured per marker-unit-year. A positive \(\alpha\) means that larger accumulated deviation from baseline is associated with a higher conditional hazard, holding the stated comparison fixed. This is an association interpretation, not a claim that changing the marker would cause the hazard to change.
Worked Example: Translating an Exposure Difference Into a Hazard Ratio
Consider two hypothetical subjects followed for 3 years. Their latent trajectories are \(m_1(t)=8+0.5t\) and \(m_2(t)=10+0.2t\), in marker-units. Suppose the fitted conditional hazard model has \(\alpha=0.12\) per marker-unit-year. Calculate the hazard ratio for subject 1 versus subject 2 at year 3, assuming their other covariates are equal.
For a linear trajectory, \(A_i(t)=b_{1i}t^2/2\). Thus, $$ A_1(3)=0.5(3^2)/2=2.25,\qquad A_2(3)=0.2(3^2)/2=0.90 $$ marker-unit-years. Their exposure difference is \(2.25-0.90=1.35\) marker-unit-years. The conditional hazard ratio is $$ \exp\{0.12(1.35)\}=\exp(0.162)\approx 1.176. $$ At year 3, subject 1’s conditional hazard is therefore about 1.176 times subject 2’s, or about 17.6% higher, under the model and the stated equal-covariate comparison. Their different baseline levels do not enter this contrast through \(A_i(t)\); only their accumulated deviations do.
Calculating the Integral From a Trajectory Model
In a joint model, \(m_i(t)\) is typically specified by a longitudinal submodel, often using subject-specific random effects. The cumulative association is then calculated from the fitted latent trajectory, rather than by treating a noisy observed marker value as the true underlying value. The integral depends on the chosen trajectory form.
For a quadratic trajectory \(m_i(t)=b_{0i}+b_{1i}t+b_{2i}t^2\), subtracting its baseline \(m_i(0)=b_{0i}\) leaves \(b_{1i}t+b_{2i}t^2\). Integrating term by term gives
Worked Example: A Quadratic Trajectory
Suppose a hypothetical subject’s latent marker trajectory is \(m(t)=6+0.4t+0.05t^2\), with time in years and marker values in marker-units. Find the baseline-adjusted and unadjusted cumulative exposures at 4 years. Then, for \(\alpha=0.07\) per marker-unit-year, compare the subject’s conditional hazard with that of a reference trajectory having \(A(4)=0\), with other covariates equal.
The baseline is \(m(0)=6\), so the deviation is \(0.4t+0.05t^2\). The adjusted integral is $$ A(4)=\frac{0.4(4^2)}{2}+\frac{0.05(4^3)}{3} =3.2+\frac{3.2}{3} \approx 4.267 $$ marker-unit-years. The unadjusted integral adds the baseline contribution \(4(6)=24\), giving \(C(4)=24+4.267=28.267\) marker-unit-years. The conditional hazard ratio relative to the stated reference is $$ \exp\{0.07(4.267-0)\}=\exp(0.2987)\approx 1.348. $$ Under the model, the subject’s conditional hazard at year 4 is about 1.348 times the reference hazard, or about 34.8% higher, for this difference in accumulated deviation.
Baseline Centering Is Not the Same as Baseline Covariate Adjustment
The subtraction of \(m_i(0)\) centers each subject’s trajectory at their own baseline. It does not force subjects with different baselines to have the same event risk. If baseline marker level may have an association with the event beyond its role in defining the accumulated deviation, it can be included explicitly, for example as a covariate \(m_i(0)\):
In this specification, \(\gamma_0\) represents the baseline marker association and \(\alpha\) represents the association with accumulated deviation, conditional on the other terms. Whether baseline should be included this way depends on the scientific question and model specification. The identity \(C_i(t)=A_i(t)+tm_i(0)\) also means that cumulative exposure definitions can encode baseline level differently; the coefficients from models using the two definitions should not be treated as interchangeable.
| Association quantity | What is accumulated? | Interpretation at a fixed time |
|---|---|---|
| Unadjusted cumulative exposure \(C_i(t)\) | The marker level \(m_i(s)\) | Accumulated marker exposure, including the baseline level’s contribution over time |
| Baseline-adjusted cumulative exposure \(A_i(t)\) | The deviation \(m_i(s)-m_i(0)\) | Accumulated excess or deficit relative to the subject’s baseline |
| Current marker \(m_i(t)\) | No accumulation | Association with marker level at the current time |
Assumptions, Diagnostics, and Common Pitfalls
The interpretation relies on a well-defined baseline time and a trajectory model that represents the latent marker over the interval being integrated. It also relies on the hazard model’s conditional structure: the proportional-hazards form and log-linear association in \(A_i(t)\) must be reasonable, and relevant covariates and dependence between the longitudinal and event processes must be handled by the joint model. A cumulative link does not automatically correct informative dropout, misspecified longitudinal trajectories, or measurement error in baseline.
Useful checks include examining whether the longitudinal model captures the observed marker patterns, checking whether the estimated trajectory is credible across follow-up, and assessing whether the assumed hazard association is adequate. Sparse measurements near baseline can make \(m_i(0)\) and hence every later deviation uncertain. A trajectory model that fits poorly can produce a misleading integral even if the integration arithmetic is correct. The baseline-adjusted quantity can also be strongly related to time and other trajectory features, so estimates may be imprecise when the data provide little variation in accumulated deviations.
- Mistake: Calling this “baseline adjustment” means baseline differences are controlled for. The definition subtracts each subject’s baseline from the trajectory; it does not necessarily include baseline marker as a covariate. Include a separate baseline term if the model and scientific question require it.
- Mistake: Interpreting \(\alpha\) as the effect of a one-unit increase in the marker. It is the log hazard ratio per one-unit increase in cumulative deviation, measured in marker-unit-years when time is in years.
- Mistake: Treating equal adjusted exposure as equal total exposure. Equal \(A_i(t)\) does not imply equal \(C_i(t)\) when baseline levels differ, because \(C_i(t)=A_i(t)+tm_i(0)\).
- Mistake: Giving a causal interpretation to the hazard ratio. The model describes a conditional association. Causal conclusions require additional design and identification assumptions.
Check Your Understanding
Try each question, then compare your reasoning with the brief solutions below.
- A linear trajectory is \(m(t)=7+0.3t\). Find its baseline-adjusted cumulative exposure at \(t=5\) years.
- For the trajectory in question 1, find the unadjusted cumulative exposure at 5 years and verify the decomposition into adjusted exposure plus baseline contribution.
- Two subjects have baseline-adjusted cumulative exposures of 3.0 and 1.5 marker-unit-years at the same time. If \(\alpha=0.10\) per marker-unit-year and other covariates are equal, calculate the conditional hazard ratio for the first subject versus the second.
- In one sentence, explain why equal baseline-adjusted cumulative exposures do not necessarily imply equal unadjusted cumulative exposures.
Solutions. (1) \(A(5)=0.3(5^2)/2=3.75\) marker-unit-years. (2) \(C(5)=7(5)+3.75=38.75\) marker-unit-years; equivalently, \(A(5)+5m(0)=3.75+35=38.75\). (3) The exposure difference is \(3.0-1.5=1.5\), so the hazard ratio is \(\exp\{0.10(1.5)\}=\exp(0.15)\approx1.162\); the first subject’s conditional hazard is about 16.2% higher. (4) The unadjusted cumulative exposure also includes the baseline contribution \(t\,m_i(0)\), which may differ across subjects.