Cumulative Association: Definition and Hazard Ratio

Why Accumulated Marker History Can Matter

A marker’s value at the current time does not always summarize the history relevant to event risk. For example, two people can have the same marker value at year four even though one has had a persistently higher marker throughout follow-up. A cumulative association represents this difference in history by integrating the subject-specific latent marker trajectory over time.

The integral is calculated from the underlying trajectory, not by treating each noisy observed measurement as the true marker value. In a joint model, the longitudinal component describes that latent trajectory, while the event component relates it to the hazard. The focus here is the meaning of the cumulative link and the hazard ratio it implies.

Definition: Let $m_i(s)$ denote subject $i$’s latent marker value at time $s$, and let $t$ be the time at which event risk is being evaluated. The cumulative exposure through $t$ is $$ A_i(t)=\int_0^t m_i(s)\,ds. $$ It is the area under the latent marker trajectory from the chosen start of follow-up to $t$. If the marker is measured in marker-units and time in years, $A_i(t)$ has units of marker-unit-years.

The integral must exist and be finite over the interval being considered. The latent trajectory must be defined over that interval; it need not be directly observed at every time. The lower integration limit is the specified start of follow-up for this model. Thus, $A_i(t)$ describes accumulated marker exposure over that defined interval, not an absolute quantity independent of the study’s time scale.

A Proportional-Hazards Model with Cumulative Exposure

One standard cumulative association model is

h_i(t\mid b_i,\mathbf{w}_i) = h_0(t)\exp\left\{\boldsymbol{\gamma}^{\mathsf T}\mathbf{w}_i+ \alpha A_i(t)\right\}.

Here, $h_i(t\mid b_i,\mathbf{w}_i)$ is subject $i$’s hazard at time $t$, conditional on being event-free just before $t$, the latent trajectory (represented in part by subject-specific random effects $b_i$), and fixed covariates $\mathbf{w}_i$. The baseline hazard $h_0(t)$ is common to subjects with the same modeled covariates and latent exposure contribution. The coefficient $\alpha$ is the cumulative association parameter.

This specification assumes a log-linear association between the cumulative exposure and the hazard: increasing $A_i(t)$ by one unit multiplies the conditional hazard by $\exp(\alpha)$, at a fixed time and with the other modeled quantities held constant. It also assumes that the cumulative exposure enters through $A_i(t)$ in this form. In particular, the parameter does not describe a separate effect of the current marker value or its slope unless those terms are included in the model as well.

For $\alpha$ to have a coherent interpretation, the units and time scale must be specified. If $m_i(s)$ is measured in marker-units and time in years, then $A_i(t)$ has marker-unit-years and $\alpha$ has units of inverse marker-unit-years. The hazard itself is measured per unit of event-time, but the exponential term is dimensionless.

Deriving the Hazard Ratio

Compare two specified latent trajectories at the same time $t$, for subjects with the same fixed covariates. Write their cumulative exposures as $A_1(t)$ and $A_0(t)$. Under the model above, their conditional hazard ratio is

\frac{h_1(t)}{h_0^{*}(t)} = \frac{h_0(t)\exp\{\boldsymbol{\gamma}^{\mathsf T}\mathbf{w}+\alpha A_1(t)\}} {h_0(t)\exp\{\boldsymbol{\gamma}^{\mathsf T}\mathbf{w}+\alpha A_0(t)\}} = \exp\left\{\alpha\left[A_1(t)-A_0(t)\right]\right\}.

In this equation, $h_0^{*}(t)$ denotes the comparison subject’s hazard; it is distinct from the baseline hazard $h_0(t)$. The common baseline hazard and covariate contribution cancel because the comparison is made at the same time and covariate values. Defining the exposure difference as $\Delta A(t)=A_1(t)-A_0(t)$, the result is

HR(t)=\exp\{\alpha\Delta A(t)\}.

This is a conditional, time-specific hazard ratio. If the cumulative exposure difference changes over time, the hazard ratio can change over time even though $\alpha$ is constant. A positive $\alpha$ means that greater cumulative exposure is associated with a higher hazard; a negative $\alpha$ means it is associated with a lower hazard. Neither sign alone establishes a causal effect.

Worked Example: A Specified Increase in Accumulated Exposure

Suppose a fitted model has $\alpha=-0.12$ per marker-unit-year. At year four, compare two otherwise identical subjects whose cumulative exposures differ by $3.5$ marker-unit-years, with the first subject having the larger exposure. Find and interpret the conditional hazard ratio.

The exposure difference is $\Delta A(4)=3.5$ marker-unit-years. Substitution gives

HR(4)=\exp\{\alpha\Delta A(4)\} =\exp\{(-0.12)(3.5)\} =\exp(-0.42) \approx 0.6570.

At year four, the subject with the larger accumulated exposure has an estimated instantaneous hazard about $0.657$ times that of the comparison subject, or about $34.3\%$ lower, conditional on being event-free at that time and on the model quantities held equal. The result concerns the hazard at year four; it is not a statement that the probability of an event by year four is $34.3\%$ lower.

Worked Examples with Changing Exposure Differences and Uncertainty

Worked Example: The Hazard Ratio Can Change Over Follow-Up

Suppose $\alpha=0.08$ per marker-unit-year. At year two, subject A’s cumulative exposure exceeds subject B’s by $4$ marker-unit-years. At year five, the difference is $9$ marker-unit-years. Assume the same covariates for both subjects and that both remain event-free at the times being compared. Calculate the hazard ratio at each time.

At year two,

HR(2)=\exp\{(0.08)(4)\}=\exp(0.32)\approx 1.377.

At year five,

HR(5)=\exp\{(0.08)(9)\}=\exp(0.72)\approx 2.054.

The model associates A’s higher accumulated exposure with a hazard about $1.377$ times B’s at year two and about $2.054$ times B’s at year five. The coefficient is unchanged; the hazard ratios differ because the exposure difference has grown. These ratios compare instantaneous hazards at their respective times, not cumulative event probabilities.

Worked Example: A Confidence Interval for a Fixed Exposure Contrast

Suppose the estimated cumulative association is $\hat{\alpha}=0.060$, with standard error $SE=0.025$, in inverse marker-unit-years. For an exposure difference of $5$ marker-unit-years, calculate the estimated hazard ratio and an approximate 95% confidence interval. Assume the large-sample normal approximation for $\hat{\alpha}$ is appropriate.

First form the approximate coefficient interval:

0.060\pm 1.96(0.025) = 0.060\pm 0.049 = (0.011,\ 0.109).

The estimated log hazard ratio for this contrast is $5(0.060)=0.300$, so the estimated hazard ratio is

\exp(0.300)\approx 1.350.

Exponentiating the coefficient limits after multiplying by the contrast gives

\left(\exp\{5(0.011)\},\ \exp\{5(0.109)\}\right) = \left(\exp(0.055),\ \exp(0.545)\right) \approx (1.057,\ 1.725).

Under the stated approximation, the 95% confidence interval for the hazard ratio associated with a $5$ marker-unit-year increase is approximately $1.057$ to $1.725$. It describes the uncertainty of the model-based estimate; it is not a 95% probability that the fixed parameter lies in this particular interval.

Assumptions, Diagnostics, and Alternatives

The hazard-ratio calculation is valid as written when the model is correctly specified for the comparison: both subjects are evaluated at the same time, the non-exposure covariates are held equal, and the cumulative exposure contrast is defined on the same time and marker scales. The latent trajectory must be estimated adequately from the longitudinal data, and the cumulative integral must be meaningful over the interval. For likelihood-based inference in a joint model, the longitudinal and event components must be appropriately specified together; censoring must also be noninformative conditional on the modeled information, or handled by a suitable censoring model.

Diagnostics should examine both components that determine the cumulative association. Check whether the longitudinal model describes the observed repeated measurements and their variation over time, and whether its fitted latent trajectories are plausible over the periods being integrated. For the event component, assess whether the assumed log-linear association is reasonable and whether the fitted event model shows systematic lack of fit. A poor trajectory fit can distort the integral even when observed measurements appear individually plausible. Sensitivity to the trajectory specification and to the time interval used for accumulation can therefore be informative.

A current-value association uses the latent marker at time $t$, while a slope association uses its rate of change at $t$. A cumulative association instead uses the history represented by the integral through $t$. These alternatives answer different scientific questions; one should not interpret a cumulative coefficient as though it were a current-value or slope coefficient. The cumulative model also imposes a particular summary of history: exposure contributes through the specified integral, with the same $\alpha$ multiplying that accumulated quantity.

Common pitfall: It is incorrect to report $\exp(\alpha)$ as the hazard ratio for any arbitrary difference in cumulative exposure. It is the hazard ratio for a one-unit increase in $A_i(t)$. For a difference of $\Delta A(t)$ units, use $\exp\{\alpha\Delta A(t)\}$, with the units and comparison time stated.

A further limitation is that an association between accumulated marker history and hazard does not, by itself, show that changing the marker would change event risk. Unmeasured confounding, model misspecification, and uncertainty in the latent trajectory can all affect interpretation. The model provides a conditional statistical association under its assumptions, not an automatic causal conclusion.

Define the trajectory and interval.
Specify the latent marker $m_i(s)$, the time scale, and the start and evaluation times.

Form the cumulative contrast.
Compute or specify $\Delta A(t)=A_1(t)-A_0(t)$ in the model’s marker-time units.

Translate it to a hazard ratio.
Use $HR(t)=\exp\{\alpha\Delta A(t)\}$, keeping the comparison at the same time and other covariates.

Interpret conditionally.
Describe the instantaneous hazard comparison among subjects event-free at $t$, and do not substitute a risk or survival-probability interpretation.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

If $\alpha=0.10$ per marker-unit-year and the exposure difference is $2$ marker-unit-years, what is the hazard ratio?
What are the units of $A_i(t)$ and $\alpha$ if the marker is measured in milligrams per liter and time in months?
Why might two subjects with equal current marker values still have different hazards under a cumulative association model?
Does a hazard ratio of $1.4$ mean that the probability of an event over follow-up is $40\%$ greater? Explain.

Solutions. (1) $HR=\exp\{(0.10)(2)\}=\exp(0.20)\approx1.221$. (2) $A_i(t)$ is milligram-per-liter-months, and $\alpha$ has inverse milligram-per-liter-months as its units. (3) Their earlier latent marker histories can differ, giving different integrals even when their marker values match at the evaluation time. (4) No. It describes a ratio of instantaneous hazards at a specified time, conditional on the modeled quantities; it is not a ratio of event probabilities over follow-up.