Time-Varying Association Parameters

Why Let the Association Change Over Time?

In a joint model, the longitudinal trajectory supplies a latent marker value at each time, while the event submodel relates that value to the instantaneous event rate. A constant association coefficient says that a one-unit difference in the current marker has the same multiplicative effect on the hazard at every time. That may be too restrictive: a marker difference could be more strongly associated with the event near the beginning or end of follow-up. A time-varying association coefficient allows that relationship to change.

The key is to distinguish how strongly marker differences are associated with the hazard from how high the baseline hazard is at a given time. Both can vary with time, but they play different roles in the model and imply different comparisons.

Model and Interpretation

Definition: Let $T_i$ be the event time for subject $i$, $w_i$ a vector of baseline covariates, and $m_i(t)$ the subject's latent longitudinal marker value at time $t$, as specified by the longitudinal submodel. A current-value hazard link with a time-varying association coefficient is $$ h_i(t\mid b_i)=h_0(t)\exp\left\{\gamma^\top w_i+\alpha(t)m_i(t)\right\}. $$ Here $b_i$ denotes the subject-specific random effects governing the latent trajectory, $h_0(t)$ is the baseline hazard, $\gamma$ is the vector of covariate coefficients, and $\alpha(t)$ is the association coefficient at time $t$.

This is a conditional hazard: it describes the instantaneous event rate at time $t$, among subjects still event-free just before $t$, given the random effects and covariates. The model assumes the association acts through the current latent marker value, not through an error-prone observed measurement at that instant. It also assumes the specified longitudinal trajectory and event-hazard forms are adequate, and that the event process is conditionally described by this hazard given the shared random effects and covariates. Right censoring must be noninformative under the analysis assumptions, often conditional on the modeled information.

The coefficient $\alpha(t)$ is a log-hazard association per one-unit difference in the marker at time $t$. If the marker is measured in units such as milligrams per liter, then $\alpha(t)$ is in inverse milligrams per liter. A positive value means higher current marker values are associated with a higher conditional hazard at that time; a negative value means they are associated with a lower hazard. This is an association conditional on the model, not by itself a causal effect.

The Time-Specific Hazard Ratio

Compare two subjects at the same time $t$, with the same $w_i$, but with latent marker values $m_A(t)$ and $m_B(t)$. Their conditional hazard ratio is obtained by dividing their hazards. The common baseline hazard and covariate contribution cancel:

\frac{h_A(t\mid b_A)}{h_B(t\mid b_B)} = \exp\left\{\alpha(t)\left[m_A(t)-m_B(t)\right]\right\}.

Thus a one-unit marker contrast has hazard ratio $\exp\{\alpha(t)\}$ at time $t$. A contrast of $\Delta$ units has hazard ratio $\exp\{\alpha(t)\Delta\}$. This is a comparison at the same time; it does not compare hazards at different times, where the baseline hazard may differ. If marker values themselves change over time, their changing difference also affects the hazard ratio, separately from changes in $\alpha(t)$.

Worked Example: A Strengthening Association Over Follow-Up

Suppose time is measured in years, the marker is measured in milligrams per liter, and the fitted association function is $\alpha(t)=0.15+0.05t$ per marker unit. Compare two subjects whose current latent marker values differ by 2 milligrams per liter at years 1 and 3. Assume their covariates are equal.

At year 1, $\alpha(1)=0.15+0.05(1)=0.20$. The time-specific hazard ratio is $ \exp\{0.20(2)\}=\exp(0.40)=1.492 $. At year 3, $\alpha(3)=0.15+0.05(3)=0.30$, giving $ \exp\{0.30(2)\}=\exp(0.60)=1.822 $.

At each time, the subject with the marker value higher by 2 milligrams per liter has the indicated conditional hazard relative to the other subject. Under this model, that same marker contrast is more strongly associated with the hazard at year 3 than at year 1. These are hazard ratios, not ratios of event probabilities over a year.

For illustration, let the baseline hazards be $0.040$ per year at year 1 and $0.060$ per year at year 3, with covariate contribution zero. If the two marker values at year 1 are 2 and 4, their hazards are $0.040\exp(0.20(2))=0.05967$ and $0.040\exp(0.20(4))=0.08902$ per year. Their ratio is $1.492$. At year 3, if their values are again 2 and 4, the hazards are $0.060\exp(0.30(2))=0.1093$ and $0.060\exp(0.30(4))=0.1992$ per year, with ratio $1.822$. The assumed $\alpha(t)$, baseline hazards, and covariate settings make the calculations comparable at each time.

Why Baseline-Hazard Changes Are Different

The baseline hazard can change freely with time in this formulation. It multiplies every subject's hazard at that time, whereas $\alpha(t)$ changes the multiplier associated with a marker contrast. A rise in $h_0(t)$ can raise all conditional hazards without changing the hazard ratio for a fixed marker contrast. Conversely, a change in $\alpha(t)$ changes that ratio even if $h_0(t)$ is unchanged.

Worked Example: A Changing Baseline With a Constant Association

Suppose $\alpha(t)=0.20$ per marker unit at both years 1 and 2, while $h_0(1)=0.040$ and $h_0(2)=0.080$ per year. Compare marker values 2 and 4, with equal covariates.

At either time, the hazard ratio for the 2-unit difference is $ \exp\{0.20(4-2)\}=\exp(0.40)=1.492 $. At year 1, the two hazards are $0.040\exp(0.40)=0.05967$ and $0.040\exp(0.80)=0.08902$ per year. At year 2, they are $0.080\exp(0.40)=0.1193$ and $0.080\exp(0.80)=0.1780$ per year.

Both hazards double between the two times because the baseline hazard doubles, but their ratio remains 1.492. The increasing absolute hazards do not show that the marker association strengthened. The calculation assumes the same marker contrast and the same association coefficient at both times; only the baseline hazard changes.

A short algebraic check makes the distinction explicit. For a fixed marker contrast $\Delta$, the log hazard ratio is

\log\left\{\frac{h_A(t)}{h_B(t)}\right\} = \log h_0(t)-\log h_0(t)+\alpha(t)\Delta = \alpha(t)\Delta.

The baseline term cancels from the contrast, while $\alpha(t)$ remains. This cancellation requires comparing subjects at the same time with the same modeled covariates. It does not mean the baseline hazard is unimportant for their absolute event rates.

Representing and Checking a Time-Varying Coefficient

In practice, $\alpha(t)$ cannot be estimated as an unrelated parameter at every possible time without imposing structure. A common approach represents it with a low-dimensional time function, for example a linear function or a spline basis:

\alpha(t)=\sum_{k=1}^{K}\theta_k B_k(t),

where $B_k(t)$ are chosen basis functions and $\theta_k$ are estimated parameters. The choice should allow meaningful variation without being so flexible that the data cannot support it. The time origin and marker scale matter for the numerical coefficients and their interpretation; the resulting $\alpha(t)$ should be interpreted on the stated time and marker scales.

Several checks are important. First, inspect the estimated function with uncertainty intervals, rather than interpreting its point estimate alone. Second, check how many subjects and events contribute at different times: a flexible function can be poorly determined near the beginning or end of follow-up. Third, check whether there is enough between-subject variation in the latent marker values at event times. If values barely vary, the data provide little information about how their association changes. Finally, assess whether the chosen trajectory and hazard forms are plausible; apparent time variation in $\alpha(t)$ may compensate for misspecification elsewhere in the joint model.

Worked Example: A Weakening Association

Suppose $\alpha(t)=0.40\exp(-0.20t)$ per marker unit, where time is in years. Compare two subjects whose current latent marker values differ by 1 unit, with equal covariates, at time 0 and time 5 years.

At time 0, $\alpha(0)=0.40$, so the hazard ratio is $\exp(0.40(1))=1.492$. At year 5, $\alpha(5)=0.40\exp(-1)=0.1472$, so the hazard ratio is $\exp(0.1472(1))=1.159$.

The fitted association implies that a one-unit higher current marker is associated with a 49.2% higher conditional hazard at time 0 and a 15.9% higher conditional hazard at year 5. This weakening describes a changing relative association for the specified marker contrast; it does not establish why the association changes. The interpretation assumes the function and marker scale are correct and that the comparison is made at the same time with covariates held equal.

Alternatives, Limitations, and Common Pitfalls

A constant association, $\alpha(t)=\alpha$, is a simpler alternative. It gives a constant hazard ratio for a fixed marker difference, even though the baseline hazard and the marker trajectories may vary. A time-varying baseline hazard is not an alternative to a time-varying association: baseline hazards are generally allowed to vary over time in either model. The scientific contrast is whether the marker's conditional log-hazard contribution per unit changes over time.

Time-varying association models have limitations. Flexible functions may be unstable when events or marker variation are sparse at some times. The result is conditional on the assumed longitudinal and event models, and the estimated association may be sensitive to marker scaling or trajectory misspecification. Also, a fitted $\alpha(t)$ describes the association at time $t$ among those still at risk under the model; it is not automatically a population-average effect or a causal effect.

Wrong reasoning: “The hazards are larger later, so the marker association must have strengthened.” This confuses the time-specific baseline hazard with the association coefficient.
Correction: Compare subjects at the same time. For a fixed marker contrast, calculate $\exp\{\alpha(t)\Delta\}$; $h_0(t)$ cancels.
Wrong reasoning: “The coefficient changed, so the hazard ratio changed by the same amount.” The coefficient is on the log-hazard scale.
Correction: Exponentiate the coefficient multiplied by the marker contrast. For a one-unit increase the ratio is $\exp\{\alpha(t)\}$, not $\alpha(t)$.
Wrong reasoning: “The model says the marker causes the event.” The model estimates a conditional association unless additional causal assumptions and design justify a causal interpretation.

Key takeaway: In a current-value link, $\alpha(t)$ is the time-specific log-hazard association per marker unit. For two subjects compared at the same time with equal covariates, the hazard ratio is $\exp\{\alpha(t)\Delta\}$. The baseline hazard controls the common time pattern in absolute hazards; it does not determine this same-time marker contrast.

Check Your Understanding

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

If $\alpha(t)=-0.10$ per unit at a given time, what is the hazard ratio for a 3-unit higher current marker value, comparing subjects at that same time and with equal covariates?
A model has constant $\alpha(t)=0.25$ but a baseline hazard that increases over time. Does the hazard ratio for a fixed 2-unit marker contrast change over time? Explain.
Suppose a marker is measured in units that are ten times as large, with all values correspondingly divided by ten. How must the numerical association coefficient change to preserve the same hazard ratios?
Why can a highly flexible estimate of $\alpha(t)$ be unreliable near the end of follow-up?

Solutions. (1) The ratio is $\exp\{(-0.10)(3)\}=\exp(-0.30)=0.7408$. The 3-unit higher value is associated with a 0.7408 times hazard, or about 25.9% lower hazard, at that time. (2) No. The ratio is $\exp\{0.25(2)\}=\exp(0.50)=1.649$ at every time. The baseline hazard may raise both hazards, but it cancels in the same-time ratio. (3) If the numerical marker values are divided by ten, the coefficient must be multiplied by ten: $\alpha_{\mathrm{new}}=10\alpha_{\mathrm{old}}$. Then $\alpha_{\mathrm{new}}\Delta_{\mathrm{new}}=\alpha_{\mathrm{old}}\Delta_{\mathrm{old}}$, preserving the hazard ratio. (4) Few subjects may remain at risk and few events may occur near the end, so there is limited information to estimate local changes in the association. Flexible functions can then have wide uncertainty or respond strongly to sparse data.