Slope Associations by Event Cause

Why the Rate of Change Can Matter

Two individuals can have the same biomarker value at a particular time but different recent trajectories: one value may be stable, while the other is rising or falling. A cause-specific hazard model with a slope association asks whether that rate of change is associated with the instantaneous rate of a particular event type, among individuals who have not yet experienced any event. The association can differ by event cause: a rising biomarker could be associated with a higher hazard of one event type and a lower hazard of another.

This is a model for an association with a latent trajectory, not a claim that changing the biomarker causes an event. It also concerns the derivative of the trajectory at the event time, not the biomarker’s current level or its accumulated history.

Definition and Model

Definition: Let \(T_i\) be the time to the first event for individual \(i\), and let \(D_i \in \{1,\ldots,K\}\) denote its cause. The cause-specific hazard for cause \(k\), conditional on the individual’s latent biomarker trajectory and covariates, is the instantaneous rate of a cause-\(k\) event at time \(t\), among individuals who are event-free just before \(t\). A slope association models this hazard using the derivative \(m_i'(t)\) of the latent biomarker trajectory \(m_i(t)\).

A slope-only association model for cause \(k\) is

\lambda_{ik}(t \mid b_i, \mathbf{w}_i) = \lambda_{0k}(t) \exp\!\left\{ \boldsymbol{\gamma}_k^\top \mathbf{w}_i + \alpha_k m_i'(t) \right\}.

Here \(\lambda_{0k}(t)\) is the cause-specific baseline hazard, \(\mathbf{w}_i\) is a vector of covariates, \(\boldsymbol{\gamma}_k\) contains their cause-specific coefficients, and \(b_i\) denotes any latent effects used to define the individual trajectory. The latent biomarker value is \(m_i(t)\), and \(m_i'(t)\) is its instantaneous rate of change. The coefficient \(\alpha_k\) is specific to cause \(k\). The model’s slope effect is log-linear and, as written, \(\alpha_k\) does not vary over time.

The units matter. If \(m_i(t)\) is measured in mg/dL and time in years, then \(m_i'(t)\) is measured in mg/dL per year. Consequently, \(\alpha_k\) is the log-hazard change per one mg/dL per year of slope difference. If the biomarker is decreasing, its slope is negative; a positive coefficient does not mean that every individual has a higher hazard, but that the hazard increases with increasing slope, all else equal.

What the Cause-Specific Slope Coefficient Means

Compare two individuals at the same time \(t\), with the same covariates and the same cause-specific baseline hazard, but with slopes \(s_A\) and \(s_B\). Dividing their cause-\(k\) hazards cancels the common baseline and covariate terms:

\frac{\lambda_{Ak}(t)}{\lambda_{Bk}(t)} = \frac{\lambda_{0k}(t)\exp\{\boldsymbol{\gamma}_k^\top\mathbf{w}+\alpha_k s_A\}} {\lambda_{0k}(t)\exp\{\boldsymbol{\gamma}_k^\top\mathbf{w}+\alpha_k s_B\}} = \exp\!\left\{\alpha_k(s_A-s_B)\right\}.

Thus a one-unit increase in slope, comparing otherwise identical trajectories at the same time, has cause-specific hazard ratio \(\exp(\alpha_k)\). For a difference of \(c\) slope units, the hazard ratio is \(\exp(c\alpha_k)\). This is a comparison of instantaneous cause-specific rates among those still event-free; it is not a comparison of event probabilities.

Worked Example: Interpreting a Cause-Specific Coefficient

Suppose a hypothetical model for cause 1 uses a biomarker measured in mg/dL and time in years. Its estimated slope coefficient is \(\hat{\alpha}_1=0.080\), with standard error \(SE=0.030\). Calculate and interpret the hazard ratio for a one-unit slope increase, and a 95% Wald confidence interval.

For a slope difference of 1 mg/dL per year, the estimated hazard ratio is

\exp(\hat{\alpha}_1)=\exp(0.080)=1.083.

The estimated cause-1 hazard is therefore about 8.3% higher per 1 mg/dL per year increase in slope, at the same time and covariate values, among individuals who remain event-free. This interpretation describes an association in the fitted model.

The coefficient-scale Wald interval is

0.080 \pm 1.96(0.030) = 0.080 \pm 0.0588 = (0.0212,\ 0.1388).

Exponentiating both endpoints gives the hazard-ratio interval

\left(\exp(0.0212),\exp(0.1388)\right) = (1.021,\ 1.149).

Under the model and the usual large-sample Wald approximation, this interval procedure has 95% coverage over repeated samples. The interval is compatible with cause-1 hazard increases of about 2.1% to 14.9% per 1 mg/dL per year slope increase. As an additional check, the Wald statistic is \(z=0.080/0.030=2.67\), giving a two-sided p-value of about 0.0077. This is evidence against \(H_0:\alpha_1=0\) under the model; it is not the probability that the null hypothesis is true.

Obtaining and Using the Latent Slope

The slope in this model is the derivative of the estimated latent longitudinal mean or trajectory, not necessarily a difference between two noisy observed measurements. For a smooth trajectory model, differentiate its fitted mean function with respect to time. For example, if the trajectory is quadratic, its slope changes with time. A single coefficient \(\alpha_k\) can therefore multiply different slope values at different event times, even though \(\alpha_k\) itself is constant.

The slope association is meaningful only with a clear time scale and biomarker scale. Changing time from years to months changes the numerical slope and hence the numerical coefficient. The implied hazard ratio for the same physical slope difference does not change if the coefficient is converted consistently.

Worked Example: Evaluating a Changing Trajectory at an Event Time

Consider hypothetical trajectories for a biomarker measured in mg/dL, with time \(t\) in years: \(m_A(t)=100+4t-0.5t^2\) and \(m_B(t)=90+t\). Suppose the cause-2 slope coefficient is \(\alpha_2=-0.25\) per mg/dL per year, and compare the trajectories at \(t=2\), holding covariates equal.

Differentiate each trajectory:

m_A'(t)=4-t,\qquad m_B'(t)=1.

At \(t=2\), the slopes are \(m_A'(2)=4-2=2\) mg/dL per year and \(m_B'(2)=1\) mg/dL per year. The slope difference is \(2-1=1\) mg/dL per year. Therefore the cause-2 hazard ratio for A relative to B is

\exp\{\alpha_2[m_A'(2)-m_B'(2)]\} = \exp\{-0.25(2-1)\} = \exp(-0.25) = 0.779.

At year 2, the model associates A’s one-unit-per-year higher slope with a cause-2 hazard about 22.1% lower than B’s, conditional on the stated comparison. This is a slope comparison at that time; it does not say that A has a lower cause-2 probability over a future interval.

Allowing the Association to Differ by Cause

A separate coefficient \(\alpha_k\) for each event cause allows the same slope contrast to have different associations with different causes. For instance, \(\alpha_1\) may be positive while \(\alpha_2\) is negative. These coefficients describe distinct cause-specific hazards, each defined among individuals who are event-free immediately before the time in question. A slope association for one cause should not be reported as though it were an association with “any event.”

A cause-specific hazard ratio alone does not determine how the probability of each event type changes over a follow-up interval. Competing causes also remove individuals from the event-free group, so absolute event probabilities depend on all relevant cause-specific hazards. The coefficient’s direct interpretation remains the cause-specific instantaneous rate comparison.

Worked Example: Opposite Slope Associations for Two Causes

Suppose a hypothetical fitted model has \(\alpha_1=0.12\) and \(\alpha_2=-0.20\), with slope measured in mg/dL per year. Compare individuals whose slopes differ by 3 mg/dL per year, with the higher-slope individual as the numerator in each hazard ratio.

For cause 1,

HR_1=\exp\{0.12(3)\}=\exp(0.36)=1.433.

The higher-slope individual has an estimated cause-1 hazard 1.433 times as large, or about 43.3% higher, under the model.

For cause 2,

HR_2=\exp\{-0.20(3)\}=\exp(-0.60)=0.549.

The same slope contrast corresponds to an estimated cause-2 hazard about 0.549 times as large, or about 45.1% lower. These different directions are not contradictory: they refer to different causes and different cause-specific hazards. Neither result alone gives the probability of experiencing either event type.

Assumptions, Diagnostics, and Limits

The interpretation above relies on the model being a reasonable description of the data. In particular, the trajectory model must represent the latent biomarker pattern well enough to estimate its derivative; the cause-specific hazard must have the specified log-linear dependence on slope; and the coefficient must be adequately represented as constant over time. The model must also represent the relevant covariate effects and cause-specific baseline hazards appropriately.

Because the trajectory is latent, the joint model must account for the relationship between longitudinal measurements and event times through the modeled latent trajectory and any included latent effects. Conditional on the modeled quantities and covariates, the remaining dependence and censoring assumptions used for estimation must also be appropriate. Event causes must be recorded consistently, and the competing-risks setup here concerns the first event, after which an individual is no longer in the event-free risk set for these cause-specific hazards.

Useful checks include inspecting longitudinal fit over time, paying particular attention to whether the fitted trajectories are smooth and supported where derivatives are evaluated. Derivatives can be unstable where observations are sparse, near the boundaries of follow-up, or when the trajectory form is poorly chosen. For the event component, assess whether the slope effect appears to vary with time, whether a log-linear slope relationship is plausible, and whether each cause has enough events to support its estimate. Uncertainty in the estimated slope should be carried through the joint-model estimation rather than treating a noisy observed difference as if it were the true slope.

Approach	Quantity linked to the cause-specific hazard	Interpretation
Slope association	Instantaneous latent rate of change \(m_i'(t)\)	Association with how quickly the biomarker is changing at time \(t\)
Current-value association	Latent biomarker level \(m_i(t)\)	Association with the biomarker level at time \(t\)
Observed-measurement approach	A recorded value or a finite difference from observed values	May reflect measurement error and visit timing, rather than the latent slope

The introductory material on current-value associations by event cause concerns the level \(m_i(t)\), which answers a different question from the derivative. A slope-only model can miss an association with the current level if one exists. Conversely, a current-value-only model does not represent an association with rate of change. A slope association also does not summarize accumulated biomarker history; that is a different modeling target.

A common mistake is to interpret \(\exp(\alpha_k)\) as the hazard ratio for a one-unit increase in the biomarker itself. That is incorrect: it is the hazard ratio for a one-unit increase in the biomarker’s slope, in the slope’s units. Another mistake is to say that a positive slope coefficient proves that raising the biomarker causes the event. It establishes a conditional model association, not a causal effect.

Key takeaway: In a cause-specific slope model, \(\exp\{\alpha_k(s_A-s_B)\}\) compares cause-\(k\) instantaneous hazards for two latent slopes at the same time and covariate values. Its direction, size, and units are specific to the event cause and slope contrast.

Check Your Understanding

Answer each question, then compare with the solutions below.

If \(\alpha_k=0.10\) per unit of slope, what is the hazard ratio for a two-unit higher slope for cause \(k\)?
A biomarker is measured in mmol/L and time in months. What are the units of its slope, and what does \(\exp(\alpha_k)\) compare?
For one cause, the fitted coefficient is negative. Does this imply that every individual with a rising biomarker has a lower event probability? Explain.
Why might substituting the difference between two noisy observed biomarker measurements for \(m_i'(t)\) be problematic?

Solutions. (1) The hazard ratio is \(\exp\{0.10(2)\}=\exp(0.20)=1.221\), so it is about 22.1% higher for the two-unit slope contrast. (2) The slope is in mmol/L per month; \(\exp(\alpha_k)\) compares cause-\(k\) hazards for a one-unit difference in that slope, at the same time and covariate values. (3) No. The coefficient describes an association in the cause-specific instantaneous hazard per slope contrast, not an individual event probability or a causal effect. (4) The difference can contain measurement error and depends on the spacing and timing of measurements; it may not represent the derivative of the latent trajectory at the event time.