Why Combine Current Value and Slope?
A biomarker may be associated with an event because of its level, because of how quickly it is changing, or both. For example, two people can have the same estimated biomarker level at a given time while one is stable and the other is rising. A hazard model using only the current value treats those profiles alike at that instant; a model that also includes the slope can distinguish them.
In a competing-risks setting, the distinction can matter separately for each event cause. A rising biomarker might be associated with a higher instantaneous hazard of one cause, while the same slope is weakly associated with, or inversely associated with, another. The model below makes these associations explicit. It describes conditional associations with cause-specific hazards; it does not by itself establish that changing the biomarker would change event risk.
Model and Conditional Interpretation
Let \(T_i\) be the time to the first event for individual \(i\), and let \(J_i\) identify its cause, with \(J_i=k\) for cause \(k\). Let \(m_i(t)\) denote the latent, error-free biomarker trajectory at time \(t\), and let \(m_i'(t)\) denote its instantaneous rate of change. The slope is the derivative of the trajectory with respect to time. Write \(X_i\) for covariates and \(b_i\) for the subject-specific random effects that determine the latent trajectory.
Precisely, the cause-\(k\) hazard is the instantaneous rate of experiencing cause \(k\) at time \(t\), given that the individual is still event-free just before \(t\), conditional on the stated trajectory and covariates. A cause-specific hazard is not the probability of eventually experiencing that cause. Individuals who experience a different cause leave the event-free risk set. Thus, the conditional interpretation includes both the covariate and trajectory values and being event-free up to the time under consideration.
In this model, \(\alpha_{k,m}\) is the change in the log cause-\(k\) hazard per one-unit increase in \(m_i(t)\), holding the slope and covariates fixed. Similarly, \(\alpha_{k,s}\) is the change in log hazard per one-unit increase in \(m_i'(t)\), holding the current value and covariates fixed. If the biomarker is measured in mg/L and time in months, then \(\alpha_{k,m}\) is per mg/L and \(\alpha_{k,s}\) is per mg/L per month. The units of each coefficient are the reciprocal of the units of its predictor.
The usual interpretation of an exponentiated coefficient is a conditional hazard ratio. For example, \(\exp(\alpha_{k,m})\) is the hazard ratio for a one-unit higher current value when slope and covariates are the same. These are conditional associations, not necessarily causal effects. In particular, changing the current value while holding the slope fixed may not correspond to a common, feasible change in the biomarker trajectory.
Assumptions and a Contrast Derivation
The model assumes a differentiable latent trajectory over the times at which the hazard is evaluated. It assumes that the log cause-specific hazard is linear in the included current-value and slope summaries, with their effects proportional over time unless interactions or time-varying coefficients are added. The baseline hazard may vary freely according to the chosen specification, but the association parameters in the displayed model do not vary with time. Covariate effects are also specified on the log-hazard scale.
As in a joint model, the longitudinal measurements inform an underlying trajectory rather than directly supplying a perfectly observed value and derivative at every instant. The model’s interpretation therefore depends on the adequacy of the longitudinal trajectory model and on representing the association between that trajectory and the event process. A slope is especially sensitive to how the trajectory is modelled and to the time scale used.
The hazard ratio for two trajectory profiles follows directly by taking a ratio of their cause-specific hazards. Consider two profiles, A and B, at the same time, with the same covariates and the same baseline hazard. Let their differences be \(\Delta m=m_A(t)-m_B(t)\) and \(\Delta s=m_A'(t)-m_B'(t)\). The baseline hazards and covariate terms cancel, giving:
This derivation shows why the combined model must be interpreted using both differences. A profile contrast that changes both level and slope has a log hazard ratio that is the sum of the two contributions. The individual coefficients are conditional effects, whereas the contrast describes the combined difference between the two specified profiles.
Worked Example: A Joint Change in Level and Slope
Suppose a hypothetical cause-1 model has \(\alpha_{1,m}=0.30\) per mg/L and \(\alpha_{1,s}=0.08\) per (mg/L per month). At a particular time, profile A has a latent level of 4 mg/L and slope of 0.2 mg/L per month. Profile B has a level of 3 mg/L and slope of \(-0.1\) mg/L per month. Assume both profiles have the same covariates and are compared at the same time.
The differences are \(\Delta m=4-3=1\) mg/L and \(\Delta s=0.2-(-0.1)=0.3\) mg/L per month. Therefore,
Under this fitted model, profile A has an estimated cause-1 hazard 1.383 times that of profile B at that time, conditional on being event-free and on the model’s conditioning variables. This is a 38.3% higher conditional cause-specific hazard, not a 38.3 percentage-point increase in the probability of cause 1. It is the combined contrast: the level and slope contributions cannot be reported as though only one had changed.
Worked Example: Different Associations for Two Causes
Consider a hypothetical model with two event causes. For cause 1, let \(\alpha_{1,m}=0.25\) per mg/L and \(\alpha_{1,s}=0.10\) per (mg/L per month). For cause 2, let \(\alpha_{2,m}=-0.15\) per mg/L and \(\alpha_{2,s}=0.20\) per (mg/L per month). Compare two profiles with the same covariates and time: one has a current value 2 mg/L higher and a slope 0.5 mg/L per month lower than the other.
For cause 1, the log hazard ratio is \( (0.25)(2)+(0.10)(-0.5)=0.50-0.05=0.45 \). Thus \(HR_1=\exp(0.45)=1.568\). The first profile has an estimated cause-1 hazard 1.568 times that of the second, conditional on the model’s covariates and event-free status.
For cause 2, the log hazard ratio is \( (-0.15)(2)+(0.20)(-0.5)=-0.30-0.10=-0.40 \). Thus \(HR_2=\exp(-0.40)=0.6703\). The same profile contrast is associated with a lower cause-2 hazard. These calculations concern separate cause-specific hazards. They do not alone determine the probability of either event cause by a future time, since that probability also depends on the hazards for all causes over time.
Correlation Between the Two Trajectory Summaries
Current value and slope often carry overlapping information. For example, if biomarker levels tend to be higher in individuals with upward trajectories, then the estimated current value and slope may be strongly correlated across the data. Including both can be scientifically appropriate, but it can make their separate coefficients imprecise. The model is then trying to distinguish the association with level from the association with rate of change using limited information about profiles that differ in one summary while remaining similar in the other.
Worked Example: Diagnosing Strong Predictor Correlation
Suppose a hypothetical analysis uses standardized current-value and slope summaries whose correlation is \(r=0.95\). As a rough design diagnostic, the variance inflation factor for one of two predictors in an ordinary regression with this correlation is
The corresponding standard-error inflation relative to uncorrelated predictors is approximately \(\sqrt{10.26}=3.20\) in that simple regression setting. This is a warning about near-redundant predictor information, not an exact standard-error formula for a joint competing-risks model. In the hypothetical fitted joint model, suppose the current-value and slope estimates are \(0.42\) and \(0.38\), with standard errors \(0.30\) and \(0.29\). The point estimates alone might suggest fairly substantial associations, but their standard errors show that the data provide limited precision for separating them. The diagnostic motivates inspecting the fitted estimates, uncertainty, and trajectory information rather than treating either point estimate as definitive.
Useful checks include examining the relationship between the fitted current values and slopes, checking whether the longitudinal model yields stable derivatives over the observed time range, and reviewing the uncertainty and dependence of the estimated association parameters. Large changes in the two coefficients when the other summary is included, wide confidence intervals, or unstable signs across reasonable model specifications can indicate that the data do not clearly separate the associations. These patterns do not prove that the model is invalid; they indicate that the individual conditional effects may be weakly determined.
Centering a predictor can make a coefficient’s reference point more interpretable, and rescaling can make coefficients easier to compare numerically. Neither operation removes the underlying correlation between current value and slope. Reporting a combined profile contrast may be more informative when the separate coefficients are imprecise, provided the contrast and its uncertainty are calculated from the fitted model. It is not appropriate to declare the two associations independent merely because both appear in the same equation.
How This Specification Differs From Alternatives
A current-value-only model includes \(m_i(t)\) but not \(m_i'(t)\); it asks whether the latent level is associated with the cause-specific hazard, without distinguishing profiles by their instantaneous trend. A slope-only model includes \(m_i'(t)\) but omits current value, so its slope association is conditional on the other terms in that model, not on level. The combined model asks whether each summary contributes conditional information given the other.
A cumulative-exposure association instead summarizes the trajectory accumulated over an interval or up to time \(t\), according to the specified exposure definition. It answers a different scientific question from the instantaneous level-and-slope model. The choice among these forms should follow the scientific meaning of the proposed association and the information supported by the observed trajectories, rather than a search for whichever specification yields the most striking coefficient.
| Specification | Trajectory summary in the hazard | Primary conditional question |
|---|---|---|
| Current value only | \(m_i(t)\) | Is the latent level associated with the cause-specific hazard? |
| Slope only | \(m_i'(t)\) | Is the instantaneous rate of change associated with the hazard? |
| Current value and slope | \(m_i(t)\) and \(m_i'(t)\) | Does each summary have an association conditional on the other? |
| Cumulative exposure | An accumulated trajectory summary | Is accumulated biomarker exposure associated with the hazard? |
Practical Limitations and Common Mistakes
The combined model is useful when both level and rate of change have a clear scientific rationale and the data contain enough trajectory variation to estimate their conditional associations. It has important limitations. Derivatives can be difficult to estimate reliably, especially with sparse or irregular measurements, and the estimated slope depends on the longitudinal model’s smoothness and time scale. Strong level-slope correlation can also leave individual association parameters uncertain even when predictions for observed profile patterns are more stable.
- Interpreting a cause-specific hazard as an event probability. A cause-specific hazard is an instantaneous rate among those still event-free, not the probability of experiencing that cause by a specified time.
- Ignoring the conditioning in each coefficient. In the combined model, \(\alpha_{k,m}\) is the level association holding slope fixed, and \(\alpha_{k,s}\) is the slope association holding level fixed. Neither is automatically the unadjusted association of that summary with the hazard.
- Reading the slope without units. A slope measured in mg/L per month has a different meaning from one measured in mg/L per year. State the time unit as well as the biomarker unit.
- Assuming that two included terms are separately well estimated. Including both terms does not eliminate their correlation. Review uncertainty and the range of observed trajectory profiles before making strong claims about either coefficient.
- Making a causal claim from an association model. Conditional hazard associations do not establish that intervening to change the biomarker would alter event risk.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In a model with biomarker units of mg/L and time measured in months, what are the units of \(\alpha_{k,m}\) and \(\alpha_{k,s}\)?
- Suppose \(\alpha_{k,m}=0.20\) per mg/L and \(\alpha_{k,s}=0.05\) per (mg/L per month). What is the hazard ratio for a profile that is 1 mg/L higher and 2 mg/L per month lower, with other terms held fixed?
- Does a cause-1 hazard ratio greater than 1 imply a higher probability of cause 1 by one year? Explain.
- Why might the separate current-value and slope estimates be imprecise when their predictors are strongly correlated?
Solutions. (1) \(\alpha_{k,m}\) is per mg/L, and \(\alpha_{k,s}\) is per mg/L per month. (2) The log hazard ratio is \((0.20)(1)+(0.05)(-2)=0.10\), so \(HR=\exp(0.10)=1.105\). The specified profile has an estimated cause-specific hazard about 1.105 times as large, conditionally. (3) No. A cause-specific hazard is an instantaneous rate among those still event-free; the probability of cause 1 by one year depends on the cause-specific hazards over time, including hazards for competing causes. (4) The data may contain few profiles that vary in level while holding slope similar, or vice versa. The model then has limited information to distinguish their conditional associations, increasing uncertainty in the separate estimates.