Covariate Effects Across Cause-Specific Hazards

Covariate Effects Are Separate From Biomarker Associations

In a competing-risks joint model, the event cause matters: a covariate can be associated with one cause-specific hazard more strongly than another. A modeling decision is whether to constrain a covariate’s hazard coefficient to be the same across causes or to estimate separate coefficients. Either choice concerns the covariate component of the hazard model. It does not determine whether the latent biomarker trajectory has a shared or cause-specific association with those hazards.

The distinction is useful both for specifying a model and for interpreting it. For example, a hazard ratio for a treatment indicator describes its multiplicative association with a particular cause-specific hazard, conditional on the model’s other predictors, including the latent biomarker trajectory. It is not automatically a hazard ratio for a cumulative incidence, nor does it summarize a pathway through a treatment-related change in the biomarker.

Definition: For subject \(i\), let \(T_i\) be the event time, \(D_i\in\{1,\ldots,K\}\) the event cause, \(X_i\) a vector of covariates, and \(m_i(t)\) the latent biomarker trajectory at time \(t\). The cause-specific hazard for cause \(k\), conditional on the modeled covariates and latent quantities, is the instantaneous event rate for cause \(k\) among subjects who have not yet experienced any event. A covariate effect is shared when its hazard coefficient is constrained to be equal across causes; it is cause-specific when its coefficient is allowed to vary by cause.

Modeling Shared and Cause-Specific Effects

A useful general form for the cause-specific hazard is

h_{ik}(t\mid b_i,X_i) = h_{0k}(t)\exp\left\{X_i^\mathsf{T}\beta_k+\eta_{ik}(t)\right\}, \qquad k=1,\ldots,K.

Here \(h_{0k}(t)\) is the baseline hazard for cause \(k\), \(\beta_k\) is the vector of covariate coefficients for that cause, and \(\eta_{ik}(t)\) collects the modeled association with the latent longitudinal process and any other specified hazard terms. For a current-value association, for example, \(\eta_{ik}(t)=\alpha_k m_i(t)\). The model may use another association structure; the key point here is to keep that term distinct from \(X_i^\mathsf{T}\beta_k\).

For a covariate with coefficient \(\beta_k\), the cause-specific hazard ratio for a one-unit increase in that covariate, holding the other covariates and the latent trajectory fixed, is \(\exp(\beta_k)\). This follows by taking the ratio of the hazards at covariate values differing by one unit: the baseline hazard and the association term cancel, leaving \(\exp(\beta_k)\). Thus a shared coefficient gives the same conditional hazard ratio for each cause, while cause-specific coefficients allow those ratios to differ.

\frac{h_{ik}(t\mid X_{ij}+1,\ \text{other terms fixed})} {h_{ik}(t\mid X_{ij},\ \text{other terms fixed})} = \exp(\beta_{kj}).

A fully shared model imposes \(\beta_{1j}=\cdots=\beta_{Kj}=\beta_j\) for the covariate \(X_{ij}\). A fully cause-specific model estimates a separate \(\beta_{kj}\) for every cause. Partial sharing is also possible: one covariate can have a common coefficient while another has cause-specific coefficients.

For a single covariate, a convenient reference-cause interaction coding is

\beta_1=\beta,\qquad \beta_k=\beta+\delta_k\quad (k=2,\ldots,K), \qquad \delta_1=0.

Under this coding, \(\beta\) is the covariate coefficient for cause 1, and \(\delta_k\) is the difference between the coefficient for cause \(k\) and that for cause 1. The effects are shared across all causes when \(\delta_2=\cdots=\delta_K=0\). This parameterization does not force sharing: it represents the unrestricted cause-specific coefficients unless those deviations are constrained to zero.

Worked Example: A Shared Covariate Effect

Suppose a hypothetical two-cause joint model includes age measured in decades, and the model constrains its coefficient to be common across the causes. The estimated coefficient is \(0.30\), with standard error \(0.12\). Interpret the estimate and calculate a 95% Wald confidence interval for the common hazard ratio per additional decade of age.

The coefficient applies to both cause-specific hazards. The estimated hazard ratio is

\exp(0.30)=1.350.

Conditional on the other modeled terms, including the latent biomarker trajectory, a ten-year increase in age is associated with an estimated 1.350-fold cause-specific hazard for each cause, or a 35.0% higher hazard. This is a conditional association, not a causal effect.

On the coefficient scale, the 95% Wald interval is \(0.30\pm1.96(0.12)=0.30\pm0.2352\), giving \((0.0648,0.5352)\). Exponentiating both endpoints gives

\left(\exp(0.0648),\exp(0.5352)\right)=(1.067,1.708).

Under the fitted model, this interval estimates a range of plausible values for the common conditional hazard ratio per decade of age. Its use for both causes follows from the equality constraint in the model.

Interpreting Differences Between Causes

When coefficients differ, the cause-specific hazard ratio for a covariate must be reported separately for each cause. The difference \(\beta_k-\beta_\ell\) compares the covariate’s log-hazard effects across causes; its exponential, \(\exp(\beta_k-\beta_\ell)\), is the ratio of the two cause-specific hazard ratios. It is not, by itself, a ratio of the cause-\(k\) hazard to the cause-\(\ell\) hazard for an individual, because those hazards also depend on their different baseline hazards and other terms.

Worked Example: Cause-Specific Effects and Their Contrast

In a hypothetical model, a one-unit increase in a binary exposure has estimated coefficients \(\hat\beta_1=0.40\) and \(\hat\beta_2=-0.20\) for causes 1 and 2. Suppose their estimated variances are \(0.0225\) and \(0.0400\), and their estimated covariance is zero. Calculate the hazard ratios and a Wald test of equal exposure effects.

For cause 1, the estimated conditional hazard ratio is \(\exp(0.40)=1.492\). For cause 2, it is \(\exp(-0.20)=0.8187\). Thus the exposure is associated with a higher cause-1 hazard and a lower cause-2 hazard, conditional on the other modeled terms.

The estimated coefficient difference is \(0.40-(-0.20)=0.60\). Under the stated covariance assumption, its standard error is

SE(\hat\beta_1-\hat\beta_2) = \sqrt{0.0225+0.0400-2(0)} = 0.250.

For \(H_0:\beta_1=\beta_2\), the Wald statistic is \(z=0.60/0.250=2.40\), giving a two-sided normal-reference \(p\)-value of approximately \(0.0164\). At significance level \(0.05\), this is evidence against equal exposure coefficients under the model. The ratio of estimated cause-specific hazard ratios is \(\exp(0.60)=1.822\): the exposure’s multiplicative association is estimated to be 1.822 times as large for cause 1 as for cause 2. This comparison is not a comparison of the two causes’ absolute hazards.

Keeping Covariate and Biomarker Terms Distinct

A coefficient on \(X_i\) and an association parameter for \(m_i(t)\) answer different conditional questions. In a current-value model, \(\beta_k\) describes how the cause-\(k\) hazard changes with \(X_i\) at a fixed biomarker trajectory, whereas \(\alpha_k\) describes how it changes with the latent biomarker value at fixed covariates. The introductory material on cause-specific association structures establishes these association terms; their inclusion does not make them covariate coefficients.

Worked Example: A Covariate Contrast Alongside a Biomarker Association

Consider a hypothetical two-cause model with a binary exposure \(X_i\), reference-cause coding \(\beta_1=0.25\) and \(\beta_2=0.25-0.45=-0.20\), and a shared current-value association \(\alpha_1=\alpha_2=0.60\). Let the biomarker be measured in standardized units. At a particular time, compare two otherwise identical subjects with the same latent biomarker value \(m_i(t)=2\), but with exposure values 1 and 0.

Because the biomarker value is held fixed, the cause-1 hazard ratio is \(\exp(0.25)=1.284\), and the cause-2 hazard ratio is \(\exp(-0.20)=0.8187\). The term \(0.60m_i(t)=1.20\) appears in both subjects’ linear predictors and cancels from each ratio. The covariate effects therefore remain distinct from the biomarker association.

For a separate contrast, a one-standardized-unit increase in the latent biomarker, with exposure fixed, has estimated hazard ratio \(\exp(0.60)=1.822\) for either cause. If instead the exposure contrast also corresponds to a \(0.30\)-unit difference in the latent biomarker at that time, the model-based predictor contrast includes both terms. For cause 1 it is \(0.25+0.60(0.30)=0.43\), giving \(\exp(0.43)=1.537\); for cause 2 it is \(-0.20+0.60(0.30)=-0.02\), giving \(\exp(-0.02)=0.9802\). These last calculations combine a covariate difference and a trajectory difference. They are not the conditional covariate hazard ratios, and they do not establish a causal or mediated effect.

Assumptions, Diagnostics, and Model Choice

The interpretations above rely on the hazard specification being appropriate. In particular, the stated exponential form assumes that each covariate has a multiplicative effect on its cause-specific hazard. A coefficient that does not depend on time imposes a proportional-effects assumption for that covariate within each cause. The hazard is conditional on the modeled latent quantities and covariates; the observed event cause is treated as correctly recorded, and censoring must be handled under an appropriate conditional independent-censoring assumption for the likelihood-based interpretation.

For a coefficient interpreted per unit increase, the covariate scale and functional form also matter. A linear term assumes a constant change in log hazard per unit. If that assumption is unsuitable, a transformation or a more flexible covariate function may be needed. The model’s hazard-ratio interpretation is always relative to the specified scale and conditioning set.

Diagnostics should examine whether the covariate effect appears compatible with proportionality for each cause, whether the covariate form is plausible, and whether estimates are unduly imprecise. Cause-specific plots or residual-based checks can reveal time patterns inconsistent with a constant coefficient. Large standard errors, unstable estimates, or strong dependence on a small number of events may signal that cause-specific effects are weakly supported by the data. A good fit for the event model does not by itself verify the longitudinal submodel or the assumed biomarker association structure.

A fully shared model is more parsimonious and can be useful when equal covariate effects across causes are scientifically defensible. An unrestricted model is more flexible but estimates more coefficients. The interaction parameterization makes the restriction explicit: for two causes, equality is \(H_0:\delta_2=0\). A likelihood-based or Wald test can assess that restriction, subject to the model and its inferential assumptions. Failure to reject equality is not proof that the effects are identical; precision and scientific plausibility matter as well.

These choices do not replace the cause-specific baseline hazards or the biomarker association terms. Nor does a cause-specific hazard ratio directly describe the cumulative incidence of a cause: cumulative incidence depends on the hazards of all competing causes. The conditional hazard ratio is best reported with its cause, covariate increment, conditioning interpretation, and uncertainty.

Common pitfall: It is incorrect to interpret a shared covariate coefficient as evidence that the biomarker association is also shared, or to interpret a cause-specific covariate hazard ratio as a cumulative-incidence ratio. The covariate coefficient and the biomarker association parameter are separate model components, and cumulative incidence reflects all competing cause-specific hazards.

Check Your Understanding

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

In a two-cause model, \(\beta_1=0.10\) and \(\beta_2=0.10\). What is the cause-specific hazard ratio for a one-unit covariate increase, and what constraint represents a shared effect?
With reference-cause coding, \(\beta_1=0.20\) and \(\delta_2=-0.30\). Find the cause-2 coefficient and its one-unit hazard ratio.
Why does \(\exp(\beta_k-\beta_\ell)\) not generally equal the cause-\(k\) hazard divided by the cause-\(\ell\) hazard?
If a current-value association is \(\alpha_k=0.40\), what is the hazard ratio for a one-unit increase in the latent biomarker, holding covariates fixed?

Solutions. (1) Both causes have hazard ratio \(\exp(0.10)=1.105\); sharing is the constraint \(\beta_1=\beta_2\). (2) \(\beta_2=\beta_1+\delta_2=0.20-0.30=-0.10\), so the hazard ratio is \(\exp(-0.10)=0.9048\). (3) The coefficient contrast compares the covariate effects, whereas the hazards also contain cause-specific baseline hazards and other cause-specific terms. (4) The biomarker hazard ratio is \(\exp(0.40)=1.492\); it is distinct from a covariate hazard ratio.