Why Correlate Random Effects Across Causes?
In a competing-risks joint model, each event cause has its own cause-specific hazard. A subject may have an unobserved propensity that raises the hazard of one cause and also raises or lowers the hazard of another. Correlated random effects provide a way to represent that subject-level dependence between latent cause-specific hazard terms, beyond any dependence already induced by the subject’s longitudinal trajectory.
The distinction matters because two kinds of dependence can appear in the same model. A shared longitudinal random effect can influence multiple cause-specific hazards through their association terms. Correlated cause-specific random effects, in contrast, describe dependence between additional latent hazard components for the causes. These are different model components, with different interpretations.
Model Definition and Assumptions
Let \(i\) index subjects and \(k=1,\ldots,K\) index event causes. Write \(\lambda_{ik}(t\mid \mathbf b_i,\mathbf u_i)\) for subject \(i\)'s cause-specific hazard at time \(t\), conditional on longitudinal random effects \(\mathbf b_i\) and cause-specific random effects \(\mathbf u_i=(u_{i1},\ldots,u_{iK})^\mathsf{T}\). A useful specification is
Here \(\lambda_{0k}(t)\) is the baseline hazard for cause \(k\), \(\mathbf x_i\) is a vector of covariates, and \(\mathbf m_i(t;\mathbf b_i)\) is the latent longitudinal trajectory or the longitudinal feature used in the association term. The cause-specific association coefficients are collected in \(\boldsymbol\alpha_k\). The term \(u_{ik}\) is a subject-level latent deviation in the log hazard for cause \(k\). The following treatment focuses on the joint distribution of the vector \(\mathbf u_i\).
A standard working specification assumes independent subjects, a common covariance matrix across subjects, and independence of \(\mathbf u_i\) from \(\mathbf b_i\) and the longitudinal measurement errors. The latter assumption is useful for separating the two dependence pathways; it is a modeling assumption, not a mathematical necessity. Conditional on the random effects and covariates, the cause-specific hazards specify the competing-event process. For an observed first event, only one cause can occur; the subject’s event-free survival depends on the sum of the cause-specific hazards.
How the Covariance Enters the Hazards
Because \(u_{ik}\) enters the log hazard additively, \(\exp(u_{ik})\) is a multiplicative hazard factor. A positive value of \(u_{ik}\) raises the hazard for cause \(k\) relative to the same subject’s hazard with \(u_{ik}=0\), holding the other terms fixed. Covariance between \(u_{ik}\) and \(u_{i\ell}\) describes whether those latent log-hazard deviations tend to move together across subjects. It is not itself a hazard ratio, and it does not directly state the association between observed event indicators.
For two causes, suppose \((u_{i1},u_{i2})^\mathsf{T}\) is bivariate normal with mean zero, variances \(\sigma_1^2,\sigma_2^2\), and covariance \(\sigma_{12}\). The correlation is \(\rho=\sigma_{12}/(\sigma_1\sigma_2)\) when both variances are positive. The covariance matrix is valid only when its variances are nonnegative and \(\sigma_{12}^2\leq \sigma_1^2\sigma_2^2\), equivalently \(-1\leq\rho\leq1\).
Worked Example: Covariance of Multiplicative Hazard Factors
Suppose two hypothetical cause-specific random effects have mean zero, variances \(\sigma_1^2=0.25\) and \(\sigma_2^2=0.36\), and correlation \(\rho=0.60\). Calculate the covariance of their multiplicative factors \(U_{i1}=\exp(u_{i1})\) and \(U_{i2}=\exp(u_{i2})\).
First, the log-hazard covariance is \(\sigma_{12}=\rho\sigma_1\sigma_2=0.60(0.50)(0.60)=0.18\). For a jointly normal pair, the moment-generating function gives \(E[\exp(u_{i1}+u_{i2})]=\exp\{(\sigma_1^2+\sigma_2^2+2\sigma_{12})/2\}\). Also, \(E[\exp(u_{ik})]=\exp(\sigma_k^2/2)\). Therefore,
Thus, the covariance of the multiplicative factors is approximately \(0.2676\). It is positive because the log-hazard effects are positively correlated. This covariance is on the scale of the hazard multipliers; it is not the correlation, which remains \(0.60\) on the random-effect scale.
The lognormal covariance formula follows directly by subtracting the product of marginal expectations from the joint expectation. Specifically, \(E[\exp(u_{i1}+u_{i2})]=\exp\{(\sigma_1^2+\sigma_2^2)/2+\sigma_{12}\}\), while the product of marginal expectations is \(\exp\{(\sigma_1^2+\sigma_2^2)/2\}\). Factoring out the latter yields the displayed expression. Since the exponential function is increasing, the sign of this covariance matches the sign of \(\sigma_{12}\).
Separating Cause Effects from Shared Longitudinal Effects
The term \(\boldsymbol\alpha_k^\mathsf{T}\mathbf m_i(t;\mathbf b_i)\) already lets a subject’s latent trajectory influence cause \(k\)'s hazard. If the same longitudinal random effects enter more than one cause-specific hazard, those hazards can be associated through \(\mathbf b_i\), even when the components of \(\mathbf u_i\) are independent. Correlated \(\mathbf u_i\) adds a separate dependence pathway: residual subject-level variation in one cause’s log hazard can be related to residual variation in another cause’s log hazard.
A compact way to see the distinction is to isolate the random part of each log hazard. With one shared longitudinal random effect \(b_i\), independent of \(\mathbf u_i\), write \(r_{ik}=\alpha_k b_i+u_{ik}\). Then \(\operatorname{Cov}(r_{i1},r_{i2})=\alpha_1\alpha_2\operatorname{Var}(b_i)+\operatorname{Cov}(u_{i1},u_{i2})\). The first term is dependence through the shared longitudinal effect; the second is dependence between cause-specific random effects. Independence of the \(u\) terms does not remove the first term.
Worked Example: Shared Trajectory Versus Correlated Cause Effects
Consider two hypothetical causes. Let the shared longitudinal random intercept satisfy \(\operatorname{Var}(b_i)=0.40\), and let its cause-specific loadings be \(\alpha_1=0.70\) and \(\alpha_2=-0.50\). Assume that \(b_i\) is independent of \(\mathbf u_i\), and that \(\operatorname{Cov}(u_{i1},u_{i2})=0\). What is the covariance of the random parts of the two log hazards?
The random parts are \(r_{i1}=0.70b_i+u_{i1}\) and \(r_{i2}=-0.50b_i+u_{i2}\). By independence and the zero covariance between the \(u\) terms,
The log-hazard random parts are negatively associated because the shared trajectory raises one cause’s hazard and lowers the other’s. Now suppose instead that the cause-specific effects have covariance \(0.08\), with all else unchanged. The total covariance becomes \(-0.14+0.08=-0.06\). The cause effects add positive dependence, but do not fully offset the negative dependence induced by the shared trajectory. Thus, the sign of the total dependence need not match the sign of either component alone.
Specifying and Checking the Covariance Structure
For two causes, the covariance matrix can be written as \(\boldsymbol\Sigma_u=\begin{pmatrix}\sigma_1^2&\sigma_{12}\\\sigma_{12}&\sigma_2^2\end{pmatrix}\). One can parameterize the off-diagonal term through a correlation, \(\sigma_{12}=\rho\sigma_1\sigma_2\), which makes the admissible range explicit. For more causes, the full matrix must be positive semidefinite; choosing pairwise correlations independently within \([-1,1]\) is not enough to guarantee this condition for the whole matrix.
Worked Example: Checking a Two-Cause Covariance Matrix
Suppose a hypothetical fitted model reports standard deviations \(0.40\) and \(0.50\) for two cause-specific random effects and correlation \(-0.30\). Find the covariance and check whether the resulting matrix is positive definite.
The covariance is \(\sigma_{12}=\rho\sigma_1\sigma_2=(-0.30)(0.40)(0.50)=-0.06\). The covariance matrix is therefore \[ \boldsymbol\Sigma_u= \begin{pmatrix} 0.16&-0.06\\ -0.06&0.25 \end{pmatrix}. \] Its determinant is \(0.16(0.25)-(-0.06)^2=0.0400-0.0036=0.0364\), which is positive. Since both diagonal entries are positive and the determinant is positive, this two-by-two matrix is positive definite. The negative correlation means subjects with higher latent log-hazard deviations for one cause tend to have lower deviations for the other; it does not imply that each subject’s two hazards must move in opposite directions over time.
Interpretation, Diagnostics, and Limitations
A fitted covariance should be interpreted on the latent random-effect scale and alongside the variance estimates and association terms for the longitudinal trajectory. Its estimate may be uncertain, especially when one cause is rare or when the data provide little information about subject-level variation across causes. A covariance estimate near its admissible boundary can also indicate weak information or a difficult estimation problem; it should not be treated as automatically meaningful evidence of a strong relationship.
Useful checks include confirming that the estimated covariance matrix is positive semidefinite, examining uncertainty intervals for variances and correlations, and checking whether estimates are stable under reasonable alternative random-effect specifications. Compare the estimated cause-effect covariance with the longitudinal pathway: a model with \(\boldsymbol\Sigma_u\) diagonal still permits dependence through shared \(\mathbf b_i\). Conversely, if the fitted model assumes \(\mathbf u_i\) is independent of \(\mathbf b_i\), that assumption should be recognized when interpreting the components.
Correlated random effects are a distributional model for latent heterogeneity. They do not by themselves identify the biological mechanism producing that heterogeneity, establish a causal relationship between causes, or imply that the observed event types are positively or negatively associated in every marginal summary. Observed competing outcomes are constrained by the fact that only one first event is observed, and marginal event probabilities also depend on baseline hazards, covariates, longitudinal association terms, and the full random-effects distribution.
Compared with independent cause-specific random effects, the correlated specification is more flexible but requires estimating additional covariance parameters. Compared with dependence represented only through shared longitudinal random effects, it can represent residual cross-cause heterogeneity not explained by the longitudinal trajectory. Neither formulation is universally preferable: the choice is a substantive and statistical modeling decision, and the assumed structure should be supported by the available information.
Check Your Understanding
Answer each question, then compare with the concise solutions below.
- For two mean-zero Gaussian cause effects with variances \(0.16\) and \(0.25\) and correlation \(0.50\), what is their covariance?
- If cause-specific random effects are independent but both hazards depend on the same longitudinal random intercept, can the log hazards still be associated? Explain.
- For a two-cause covariance matrix with variances \(0.09\) and \(0.16\), what is the largest allowable absolute covariance?
- Does positive covariance between cause-specific random effects mean the observed event indicators for the two causes must be positively associated? Why or why not?
Solutions. (1) The covariance is \(0.50(0.40)(0.50)=0.10\). (2) Yes. The shared intercept contributes \(\alpha_1\alpha_2\operatorname{Var}(b_i)\) to the covariance of the log-hazard random parts, which can be nonzero. (3) The bound is \(\sqrt{0.09(0.16)}=0.12\), so the absolute covariance cannot exceed \(0.12\). (4) No. This covariance concerns latent log-hazard deviations; observed competing outcomes also depend on the event process and only one first event can be observed.