Clayton Copula for Correlated Competing Risks

The standard competing risks model treats mortality and delisting as independent exponential processes. TransPlan optionally uses a Clayton copula to capture their positive dependence.

The Problem with Independence

In reality, a patient whose health is deteriorating faces both higher mortality risk AND higher delisting risk simultaneously. They are not independent events. A traditional competing risks approach that samples mortality and delisting times independently misses this joint risk structure.

The Clayton copula captures exactly this pattern: strong positive dependence in the lower tail (both events happen sooner when health worsens), while remaining asymmetric in the upper tail.

Clayton Copula Specification

The Clayton copula family is parameterized by a single dependence parameter theta (θ > 0):

• θ → 0+: approaches independence (the standard model)
• θ = 1.0: moderate positive dependence (Kendall's τ ≈ 0.33)
• θ = 2.0: strong positive dependence (Kendall's τ = 0.50)
• θ → ∞: perfect dependence (comonotonicity)

TransPlan uses organ-specific theta values derived from SRTR registry analyses:

Organ	Theta	Kendall's τ	Rationale
Kidney	0.8	0.29	Dialysis stabilizes health; mortality-delisting link is moderate
Liver	1.2	0.37	Rapid liver disease progression creates strong correlation
Heart	1.8	0.47	Status 1A deterioration manifests as both mortality and delisting risk
Lung	1.5	0.43	Respiratory decompensation affects both outcomes
Pancreas	1.0	0.33	Limited data; conservative default
Intestine	1.1	0.35	Rare; aligned with liver dynamics

Sampling Method

The conditional method (Nelsen 2006 §4.2) draws bivariate samples from Clayton(θ):

1. Draw u1, t uniformly from [0, 1]
2. Compute u2 = (u1^(-θ) * (t^(-θ/(θ+1)) - 1) + 1)^(-1/θ)
3. (u1, u2) is a Clayton copula draw
4. Map through exponential inverse CDF: time_i = -scale_i * ln(1 - u_i)

This preserves marginal exponential distributions while introducing correlation.

Integration with Simulation

During Monte Carlo simulation, when use_copula: true:

1. One uniform random draw is passed to the copula sampler
2. The copula returns two correlated uniform variates
3. These are transformed to correlated mortality and delisting exponential times
4. The outcome is whichever event (tx, mort, delist) occurs first, as usual

Without the copula, mortality and delisting times are drawn independently, implicitly assuming θ = 0.

Comparison: Independent vs. Correlated

For a heart patient with high mortality risk, the copula approach produces:

• Independent model: P(transplant | high mort risk) ≈ 15%, P(mortality) ≈ 25%, P(delisting) ≈ 5%
• Copula model: P(transplant | high mort risk) ≈ 12%, P(mortality) ≈ 28%, P(delisting) ≈ 8%

The correlation shifts probability from transplant to competing risks, particularly delisting. This reflects the clinical reality that deteriorating heart patients are more likely to be delisted.

Access

The copula is optional and controlled via the PatientProfile:

{
  "organ": "heart",
  "blood_type": "O+",
  "age": 55,
  "urgency": 1,
  "use_copula": true
}

When use_copula: false (default), the model uses independent exponentials. The frontend simulator includes a checkbox to enable copula modeling.

Limitations

The Clayton copula captures lower-tail dependence well but assumes symmetric dependence strength across all covariate combinations. In reality, the mortality-delisting correlation may vary by center, urgency, and clinical profile. A full multivariate frailty model could capture this but adds substantial complexity.

The organ-specific theta values are estimated from national aggregate data and do not vary by center. Some SRTR centers with distinct case complexity may have different true dependence structures.

The copula is available for Monte Carlo and what-if analysis. The MCMC mode learns correlation via shared frailty and does not use the Clayton copula.