Joint Time-To-Event Outcomes

Why Model Efficacy and Toxicity Times Jointly?

In dose finding, efficacy and toxicity are often recorded as times from treatment to an event. A model for each time separately can describe how quickly efficacy and toxicity occur at a given dose. But separate models do not, by themselves, say whether patients who experience efficacy early are also more or less likely to experience toxicity early. That dependence can change the probability of a clinically relevant combination, such as efficacy by a specified time without toxicity by that time.

The distinction matters even when the marginal event risks are well estimated. A dose comparison based only on those risks may not determine the probability of a joint outcome. This tutorial focuses on that gap: what marginal hazard models specify, how a shared frailty links the two event times, and what assumptions are involved in using the resulting joint probabilities for early dose decisions.

A pair of marginal hazards determines the two marginal survival functions, using \(S_k(t\mid d)=\exp\{-\int_0^t h_k(u\mid d)\,du\}\) when the event-time distribution is continuous and there is no point mass at time zero. It does not determine the joint distribution of \((T_E,T_T)\). In particular, it does not determine \(P(T_E\le t,T_T\le t\mid d)\), or \(P(T_E\le t,T_T>t\mid d)\). Those require an assumption or model for dependence. Assuming independence is one such model, not a consequence of fitting two marginal hazards.

Marginal Hazards and the Independence Assumption

A simple marginal model at a fixed dose assumes constant hazards: \(h_E(t\mid d)=\lambda_E(d)\) and \(h_T(t\mid d)=\lambda_T(d)\), with positive rates measured per unit time. Then \(S_E(t\mid d)=e^{-\lambda_E(d)t}\), and the corresponding efficacy probability by time \(t\) is \(1-e^{-\lambda_E(d)t}\); the same expressions apply to toxicity with its own rate. To obtain joint probabilities from these marginal models, an analyst must specify how the event times are related.

Under independence, \(P(T_E>t,T_T>t\mid d)=S_E(t\mid d)S_T(t\mid d)\). Inclusion-exclusion then gives the probability of both events by \(t\) as \(1-S_E(t\mid d)-S_T(t\mid d)+S_E(t\mid d)S_T(t\mid d)\). Independence may be a useful reference model, but it is a substantive assumption. Patients can differ in underlying susceptibility, disease severity, or other features that affect both event processes.

A Shared-Frailty Model

A shared frailty model represents unobserved patient-to-patient heterogeneity by a positive random variable \(U\). Conditional on \(U\) and dose \(d\), the efficacy and toxicity event times are assumed independent, with hazards scaled by the same \(U\). A patient with larger \(U\) has faster event rates for both endpoints under this formulation. Conditional independence does not imply marginal independence: once \(U\) is averaged out, the event times are associated.

For constant conditional baseline rates \(\lambda_E(d)\) and \(\lambda_T(d)\), conditional survival to times \(t\) and \(s\) is \(\exp\{-U[\lambda_E(d)t+\lambda_T(d)s]\}\). Averaging over \(U\) gives a joint survival function. The derivation uses the gamma frailty Laplace transform; writing \(q=\lambda_E(d)t+\lambda_T(d)s\), the gamma density has shape \(1/\theta\) and rate \(1/\theta\), so its expectation is:

The last equality follows by the gamma integral \(\int_0^\infty u^{a-1}e^{-bu}\,du=\Gamma(a)b^{-a}\), for \(a,b>0\). Setting one time to zero gives each marginal survival function; keeping both times gives the joint survival function.

The shared frailty changes the marginal hazards as well as the dependence. For example, with constant conditional rate \(\lambda_E(d)\), the marginal efficacy hazard is \(\lambda_E(d)/[1+\theta\lambda_E(d)t]\), not the constant conditional rate. As time passes, patients with larger frailty tend to experience the event earlier and leave the event-free group, reducing the average hazard among those remaining event-free. The same reasoning applies to toxicity.

Using Joint Probabilities in an Early Dose Comparison

A dose decision may depend on more than the marginal probability of efficacy or toxicity. For example, one may compare the chance of efficacy by a chosen horizon without toxicity by that horizon. For any joint survival model, this probability is \(P(T_E\le t,T_T>t\mid d)=S_T(t\mid d)-P(T_E>t,T_T>t\mid d)\). A joint model supplies the second term; marginal models alone do not.

Assumptions, Diagnostics, and Alternatives

The shared-frailty calculations are interpretable only under their assumptions. The same nonnegative frailty multiplies both conditional hazards; the event times are independent conditional on that frailty; the frailty distribution is gamma with mean one; and the chosen dose-specific baseline hazards are appropriate. The constant-rate examples add the stronger assumption that each conditional hazard is constant over time. A fitted analysis also needs a defensible approach to incomplete event-time observation, including any censoring; the formulas above do not by themselves provide that approach.

Useful checks include comparing predicted and observed marginal event risks at clinically relevant horizons, checking whether the assumed time pattern of each marginal hazard is plausible, and examining whether predictions for joint outcome categories agree with the observed patterns. Large discrepancies in either marginal calibration or joint event patterns are warnings, not problems that can be repaired merely by changing the frailty variance. In small dose-finding samples, the dependence parameter may be weakly informed by the data, so uncertainty in \(\theta\) and sensitivity to plausible values should be made visible in decisions.

A shared frailty is not automatically a causal explanation for dependence: it is a statistical representation of latent heterogeneity. It also imposes a particular dependence pattern, which may not fit the clinical process. A copula offers a different way to link margins, while independence is a simpler special case. None is universally preferable; the choice should reflect the scientific question, data support, and sensitivity of the dose comparison to alternative plausible dependence assumptions.

Common Pitfall

• Wrong reasoning: “I have fitted an efficacy hazard and a toxicity hazard, so I can multiply their event probabilities to obtain the probability of both events.”
• Correction: Multiplication is justified only under independence of the relevant event indicators (or a model that entails that independence). Separate marginal hazards do not establish it. State the dependence assumption, calculate joint probabilities from the specified joint model, and examine whether a dose comparison changes under plausible alternatives.