Time-to-Event Analysis With External Controls

Why Time Zero Matters

A time-to-event comparison depends on more than the event dates. It also depends on when follow-up begins, which patients are eligible at that point, how the event is defined, and how censoring is handled. These choices are especially consequential when trial participants are compared with patients assembled from a separate registry or routine-care data source. If the sources start follow-up at different clinical milestones, the estimated contrast can reflect that misalignment rather than differences between treatment strategies.

For example, starting external-control follow-up at diagnosis while starting trial follow-up at randomization gives the two groups different periods at risk. A patient must survive long enough after diagnosis to enter a trial; counting that earlier period only for the external control can create a misleading advantage or disadvantage. The time origin should represent the same decision point in both sources, such as meeting eligibility criteria and beginning the treatment strategy being compared.

Adjusted Bayesian Survival Models

Let \(A=1\) denote trial treatment and \(A=0\) denote the external control. Let \(X\) denote measured baseline covariates, such as age, disease severity, or prior treatment. A survival function \(S_a(t\mid X)\) is the probability of remaining event-free beyond time \(t\), conditional on treatment group \(a\) and covariates \(X\). The hazard \(h_a(t\mid X)\) is the instantaneous event rate at time \(t\), conditional on having remained event-free until then.

One useful model is a piecewise exponential proportional-hazards model. Divide follow-up into prespecified intervals, indexed by \(j\), and let the baseline hazard be constant within each interval. The model is

Here \(\lambda_{0j}\) is the baseline hazard in interval \(j\), \(\beta\) is the treatment coefficient, and \(\gamma\) contains covariate coefficients. Under this model, the treatment hazard ratio is \(\exp(\beta)\), conditional on the included covariates, and is assumed constant over time. A Bayesian analysis assigns priors to the unknown coefficients and baseline hazards, then combines the likelihood with those priors to obtain a posterior distribution. For example, a regularizing prior might be \(\beta\sim N(0,1.5^2)\); the baseline-hazard priors must also keep hazards positive.

For participant \(i\), let \(T_i\) be observed follow-up time and \(\delta_i=1\) if the event is observed, or 0 if the observation is censored. Conditional on the model parameters, the contribution to the likelihood is

An observed event contributes its hazard at the event time as well as survival up to that time. A censored observation contributes survival only up to its censoring time. This likelihood requires censoring to be independent of the event time conditional on treatment and included covariates. If loss to follow-up depends on unobserved prognosis even after conditioning, this model does not automatically correct the resulting bias.

The treatment coefficient also needs careful interpretation. Trial assignment may be randomized within the trial, but membership in the external-control group is not thereby randomized. A causal interpretation requires, among other conditions, that important confounders of source or treatment group and outcome are measured and adjusted, and that comparable patients have a positive chance of appearing in each group. It also requires compatible treatment definitions, endpoint ascertainment, and time origins. These are substantive assumptions, not consequences of Bayesian estimation. The earlier material on Evaluating External-Control Selection Bias is relevant to assessing whether the groups are plausibly comparable.

From Conditional Models to Standardized Survival

A conditional hazard ratio is not the only useful summary, and it may not be the most clinically direct one. Survival probabilities at a specified time can be standardized to a target population, such as the trial-eligible population. Standardization means predicting survival under each treatment for the covariate values in that target population and then averaging those predictions using the target population's covariate distribution.

For posterior draw \(m\), let \(S_a^{(m)}(t\mid X_i)\) be the predicted survival for target-population participant \(i\) under treatment \(a\). With \(N\) target participants and equal target weights, the standardized survival is

The standardized survival difference at time \(t\) for that draw is \(S_1^{(m)}(t)-S_0^{(m)}(t)\). Repeating this calculation over posterior draws produces a posterior distribution for the difference, from which a posterior mean, credible interval, or posterior probability of a positive difference can be reported. The result is tied to the named target population and time point; changing either can change the estimate.

The piecewise exponential model also gives a direct justification for the survival calculation. For a hazard \(h(t\mid X)\), survival through time \(t\) is the exponential of the negative cumulative hazard, where the cumulative hazard is the integrated hazard from 0 to \(t\):

Within an interval with constant hazard \(\lambda\) lasting \(d\) time units, the cumulative hazard is \(\lambda d\), so survival across that interval is \(\exp(-\lambda d)\). Across successive intervals, cumulative hazards add, giving survival equal to the exponential of minus their sum. This derivation explains why interval-specific hazards and interval boundaries affect predicted survival, even when the fitted model is otherwise well behaved.

Checking Proportional Hazards and Other Failure Modes

The proportional-hazards assumption says that the treatment hazard ratio is constant over follow-up, conditional on the modeled covariates. It can be questionable when effects are delayed, diminish, or change direction. A single hazard ratio can then conceal clinically important time patterns. Diagnostics should include plots of estimated survival by group, examination of time-specific contrasts, and assessment of whether adding treatment-by-time terms changes the fit or the substantive conclusions. In a Bayesian analysis, posterior uncertainty in these quantities should be shown rather than treating a noisy pattern as definitive evidence of a violation.

If proportional hazards is not plausible, a time-varying treatment coefficient can be considered, for example \(\beta(t)\), or a more flexible survival model can be fitted. The reported comparison should then emphasize time-specific standardized survival or another clearly defined time-based summary rather than forcing a constant hazard ratio. The choice of intervals in a piecewise model should also be examined: too few intervals can miss changes in hazard, while too many can leave sparse intervals poorly informed by events.

Other important checks concern data and design. Confirm that event dates, censoring dates, endpoint definitions, eligibility, and covariates are measured comparably across sources. Inspect whether follow-up is truncated differently, whether some target covariate profiles lack external controls, and whether conclusions change under reasonable modeling choices. A model cannot recover information for covariate patterns with no meaningful overlap or remove bias from unmeasured confounding merely by including a prior.

A related alternative is restricted mean survival time through a fixed horizon: the expected event-free time accumulated up to that horizon. It can provide an interpretable time-scale contrast when hazards are not proportional, but it is a different estimand from a hazard ratio or survival difference at a single time. The horizon must be chosen where follow-up supports estimation.

Common Mistakes and Limitations

• Using different time origins. A comparison of diagnosis-to-event time with randomization-to-event time does not represent the same follow-up period. Define and implement a common eligibility and start point first.
• Reading a hazard ratio as a survival difference. A hazard ratio compares instantaneous rates among those still event-free; it does not directly state how many additional patients survive to a given time.
• Treating adjustment as randomization. Covariate adjustment addresses measured variables under a correctly specified analysis. It does not guarantee exchangeability or eliminate unmeasured confounding in external controls.
• Reporting a constant hazard ratio despite clear time variation. Check proportional hazards and use time-specific summaries or a flexible model when a constant effect is not credible.
• Assuming censoring is harmless. Censoring contributes valid information under the stated conditional independence assumption. Informative loss to follow-up can bias results and requires additional modeling or sensitivity analysis.

These methods compare time to a specified event. If a different event prevents the event of interest from occurring, treating that event as ordinary censoring can answer the wrong probability question. The event structure and estimand must be defined before choosing the survival analysis.