Abstract
Augmenting randomised trials with external controls is increasingly common in oncology and rare diseases. This paper reviews the methodology of externally controlled trials - estimands, identification assumptions, estimation, efficiency and sensitivity analysis - for right-censored survival outcomes, and organises it around a single question: what quantity does a borrowing analysis actually deliver? We distinguish the intended estimand, fixed in advance by the clinical question and the target population, from the borrowing-dependent estimand, the quantity to which a given borrowing rule is consistent. The two coincide under source-conditional exchangeability and separate otherwise, so that the borrowing-dependent estimand names, and makes estimable, a discrepancy that is otherwise reported only as unquantified bias. Within a nonparametric potential-outcomes framework we (i) define borrowing-weighted causal contrasts indexed by a weighting kernel and set out when such a contrast is, and is not, a scientifically meaningful target in its own right; (ii) establish nonparametric identification and derive the efficient influence function; (iii) construct a doubly robust, locally efficient estimator with n-consistency; (iv) introduce a sensitivity-analysis framework yielding closed-form bias bounds, E-values, and tipping points; and (v) characterise the sensitivity of the delivered estimand to the pre-specified borrowing rule via Fréchet differentiability. Simulations across 12 factorial scenarios confirm up to 49% RMSE reduction under compatibility, show that the estimators remain unbiased for the borrowing-dependent estimand when compatibility fails, and validate the sensitivity bounds. A worked example in metastatic colorectal cancer illustrates the complete analytical workflow.