A decomposition of Fisher's information to inform sample size for developing fair and precise clinical prediction models -- Part 2: time-to-event outcomes

Background: When developing a clinical prediction model using time-to-event data, previous research focuses on the sample size to minimise overfitting and precisely estimate the overall risk. However, instability of individual-level risk estimates may still be large. Methods: We propose a decomposition of Fisher's information matrix to examine and calculate the sample size required for developing a model that aims for precise and fair risk estimates. We propose a six-step process which can be used before data collection or when an existing dataset is available. Steps (1) to (5) require researchers to specify the overall risk in the target population at a key time-point of interest; an assumed pragmatic 'core model' in the form of an exponential regression model; the (anticipated) joint distribution of core predictors included in that model; and the distribution of any censoring. Results: We derive closed-form solutions that decompose the variance of an individual's estimated event rate into Fisher's unit information matrix, predictor values and total sample size; this allows researchers to calculate and examine uncertainty distributions around individual risk estimates and misclassification probabilities for specified sample sizes. We provide an illustrative example in breast cancer and emphasise the importance of clinical context, including risk thresholds for decision making, and examine fairness concerns for pre- and post-menopausal women. Lastly, in two empirical evaluations, we provide reassurance that uncertainty interval widths based on our approach are close to using more flexible models. Conclusions: Our approach allows users to identify the (target) sample size required to develop a prediction model for time-to-event outcomes, via the pmstabilityss module. It aims to facilitate models with improved trust, reliability and fairness in individual-level predictions.