A decomposition of Fisher's information to inform sample size for developing or updating fair and precise clinical prediction models for individual risk-part 1: binary outcomes.

Background: When using a dataset to develop or update a clinical prediction model, small sample sizes increase concerns of overfitting, instability, poor predictive performance and a lack of fairness. For models estimating the risk of a binary outcome, previous research has outlined sample size calculations that target low overfitting and a precise overall risk estimate. However, more guidance is needed for targeting precise and fair individual-level risk estimates.

Methods: We propose a decomposition of Fisher's information matrix to help examine sample sizes required for developing or updating a model, aiming for precise and fair individual-level risk estimates. We outline a five-step process for use before data collection or when an existing dataset or pilot study is available. It requires researchers to specify the overall risk in the target population, the (anticipated) distribution of key predictors in the model and an assumed 'core model' either specified directly (i.e. a logistic regression equation is provided) or based on a specified C-statistic and relative effects of (standardised) predictors.

Results: We produce closed-form solutions that decompose the variance of an individual's risk estimate into the Fisher's unit information matrix, predictor values and the total sample size. This allows researchers to quickly calculate and examine the anticipated precision of individual-level predictions and classifications for specified sample sizes. The information can be presented to key stakeholders (e.g. health professionals, patients, grant funders) to inform target sample sizes for prospective data collection or whether an existing dataset is sufficient. Our proposal is implemented in our new software module pmstabilityss. We provide two real examples and emphasise the importance of clinical context, including any risk thresholds for decision making and fairness checks.

Conclusions: Our approach helps researchers examine potential sample sizes required to target precise and fair individual-level predictions when developing or updating prediction models for binary outcomes.