A decomposition of Fisher's information to inform sample size for developing or updating fair and precise clinical prediction models -- Part 3: continuous outcomes

Background
Clinical prediction models enable healthcare professionals to estimate individual outcomes using patient characteristics. Current sample size guidelines for developing or updating models with continuous outcomes aim to minimise overfitting and ensure accurate estimation of population-level parameters, but do not explicitly address the precision of predictions. This is a critical limitation, as wide confidence intervals around predictions can undermine clinical utility and fairness, particularly if precision varies across subgroups.

Methods
We propose methodology for calculating the sample size required to ensure precise and fair predictions in models with continuous outcomes. Building on linear regression theory and the Fisher’s unit information matrix, our approach calculates how sample size impacts the epistemic (model-based) uncertainty of predictions and allows researchers to either (i) evaluate whether an existing dataset is sufficiently large, or (ii) determine the sample size needed to target a particular confidence interval width around predictions. The method requires real or synthetic data representing the target population. To assess fairness, the approach can evaluate prediction precision across subgroups. Extensions to prediction intervals are included to additionally address aleatoric uncertainty.

Results
We demonstrate the methodology by examining the sample size required to develop or update a model for predicting Forced Expiratory Volume (FEV) using age, height, and sex. Existing guidance suggests a minimum sample size of 237 participants for this setting. We show this corresponds to an anticipated mean confidence interval width of 0.206 litres across all participants in the target population, and that widths may be considerably larger for some individuals. Our new approach calculates a minimum of 694 patients would be needed to ensure all anticipated interval widths were ≤0.3. Subgroup analysis showed comparable anticipated precision across sex subgroups. Sensitivity analysis assuming conditional independence among predictors yielded consistent results. For prediction intervals, the magnitude of the residual variance imposes a lower bound on interval width, even with very large samples.

Conclusions
Our methodology provides a practical framework for examining required sample sizes when developing or updating prediction models with continuous outcomes, focusing on achieving precise and equitable predictions. It supports the development of more reliable and fair models, enhancing their clinical applicability and trustworthiness.