Clinical prediction models are increasingly used to estimate individual outcomes from patient characteristics, but a new study argues that the sample size rules most researchers rely on may quietly leave predictions too imprecise for confident clinical use.
Current guidance for developing or updating models that predict continuous outcomes, such as blood pressure or lung volume, focuses on minimising overfitting and on precisely estimating population-level quantities: the overall mean outcome and the residual variance. The authors point out a critical limitation of this approach. It does not explicitly address the precision of predictions for individual patients. Wide confidence intervals around a predicted value can undermine clinical utility, and if that imprecision falls unevenly across subgroups, it raises fairness concerns that existing formulas never measure.
The new methodology builds on Fisher’s unit information matrix theory, a decomposition that separates the effect of sample size from the effects of predictor distribution and residual variance. In linear regression estimated by maximum likelihood, the variance-covariance matrix of the estimated regression coefficients can be written as the inverse of the Fisher’s unit information matrix divided by the sample size. The unit information matrix represents the expected statistical information contributed by a single observation, and because observations are assumed independent and identically distributed, total information scales directly with the number of participants. This decomposition is the mathematical trick that makes the sample size calculation tractable.
To apply the method, researchers first identify the core predictors expected in the model, typically variables with well-established predictive value in the clinical context. They then need a dataset, real or simulated, that reflects the joint distribution of those predictors in the target population. If individual-level data are unavailable, the authors show how published regression coefficients and summary statistics can be used to reconstruct the variance-covariance matrix of the predictors, or how a dataset can be simulated under an assumption of conditional independence between predictors. The Fisher’s unit information matrix is then derived from this dataset together with either an assumed residual variance or an anticipated R-squared value for the model.
The framework supports two practical options. Researchers with an existing dataset can calculate the anticipated confidence interval widths for predictions across all individuals at a given sample size, checking whether the data already in hand is fit for purpose. Alternatively, when planning new data collection, the formula can be rearranged to determine the minimum sample size needed to keep all anticipated interval widths below a specified target. The calculations can also be run separately within subgroups, allowing researchers to check whether one group, such as an underrepresented population, would systematically experience wider and therefore less reliable intervals.
The authors demonstrate the approach with a model predicting Forced Expiratory Volume in children and adolescents from age, height, and sex, using an openly available dataset of 654 children collected in East Boston during the 1970s. Existing guidance, implemented in the pmsampsize package, suggests a minimum of 237 participants for this setting, assuming an R-squared of 0.77, a mean FEV of 2.64 litres, and a standard deviation of 0.87 litres. The new calculations show that 237 participants corresponds to an anticipated mean confidence interval width of 0.206 litres across the target population, with widths for some individuals considerably larger, reaching up to roughly 0.51 litres in the observed dataset.
When the team instead targeted a maximum interval width of 0.3 litres, considered reasonably acceptable for this clinical scenario, the required sample size rose to 694 participants using the real dataset, and 733 when using simulated data. At the existing dataset’s full size of 654, the mean anticipated interval width was 0.124 litres, ranging from 0.089 to 0.309. A subgroup analysis by sex found comparable precision for females and males, with mean widths of 0.122 and 0.125 litres respectively, which the authors describe as reassuring for fairness. A sensitivity analysis assuming the predictors were conditionally independent yielded results highly consistent with those from the true joint distribution, suggesting the method remains usable when only limited summary information is available.
The paper also extends the framework to prediction intervals, which capture not only epistemic uncertainty, the reducible model-based error that shrinks with more data, but also aleatoric uncertainty, the irreducible residual variability that no sample size can eliminate. The analysis reveals a hard mathematical constraint: the residual variance imposes a lower bound on prediction interval width regardless of how large the sample becomes. In the FEV example, the residual variance was 0.173, and the mean 95% prediction interval width was 1.634 litres at 654 participants. Even with a sample of 100 million, eradicating epistemic uncertainty entirely, the mean width fell only marginally to 1.629 litres. For this reason, the authors generally recommend focusing sample size calculations on confidence interval width, and note that when residual variability is substantial, improving predictions may require identifying more informative predictors rather than simply recruiting more patients.
The authors emphasise that their method assumes unbiased, well-calibrated predictions and should be used alongside existing tools such as pmsampsize, which address overfitting and population-level calibration. They also note limitations: the derivations apply specifically to linear regression, missing data uncertainty is not formally incorporated, and consistency of interval widths across subgroups is only one aspect of fairness that does not guarantee a model will be fully equitable. Internal validation remains essential once data collection and analysis are underway. Even so, the framework offers researchers a practical way to justify sample sizes for grant applications and study planning, to check whether existing datasets are adequate before analysis, and to identify prospective precision disparities across patient groups, supporting the development of prediction models that are both more reliable and more equitable.
Subject of Research: Sample size methodology for precise and fair clinical prediction models with continuous outcomes
Article Title: A decomposition of Fisher’s information to inform sample size for developing or updating fair and precise clinical prediction models – part 3: continuous outcomes
Article References: Whittle, R., Riley, R. D., Archer, L., Collins, G. S., Legha, A., Snell, K. I., & Ensor, J. (2026). A decomposition of Fisher’s information to inform sample size for developing or updating fair and precise clinical prediction models – part 3: continuous outcomes. Diagnostic and Prognostic Research, 10(1), Article 11. https://doi.org/10.1186/s41512-026-00228-9
Image Credits: AI Generated
DOI: 10.1186/s41512-026-00228-9
Keywords: clinical prediction models, sample size, Fisher's information, continuous outcomes, linear regression, prediction precision, confidence intervals, fairness, aleatoric uncertainty, pmsampsize, forced expiratory volume, model updating
News Source: Ophelia Keating. (October 30, 2026). Sample Size Formula Targets Precise and Fair Predictions in Clinical Models with Continuous Outcomes. Scienmag.
