Fisher Information Framework Enhances Sample Size Calculations for Clinical Prediction Models

Methodologists have introduced an analytical framework based on the mathematical decomposition of Fisher’s information matrix to calculate the sample sizes required for developing and updating clinical prediction models for time-to-event outcomes. Writing in Diagnostic and Prognostic Research, a research team led by Richard D. Riley and Joie Ensor demonstrates how statistical instability in individual-level survival probabilities can remain substantial even when traditional sample size criteria for population-level calibration and overall risk are fully satisfied. Prior sample size guidance implemented in tools such as pmsampsize focused on avoiding regression overfitting, limiting shrinkage, and bounding the absolute margin of error around the overall population event probability. However, when risk predictions inform critical medical choices for individual patients, broad confidence intervals around individual predictions can lead to inconsistent classifications across clinical decision thresholds. The newly derived methodology addresses this challenge by directly disentangling total sample size from Fisher’s unit information matrix, providing closed-form solutions that allow investigators to quantify individual estimation uncertainty, evaluate predictive fairness across demographic subgroups, and identify required development cohort sizes prior to prospective recruitment.

The mathematical foundation of the framework relies on expressing the variance-covariance matrix of the estimated regression parameters as the product of the inverse of the total sample size and the inverse of Fisher’s unit information matrix. In their formulation, the authors express the underlying predictive architecture as an exponential proportional hazards regression model, which equivalently takes the form of an accelerated failure time model where the log of individual survival time depends linearly on a core set of predictor variables and an extreme-value error term. By establishing this analytical structure, the unit information matrix corresponds to the expected value of the cross-product of the predictor design matrix weighted by an individual exponential term that combines the linear predictor value and the observed log follow-up time, defined as the minimum of the log event time and the log censoring time. Because the variance of an individual patient’s estimated log hazard rate is expressed as a quadratic form involving their unique predictor vector and the inverted unit information matrix divided by the total sample size, researchers can project prediction variance across diverse patient profiles before analyzing new data.

To guide practical implementation, the authors outline a structured six-step protocol that researchers can follow during the design phase of a cohort study or when evaluating an existing healthcare dataset. In the first step, investigators define a core set of well-established clinical prognostic predictors, identify demographic variables relevant to algorithmic fairness evaluations, and specify a definitive clinical time point for survival probability forecasting. Step two involves obtaining or simulating an empirical dataset that accurately captures the joint distribution of these core predictors, which can be accomplished through an existing cohort, a pilot investigation, or a synthetic dataset created by external data holders. In the third step, the user specifies a core exponential model expressing how baseline event rates depend on the chosen predictors, which can be achieved either by setting explicit parameter coefficients or by defining the overall target population risk alongside an anticipated concordance index and relative predictor weightings. The remaining steps derive observed or simulated event and censoring times, compute Fisher’s unit information matrix by averaging across the patient sample, and project the expected width of uncertainty intervals or solve directly for the sample size required to satisfy targeted precision criteria.

The study highlights how this approach bridges prospective design with decision-theoretic principles and clinical thresholds. Under maximum likelihood theory, the anticipated ninety-five percent uncertainty intervals around an individual’s estimated log event rate can be transformed directly to the cumulative event probability scale at the preselected follow-up horizon. Unlike traditional metrics that average error across the entire population, this individual-level framework evaluates the probability of misclassification when clinical actions hinge on explicit risk thresholds, such as deciding whether to initiate systemic adjuvant therapy or continue long-term anticoagulation. By plotting each individual patient’s true model risk against their anticipated uncertainty interval width or against the proportion of their risk distribution that falls on the opposite side of a decision cut-off, investigators and clinical stakeholders can evaluate whether a prospective sample size yields actionable precision. When uncertainty intervals broadly straddle crucial decision thresholds, the framework enables researchers to calculate the expanded sample size needed to contract prediction intervals until misclassification probabilities drop below clinically acceptable targets.

The authors demonstrate the practical utility of their methodology using empirical data from the German Breast Cancer Study Group, examining five-year recurrence risks among two hundred and twenty participants with node-positive, estrogen-receptor-positive disease who received tamoxifen treatment. The core model incorporated five standard prognostic predictors represented by six parameters: patient age, tumor size, number of positive lymph nodes, menopausal status, and histological grade. Applying established population-level sample size rules via pmsampsize suggested that a minimum cohort of three hundred and fifty-five participants, with one hundred and twenty-six anticipated recurrences over twelve hundred and sixty-eight person-years, would suffice to restrict overfitting and estimate overall risk precisely. However, evaluating this baseline sample size within the new framework revealed that while the overall mean uncertainty interval width was zero point twenty-three and mean absolute prediction error was zero point zero forty-six, patients with true five-year recurrence risks of thirty percent or less faced mean misclassification probabilities of zero point eighteen around a twenty percent clinical decision threshold, with individual misclassification probabilities extending up to zero point fifty.

To resolve this localized instability around therapeutic decision boundaries, the authors applied their rearrangement formula to determine the cohort size necessary to ensure that ninety-five percent uncertainty interval widths remained below zero point twenty for all individuals with true recurrence risks at or below thirty percent. The calculation revealed that achieving this refined precision threshold required expanding the total development sample size from three hundred and fifty-five to approximately nine hundred and twenty participants. Under this expanded enrollment, the mean misclassification probability for patients in the sensitive lower-risk stratum decreased from zero point eighteen to zero point twelve. The authors note that this analytical capacity allows study planners to discuss trade-offs explicitly with patient and public involvement groups and clinical panels, identifying specific regions of the prognostic spectrum where precision is vital and determining the exact sample size needed to prevent patients from receiving inappropriate clinical recommendations due to statistical imprecision.

The investigation also evaluates scenarios where patient-level developmental data cannot be directly accessed prior to formal project funding due to regulatory or confidentiality constraints, requiring reliance on synthetic data. Using the synthpop package in R, the researchers generated a synthetic cohort of ten thousand individuals designed to mirror the multivariable conditional distributions of the German Breast Cancer Study Group variables. When the six-step sample size procedure was applied using this synthetic cohort with simulated follow-up and censoring times, the resulting precision metrics closely tracked those derived from the original dataset, yielding a mean absolute prediction error of zero point zero forty-six and a mean uncertainty interval width of zero point twenty-two under the three hundred and fifty-five participant scenario. Even when synthetic data lacked follow-up information entirely, requiring uniform censoring assumptions between two and seven point twenty-eight years, the derived sample size requirements remained highly consistent, demonstrating that synthetic datasets generated by external databanks offer a reliable vehicle for executing rigorous pre-study sample size planning.

Algorithmic fairness evaluations represent another key capability integrated directly into the sample size decomposition framework, aligning with reporting guidelines such as TRIPOD-plus-AI. By incorporating demographic and protected characteristics into the initial predictor distribution, researchers can assess whether a candidate sample size produces equitable predictive stability across patient subgroups, regardless of whether those characteristics are retained as active predictors in the final regression equation. In the breast cancer application, evaluating the expanded cohort of nine hundred and twenty participants across menopausal categories revealed that premenopausal women, who constituted only twenty-six percent of the underlying cohort, exhibited slightly wider uncertainty distributions than postmenopausal women, with a mean absolute prediction error of zero point zero thirty-two compared to zero point zero twenty-seven and a mean misclassification probability of zero point zero fifty-nine versus zero point zero thirty. Because both values remained low and the absolute divergence was modest, the sample size was judged sufficient to avoid critical fairness disparities, though the authors note that larger disparities could prompt researchers to intentionally enrich prospective recruitment within underrepresented groups.

Addressing the methodological assumption of a constant baseline hazard inherent to exponential modeling, the researchers conducted empirical validation studies comparing exponential formulations against Weibull and flexible parametric survival models utilizing restricted cubic splines with four internal knots. In both the primary breast cancer dataset and a secondary prostate cancer cohort comprising five hundred and two patients followed for twelve-month mortality across four predictors, the empirical baseline hazards exhibited distinct non-constant behavior over time. Despite these underlying distributional deviations, the ninety-five percent prediction uncertainty intervals calculated under the closed-form exponential framework aligned remarkably well with the standard errors estimated numerically via non-linear post-estimation commands under flexible spline models. In the breast cancer dataset, for instance, the mean interval width under the Weibull specification was zero point two hundred and ninety-one, compared to zero point two hundred and eighty-six under the exponential approach, confirming that the analytical tractability of the exponential assumption provides robust approximations for study sizing without demanding complex numerical integration of ancillary parameters.

The proposed framework, implemented computationally within the open-source pmstabilityss module for Stata with forthcoming extensions in R and Python, provides clinical epidemiologists and biostatisticians with a formal mathematical tool to align study recruitment with individual predictive reliability. The authors emphasize that while their equations focus on epistemic uncertainty arising from finite sample parameter estimation in unpenalized models, further investigations are required to extend the decomposition principles to high-dimensional datasets requiring variable selection, shrinkage estimators, and non-parametric machine learning algorithms like random survival forests. By shifting the sample size paradigm from purely population-level stability to the rigorous control of individual prediction variance and classification error, the methodology ensures that future diagnostic and prognostic algorithms can be developed with sufficient statistical power to support transparent, equitable, and dependable medical decision-making across diverse clinical populations.

Subject of Research: Sample size methodology for developing precise and fair time-to-event clinical prediction models

Article Title: A decomposition of Fisher’s information to inform sample size for developing or updating fair and precise clinical prediction models — part 2: time-to-event outcomes

Article References: Riley, R. D., Collins, G. S., Archer, L., Whittle, R., Legha, A., Kirton, L., Dhiman, P., Sadatsafavi, M., Adderley, N. J., Alderman, J., Martin, G. P., & Ensor, J. (2025). A decomposition of Fisher’s information to inform sample size for developing or updating fair and precise clinical prediction models — part 2: time-to-event outcomes. Diagnostic and Prognostic Research, 9(1), Article 33. https://doi.org/10.1186/s41512-025-00204-9

Image Credits: AI Generated

DOI: 10.1186/s41512-025-00204-9

Keywords: Clinical Prediction Models, Sample Size Determination, Time-to-Event Analysis, Fisher Information, Algorithmic Fairness, Survival Analysis, Biostatistics, Diagnostic and Prognostic Research, decomposition, Fisher, information, inform