Holdouts set for safe predictive model updating

TL;DR

This work introduces a principled holdout-set updating framework for predictive risk scores that drift over time and trigger interventions, addressing bias that arises when updates learn from intervention-altered outcomes. The authors formalize the problem, prove the existence of an optimal holdout size (OHS) under plausible assumptions, and develop two estimation methods—parametric and Bayesian emulation—to compute the OHS. Through simulations and a concrete ASPRE preeclampsia case study, they demonstrate that holding out a subpopulation to update the model yields asymptotically near-oracle performance, with a recommended holdout size in the several- to ten-thousand range depending on population scale. The approach provides a practical, ethically justifiable strategy for safe updating of risk scores in complex, intervention-informed settings and offers concrete guidance for planning model lifecycles in health analytics.

Abstract

Predictive risk scores for adverse outcomes are increasingly crucial in guiding health interventions. Such scores may need to be periodically updated due to change in the distributions they model. However, directly updating risk scores used to guide intervention can lead to biased risk estimates. To address this, we propose updating using a `holdout set' - a subset of the population that does not receive interventions guided by the risk score. Balancing the holdout set size is essential to ensure good performance of the updated risk score whilst minimising the number of held out samples. We prove that this approach reduces adverse outcome frequency to an asymptotically optimal level and argue that often there is no competitive alternative. We describe conditions under which an optimal holdout size (OHS) can be readily identified, and introduce parametric and semi-parametric algorithms for OHS estimation. We apply our methods to the ASPRE risk score for pre-eclampsia to recommend a plan for updating it in the presence of change in the underlying data distribution. We show that, in order to minimise the number of pre-eclampsia cases over time, this is best achieved using a holdout set of around 10,000 individuals.

Figures (10)

• Figure 1: Dynamics of a risk model, $\rho$, across three epochs under holdout set updating. Each column corresponds to an epoch, denoted by subscripts $0,1,2$, representing consecutive intervals of continuous time $(0,1], (1,2],$ and $(2,3]$ respectively. Ellipses containing $X$ or $Y$ correspond to covariates and outcomes respectively, with superscripts $i$ and $h$ denoting the partition into intervention and holdout sets respectively.
• Figure 2: Cost per sample per unit of time for no-update, naïve-update, holdout-update, oracle, and alternative-update (using a treatment indicator as a covariate) strategies for a simulated example with population drift in risk. The costs associated with naïve updating will decrease during an epoch if $f_t$ drifts towards the fitted risk score (that is, $\xi_t^2(f_t,\rho_t)$ decreases with $t$) and increase if it drifts away (that is, $\xi_t^2(f_t,\rho_t)$ increases with $t$). Supplementary Figure \ref{['supp_fig:cumcost']} shows the cumulative cost over time.
• Figure 3: Convergence rates with parametric (top) and emulation (bottom) algorithms, using either a random (black) or greedy (gray) methods to select the next value, $\mathbf{n}$, with parametric assumptions satisfied (left) or unsatisfied (right). Simulations were run for 200 datasets from each underlying model. In larger panels, horizontal lines show true optimal holdout set (OHS) size; the OHS results from all simulation runs are discretised on a 1000-resolution grid, with vertical lines indicating OHS values that occurred in at least 2.5% of simulations (i.e., 5 occurrences). Smaller panels show root mean-square error between total costs from simulations and minimal total cost under random/greedy methods. Note variable axis scaling under the two models.
• Figure 4: Estimation of cost functions (black lines), OHS (black dots), error using parametric (middle left) and emulation (leftmost; outer lines $\mu(n) + 3\sqrt{\Psi(n)}$) algorithms, change in estimated OHS and error with number of sample points $|\mathbf{n}|$ (middle right; solid lines: parametric, dashed lines: emulation), and overview of PRE risk per patient in final updating strategy (rightmost; solid black curve $k_2(n)$; heavy black line minimum possible risk $\min_n k_2(n)$; upper dashed line risk in holdout set; lower dashed line risk in intervention set; horizontal narrow solid line overall average risk). Note that the 'best' points (black dots) to optimize parametric estimation are spread-out to estimate $\theta$ well, but for emulation they are clustered for accurate local approximation. Error measures for OHS in parametric and emulation algorithms (red/blue shaded respectively) have different meanings and are not comparable.
• Figure S10.1: Cumulative costs of updating strategies, defined analogously to figure \ref{['fig:holdout_dominance']}

Theorems & Definitions (36)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Corollary 1
• Theorem 5
• Theorem 6
• proof
• proof
• Lemma S1