Counterfactual Explanations for Hypertension Treatment Recommenders Under Sparse Laboratory Monitoring: A Healthcare Decision Stability Estimation

Counterfactual explanations are increasingly proposed as a route to clinical trust in treatment recommender systems, yet their stability under the sparse laboratory monitoring typical of hypertension care has not been characterised. The recommender is modelled as a threshold function over a latent risk score, and the counterfactual explanation is defined as the minimal perturbation of the feature vector that crosses the decision boundary. Under sparse monitoring, the risk score is observed with a measurement error whose variance is a decreasing function of the number of laboratory panels, so the explanation inherits a sampling distribution. Three propositions are derived: the explanation variance is bounded below by a term proportional to the inverse of the number of panels; the probability that the explanation changes sign under resampling is an increasing function of the intraclass correlation; and the minimum detectable instability at 80% power is a decreasing convex function of the number of clusters. A design-based estimation procedure is specified and its operating characteristics are characterised across a declared variance structure with between-cluster SD 6.0, residual SD 4.5 and intraclass correlation 0.64, using 2000 replications per design cell over cluster counts of 8, 16, 24 and 32 and effect sizes of 3, 5 and 8 percentage points. The design requires 56 clusters to detect a 5 percentage-point instability at 80% power. The contribution is a formal account of why counterfactual explanations in hypertension recommenders are intrinsically unstable under sparse monitoring, and a design-based procedure for estimating that instability before deployment.