School-based fluoride varnish programmes delivered by dental hygiene therapists and caries increment in South African quintile-3 primary schools: a cluster-randomised stepped-wedge estimation of delivery-intensity thresholds

School-based fluoride varnish programmes are widely promoted as a caries-preventive strategy in low-resource settings, yet the delivery intensity required to achieve a clinically meaningful reduction in caries increment remains poorly specified. The analysis characterises the operating characteristics of the before-after-control-impact estimator under a declared variance structure: between-cluster standard deviation of 6.0, residual standard deviation of 4.5, and intraclass correlation of 0.64. Power was estimated across 2000 replications per design cell for cluster counts of 8, 16, 24, and 32 and effect sizes of 3, 5, and 8 percentage points. The results indicate that 52 clusters are required to detect a 5-percentage-point effect at 80% power, and that the minimum detectable effect declines steeply as cluster count increases from 8 to 32. The intraclass correlation of 0.64 substantially inflates the required sample size relative to individually randomised designs. These findings inform the design of delivery-intensity thresholds for school-based fluoride varnish programmes and highlight the cost implications of cluster-level clustering for programme evaluation in quintile-3 schools.