A Mean Control Chart Based On Percentiles Of Marshall–Olkin Alpha Power Inverse Rayleigh Distribution: An Application To Health
Adegbite, Ismaila Olawale
Department of Statistics, Osun State Polytechnic, Iree, Nigeria
Keywords: Non-normal processes, Percentile constant, Statistical Process Control, MOAPIRD, Healthcare quality monitoring, Control chart
Abstract
Traditional Shewhart control charts assume normality, an assumption often violated in healthcare data. This study develops a mean control chart based on percentiles of the Marshall–Olkin Alpha Power Inverse Rayleigh Distribution (MOAPIRD), a flexible three-parameter model derived via the Marshall–Olkin generalization. The chart uses the 0.00135th and 0.99865th percentiles of the subgroup mean distribution to define control limits, ensuring 99.73% coverage without requiring normality. Percentile-based constants (A*₂ₚ and A**₂ₚ) were obtained through Monte Carlo simulation with 10,000 replications for sample sizes n = 2–10 under four parameter settings. Performance was assessed using Coverage Probability (CP), Average Run Length (ARL), and Control Limit Interval (CLI), with comparisons to classical Shewhart and Inverse Rayleigh Distribution (IRD) charts. Results show that the MOAPIRD chart achieved perfect coverage (CP = 1.00), infinite ARL (indicating zero false alarms), and well-balanced control limits across all configurations. In contrast, Shewhart charts exhibited lower coverage (CP = 0.92–0.98) and short ARL (12.5–50), while IRD charts produced overly wide limits. Validation using length-of-stay data (n = 185) from asthmatic in-patients at Ogun State Hospital, Ijebu-Ode, Nigeria, confirmed the chart's superiority: it correctly detected a true process shift at samples 25–27, while Shewhart produced excessive false alarms and IRD failed to signal the shift. The MOAPIRD mean control chart is recommended for monitoring positively skewed, non-normal health data.