Phase I control chart for individual autocorrelated data: application to prescription opioid monitoring

Phase I or retrospective process monitoring plays a key part in an overall statistical process monitoring (SPM) regime and is increasingly emphasized in the recent literature. At present, a lot of the data in a variety of settings (public and private sector organizations) are collected individually and sequentially and thus are serially correlated (or autocorrelated). Though a reasonable amount of work is available in the control charting literature for prospective (Phase II) autocorrelated data monitoring, very little work exists for the retrospective phase (Phase I). In this article, we present a Shewhart-type control chart for Phase I monitoring of individual autocorrelated data, assuming normality, with estimated parameters. The methodology, while developed and presented for the first-order autoregressive (AR(1)) model for simplicity, may be adapted to more general time series models. The correct charting constants, adjusted for autocorrelation and parameter estimation, are derived, and tabulated for a nominal in-control (IC) false alarm probability (FAP). Simulation results show that the proposed chart is favorably IC FAP robust and effective for reasonably small sample sizes, moderate autocorrelation, and some model miss-specifications, compared to other approaches. An illustration using some public health data involving prescription fentanyl transactions is provided to show the potential for broader areas of applications of the proposed methodology. Along with a summary and recommendations, some future research areas are indicated. An R package is developed and made available for implementing the proposed methodology on demand.