Use of risk-adjusted CUSUM and RSPRT charts for monitoring in medical contexts

In this paper we discuss the use of charts derived from the sequential probability ratio test (SPRT): the cumulative sum (CUSUM) chart, RSPRT (resetting SPRT), and FIR (fast initial response) CUSUM. The theoretical development of the methods is described and some considerations one might address when designing a chart, explored, including the approximation of average run lengths (ARLs), the importance of detecting improvements in a process as well as detecting deterioration and estimation of the process parameter following a signal. Two examples are used to demonstrate the practical issues of quality control in the medical setting, the first a running example and the second a fully worked example at the end of the paper. The first example relates to 30-day mortality for patients of a single cardiac surgeon over the period 1994-1998, the second to patient deaths in the practice of a single GP, Harold Shipman. The charts' performances relative to each other are shown to be sensitive to the definition of the 'out of control' state of the process being monitored. In light of this, it is stressed that a suitable means by which to compare charts is chosen in any specific application.