Stability of Approximations of Average Run Length of Risk-Adjusted CUSUM Schemes Using the Markov Approach: Comparing Two Methods of Calculating Transition Probabilities

Risk-adjusted CUSUM schemes are designed to monitor the number of adverse outcomes following a medical procedure. An approximation of the average run length (ARL), which is the usual performance measure for a risk-adjusted CUSUM, may be found using its Markov property. We compare two methods of computing transition probability matrices where the risk model classifies patient populations into discrete, finite levels of risk. For the first method, a process of scaling and rounding off concentrates probability in the centre of the Markov states, which are non-overlapping sub-intervals of the CUSUM decision interval, and, for the second, a smoothing process spreads probability uniformly across the Markov states. Examples of risk-adjusted CUSUM schemes are used to show, if rounding is used to calculate transition probabilities, the values of ARLs estimated using the Markov property vary erratically as the number of Markov states vary and, on occasion, fail to converge for mesh sizes up to 3,000. On the other hand, if smoothing is used, the approximate ARL values remain stable as the number of Markov states vary. The smoothing technique gave good estimates of the ARL where there were less than 1,000 Markov states.