A CUSUM Chart for Monitoring a Proportion with Autocorrelated Binary Observations

When traditional control charts are used to monitor a proportion p, it is assumed that the binary observations are independent. This paper investigates the problem of monitoring p when there is a continuous stream of autocorrelated binary observations that follow a two-state Markov chain model with first-order dependence. It is shown that both the Shewhart p-chart and the most efficient chart for independent observations, the Bernoulli CUSUM chart, are not robust to autocorrelation, and that adjusting the control limits of these traditional charts to account for the autocorrelation is not an efficient approach. Here we construct a Markov binary CUSUM (MBCUSUM) chart based on a log-likelihood-ratio statistic and show that this chart can be well approximated by using a Markov chain model, for which exact properties are calculable. Numerical results show that the MBCUSUM chart will detect most increases in p faster than competing charts. The effect of the size of the Phase I data set used in setting up the MBCUSUM chart is also investigated.