CUSUM chart to monitor autocorrelated counts using Negative Binomial GARMA model

Cumulative sum control charts have been used for health surveillance due to its efficiency to detect soon small shifts in the monitored series. However, these charts may fail when data are autocorrelated. An alternative procedure is to build a control chart based on the residuals after fitting autoregressive moving average models, but these models usually assume Gaussian distribution for the residuals. In practical health surveillance, count series can be modeled by Poisson or Negative Binomial regression, this last to control overdispersion. To include serial correlations, generalized autoregressive moving average models are proposed. The main contribution of the current article is to measure the impact, in terms of average run length on the performance of cumulative sum charts when the serial correlation is neglected in the regression model. Different statistics based on transformations, the deviance residual, and the likelihood ratio are used to build cumulative sum control charts to monitor counts with time varying means, including trend and seasonal effects. The monitoring of the weekly number of hospital admissions due to respiratory diseases for people aged over 65 years in the city São Paulo-Brazil is considered as an illustration of the current method.