CUSUM chart to monitor autocorrelated counts using Negative Binomial GARMA model

Abstract

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.

Effect of neglecting autocorrelation in regression EWMA charts for monitoring count time series

“…Note that values of L R t yielded by may be negative (the last fraction in the logarithm operator is <1) mainly calculated with high values of y t (as in real count time series analyzed in Albarracin et al). Additionally, the values of L R t are not independent for t = 1,…, n (as they are function of y t ).…”

“…The weekly data from the previous years 2006‐2010 were used to calculate in‐control expectation, μ 0, t , once it is a nonepidemic period. In Albarracin et al, an NB‐GARMA(2,0) model including linear trend and seasonal components was fitted to the weekly data from January 2006 to December 2010. Additionally, the logarithmic function expressed in was chosen as the link function with where t is the number of each week, t = 1,…,261.…”

“…Figure shows the observed weekly number of admissions, its predicted values using the NB‐GARMA(2,0) model, and the forecasts for 2011. The estimates of all coefficients are in Table as presented in Albarracin et al…”

“…A key assumption usually made on count process monitoring is the independence of observations. However, the autocorrelation may be present in the observations and may have a strong impact on the properties of control charts as discussed in Chao‐Wen and Reynolds Jr and Albarracin et al…”

Generalized autoregressive and moving average models: control charts, multicollinearity, and a new modified model

“…For count time series, the generalized autoregressive moving average (GARMA) and the generalized linear autoregressive moving average (GLARMA) (Dunsmuir et al, 2015) models extend the univariate Gaussian ARMA time series model to a flexible observation-driven model for non-Gaussian time series allowing to model discrete and continuous time series. These classes of models are widely used in areas of surveillance (Dugas et al, 2013;Albarracin et al, 2017) where it is necessary to model discrete response time series in terms of covariates. In this paper, we focus on the GARMA model.…”