The current analysis of monthly mortality series confirms the existence of autocorrelation and seasonality in both the raw and expected series at the individual ICU level, avoiding any potential confounding at the aggregate level due to Simpson’s paradox. We thus concur with the findings of Alwan [13, 50] and Bisgaard and Kulahci , who documented the pervasiveness of autocorrelation in a variety of series, industrial and non-industrial. We also established that out-of-control signalling of the raw mortality series with respect to both 3 standard error risk-adjusted and RA-EWMA control limits was not evident with analysis of the residuals from the GARCH time series model. Thus the identification of (G)ARCH processes is an important issue for SPC .
As our focus was directed to an understanding of the underlying data-generating process  and the performance of the RA-EWMA control limits under conditions of autocorrelation, we deemed it appropriate to also subject the expected series, from which the control limits for the raw mortality series were established, to formal time-series estimation. Not surprisingly, as the underlying mortality estimates from a random coefficient model are obligatorily “smoothed” (see also Figure 1), no ARCH effects, representing “volatility”, were demonstrated and a relatively simple seasonal autoregressive model was established (Table 1). As the EWMA is based upon an ARIMA(0,1,1), that is an integrated moving average process [45, 52], it has been applied to autocorrelated data , although the majority of studies have used relatively simple non-seasonal autoregressive models (AR(1) or AR(2)) with fixed λ (usually 0.2, which is the default for the SPC model of Statgraphics software). Residual-EWMA charts, in the context of time series modelling, would appear to be more robust than EWMA applied to the original (autocorrelated) data [53, 54]. Reynolds and Lu have recommended that under autocorrelation “…traditional control chart methodology should not be applied without modification…”  and Human et al. have recently sounded a cautionary note about the robustness of the conventional EWMA .
For the classical SPC model, a process is in control if the mean and standard deviation estimate remain within prescribed control limits , usually three-sigma; that is, for a normally distributed series, 99.7% of observations should lie within the limits  and there a probability of 0.0027 that any point exceeds the control limit [32, 59]. However this definition does not necessarily entail the formal time-series notion of stationarity (strict or weak), where the requirement for stationarity is that the first two moments (mean and variance [45, 50]) and the autocorrelation function are time-invariant, albeit a stationary processes may be auto-correlated . In the industrial/engineering sphere, practitioner response to process autocorrelation  was to embrace a time-series paradigm and apply SPC methods to the residuals of a formal time-series model [45, 62], albeit there were different tactical approaches [15, 63]; or to develop modified control limit schemes [64–66]. It is instructive to note that the non-model based EWMAST chart (EWMA chart for stationary processes ), recommended by Winkel and Zhang , pre-supposes a stationary (not “in-control”) process. In a systematic review of the application of statistical process control in healthcare, Thor et al.  adduced only one literature reference  and a calendar year 2003 monograph which discussed autocorrelation in medical series. As argued by Alwan and Roberts , systematic non-random patterns in series make separation of the classic common and special causes difficult, as departures from control, nominally traceable to special causes, are confounded by autocorrelation and, in the current series, seasonality. Two further concerns were raised by the authors; first, the undue emphasis placed upon normality and the (erroneous) assumption that “approximate normality” implies a state of statistical control; and second, in the presence of a well-fitting time series model with residuals consistent with white-noise (“randomness”), it is “…futile to search for departures from statistical control and their corresponding special causes…”. The latter caution resonates with the current finding of frequent signalling of the raw mortality series compared with in-control residuals from an apposite time series model; with respect to the error process, such signalling represents false positivity .
Cook and co-workers, “…explicitly compare[d] EWMA(observed) and EWMA(predicted) …[with] thresholds around the EMWA(predicted)…”, employing the EWMA (λ = 0.005-0.020) to “…effectively attenuate noise in the data and smooth an erratic but unbiased risk model” , although no criteria of “erratic” were provided. Smoothed control limits for the expected series were also utilised in a review paper by Cook et al. () and Pilcher et al. (, λ = 0.005), albeit the data structure differed; sequential plotting of each patient admission versus monthly mortality rates in the current paper. Our focus and methodology were different, in that we were concerned to both understand and formally model the “noise in the data”. This being said, in the current series, the smoothed EWMA (λ = 0.51) raw series (Figure 4) was demonstrated to signal using 3 standard error expected mortality control limits.
The sophistication of time-series modelling in standard statistical software packages makes the formal analyses of the current study feasible; in particular, automated routines for application of time series models . However, for the application of appropriate SPC to mortality series from multiple ICUs in a data-base, there are unresolved statistical issues [71, 72]. From the perspectives of this study, a multivariate approach may be established using more conventional estimators (multivariate GARCH  and vector autoregression models ) or by newly described hierarchical/functional time series [75, 76].