2.3.2. Autoregressive Moving Average (ARMA)

The “natural” variability of the control data is captured and used to establish the in-control distribution. In our study, the time process of interest formed a time series with seasonal variations, meaning that a regression model with an autoregressive moving average (ARMA) with (1, 1) error terms was required to fit the data during the control period preceding the onset of the pandemic in our geographical location (i.e., between 2011 and 2019). A double square root transformation of the daily number of ED visits was made to obtain normality wt=yt4 where wt is the transformed response variable and yt denotes the number of respiratory syndrome ED visits at day t. The standard format of the regression model was:

where μwt is the mean response, which depended on a set of predictor variables (e.g., the month of the year, the day of the week, holidays, or trend effects), and εt is an error term that follows an ARMA (1, 1) process:

where ϕ1 and θ1 are the AR (autoregressive) and MA (moving average) coefficients, respectively.

The predictors’ effects were estimated for the following dummy variables: (i) Month of the year: M1 to M11 stand for January to December, with July as a reference; (ii) day of the week: D1 to D6 stand for Monday to Sunday, with Wednesday as a reference; and (iii) holidays: C1, C2, C3, etc., refer to the holiday period and a day either side of this period, with other days as a reference. The sine and cosine functions were used to allow for seasonal effects. Finally, we included the trend variable t. In summary, the mean response was modelled as: