ARIMA Time Series Modelling Of Excess Mortality 1975 – 2021: outlier method
England & Wales monthly excess mortality by quinary age band & sex (rev 1.1)
In a recent newsletter I undertook statistical modelling of this dataset and opted to share the sizeable output file with subscribers instead of pasting 36 slides and copious tabulations in a series of lengthy posts. I haven’t had any feedback from subscribers as yet but I thought I’d bravely plough on and try this method of dissemination once again for a differing methodology.
Outlying Months
The idea this time round was to junk my raft of 13 monthly indicator variables and use the ARIMA time series procedure itself to flag up months between Jan 1975 and Dec 2021 where excess mortality was found to differ markedly from the 5-year baseline ‘norm’.
Source Files
Paying subscribers wishing to take a peek at these results should ideally start by reading this newsletter first, followed by this newsletter. The file you require once again is EW DOD excess monthly mortality 1975 – 2021 (ARIMA).spv, which may be found sitting in a shared Google Drive folder which may be accessed here.
What Constitutes An Outlier?
A darn good question. The ARIMA forecasting module within SPSS defines no less than 7 different types:
The fully automated expert modeller was utilised as before to ensure consistency in model identification. In practice I am usually able to develop a superior model manually but this takes time and patience and the automated models always pass muster. Given that I was facing the tasking of identifying and developing 36 ARIMA models with little spare time this morning it was pretty much a no-brainer to press the big button and sit back!
Outlier Method Males
This is the section you’ll need to open out in the SPV file. Keen types might like to eyeball the Model Description summary and compare model structures with those found previously. Many are identical but there are some differences worth cogitating. Also worth cogitating are changes to goodness-of-fit statistics found in Model Fit. In the Model Statistics summary table we get to see the number of outlying months identified for each of the 18 age bands. This alone is darn useful information because it gives a snapshot of the overall situation with just a couple of outlying months identified over the period 1975 - 2021 for males up to 44 years of age.
The next table of interest is Outliers where we get to see what months were considered rogue, and in what way. There are many ways to approach this information but a good starting point might be to check for mention of months during 2020/’21 and see whether the Estimate is positive or negative. The first instance my eyeballs pick up is Aug 2020 for 20 - 24y males, where we find a negative local trend estimated at -0.167 excess deaths per 100k target population (p<0.001). Thus, for some reason, young males had a good run of health during late summer. The model structure for this group is ARIMA(1,0,1)(0,0,0) so this isn’t anything seasonal (P=0, D=0, Q=0) nor subject to changes of an underlying trend (d=0), it is simply a run of unusually good health.
We observe a similar spate of apparent good health for Mar 2021 for 25 - 29y, 30 - 34y and 35 -39y males, and for Feb 2021 for 40 - 44y males. We cannot be sure that these dips are genuine for it certainly appears that ONS are well behind in processing the death certificates of younger generations (a handy newsletter to read is this one). In this regard please note the unusual and somewhat unseasonable negative excess for Dec 2021 for males aged 50 - 74y. Again this is likely artefact arising from certificate processing (lack thereof).
Evidence of a pandemic ‘proper’ starts with 40 - 44y males, where we observe an estimate of +6.790 excess deaths per 100k target population (p<0.001) for an innovational outlying Apr 2020. The progression in estimates for Apr 2020 is quite something, ending with +1312.871 excess deaths per 100k target population for males aged 85 years and over (p<0.001). It will be interesting to compare these estimates with those derived by other researchers. A similar progression can be found for Jan 2021 in males aged 50 years and over. Aside from Apr 2020 and Jan 2021 there is absolutely no sign of anything amiss within the male population. This is a mighty peculiar pandemic!
Outlier Method Females
What first caught my eye with the summary table for the Model Description are the non-seasonal structures with moving average (MA) parameters running to q=8, q=9, q=10 & q=11 months for 6 of the 18 age groups. This may be a quirk of the expert modelling system otherwise it would suggest females in these age bands are subject to random ‘shocks’ for a considerable portion of the year; that is to say, their deaths are less predictable than males. This relative lack of predictability for females is also reflected in the lower goodness-of-fit statistics in the summary table for Model Fit.
A quick squint at the summary table for Model Statistics reveals remarkably few outlying months for ages up to 59y, we then have a burst of 11 and 13 months for 60 - 64y and 65 - 69y respectively before things settle down for 70 - 79y females, with another burst of 12 outlying months for 80 - 84y females. This is utterly fascinating and I view this as being indicative of a different ‘death process’.
Pandemic-wise the first evidence of something amiss arises with 45 - 49y females during Apr 2020, with an estimated leap of +6.064 excess deaths per 100k target population (p<0.001). We then observe a steady progression similar to that for males. There are some interesting runs of apparent good health for Feb, Mar, May and Oct 2021 for females aged 20 - 49y and these once again arouse suspicion that deaths certificates are not being processed as they should - the progression of unusual and unseasonable negative excess for Dec 2021 for females aged 60 - 69y certainly gives credence to this notion. Once more we find that, aside from Apr 2020 and Jan 2021, there is absolutely no sign of anything amiss within the female population.
Graphical Summary of Results
Herewith two of those coloured matrix diagrams that denote ARIMA model structure and months standing out as detected outliers, with red denoting an increase in excess mortality and blue denoting a decline. The blue beginning to 2021 is rather evident and I suggest this is due to delays in death certificate processing.
