Supplementary text S1
Description of the method to calculate confidence limits for excess deaths and age
adjusted excess deaths rates
Let consider:



𝑦𝑡𝑎𝑔 the number of observed deaths in month t (1…12) of flu-year a (1…24) and age
group g (1…8);
𝐸𝑎 the estimate of the epidemic period of flu-year a
𝐷𝑎𝑔 the periods of months where an excess of deaths in 𝑦𝑡𝑎𝑔 is attributed to the
epidemic period of flu-year a
1. Compute the monthly rate of deaths adjusted for a 30.4 days month:
𝑦𝑡𝑎
𝑟𝑡𝑎𝑔 = (
30.4)⁄𝑁𝑎𝑔
𝑚𝑡
2. Compute the time series 𝑧𝑡𝑎𝑔 = 𝑙𝑛(𝑟𝑡𝑎𝑔 ) in order to stabilize de variance;
∗
3. Compute 𝑧𝑡𝑎𝑔
the time series of natural logarithm of the death rare in month t (1…12)
of flu-year a (1…24) and age group g (1…8) without the epidemic periods 𝐸𝑎 , i.e.
∗
𝑧𝑡𝑎𝑔
= {(𝑡, 𝑎) ∉ 𝐸𝑎 };
∗
4. For each age group g (1…8) a cyclical regression model is fitted to 𝑧𝑡𝑎𝑔
, the model is
′
then used to predict the number of deaths for the 𝐸𝑎 periods. A new time series 𝑧𝑡𝑎𝑔
is
∗
then build by inputting the missing values of 𝑧𝑡𝑎𝑔 with cyclical regression model
predictions;
′
5. For each age group g an seasonal ARIMA model is fitted to the time series 𝑧𝑡𝑎𝑔
. Then
′
compute the time series 𝑧̂𝑡𝑎𝑔 representing the fitted values using the adjusted seasonal
ARIMA model will represent the natural logarithm of the monthly death rate baseline
without the effect of the epidemic periods 𝐸𝑎 and the respective upper 95%
′
confidence limit given by 𝑧̂𝑡𝑎𝑔
+ 𝑝0.975 𝜎̂𝑒𝑔 , where 𝜎̂𝑒𝑔 is the standard deviation of the
seasonal ARIMA model residuals and 𝑝0.975 the 0.975 percentile of the standard
normal distribution;
6. Log baseline and upper 95% confidence limits are anti log:
′
′
𝑟𝑡𝑎𝑔
= exp⁡(𝑧̂𝑡𝑎𝑔
)
′
′
𝑟𝑢𝑝𝑡𝑎𝑔
= exp⁡(𝑧̂𝑡𝑎𝑔
+ 𝑝0.975 𝜎̂𝑒𝑔 )
7. The 𝐷𝑎𝑔 are then obtained as the periods included in 𝐸𝑎 where
′
𝑟𝑡𝑎𝑔 > 𝑟𝑢𝑝𝑡𝑎𝑔
8. Compute excess rate and absolute excess deaths attributable to influenza epidemics
for (𝑡, 𝑎, 𝑔) ∈ 𝐷𝑎𝑔 :
′
Excess rate - 𝑒𝑥𝑐_𝑟𝑡𝑎𝑔 = 𝑟𝑡𝑎𝑔 − 𝑟𝑡𝑎𝑔
Absolute excess deaths - 𝑒𝑥𝑐_𝑦𝑡𝑎𝑔 = 𝑒𝑥𝑐_𝑟𝑡𝑎𝑔 × 𝐴𝑡𝑎𝑔
where 𝐴𝑡𝑎𝑔 =
𝑚𝑡 ×𝑁𝑎𝑔
30.4
9. Compute total excess deaths in the epidemic period 𝐸𝑎 for age group g, i.e. in period
𝐷𝑎𝑔 :
Absolute excess deaths in 𝐷𝑎𝑔 : 𝑒𝑥𝑐_𝑦𝑎𝑔 = ∑𝑡∈𝐷𝑎𝑔 𝑒𝑥𝑐_𝑦𝑡𝑎𝑔
10. Compute total excess deaths in epidemic period 𝐸𝑎 (all age groups)
8
𝑒𝑥𝑐_𝑦𝑎 = ∑
𝑔=1
𝑒𝑥𝑐_𝑦𝑎𝑔
11. Compute age-standardized excess rates for epidemic period 𝐸𝑎 :
8
𝑒𝑥𝑐_𝑦𝑎𝑔
𝑒𝑥𝑐_𝑟𝑎 = ∑
𝑤𝑔 ×
𝑁𝑎𝑔
𝑔=1
Where 𝑤𝑔 is the weight of age group g in the reference population used (world population
2000);
12. Confidence interval for the total excess deaths in epidemic period 𝐸𝑎 - 𝑒𝑥𝑐_𝑦𝑎 :
8
𝑒𝑥𝑐_𝑦𝑎 = ∑
𝑔=1
𝑒𝑥𝑐_𝑦𝑎𝑔
Let start by finding the distribution of 𝑒𝑥𝑐_𝑦𝑎𝑔 , if
′
′
𝑒𝑥𝑐_𝑦𝑎𝑔 = ∑ (𝑟𝑡𝑎𝑔 − 𝑟𝑡𝑎𝑔
) × 𝐴𝑡𝑎𝑔 = ∑ 𝑟𝑡𝑎𝑔 𝐴𝑡𝑎𝑔 − ∑ 𝑟𝑡𝑎𝑔
𝐴𝑡𝑎𝑔
𝑡,𝑎∈𝐷𝑎𝑔
𝑡,𝑎∈𝐷𝑎𝑔
𝑡,𝑎∈𝐷𝑎𝑔
Assuming that the observed rates 𝑟𝑡𝑎𝑔 are fixed, we only need to find the distribution of
′
∑𝑡,𝑎∈𝐷𝑎𝑔 𝑟𝑡𝑎𝑔
𝐴𝑡𝑎𝑔 .
′
′
2
From the seasonal ARIMA model we know that 𝑧̂𝑡𝑎𝑔
∼ 𝑁(𝑧𝑡𝑎𝑔
, 𝜎𝑒𝑔
) so
′
′
′
2
𝑟𝑡𝑎𝑔
= 𝑒 𝑧̂𝑡𝑎𝑔 ∼ 𝐿𝑜𝑔 − 𝑁(𝑧𝑡𝑎𝑔
, 𝜎𝑒𝑔
)
and
′
′
2
𝐴𝑡𝑎𝑔 𝑟𝑡𝑎𝑔
∼ 𝐿𝑜𝑔 − 𝑁(𝑙𝑛(𝐴𝑡𝑎𝑔 ) + 𝑧𝑡𝑎𝑔
, 𝜎𝑒𝑔
)
And by the Fenton and Wilkinson approximation
2
′
′
′ ,𝜎 ′ )
∑ 𝑟𝑡𝑎𝑔
𝐴𝑡𝑎𝑔 = ∑ 𝑦𝑡𝑎𝑔
∼ ⁡𝐿𝑜𝑔 − 𝑁 (𝜇𝑦𝑎𝑔
𝑦𝑎𝑔
𝑡,𝑎∈𝐷𝑎𝑔
𝑡,𝑎∈𝐷𝑎𝑔
Where
′
𝜎𝑦2𝑎𝑔
(𝑒
′ = 𝑙𝑛
2
𝜎𝑒𝑔
− 1)
∑𝑡,𝑎∈𝐷𝑎𝑔 𝑒 2(𝑙𝑛(𝐴𝑡𝑎𝑔 )+𝑧𝑡𝑎𝑔)
[∑𝑡,𝑎∈𝐷𝑎𝑔 𝑒
(
′
𝑙𝑛(𝐴𝑡𝑎𝑔 )+𝑧𝑡𝑎𝑔
]
2
+1
)
and
′ = 𝑙𝑛 (∑𝑡,𝑎∈𝐷
𝜇𝑦𝑎𝑔
𝑒
𝑎𝑔
′
𝑙𝑛(𝐴𝑡𝑎𝑔 )+𝑧𝑡𝑎𝑔
2
𝜎𝑒𝑔
⁡) +
2
−
𝜎2′
𝑦𝑎𝑔
2
Consider now:
8
𝑒𝑥𝑐_𝑦𝑎 = ∑
8
𝑒𝑥𝑐_𝑦𝑎𝑔 = ∑
𝑔=1
[∑
𝑔=1
𝑡,𝑎∈𝐷𝑎𝑔
8
8
=∑
∑
𝑔=1
𝑡,𝑎∈𝐷𝑎𝑔
𝑦𝑡𝑎𝑔 − ∑
𝑦𝑡𝑎𝑔 − ∑
∑
𝑔=1
𝑡,𝑎∈𝐷𝑎𝑔
𝑡,𝑎∈𝐷𝑎𝑔
′
𝑟𝑡𝑎𝑔
𝐴𝑡𝑎𝑔 ] =
′
𝑟𝑡𝑎𝑔
𝐴𝑡𝑎𝑔
Considering that ∑8𝑔=1 ∑𝑡,𝑎∈𝐷𝑎𝑔 𝑦𝑡𝑎𝑔 is fixed then we must find the distribution of
′
′
′
∑8𝑔=1 ∑𝑡,𝑎∈𝐷𝑎𝑔 𝑟𝑡𝑎𝑔
𝐴𝑡𝑎𝑔 = ∑8𝑔=1 ∑𝑡,𝑎∈𝐷𝑎𝑔 𝑦𝑡𝑎𝑔
= ∑8𝑔=1 𝑦𝑎𝑔
.
2
′
′ , 𝜎 ′ ) so:
From the results above we know that 𝑦𝑎𝑔
∼ ⁡𝐿𝑜𝑔 − 𝑁 (𝜇𝑦𝑎𝑔
𝑦𝑎𝑔
8
∑
′
𝑦𝑎𝑔
∼ 𝐿𝑜𝑔 − 𝑁 (𝜇𝑦𝑎′ , 𝜎𝑦2𝑎′ )
𝑔=1
Where by the Fenton and Wilkinson approximation
𝜎𝑦2𝑎′
∑8𝑔=1 𝑒
= 𝑙𝑛
[
2𝜇𝑦′ +𝜎2′
𝑎𝑔
𝑦𝑎𝑔
(𝑒
𝜎2′
𝑦𝑎𝑔
− 1)
2
𝜇 ′ +𝜎2′ ⁄2
(∑8𝑔=1 𝑒 𝑦𝑎𝑔 𝑦𝑎𝑔 )
8
𝜇𝑦𝑎′ = 𝑙𝑛 (∑
𝑒
𝑔=1
𝜇𝑦′ +𝜎2′ ⁄2
𝑦𝑎𝑔
𝑎𝑔
)−
+1
]
𝜎𝑦2𝑎′
2
The 95%confidence limits for 𝑒𝑥𝑐_𝑦𝑎 are computed using the upper 0.975 and lower 0.025
probability quantiles of the Log-N distribution with 𝐿𝑜𝑔 − 𝑁 (𝜇𝑦𝑎′ , 𝜎𝑦2𝑎′ )
13. –Confidence interval for standardized excess rates for epidemic period 𝐸𝑎 :
8
8
𝑒𝑥𝑐_𝑦𝑎𝑔
𝑒𝑥𝑐_𝑟𝑎 = ∑
𝑤𝑔 ×
= ∑ 𝑊𝑎𝑔 × 𝑒𝑥𝑐_𝑦𝑎𝑔 =
𝑁𝑎𝑔
𝑔=1
𝑔=1
𝑤𝑔
Where 𝑊𝑎𝑔 = 𝑁
𝑎𝑔
8
=∑
8
𝑊𝑎𝑔 𝑦𝑎𝑔 − ∑
𝑔=1
′
𝑊𝑎𝑔 𝑦𝑎𝑔
=
𝑔=1
2
′
′ ,𝜎 ′ )
𝑊𝑎𝑔 𝑦𝑎𝑔
~𝐿𝑜𝑔 − 𝑁 (𝑙𝑛(𝑊𝑎𝑔 ) + 𝜇𝑦𝑎𝑔
𝑦𝑎𝑔
Where by the Fenton and Wilkinson approximation
𝑒𝑥𝑐_𝑟𝑎 ~𝐿𝑜𝑔 − 𝑁(𝜇𝑟 𝑎 , 𝜎𝑟2𝑎 )
𝜎𝑟2𝑎
∑8𝑔=1 𝑒
= 𝑙𝑛
[
2𝑙𝑛(𝑊𝑎𝑔 )+𝜇𝑦′ +𝜎2′
𝑦
𝑎𝑔
𝑎𝑔
(∑8𝑔=1 𝑒
𝜎2′
𝑦𝑎𝑔
𝑙𝑛(𝑊𝑎𝑔 )+𝜇𝑦′ +𝜎2′ ⁄2
𝑦
𝑎𝑔
𝑎𝑔
8
𝜇𝑟 𝑎 = ⁡𝑙𝑛 (∑
(𝑒
𝑒
𝑔=1
− 1)
2
+1
)
𝑙𝑛(𝑊𝑎𝑔 )+𝜇𝑦′ +𝜎2′ ⁄2
𝑦𝑎𝑔
𝑎𝑔
]
𝜎𝑟2𝑎
)−
2
The 95%confidence limits for 𝑒𝑥𝑐_𝑦𝑎 are computed using the upper 0.975 and lower
0.025 probability quantiles of the Log-N distribution with 𝐿𝑜𝑔 − 𝑁 (𝜇𝑦𝑎′ , 𝜎𝑦2𝑎′ )