The model adopted to estimate the trend of crude death ratios[1] with the Italian data updated on April 17,2020 is a semiparametric model (see Wood 2017). Let y be the crude death ratio per 100000 population, i=1,2,3,4 denote one of the following 4 groups of Italian regions:
(G1) Lombardia, Valle d’Aosta; (G2) Piemonte, Trentino Alto Adige, Emilia Romagna, Liguria, Marche; (G3) Veneto, Abruzzo, Friuli Venezia Giulia, Toscana; (G4) Basilicata, Calabria, Campania, Lazio, Molise, Puglia, Sicilia, Sardegna, Umbria.
The model:
where fi(t) is a smooth term (thin plate regression spline) describing the nonlinear relation between the crude death ratio and time by group, the error is assumed to be Gaussian. Estimates are obtained using the package mgcv in the R software and group G1 is chosen as a corner point.
The following table summarizes the detailed results and in figure 1 the plots of the 4 smooth terms are reported
Table 1 Model estimates
Parametric coefficients:
| ||||
Estimate
|
Std. Error
|
t value
|
Pr(>|t|)
| |
b0
 |
0.12223
|
1.39013
|
0.08800
|
0.9300
|
b1
|
1.32071
|
0.05269
|
25.06600
|
<2e-16
|
b variation G2
|
-0.66093
|
0.03635
|
-18.18500
|
<2e-16
|
b variation G3
|
-1.09570
|
0.02831
|
-38.69900
|
<2e-16
|
b variation G4
|
-1.24006
|
0.02078
|
-59.67100
|
<2e-16
|
Approximate significance of smooth terms
| ||||
edf
|
Ref.df
|
F
|
p-value
| |
f1(t)
|
5.668
|
6.826
|
205.890
|
< 2e-16
|
f2(t)
|
4.409
|
5.470
|
123.487
|
< 2e-16
|
f3(t)
|
2.500
|
3.192
|
16.969
|
3.99e-11
|
f4(t)
|
2.169
|
2.784
|
6.336
|
0.000398
|
R-sq.(adj) = 0.961 Deviance explained = 96.2%
Figure 1 Estimated Smooth terms
Wood S.N. (2017) Generalized Additive Models: An Introduction with R (2nd edition). Chapman and Hall/CRC Press.
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