COVID-19 Data Analysis

Efficacia Vaccinale

Stima dell'efficacia vaccinale dai Report ISS.

Max Pierini


Sono usati i report dal 2021-07-14 al 2021-11-03.

  • Gruppo di controllo: non vaccinati
  • Gruppo sperimentale: vaccinati con ciclo completo

NB: Attualmente sono considerati i vaccinati completi non differenziati; successivamente, quando i dati lo permetteranno, sarà stimata anche la differenza tra ciclo completo entro 6 mesi, superiore a 6 mesi e vaccinati con dose aggiuntiva/booster, introdotta successivamente al 2021-11-03

Metodo

Sappiamo che l'efficacia $E$ è stimata come Riduzione del Rischio Relativo (RRR), pari al complementare del Rischio Relativo (RR)

$$ E = \mathrm{RRR} = 1 - \mathrm{RR} = 1 - \frac{\mathrm{EER}}{\mathrm{CER}} $$

dove EER (Experimental Event Rate) e CER (Control Event Rate) sono rispettivamente le incidenze di eventi nel gruppo sperimentale e di controllo, ovvero

$$ \mathrm{EER} = \frac{\mathrm{EE}}{\mathrm{EN}} \\ \mathrm{CER} = \frac{\mathrm{CE}}{\mathrm{CN}} $$

dove EE (Experimental Events) e CE (Control Events) sono gli eventi, EN (Experimental Number) e CN (Control Number) sono la popolazione, rispettivamente nel gruppo sperimentale e di controllo.

Possiamo quindi riscrivere

$$ E = 1 - \frac{ \frac{\mathrm{EE}}{\mathrm{EN}} }{ \frac{\mathrm{CE}}{\mathrm{CN}} } = 1 - \frac{\mathrm{EE}}{\mathrm{CE}} \frac{\mathrm{CN}}{\mathrm{EN}} $$

da cui

$$ \frac{\mathrm{EE}}{\mathrm{CE}} \frac{\mathrm{CN}}{EN} = 1 - E $$
$$ \frac{\mathrm{EE}}{\mathrm{CE}} = (1 - E) \frac{\mathrm{EN}}{\mathrm{CN}} $$
$$ \frac{\mathrm{EE}}{\mathrm{CE}} = \frac{\mathrm{EN}}{\mathrm{EN}} - E \frac{\mathrm{EN}}{\mathrm{CN}} $$
$$ \frac{\mathrm{EE}}{\mathrm{CE}} - \frac{\mathrm{EN}}{\mathrm{CN}} = -E \frac{\mathrm{EN}}{\mathrm{CN}} $$
$$ \frac{\mathrm{EN}}{\mathrm{CN}} - \frac{\mathrm{EE}}{\mathrm{CE}} = E \frac{\mathrm{EN}}{\mathrm{CN}} $$

e quindi indicando

$$ y = \frac{\mathrm{EN}}{\mathrm{CN}} - \frac{\mathrm{EE}}{\mathrm{CE}} \\ x = \frac{\mathrm{EN}}{\mathrm{CN}} $$

otteniamo

$$ y = E \cdot x $$

ovvero l'efficacia $E$ è il parametro di pendenza (slope) di una retta passante per l'origine.

Possiamo pertanto utilizzare i dati di tutti i report ISS (eventi osservati e popolazione per fascia d'età, in ogni periodo di 30 giorni considerato) per stimare l'efficacia vaccinale tra il gruppo sperimentale (vaccinati a ciclo completo) e di controllo (non vaccinati) con una regressione lineare semplice (OLS, Ordinary Least Square) e distribuire il parametro $E \sim \mathbf{Normal}(\mu, \sigma)$ dove $\mu$ è il valore medio del parametro e $\sigma$ l'errore standard ottenuti dalla regressione lineare.

Osservazioni

OLS

/root/anaconda3/envs/covid_ok/lib/python3.8/site-packages/scipy/stats/stats.py:1603: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=17
  warnings.warn("kurtosistest only valid for n>=20 ... continuing "
EFFICACIA su diagnosi nella fascia 12-39 (Intervalli di confidenza al 99%)
Dep. Variable: y R-squared (uncentered): 0.999
Model: OLS Adj. R-squared (uncentered): 0.999
Method: Least Squares F-statistic: 2.120e+04
Date: Wed, 01 Dec 2021 Prob (F-statistic): 2.06e-26
Time: 18:32:35 Log-Likelihood: 32.924
No. Observations: 17 AIC: -63.85
Df Residuals: 16 BIC: -63.01
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.005 0.995]
EFFICACIA 0.8254 0.006 145.595 0.000 0.809 0.842
Omnibus: 2.295 Durbin-Watson: 0.265
Prob(Omnibus): 0.318 Jarque-Bera (JB): 1.632
Skew: 0.577 Prob(JB): 0.442
Kurtosis: 2.013 Cond. No. 1.00


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

EFFICACIA su ricoveri nella fascia 12-39 (Intervalli di confidenza al 99%)
Dep. Variable: y R-squared (uncentered): 1.000
Model: OLS Adj. R-squared (uncentered): 1.000
Method: Least Squares F-statistic: 1.591e+05
Date: Wed, 01 Dec 2021 Prob (F-statistic): 2.05e-33
Time: 18:32:35 Log-Likelihood: 47.706
No. Observations: 17 AIC: -93.41
Df Residuals: 16 BIC: -92.58
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.005 0.995]
EFFICACIA 0.9479 0.002 398.894 0.000 0.941 0.955
Omnibus: 2.218 Durbin-Watson: 0.423
Prob(Omnibus): 0.330 Jarque-Bera (JB): 1.555
Skew: 0.546 Prob(JB): 0.459
Kurtosis: 1.999 Cond. No. 1.00


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

EFFICACIA su intensive nella fascia 12-39 (Intervalli di confidenza al 99%)
Dep. Variable: y R-squared (uncentered): 1.000
Model: OLS Adj. R-squared (uncentered): 1.000
Method: Least Squares F-statistic: 6.526e+04
Date: Wed, 01 Dec 2021 Prob (F-statistic): 2.56e-30
Time: 18:32:35 Log-Likelihood: 39.431
No. Observations: 17 AIC: -76.86
Df Residuals: 16 BIC: -76.03
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.005 0.995]
EFFICACIA 0.9877 0.004 255.468 0.000 0.976 0.999
Omnibus: 0.093 Durbin-Watson: 0.259
Prob(Omnibus): 0.955 Jarque-Bera (JB): 0.272
Skew: 0.134 Prob(JB): 0.873
Kurtosis: 2.441 Cond. No. 1.00


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

EFFICACIA su decessi nella fascia 12-39 (Intervalli di confidenza al 99%)
Dep. Variable: y R-squared (uncentered): 0.991
Model: OLS Adj. R-squared (uncentered): 0.990
Method: Least Squares F-statistic: 1703.
Date: Wed, 01 Dec 2021 Prob (F-statistic): 1.11e-17
Time: 18:32:35 Log-Likelihood: 10.100
No. Observations: 17 AIC: -18.20
Df Residuals: 16 BIC: -17.37
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.005 0.995]
EFFICACIA 0.8959 0.022 41.270 0.000 0.832 0.959
Omnibus: 21.385 Durbin-Watson: 1.101
Prob(Omnibus): 0.000 Jarque-Bera (JB): 28.430
Skew: -1.852 Prob(JB): 6.71e-07
Kurtosis: 8.139 Cond. No. 1.00


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

EFFICACIA su diagnosi nella fascia 40-59 (Intervalli di confidenza al 99%)
Dep. Variable: y R-squared (uncentered): 0.998
Model: OLS Adj. R-squared (uncentered): 0.998
Method: Least Squares F-statistic: 7236.
Date: Wed, 01 Dec 2021 Prob (F-statistic): 1.10e-22
Time: 18:32:35 Log-Likelihood: 16.117
No. Observations: 17 AIC: -30.23
Df Residuals: 16 BIC: -29.40
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.005 0.995]
EFFICACIA 0.7597 0.009 85.064 0.000 0.734 0.786
Omnibus: 2.577 Durbin-Watson: 0.274
Prob(Omnibus): 0.276 Jarque-Bera (JB): 1.296
Skew: -0.320 Prob(JB): 0.523
Kurtosis: 1.808 Cond. No. 1.00


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

EFFICACIA su ricoveri nella fascia 40-59 (Intervalli di confidenza al 99%)
Dep. Variable: y R-squared (uncentered): 1.000
Model: OLS Adj. R-squared (uncentered): 1.000
Method: Least Squares F-statistic: 2.793e+05
Date: Wed, 01 Dec 2021 Prob (F-statistic): 2.28e-35
Time: 18:32:35 Log-Likelihood: 43.321
No. Observations: 17 AIC: -84.64
Df Residuals: 16 BIC: -83.81
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.005 0.995]
EFFICACIA 0.9527 0.002 528.506 0.000 0.947 0.958
Omnibus: 1.785 Durbin-Watson: 0.387
Prob(Omnibus): 0.410 Jarque-Bera (JB): 1.322
Skew: -0.487 Prob(JB): 0.516
Kurtosis: 2.042 Cond. No. 1.00


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

EFFICACIA su intensive nella fascia 40-59 (Intervalli di confidenza al 99%)
Dep. Variable: y R-squared (uncentered): 1.000
Model: OLS Adj. R-squared (uncentered): 1.000
Method: Least Squares F-statistic: 2.068e+05
Date: Wed, 01 Dec 2021 Prob (F-statistic): 2.52e-34
Time: 18:32:35 Log-Likelihood: 40.443
No. Observations: 17 AIC: -78.89
Df Residuals: 16 BIC: -78.05
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.005 0.995]
EFFICACIA 0.9711 0.002 454.793 0.000 0.965 0.977
Omnibus: 1.204 Durbin-Watson: 0.655
Prob(Omnibus): 0.548 Jarque-Bera (JB): 1.047
Skew: -0.457 Prob(JB): 0.592
Kurtosis: 2.199 Cond. No. 1.00


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

EFFICACIA su decessi nella fascia 40-59 (Intervalli di confidenza al 99%)
Dep. Variable: y R-squared (uncentered): 1.000
Model: OLS Adj. R-squared (uncentered): 1.000
Method: Least Squares F-statistic: 1.399e+05
Date: Wed, 01 Dec 2021 Prob (F-statistic): 5.76e-33
Time: 18:32:35 Log-Likelihood: 37.291
No. Observations: 17 AIC: -72.58
Df Residuals: 16 BIC: -71.75
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.005 0.995]
EFFICACIA 0.9612 0.003 373.983 0.000 0.954 0.969
Omnibus: 0.028 Durbin-Watson: 1.045
Prob(Omnibus): 0.986 Jarque-Bera (JB): 0.179
Skew: 0.079 Prob(JB): 0.914
Kurtosis: 2.523 Cond. No. 1.00


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

EFFICACIA su diagnosi nella fascia 60-79 (Intervalli di confidenza al 99%)
Dep. Variable: y R-squared (uncentered): 0.998
Model: OLS Adj. R-squared (uncentered): 0.998
Method: Least Squares F-statistic: 8034.
Date: Wed, 01 Dec 2021 Prob (F-statistic): 4.79e-23
Time: 18:32:35 Log-Likelihood: 2.9774
No. Observations: 17 AIC: -3.955
Df Residuals: 16 BIC: -3.122
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.005 0.995]
EFFICACIA 0.7618 0.008 89.631 0.000 0.737 0.787
Omnibus: 2.399 Durbin-Watson: 0.387
Prob(Omnibus): 0.301 Jarque-Bera (JB): 1.387
Skew: -0.414 Prob(JB): 0.500
Kurtosis: 1.872 Cond. No. 1.00


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

EFFICACIA su ricoveri nella fascia 60-79 (Intervalli di confidenza al 99%)
Dep. Variable: y R-squared (uncentered): 1.000
Model: OLS Adj. R-squared (uncentered): 1.000
Method: Least Squares F-statistic: 9.378e+04
Date: Wed, 01 Dec 2021 Prob (F-statistic): 1.41e-31
Time: 18:32:35 Log-Likelihood: 20.610
No. Observations: 17 AIC: -39.22
Df Residuals: 16 BIC: -38.39
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.005 0.995]
EFFICACIA 0.9225 0.003 306.241 0.000 0.914 0.931
Omnibus: 4.125 Durbin-Watson: 0.284
Prob(Omnibus): 0.127 Jarque-Bera (JB): 1.436
Skew: -0.209 Prob(JB): 0.488
Kurtosis: 1.639 Cond. No. 1.00


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

EFFICACIA su intensive nella fascia 60-79 (Intervalli di confidenza al 99%)
Dep. Variable: y R-squared (uncentered): 1.000
Model: OLS Adj. R-squared (uncentered): 1.000
Method: Least Squares F-statistic: 3.307e+05
Date: Wed, 01 Dec 2021 Prob (F-statistic): 5.89e-36
Time: 18:32:35 Log-Likelihood: 30.764
No. Observations: 17 AIC: -59.53
Df Residuals: 16 BIC: -58.69
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.005 0.995]
EFFICACIA 0.9534 0.002 575.076 0.000 0.949 0.958
Omnibus: 1.609 Durbin-Watson: 0.913
Prob(Omnibus): 0.447 Jarque-Bera (JB): 1.047
Skew: -0.302 Prob(JB): 0.593
Kurtosis: 1.945 Cond. No. 1.00


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

EFFICACIA su decessi nella fascia 60-79 (Intervalli di confidenza al 99%)
Dep. Variable: y R-squared (uncentered): 1.000
Model: OLS Adj. R-squared (uncentered): 1.000
Method: Least Squares F-statistic: 9.699e+04
Date: Wed, 01 Dec 2021 Prob (F-statistic): 1.08e-31
Time: 18:32:35 Log-Likelihood: 20.435
No. Observations: 17 AIC: -38.87
Df Residuals: 16 BIC: -38.04
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.005 0.995]
EFFICACIA 0.9479 0.003 311.437 0.000 0.939 0.957
Omnibus: 0.964 Durbin-Watson: 0.359
Prob(Omnibus): 0.618 Jarque-Bera (JB): 0.236
Skew: -0.283 Prob(JB): 0.889
Kurtosis: 3.113 Cond. No. 1.00


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

EFFICACIA su diagnosi nella fascia 80+ (Intervalli di confidenza al 99%)
Dep. Variable: y R-squared (uncentered): 0.999
Model: OLS Adj. R-squared (uncentered): 0.999
Method: Least Squares F-statistic: 1.709e+04
Date: Wed, 01 Dec 2021 Prob (F-statistic): 1.15e-25
Time: 18:32:35 Log-Likelihood: -5.3576
No. Observations: 17 AIC: 12.72
Df Residuals: 16 BIC: 13.55
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.005 0.995]
EFFICACIA 0.7840 0.006 130.739 0.000 0.767 0.802
Omnibus: 3.170 Durbin-Watson: 0.569
Prob(Omnibus): 0.205 Jarque-Bera (JB): 1.737
Skew: -0.515 Prob(JB): 0.420
Kurtosis: 1.820 Cond. No. 1.00


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

EFFICACIA su ricoveri nella fascia 80+ (Intervalli di confidenza al 99%)
Dep. Variable: y R-squared (uncentered): 1.000
Model: OLS Adj. R-squared (uncentered): 1.000
Method: Least Squares F-statistic: 8.908e+04
Date: Wed, 01 Dec 2021 Prob (F-statistic): 2.12e-31
Time: 18:32:35 Log-Likelihood: 6.7580
No. Observations: 17 AIC: -11.52
Df Residuals: 16 BIC: -10.68
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.005 0.995]
EFFICACIA 0.8776 0.003 298.466 0.000 0.869 0.886
Omnibus: 7.721 Durbin-Watson: 0.700
Prob(Omnibus): 0.021 Jarque-Bera (JB): 4.684
Skew: -1.154 Prob(JB): 0.0962
Kurtosis: 4.135 Cond. No. 1.00


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

EFFICACIA su intensive nella fascia 80+ (Intervalli di confidenza al 99%)
Dep. Variable: y R-squared (uncentered): 0.999
Model: OLS Adj. R-squared (uncentered): 0.999
Method: Least Squares F-statistic: 1.270e+04
Date: Wed, 01 Dec 2021 Prob (F-statistic): 1.24e-24
Time: 18:32:35 Log-Likelihood: -10.351
No. Observations: 17 AIC: 22.70
Df Residuals: 16 BIC: 23.53
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.005 0.995]
EFFICACIA 0.9065 0.008 112.690 0.000 0.883 0.930
Omnibus: 7.341 Durbin-Watson: 0.169
Prob(Omnibus): 0.025 Jarque-Bera (JB): 4.699
Skew: -1.243 Prob(JB): 0.0954
Kurtosis: 3.673 Cond. No. 1.00


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

EFFICACIA su decessi nella fascia 80+ (Intervalli di confidenza al 99%)
Dep. Variable: y R-squared (uncentered): 1.000
Model: OLS Adj. R-squared (uncentered): 1.000
Method: Least Squares F-statistic: 7.904e+04
Date: Wed, 01 Dec 2021 Prob (F-statistic): 5.53e-31
Time: 18:32:35 Log-Likelihood: 4.7608
No. Observations: 17 AIC: -7.522
Df Residuals: 16 BIC: -6.688
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.005 0.995]
EFFICACIA 0.9297 0.003 281.135 0.000 0.920 0.939
Omnibus: 0.990 Durbin-Watson: 0.228
Prob(Omnibus): 0.610 Jarque-Bera (JB): 0.838
Skew: -0.477 Prob(JB): 0.658
Kurtosis: 2.479 Cond. No. 1.00


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.


© 2020 Max Pierini. Thanks to Sandra Mazzoli & Alessio Pamovio. ipynb-website © 2017 Peter Carbonetto & Gao Wang

Exported from Italia/Vaccini_Efficacia.ipynb committed by maxdevblock on Wed Dec 1 17:35:27 2021 revision 52, c97a1ef0