Covid-19 Vaccines

Author

Russell Almond

Published

November 18, 2020

Good News! In November, both Pfizer and Moderna announce Phase 3 Vaccine Trials with promising results.

With over 30,000 participants in each study they reported the following data.

Treatment Pfizer ModernaAll ModernaSevere N
Placebo 90 95 11 15000
Vaccine 5 5 0 15000

Measures of Effectiveness

We start with a cross-tab

Treatment Sick Healthy Total
Placebo SP HP NP
Vaccine SV HV NV
Total NS NH N

Odds of getting sick

Placebo: $ SP/HP $ Vaccine: $ SV/HV $

Cross Product (Odds) Ratio

\[ OR = \frac{SP/HP}{SV/HV} \] How much does your odds of getting sick increase if you get the placebo instead of the vaccine.

Risk Ratio

\[ RR = \frac{SP/NP}{SV/NV} \] How much does your probiliby of getting sick increase if you get the placebo instead of the vaccine.

Vaccine Effectiveness

\[ VE = 100 (1 - \frac{1}{RR}) \]

Chi-square test

Null hypothesis is that getting the disease is independent of the vaccine. In other words, \(OR=RR=1\).

\[ SV/NV = SP/NP \]

Large chi-squared value incidates that cross product rate is not 1 (but doesn’t tell if placebo or vaccine is better!

Z-score test

Another way to work with these data is to calculate probabilities of infection for each group and the standard errors. Then can use the \(z\)-test to compare.

\[p_V = p(S|V) = SV/NV \qquad SE(p_V) = \sqrt{p_V(1-p_V)/NV} \] \[p_P = p(S|P) = SP/NP \qquad SE(p_P) = \sqrt{p_P(1-p_P)/NP} \] \[ z = \frac{p_P-p_V}{\sqrt{SE(p_V)^2 + SE(p_P)^2}}\]

Pfizer Vaccine

There were around 30,000 volunteers in the Phase 3 trials; 15,000 in each arm.

Treatment Pfizer p.Pfizer s.Pfizer
Placebo 90 6e-03 0.00063
Vaccine 5 3e-04 0.00015

X^2 – Pfizer

In SPSS this is done by producing a cross-tab. We don’t have the number of negative cases in each arm of the study, but up to rounding error it is just the sample size, so we will use that.

Pfizer Cross Tab

Phizer X2

    Pearson's Chi-squared test with Yates' continuity correction

data:  as.matrix(select(covidVaccines, Pfizer, N))
X-squared = 74.034, df = 1, p-value < 2.2e-16
Z =  8.75 p =  0

Pfizer Risk Ratio
Risk Ratio:  18 Vaccine Effectiveness:  94.4

Moderna Vaccine – All Cases

There were around 30,000 volunteers in the Phase 3 trials; 15,000 in each arm.

Treatment ModernaAll p.ModernaAll s.ModernaAll
Placebo 95 0.0063 0.00065
Vaccine 5 0.0003 0.00015

X^2 – Moderna (All Cases)

In SPSS this is done by producing a cross-tab. We don’t have the number of negative cases in each arm of the study, but up to rounding error it is just the sample size, so we will use that.

Moderna All Cases Cross Tab

Moderna All Cases X2

    Pearson's Chi-squared test with Yates' continuity correction

data:  as.matrix(select(covidVaccines, ModernaAll, N))
X-squared = 78.942, df = 1, p-value < 2.2e-16
Z =  9.03 p =  0

Moderna All Cases Risk Ratio
Risk Ratio:  19 Vaccine Effectiveness:  94.7

Moderna Vaccine – Severe Cases

Treatment ModernaSevere p.ModernaSevere s.ModernaSevere
Placebo 11 7e-04 0.00022
Vaccine 0 0e+00 0.00000

X^2 – Moderna (All Cases)

In SPSS this is done by producing a cross-tab. We don’t have the number of negative cases in each arm of the study, but up to rounding error it is just the sample size, so we will use that.

Moderna Severe Cases Cross Tab

Moderna Severe Cases X2

    Pearson's Chi-squared test with Yates' continuity correction

data:  as.matrix(select(covidVaccines, ModernaSevere, N))
X-squared = 9.0869, df = 1, p-value = 0.002574
Z =  3.32 p =  0 
Risk Ratio:  Inf Vaccine Effectiveness:  100

Yikes! The estimate for the chances of getting Severe Covid-19 with the virus is 0. Divide by zero error!

But probability zero means impossible. That is not right!

Continuity Correction

Fix this by adding a conditinuity correction. We add 1/2 to all of the entries in the table.

In particular, this makes the estimated rate for getting severe COVID-19 \(\frac{1}{2}/(N+1)\).

Treatment ModernaSevere p.ModernaSevere s.ModernaSevere
Placebo 11 0.00077 0.00023
Vaccine 0 0.00003 0.00005 

Risk Ratio:  23 Vaccine Effectiveness:  95.7

References:

  • StatNews article on Pfizer vaccine: https://www.statnews.com/2020/11/09/covid-19-vaccine-from-pfizer-and-biontech-is-strongly-effective-early-data-from-large-trial-indicate/

  • Official Protocol document from Pfizer: https://www.pfizer.com/science/coronavirus

  • Pfizer Press Release: https://www.pfizer.com/news/press-release/press-release-detail/pfizer-and-biontech-announce-vaccine-candidate-against

  • Moderna Press Release: https://investors.modernatx.com/news-releases/news-release-details/modernas-covid-19-vaccine-candidate-meets-its-primary-efficacy

  • Entries from Andrew Gelman’s Blog: https://statmodeling.stat.columbia.edu/2020/11/16/estimating-efficacy-of-the-vaccine-from-95-true-infections/

https://statmodeling.stat.columbia.edu/2020/11/11/the-pfizer-biontech-vaccine-may-be-a-lot-more-effective-than-you-think/

  • How to use SPSS to obtain Odd Ratio and Relative Risk http://brahms.emu.edu.tr/icetin/spss8-RR-OR.pdf