Efficacy of COVID vaccines

Article about Efficacy of COVID vaccines

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Jul 7, 2021

With the communication regarding vaccination efficacy, everyone who had some statistics knows that a Bayesian model approach for estimating the effects of Covid-19 vaccines would be required (see figure 2).


Unfortunately, the provided data for vaccination efficacy is often insufficient and the metrics not consistent. We would be interested in the rue posterior vaccination efficacy of the different vaccine regimes (and also the side effects).

As a short summary:

What is Vaccine Efficacy

Before moving to Bayesian statistics, let me explain how to estimate Vaccine Efficacy. First, we need to randomly assign study participants into two groups: vaccine and placebo groups. People in the first group called the vaccine group, receive vaccines, and those in the placebo group received a placebo. Neither participants nor personnel who apply knows if a participant is getting a vaccine or placebo. Then, when the study is finished, the Vaccine Efficacy is estimated by the formula

\\
\

where IRR is the Incidence Rate Ratio given by the formula

\\
\\

Vaccine Incidence Rate is the ratio of confirmed cases of Covid-19 illness per number of people in the vaccine group and Placebo Incidence Rate is the same for the placebo group



Most people remember though the Bayes theorem


As a short reminder:

Bayes theorem is used in Bayesian methods to update probabilities, which are degrees of belief, after obtaining new data. Given two events {\displaystyle A}A and {\displaystyle B}B, the conditional probability of {\displaystyle A}A given that {\displaystyle B}B is true is expressed as follows:

{\displaystyle P(A\mid B)={\frac {P(B\mid A)P(A)}{P(B)}}}{\displaystyle

where {\displaystyle P(B)\neq 0}{\displaystyle. Although Bayes theorem is a fundamental result of probability theory, it has a specific interpretation in Bayesian statistics. In the above equation, {\displaystyle A}A usually represents a proposition (such as the statement that a coin lands on heads fifty percent of the time) and {\displaystyle B}B represents the evidence, or new data that is to be taken into account (such as the result of a series of coin flips). {\displaystyle P(A)}P(A) is the prior probability of {\displaystyle A}A which expresses ones beliefs about {\displaystyle A}A before evidence is taken into account. The prior probability may also quantify prior knowledge or information about {\displaystyle A}A{\displaystyle P(B\mid A)}P(B\mid is the likelihood function, which can be interpreted as the probability of the evidence {\displaystyle B}B given that {\displaystyle A}A is true. The likelihood quantifies the extent to which the evidence {\displaystyle B}B supports the proposition {\displaystyle A}A{\displaystyle P(A\mid B)}P(A\mid is the posterior probability, the probability of the proposition {\displaystyle A}A after taking the evidence {\displaystyle B}B into account. Essentially, Bayes theorem updates ones prior beliefs {\displaystyle P(A)}P(A) after considering the new evidence {\displaystyle B}B.[1]

The probability of the evidence {\displaystyle P(B)}P(B) can be calculated using the law of total probability. If {\displaystyle \{A_{1},A_{2},\dots ,A_{n}\}}{\displaystyle is a partition of the sample space, which is the set of all outcomes of an experiment, then,

{\displaystyle P(B)=P(B\mid A_{1})P(A_{1})+P(B\mid A_{2})P(A_{2})+\dots +P(B\mid A_{n})P(A_{n})=\sum _{i}P(B\mid A_{i})P(A_{i})}{\displaystyle

When there are an infinite number of outcomes, it is necessary to integrate over all outcomes to calculate {\displaystyle P(B)}P(B) using the law of total probability. Often, {\displaystyle P(B)}P(B) is difficult to calculate as the calculation would involve sums or integrals that would be time-consuming to evaluate, so often only the product of the prior and likelihood is considered, since the evidence does not change in the same analysis. The posterior is proportional to this product:

{\displaystyle P(A\mid B)\propto P(B\mid A)P(A)}{\displaystyle

The maximum a posteriori, which is the mode of the posterior and is often computed in Bayesian statistics using mathematical optimization methods, remains the same. The posterior can be approximated even without computing the exact value of {\displaystyle P(B)}P(B) with methods such as Markov chain Monte Carlo or variational Bayesian methods.


So it would be very helpful to take the appropriate data and do the calculation as outlined in the Bayesian statistics to get the rue picture. As for people in favor and against vaccinations, the communication is usually biased.


Some help would be here if the data is available.


Martin Signer activities: Executive, Business Development/Sales, Capital Markets, Investor Relations/Marketing, Business Development/Sales, Capital Markets, Events, Investor Relations/Marketing, Human Resources, Executive, Business Development/Sales, Capital Markets, Investor Relations/Marketing, Business Development/Sales, Capital Markets, Events, Investor Relations/Marketing, Human Resources, Executive, Business Development/Sales, Business Development/Sales, Executive, Business Development/Sales, Business Development/Sales, Executive, Business Development/Sales, Business Development/Sales, Executive, Business Development/Sales, Business Development/Sales, Executive, Business Development/Sales, Business Development/Sales, Executive, Business Development/Sales, Business Development/Sales, Executive, Business Development/Sales, Business Development/Sales, Executive, Business Development/Sales, Business Development/Sales, Managing Director, IR, test, Testing, Testing, Participant, Partner, Media partner, Media partner, Event manager, Events, Tester, Event listing, Tester, IR, IR, Test, Managing Director, IR, IR, IR, IR Partner, Partner, Employer, IR, IR, IR, IR, IR, IR, IR, IR, Managing Director, Managing Director, IR, IR, IR, Promotion, Promotion, Promotion, Promotion, Promotion, Promotion, Promotion, Promotion, Promotion, Promotion, Promotion, Marketing, Marketing, Marketing, Marketing, Events, Marketing, Marketing, Marketing, Marketing, Marketing, Marketing, Marketing, Marketing, Testing, Marketing partner, tester, tester, Job position, Event organizer, Marketing, Event organizer, Event organizer, Event organizer, Event organizer, Event organizer, Event organizer, Event organizer, Marketing, Marketing, Tester, Marketing, Event organizer, Marketing, Testing, Testing, Testing, Testing, Testing, Marketing, Co-organizer
Martin Signer
Jul 7, 2021 01:25

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