How Real is the Effectiveness?

Introduction

Or rather: how effective is reality? No, how real is reality? What exactly is effectiveness?

Imagine an experiment with a certain number of people. 50% of the participants are in the experimental group (vaccination) and 50% in the control group (no vaccination). After a certain period, the number of infections is checked.

If there were 100 infected individuals in the control group, and we denote the effectiveness in % as \(W\), then the number of infected individuals in the experimental group is:

\[100 - W\]
Example: 100 infected in the non-vaccinated group and an effectiveness of 90% means we would expect 10 infected in the vaccinated group (note that there are equal numbers of vaccinated and non-vaccinated). If an effectiveness is provided, one can immediately calculate \(100 - \mathrm{effectiveness}\) and get a good idea of what that means.

A small test: If a vaccine has an effectiveness of 46%, how many infected individuals are expected in the experimental group if there are 1000 infected in the control group (with equally sized groups)?

If the total number of infected \(I\) and the effectiveness \(W\) in % are known, it can be calculated how many were infected in the control group (\(K\)) and how many in the experimental group (\(E\)).

\[K = 100I/(200-W)\]
\[E = I - K\]
For the example from Unstatistik (https://www.rwi-essen.de/unstatistik/109/) with 90% effectiveness (\(W\)) and 94 confirmed infections (\(I\)):

100 * 94 / (200 - 90)

[1] 85.45455

94 - 100 * 94 / (200 - 90)

[1] 8.545455

There isn’t much more one needs to know about effectiveness, but for those interested, the following section provides a more detailed explanation.

Formula for Effectiveness

\[ (1 - E/K) 100 = W\]

\(W\) is the effectiveness in %, \(E\) is the number of infected in the experimental group (vaccinated), and \(K\) is the number of infected in the control group (unvaccinated).

To illustrate an effectiveness of 90%, let’s consider the hypothetical case where there are 100 infected in the control group. How many would there be in the experimental group?

\[ (1 - E/100) 100 = 90\]
\[ 100 - E = 90\]
\[ E = 100 - 90 = 10\]
For 100 infected in the control group, there are only 10 infected in the experimental group.

In general (for 100 infected in the control group):

\[ E = 100 - W \]

CureVac

The case of CureVac is interesting. The stock price dropped after an effectiveness study showed only 47% effectiveness, significantly lower than competitors.

An effectiveness of 47% means that for 100 infected in the control group, there are 53 in the experimental group. This isn’t as bad as 47% might sound. CureVac is effective; it’s much better to be vaccinated with CureVac than not vaccinated at all. Any positive effectiveness is good; only at 0% does the vaccine do nothing.

We would need to account for the costs of vaccination, although we assume for now that there are none. The benefit of preventing illness also needs to be quantified. And, of course, higher effectiveness is always desirable. However, this doesn’t automatically mean that 47% is absolutely bad, just relatively.

Given Number of Infected

If the number of infected individuals and the effectiveness are known, it’s possible to calculate how many became ill in the experimental and control groups:

\[K + E = I\]

\[(1 - E/K) 100 = W\]

\[(1 - (I-K)/K) 100 = W\]

\[(K-I+K) 100 = WK\]
\[200K-100I = WK\]
\[200K-WK = 100I\]
\[K(200-W) = 100I\]
\[K = 100I/(200-W)\]
\[E = I - K\]