Introduction and Examples

Abstract

Vaccine efficacy and vaccine effectiveness, VE, are generally estimated as one minus some measure of relative risk, RR, in the vaccinated group compared to the unvaccinated group:

$$VE = 1-RR.$$

(1.1)

Historically, interest focused on evaluating protective effects of vaccination. Study designs and statistical analysis played a role even with early immunizations. In the November 5, 1904, issue of the British Medical Journal, Karl Pearson published a criticism of the Antityphoid Committee’s report on the anti-typhoid inoculation statistics from the South African War and from India. The Committee’s report had recommended continued use of anti-typhoid inoculation. Armed with the correlation coefficient and the Theory of Correlation, Pearson reanalyzed the data and claimed that the correlations between protection against disease and inoculation ranged from 0.021 and 0.445, mostly around 0.1, with the correlations against mortality in a similar range. He compared these values with his analysis of the relation of recovery from smallpox with smallpox vaccination, which were in the range 0.576 and 0.769. Although he demurred somewhat due to his lack of knowledge about typhoid, he wrote that the results were such as “would justify suspension of [anti-typhoid inoculation] as a routine method” (p. 1244). He suggested “that a more effective serum or effective method of administration must be found before inoculation ought to become a routine practice in the army” (p. 1245). The immunologist A. E. Wright countered the following week, writing that although he did not understand the correlation coefficient, the mortality was reduced four- to sixfold, so that Pearson’s conclusion must be wrong and that the Medical Advisory Board, who had heeded the criticism, could not “hide behind Professor Pearson’s petticoats” (p. 1345). In addition to low correlation, Pearson was also arguing about the variability of the data. The argument continued in the British Medical Journal weekly for a full nine weeks until December 31, 1904, when Pearson finally gave up continuing the controversy after Wright refused to deal with what he had called “statistical minutiae” (p. 1614) and the “mathematical expression” (p. 1727).