Bounds on the effect of vaccine induced immune response on outcome

A major goal of vaccine development is the identification of immune responses that are responsible for vaccine efficacy. In theory, modest vaccines could be successfully improved by increasing such immune responses. And for a vaccine with a great benefit in one population, inducing such immune response in a different population could help one conclude the vaccine would have great benefit there. Such identification is tricky because the immune response to vaccination can only be measured in the vaccine group and thus immune responses might only be identifying individuals with a constitutional ability to remain uninfected, rather than being causal. Define the vaccine induced immune response as X(1). The value X(1) is a potential outcome; it is measured directly in vaccinees but unobserved in the placebo group. Our goal is to regress outcome on X(1) separately in the vaccine and placebo groups and to see if the vaccine effect varies with X(1). Put another way, our goal is to see if there is a vaccine by X(1) interaction. Regression of outcome on X(1) is easy to do in the vaccine group, but difficult in the placebo group as X(1) is not observed. In this paper we derive bounds on the regression curve in the placebo group. For a continuous endpoint these bounds can be unhelpful, or can help modestly temper our enthusiasm for a role of X(1) on the vaccine effect. For binary outcomes with 100% placebo infection the bound is very tight but unhelpful as 100% infection precludes identification of any covariate with a differential effect on placebo infection. We apply these methods to experiments of anthrax vaccine in rabbits with survival to challenge as the outcome and demonstrate how to extrapolate the model to humans.