Estimation of vaccine efficacy in a repeated measures study under heterogeneity of exposure or susceptibility to infection

Abstract

Vaccine efficacy (VE) is commonly estimated through proportional hazards modelling of the time to first infection or disease, even when the event of interest can recur. These methods can result in biased estimates when VE is heterogeneous across levels of exposure and susceptibility in subjects. These two factors are important sources of unmeasured heterogeneity, since they vary within and across areas, and often cannot be individually quantified. We propose an estimator of VE per exposure that accounts for heterogeneous susceptibility and exposure for a repeated measures study with binary recurrent outcomes. The estimator requires only information about the probability distribution of environmental exposures. Through simulation studies, we compare the properties of this estimator with proportional hazards estimation under the heterogeneity of exposure. The methods are applied to a reanalysis of a malaria vaccine trial in Brazil.

Figure 1

Population VE based on the hazard ratio comparing vaccinated with unvaccinated subjects as a function of time and individual VE of 50%. (a) Exposure was generated assuming a mixing distribution of two Poisson distributions with means λ₁=12 and λ₂=2, with the probability of occurrence of λ₂ from 0.1 to 0.9, as specified. Triple dot-dashed line, 0.1; dashed line, 0.3; dot-dashed line, 0.5; dotted line, 0.7; solid line, 0.9. (b) Exposure was generated assuming a continuous mixture of Poisson distributions (negative binomial) with different means and variances (means and variances were chosen to mimic the mean and variance of the two point distribution). Triple dot-dashed line, mean=11 and variance=20; dashed line, mean=9 and variance=30; dot-dashed line, mean=7 and variance=32; dotted line, mean=5 and variance=26; solid line, mean=3 and variance=12.

Figure 2

Comparison of the per cent bias in VE under heterogeneity of the intensity of exposure, as a function of the expected value of the mixing distribution of exposure, in 250 simulations each with a sample size of 1200 subjects. VE was estimated via modelling of time to first infection/disease (using proportional hazard and Gaussian frailty models), via modelling of time to all infection/disease (using Anderson–Gill model) and via a repeated measures model with a complementary log–log link and GEE approach. Exposure was generated assuming a continuous mixture of Poisson distributions with the specified mean intensity λ (and variance ϕλ). Dashed line, first event frailty model; dot-dashed line, repeated measures; dotted line, Anderson–Gill; solid line, first event proportional hazard.