Selection and quantification of infection endpoints for trials of vaccines against intestinal helminths

Abstract

Vaccines against human helminths are being developed but the choice of optimal parasitological endpoints and effect measures to assess their efficacy has received little attention. Assuming negative binomial distributions for the parasite counts, we rank the statistical power of three measures of efficacy: ratio of mean parasite intensity at the end of the trial, the odds ratio of infection at the end of the trial, and the rate ratio of incidence of infection during the trial. We also use a modelling approach to estimate the likely impact of trial interventions on the force of infection, and hence statistical power. We conclude that (1) final mean parasite intensity is a suitable endpoint for later phase vaccine trials, and (2) mass effects of trial interventions are unlikely to appreciably reduce the force of infection in the community - and hence statistical power - unless there is a combination of high vaccine efficacy and a large proportion of the population enrolled.

Fig. 1

Relation between the proportion parasite-positive at the end of the trial, incidence rate, and the two parameters of the negative binomial distribution: the mean parasite intensity (μ), and the k dispersion parameter.

Fig. 2

Surface and contour plots of Z statistics from four candidate effect measures. The horizontal axes in the surface plots, and both axes in the contour plots, show possible values of mean infection intensity (number of parasites seen per person) in the two arms (‘0’ and ‘1’) of a trial with 100 people per arm and a fixed value of 0.5 for negative binomial dispersion parameter (k). The vertical axis in the surface plot, and the contour values in the contour plot, show Z statistics: the expected value of the test statistic divided by its standard error. A Z statistic of 2 (or −2) corresponds to a two-sided p value of 0.05, and Z = 4 corresponds to p = 0.00006. Comparing the effect measures, a higher Z value for the same means indicates greater statistical power. For any combination of mean infection intensity, the ratio of means has the highest absolute value of Z, and so is the most powerful effect measure, followed by the rate ratio. The ratio of means achieves Z values more than 20, while the others do not reach Z = 10. Whether the rate estimation is based on three examination rounds, or continuous surveillance, makes little difference, as indicated by comparing their Z values for any given combination of mean infection intensities in the two arms. Finally, the odds ratio is slightly less powerful than the rate ratio. This can be seen, for example, by noting that the latter reaches values of Z = ±6 for slightly smaller differences in means: these contours are further in from the upper left and bottom right corners.

Fig. 3

For small values of k, the ratio of means is not always the most powerful efficacy measure. In this example, k is proportional to the square root of the mean infection intensity, and equal to 0.5 when the mean is 100 (the highest value in the range considered). Other parameters are as in Fig. 2. In the current figure, the rate ratio is the most powerful efficacy measure. To see this, note that, for example, the rate ratio efficacy measures reach Z values of 10 for some combinations of means, while the other efficacy measures do not.

Fig. 4

Examples of model predictions of the change in mean parasite intensity over time. People in the trial, whether in the vaccine or placebo arm, are assumed to have any initial infections cleared at baseline.

Fig. 5

The effect of vaccine efficacy and proportion of population in the trial on (a) the post-trial equilibrium mean in the control arm (left panel) and (b) Z statistic for the ratio of mean parasite intensity in vaccine/control arms (right panel).