The Effects of Herd Immunity on the Power of Vaccine Trials

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1 University of California, Los Angeles From the SelectedWorks of Ron Brookmeyer 2009 The Effects of Herd Immunity on the Power of Vaccine Trials Blake Charvat Ron Brookmeyer, Johns Hopkins Bloomberg School of Public Health Jay Herson, Johns Hopkins Bloomberg School of Public Health Available at:

2 The Effects of Herd Immunity on the Power of Vaccine Trials BLAKE CHARVAT, RON BROOKMEYER, and JAY HERSON We evaluate the effects of herd immunity on the power of vaccine trials. We consider large-scale trials in which persons are individually randomized to either placebo or vaccine. We evaluate the adequacy of naive power calculations that ignore the effects of herd immunity such as those based on the comparison of two independent binomials. We developed a simulation design to evaluate the quantitative effects of herd immunity on power. The simulation design accounted for nonhomogeneous mixing. We found that naive power calculations that ignore the effects of herd immunity can seriously overestimate power. In fact, we found that as sample size increases it is possible for the power to actually decrease. The reason is that herd immunity reduces the overall number of infections. In the situations we considered, power may eventually begin to decrease once the proportion of the population enrolled in the trial exceeds about 25%. We discuss the findings in the context of a pneumococcal vaccine trial for children. Our results serve as a cautionary note that naive sample size calculations for larger scale vaccine trials that ignore the impact of herd immunity can yield underpowered studies. Simulations such as that suggested here can help alert investigators to situations where significant dilution of power could result from ignoring the effects of herd immunity. Key Words: Clinical trial; Power; Simulation. 1. Introduction Vaccination for a communicable disease reduces disease incidence through two mechanisms, the direct and indirect effects of the vaccine (Halloran 1998; Halloran, Struchiner, and Longini 1997). The direct effect refers to the reduction in disease incidence because the vaccine acts to reduce the probability of transmission when an uninfected individual is exposed to an infected individual. We shall also refer to the direct effect of a vaccine as its biological effect. The indirect effect refers to the reduction in disease incidence due to factors other than the direct reduction in the transmission probability to an uninfected individual from a single exposure. The main indirect effect is known as herd immunity, which refers to incidence reduction due to the removal of would-be infectious individuals from the pool of infectious carriers. This results in decreased opportunity for uninfected individuals (both unvaccinated and vaccinated) to come in contact with infected individuals. Herd immunity derives its name from the effects at the population or herd level (Fine 1993). There have been important methodological advances in estimating and disentangling the direct and indirect effects of vaccines (see, e.g., Longini, Halloran, and Nizam 2002; Longini et al. 1998; and Moulton et al. 2006). It is widely recognized that herd immunity is an important beneficial effect of widespread vaccination. However, what does not appear to be widely appreciated is that herd immunity is an important factor to consider when designing large-scale vaccine trials, and in particular when determining power and sample size requirements. Herd immunity can reduce the statistical power of studies because the number of study endpoints (the disease incidence) is reduced. In this article, we study the effect of herd immunity on the power of large-scale vaccination trials in which participants are individually randomized to either the placebo or vaccine. We show c American Statistical Association Statistics in Biopharmaceutical Research February 2009, Vol. 1, No. 1 DOI: /sbr

3 Statistics in Biopharmaceutical Research: Vol. 1, No. 1 that failure to account for herd immunity can result in underpowered trials. Moreover, we report that, under some conditions, increases in the sample size of a trial could actually lead to decreases in statistical power because of the effects of herd immunity. This work was motivated by a large-scale pneumococcous vaccine trial in children similar to that reported by Black et al. (2000). The trial involves the individual randomization of children to either the vaccination or placebo arm in several communities. Given the increase in antibiotic-resistant strains of pneumococcous (McCormick et al. 2003) development of a vaccine is important for control of pneumonia especially (Dagan et al. 2001). A significant fraction of all children living in the communities were expected to be enrolled in the study. Section 2 introduces the key definitions and model considerations. Section 3 describes a simulation design for evaluating the effects of herd immunity on statistical power of vaccine trials. Section 4 gives the results. Section 5 illustrates the approaches, and Section 6 discusses the findings and implications for vaccine trial design. 2. Preliminary Considerations This section considers the problem of designing an individually randomized placebo-controlled vaccine trial. The goal is to evaluate the efficacy of a vaccine for protecting against an infectious agent that is spread by person-to-person transmission. The focus of this article is on large-scale randomized placebo controlled vaccine trials in which a large proportion of the population in a community will be enrolled in the study. In such situations the effect of herd immunity on power must be considered. Herd immunity reduces the incidence of infection not only among vaccinated children but also among unvaccinated children because the number of infectious exposures is reduced. Herd immunity is an indirect effect of the vaccine because the risk of disease is altered even among persons who have not directly received the vaccine intervention. We focus in this article on the evaluation of vaccines in which the protective benefit arises from reducing the susceptibility to infection without reducing infectiousness. We are interested in designing a trial with adequate sample size to have sufficient power. Consider a community composed of N persons. A subset n of the N persons in the community is selected for inclusion in the trial. Suppose n 0 persons are individually randomized to the placebo group and n 1 persons are randomized to the vaccine group (where n 0 + n 1 = n). We follow each individual in the study for T days. We observe the cumulative number of cases of infection over the course of the trial in the placebo and vaccinated group which are called X 0 and X 1, respectively. The sample sizes of such a vaccine trial are typically determined so that the trial achieves high power (e.g., 90%). A traditional power calculation is usually based on a test for two independent binomial proportions (ˆr 1 = X 1 /n 1 and ˆr 0 = X 0 /n 0 ). An a priori estimate of r 0 is needed for the power calculation, which is typically derived from historical data on the background cumulative incidence rate of infection in a completely unvaccinated population. The sample sizes are calculated to provide adequate statistical power to achieve a specified reduction in the incidence rate. However, as we discuss, this may be a naive approach that can significantly overestimate power due to the failure to account for herd immunity. The power of a study is determined not only by the sample size (n) but also by the sampling fraction n/n which is the proportion of the community enrolled in the study. If a significant fraction of the population is enrolled in the study and the vaccine is indeed efficacious, then herd immunity lowers the incidence rate in the unvaccinated study participants below the historical background rate. To understand and quantify the role of herd immunity and its impact on statistical power, we must first introduce some key probabilistic concepts to characterize the transmission of the infection. We consider a vaccine that reduces the probability of transmission per infectious contact. We define the transmission probabilities per infectious exposure as follows. Let p 0 represent the conditional probability of infection if an uninfected unvaccinated individual is exposed to an infectious individual. Let p 1 represent the similar probability for a vaccinated individual. We denote the cumulative probabilities that a vaccinated and unvaccinated person becomes infected over the course of the T days of the study by r 0 and r 1, respectively. The probabilities r 0 and r 1 represent the cumulative risk of infection over the course of the study while p 0 and p 1 represent the transmission probabilities per exposure to the infectious agent. The cumulative risks of infection depend on the daily numbers of exposures to infection. We define a single exposure as the contact with a single infected individual over the course of a day. We let Y t represent the number of such exposures on day t, where Y t is a stochastic process. If the risks of transmission are independent from one exposure to the next and homogenous in the sense that the transmission probabilities are the same for all infectious contacts, then conditional on the stochastic process Y t we have r i = 1 T (1 p i ) Y t, t=1 where the subscripts i = 0 and i = 1 refer to the placebo and vaccinated groups, respectively. 2

4 The Effects of Herd Immunity on the Power of Vaccine Trials Figure 1. Schematic illustration of course of infectiousness in an infected person. The biological efficacy of the vaccine is defined to be the proportionate reduction in the transmission probability per exposure. Specifically, the biological efficacy of the vaccine is VE B = 1 p 1 p 0. In a vaccine trial we can measure the cumulative incidence (or attack) rates (see, e.g., Greenwood and Yule 1915; Halloran, Struchiner, and Longini 1997). We define VE CI to be the proportionate reduction in the cumulative incidence rates of infection. Specifically, the vaccine efficacy based on the cumulative incidence rates is defined to be VE CI = 1 r 1 r 0. The quantity VE CI incorporates both direct and indirect effects of the vaccine, and also depends on the study duration T. In the Appendix we show that VE CI is bounded above by the biological vaccine efficacy; that is VE CI = VE B if VE B > 0. A vaccine trial can provide direct information about VE CI since we can observe the quantities X 0 and X 1. In contrast, VE B is not directly observable and is difficult to disentangle from a vaccine trial without additional modeling assumptions (e.g., Longini et al. 1998), expert scientific judgment, or alternative designs such as community randomized studies (e.g., Moulton et al. 2006). A newly infected person is not necessarily immediately infectious to others. Figure 1 illustrates the time course of infection within an individual. A newly infected person has a latency period during which the individual is not infectious. Subsequently the individual becomes infectious for a period of time following which they are no longer infectious to others. In general, the latency and infectious periods are random quantities. We will be assuming that once an infected individual is no longer infectious he or she is considered immune to reinfection at least for the remainder of the trial. 3. Simulation Design In this section we explain the simulation methodology used to evaluate the effect of herd immunity on the power of a vaccine trial. The simulation allows for heterogeneity in that we will not necessarily assume that the transmission probabilities are the same across all exposure contacts in the population. Our simulation design was motivated by the work of Halloran et al. (2002), whose framework assigns individuals higher probabilities of infection on a given day when interacting with close contacts (e.g., members of immediate family) than with more distant contacts (e.g., acquaintances from a different neighborhood). The structure of our simulation involves generating a population of N persons who are subdivided into M clusters of equal size (50 persons per cluster). Our choice of the cluster size (50 persons) was motivated by the pneumococcous vaccine trial in children and the transmission dynamics that occur through day care facilities. In our simulation, we consider vaccines that reduce the susceptibility to infection. Vaccines could also have other beneficial effects such as reducing the infectiousness of infected persons or shortening the duration of the infectious period, although those effects are not considered in this simulation design. The simulation is run in discrete time by simulating the numbers of new persons who become infected each day. The risks of infection depend on the number of contacts with infectious persons and the transmission probabilities (Daley and Gani 1999). A discrete numerical distance is defined between each cluster so that each individual is a measured distance from each other individual (distance is 0 for individuals in the same cluster). The probability per day that an unvaccinated and uninfected individual becomes infected upon contact with an infectious individual at distance d is p 0d. The corresponding probability for a vaccinated individual is p 1d. In general, the probability that an individual of vaccination status i (i = 0 for control and i = 1 for vaccinated) becomes infected on day t is α i (t) = 1 d (1 p id ) Y td, (1) whereăy td are the numbers of infectious persons on day t who are a distance d away from the individual. Equation (1) assumes that a person has contact with persons at all distances. We recognize that there is additional heterogeneity in a population in that persons have differential affinities for traveling between clusters. A person may have no contact with persons from a distant cluster. To account for this, we assign to each person a maximum traveling distance to neighboring clusters which is a random variable (D Discrete Uniform[0,5]) which characterizes the person s propensity to travel. A person travels to all clusters that are within this fixed dis- 3

5 Statistics in Biopharmaceutical Research: Vol. 1, No. 1 Figure 2. Schematic illustration of simulation design incorporating contacts within and between clusters. A person is allowed to travel to clusters within a distance D of their home cluster. tance D each day, and has contact with all persons in such clusters each day. Figure 2 is a schematic figure that illustrates the clusters and the allowance for traveling among clusters in our simulation. The maximum traveling distance D is assumed fixed for any person over the course of the study but D is allowed to vary among persons. That is, the maximum traveling distance D is fixed for an individual from day to day, however, D is allowed to vary from individual to individual. Then, the probability that an individual who has maximum daily traveling distance D and vaccination status i becomes infected on day t is α i D (t) = 1 d D (1 p id ) Y td. We assume the transmission probabilities declines linearly with distance, that is, p id = p i0 (p i0 ) d D + 1. In summary, the structure of our simulation is as follows. The simulation is run daily in discrete time for T = 365 days. Each person is assigned a random maximum daily traveling distance (D) which remains fixed for that individual throughout the study. We simulate the event that an individual becomes infected on day t by a Bernoulli trial with event probability α i D (t). At day 0, we seed the population with initial infections. Each of the N persons has a probability of of being initially infected on day 0. We will assume that the length of the latency and infectious periods follow independent exponential distributions with mean of two days and four days, respectively. We chose the value of p 00 to yield a one-year cumulative incidence rate in an unvaccinated population of 1.5 cases per 100 individuals (i.e., p 00 was determined, by simulation, so that r 0 = in a completely unvaccinated population). Figure 3 illustrates a simulated epidemic in an unvaccinated population using these parameters. Thus, the simulation allows susceptible persons to come into contact with persons from many different clusters. The probability they become infected is determined by the transmission rates (p id ), and the number of contacts made with infectious persons from different distanced clusters (Y td ). We performed 500 simulations for each set of conditions. We varied the sample size (n), the population size N (and thus the sampling fraction f = n/n), as well as the biological efficacy of the vaccine VE B = 1 p 1d /p 0d. In each run of the simulation we calculated the chi-squared statistic to test the null hypothesis of equality of the cumulative infection rates in the placebo group and in the 4

6 The Effects of Herd Immunity on the Power of Vaccine Trials Figure 3. Example of a single simulated epidemic (daily incidence) in an unvaccinated population. vaccine group (H 0 : r 0 = r 1 ): χ 2 = ) 2 (ˆr0 ˆr 1 ( ), ˆr(1 ˆr) 1 n0 + n 1 1 where ˆr = (X 1 + X 0 ) /n. The simulated power was the fraction of simulations in which the null hypothesis was rejected at the 0.05 level of significance. We also calculated the estimate of VE CI in each simulation as well as its mean and standard deviation over all replications of the simulation. 4. Results We present two sets of results. In the first set we varied the sample size (n = n 1 + n 2, where n 1 = n 2 ) keeping the total population N fixed, while in the second set we fixed the sample size and varied the population size N of the underlying community. We first consider the simulations results when N is held fixed and we vary n. We considered a population of size N = 35,250 persons who were subdivided into 650 clusters. Figure 4 shows the power results for VE B = 0.50, 0.80, and We find that as sample size increases each of the power curves initially increases but then ultimately decreases. The behavior of the power curves appear to plateau when roughly 25% of the population is enrolled in the vaccine trial. We find the surprising and somewhat counterintuitive result that as n continues to increase, the power ultimately begins to decrease. Essentially, what is happening is that the power increases derived from increases in n is counterbalanced by a decrease in power because of a lower incidence of infection that results from herd immunity when an efficacious vaccine is given to a significant fraction of a population. Accordingly, we find that there is a point of diminishing returns whereby increasing sample size in a large-scale vaccine trial can actually lead to decreases in statistical power. We now consider the simulation results when the sample size of the vaccine trial n is held fixed, but we vary N (and thus the sampling fraction). The objective of this set of results is to show that the power depends not only on the sample size of the trial but also the sampling fraction f. We fixed the sample size at n = 8,460 (n 0 = n 1 = 4,230; as discussed in the next section). This sample size was chosen because a naive power calculation based on two binomials shows that such sample sizes would have 90% power to detect a VE CI = 0.50 if the background cumulative incidence in the control group is 0.015) Table 1 shows the results when VE B = 0.80 with the sample size fixed at n = 8,460. This table shows that as the sampling fraction enrolled in the study (f ) decreases, the power increases. In fact, the power ranges from 0.30 when f = 1 to a power of 0.99 when f = The surprising result here is that the power varies so dramatically even though the sample size of the trial was held fixed. Insight into why the power varies so strongly with f can 5

7 Statistics in Biopharmaceutical Research: Vol. 1, No. 1 Figure 4. Simulated power versus sample size for a vaccine trial conducted in a population of N = 35,250 persons. be gleaned by examining the mean numbers of infections (cases) that occurred over the course of the simulated trials. When f = 0.04 the mean number of infections was 75.2 but when f = 1, the mean number of infections (cases) was only The reason for this variation is herd immunity. As an increasingly larger fraction of the population is enrolled in a trial with an efficacious vaccine, then that lowers the total number of infections that occur, which in turn lowers the statistical power of the study. Table 1 also show the mean of the estimated values of VE CI which we find decreases as f increases. Figure 5 shows the power curves as a function of f for VE B = 0.80, 0.50, and 0.33 when the sample size n is fixed (n = 8,560). The figure illustrates that power decreases as the enrollment fraction (f ) increases because of the effects of herd immunity. In fact, even for the lowest value of f considered ( f = 0.04) we find a significant diminution of power. For example, when VE B = 0.50, we find that the power was only 0.80 even though a naive sample size calculation that ignored the effects of herd immunity would have suggested a power of Illustration We illustrate the simulation results presented in Section 4 in the context of a specific example. Suppose we wish to design a two-arm, placebo-controlled pneumococcal vaccine trial in children to determine the sample size requirements. Suppose it is also known from historical data that the annual cumulative incidence rate of infection among a population of unvaccinated children is Furthermore, the study investigators believe that the proposed vaccine would reduce the transmission probability per contact by half, that is, the biological efficacy of the vaccine is VE B = The problem is to determine the sample size to detect such a vaccine efficacy with power of 0.90 using a two-sided test with Type I error rate of Suppose the size of the population of children in the community in which the trial is to be conducted is N = 35,250. As indicated in the previous section, a naive sample size requirement based on two binomial proportions indicates that a sample size of n = 8,460 would be adequate to distinguish a proportion of from However, the simulation design described in Section 3 suggests that such a sample size would not be adequate. The simulated power is only 0.59 (see Figure 4) with n = 8,460. Thus, a sample size of 8,460 is inadequate to achieve the desired statistical power. The reason the naive sample size calculation is misleading is because it ignores herd immunity. Herd immunity has a significant impact because 24% of the population was enrolled in the trial. We can obtain more insight into why the naive sample size calculations fail if we closely examine the simulation results. The average simulated numbers of infections in the placebo and vaccinated groups were 44.1 and 25.3, respectively. Thus, the average simulated cumulative incidence in the placebo group was only 44.1/4,230 = , which was over 30% lower than the rate of that was assumed by the naive sample size calculation. 6

8 The Effects of Herd Immunity on the Power of Vaccine Trials Table 1. Output from simulations (500 replications each) with VE B = 80%, n = 8,460. Avg # of Avg # of placebo vaccinated Avg Population Sample Percent cases cases VE% size size enrolled (sd) (sd) (sd) Power% 8,460 8, (10.0) (3.5) (56.8) 14, (14.8) (4.3) (40.7) 35, (20.0) (5.7) (21.1) 211, (12.9) (5.5) (9.6) Figure 5. Simulated power versus fraction of the population enrolled in the trial ( f ) for a fixed trial sample size of n = 8,560 with VE B = 0.80, 0.50 and

9 Statistics in Biopharmaceutical Research: Vol. 1, No. 1 It is true that in a completely unvaccinated population we would expect a cumulative incidence of 0.015, but because a significant fraction of the population received vaccine (24%), herd immunity depresses the cumulative incidence among the unvaccinated to only As a result, power declines. Furthermore, the mean simulated VE CI was actually only 0.44 which is less than VE B = Indeed, the lower value of VE CI, compared to VE B, was predicted by the theoretical finding in the Appendix. The naive sample size calculation used a vaccine efficacy of 0.50 that was based on the assumed biological efficacy, rather than VE CI (which was actually only 0.44). Even if the value of 0.44 was used, the naive sample size calculation still would have overestimated power because the infection rate in the unvaccinated group was assumed too large. It would have been difficult to predict by how much the infection rate among the unvaccinated would have decreased due to herd immunity without performing the simulation study. How much larger would the sample size have to be to achieve a power of 0.90? Figure 4 shows that in fact a power of 0.90 cannot be achieved under the assumed infection transmission dynamics used in the simulation. The maximum achievable power is only 0.66 which would require enrolling a sample size of 26,910. As sample size increases, infection rates in the unvaccinated group declines because more persons in the population are vaccinated which increases the magnitude of the herd immunity phenomena. We recognize that these results depend on the particular simulation design (e.g., numbers of clusters; the assumed mixing of children between clusters). However, the simulation approach allows investigators to study the sensitivity of the power to a range of assumptions. This example serves as a cautionary note that naive sample size calculations that ignore the potential impact of herd immunity can be severely misleading and yield underpowered studies. Simulations such as that suggested here can help alert investigators when designing trials to situations when significant dilution of power could result from herd immunity. 6. Discussion The objective of this article was to evaluate the effects of herd immunity on the power and sample size requirements of a large-scale vaccine trial. We used a simulation study to quantify the potential effects of herd immunity. We obtained the surprising and somewhat counterintuitive finding that increases in sample size do not necessarily lead to increases in statistical power. Under the particular simulation conditions that we studied, we found that the power curve plateaus and then decreases when the population enrolled in the trial exceeds roughly about 25%. We also found that naive sample size calculations that ignore the effects of herd immunity can result in overestimation of the power of a study. Even when only 4% of the population is enrolled in a trial we still detected significant reductions in power resulting from herd immunity. Our results serve as a cautionary note to account for the effects of herd immunity when designing large-scale vaccine trials. We evaluated the power of a trial using a test for two proportions. Alternatively, the simulations could have been carried out based on a comparison of two Poisson variables using a time-to-event analysis that accounts for person time of follow-up. In that setting, we would expect to obtain similar results to that reported here. Furthermore, it might also be useful to evaluate the power of statistics that test for noninferiority. For example, it would be interesting to evaluate the performance of procedures for establishing minimum vaccine efficacy, as developed by Farrington and Manning (1990) that address the issue of herd immunity. We evaluated the effects of herd immunity through simulation because it is not possible to obtain analytically tractable solutions. However, the simulation design depends on a number of important assumptions and specifications. For example, the power clearly depends on the assumed biological efficacy of the vaccine which is an important source of uncertainty. Another important input factor is the initial condition, that is, the expected numbers of persons in the community who are infected at the start of the trial. The initial number infected could depend on when a trial is commenced during an epidemic season. Another important input parameter is the background disease incidence. As the disease incidence increases, the power increases. One measure of the effect of herd immunity, h, is the proportionate reduction in the event rate among the unvaccinated portion of a population compared to the event rate in a completely unvaccinated population. For example, with f = 0.60, n = 8,460, and N = 14,100, the herd effects, h, with annual incidence rates of 1.5% and 30% are 0.69 and 0.90, respectively. Thus we find that, with this measure of the herd immunity effect, the effect becomes more pronounced with increasing incidence. Another interesting design parameter is the randomization ratio. We performed our simulations using 1:1 randomization. If more persons are randomized to the vaccine, one would expect that the effect of herd immunity becomes more pronounced because a larger fraction of the population is given the vaccine. On the other hand, if more persons are randomized to the placebo the effect of herd immunity may be mitigated. It would be interesting to investigate to what extent changing the randomization ratio can alleviate the effects of herd immunity. 8

10 The Effects of Herd Immunity on the Power of Vaccine Trials Simulation allows investigation of power to a range of conditions and assumptions. Alternative simulation designs from that considered here could account for different heterogeneities and contact mixing behaviors within a population and other sources of heterogeneity (Lekone and Finkenstadt 2006). The design of such simulations should account for knowledge of the transmission dynamics of the disease and the demographic and social structure of the population. Other complexities such as staggered enrollment of patients into the trial and allowance for seasonal variation of incidence could also be taken into account in the simulation. A. Appendix We show here that for an effective vaccine where VE B >0, that VE CI VE B. We first consider the case of a homogeneous model where the transmission probabilities do not depend on distance. Let K represent the total number of infectious to noninfectious contacts over the course of the trial, that is, K is the random sum of the stochastic process (Y t ) over the course of the T days of observation. The associated probability mass function, g, is ( T ) g(k) = P Y t = k. t=1 The cumulative probabilities of infection (r 0, r 1 ) are r i = 1 T (1 p i ) Y t = 1 (1 p i ) K, K = t=1 T Y t. t=1 First we show that the inequality holds when the number of cumulative infectious to noninfectious contacts is conditioned upon K = k, where we denote the fact that VE is conditional by VE(k). Thus, we want to show that, conditional on k, This equivalently holds if 1 r 1 r 0 1 p 1 p (1 p 1) k 1 (1 p 0 ) k 1 p 1 p 0, which itself can be re-expressed as Set 1 (1 p 1 ) k 1 (1 p 0 ) k p 1 p 0. h(k) = 1 (1 p 1) k 1 (1 p 0 ) k. (A.1) When k is 1, then the two quantities are in fact equal. The result follows now if h is a nondecreasing function of k, which is shown by induction. When k is 2, then the inequality to be confirmed is h(2) is greater than or equal to h(1), which means that or, after simplification, that 1 (1 p 1 ) 2 1 (1 p 0 ) 2 p 1 p 0 2 p 1 2 p 0 1. The latter is true because 0 < p 1 p 0 < 1, which in turn implies 2 > 2 p 1 2 p 0 > 1. Next follows the induction step we assume that the inequality holds for a given k(> 2): which implies or that 1 (1 p 1 ) k 1 (1 p 0 ) k 1 (1 p 1) k 1 1 (1 p 0 ) k 1 1 (1 p 0 ) k 1 1 (1 p 0 ) k 1 (1 p 1) k 1 1 (1 p 1 ) k, 1 (1 p 0 ) j 1 (1 p 0 ) j+1 1 (1 p 1) j. (A.2) 1 (1 p 1 ) j+1 The above inequalities are assumed to be true, and we must next show that 1 (1 p 1 ) k+1 1 (1 p 0 ) k+1 1 (1 p 1) k 1 (1 p 0 ) k, which can be equivalently re-expressed in the same manner as above, so that 1 (1 p 0 ) k 1 (1 p 0 ) k+1 1 (1 p 1) k 1 (1 p 1 ) k+1, where k > 2; this itself holds from letting k = j in (A.2) above. Thus h is a strictly nondecreasing function, and hence the inequality in fact holds conditional on the cumulative number of infectious contacts. Using this information as well as the pdf for K, we get that VE CI = k VE CI (k) g(k) k = VE B g(k) k = VE B g(k) = VE B. k VE B (k) g(k) The extension to the nonhomogeneous case in which the transmission probabilities depend on distance is analogous to the derivation for homogenous case except we condition both on k and distance (the details are omitted). 9

11 Statistics in Biopharmaceutical Research: Vol. 1, No. 1 Acknowledgments The authors gratefully acknowledge the technical assistance of Ann Vanden Langenberg in preparing the figures. [Received September Revised December 2007.] References Black, S., Shinefield, H., Fireman, N., et al. (2000), Efficacy, Safety and Immunogenicity of Heptavalent Pneumococcal Conjugate Vaccine in Children, Pediatric Infectious Disease Journal, 19, Daley, D. J., and Gani, J. (1999), Epidemic Modelling: An Introduction, New York: Cambridge University Press. Dagan, R. et al. (2001), Conjugate Vaccines: Potential Impact on Antibiotic Use? International Journal of Clinical Practice, 18, Farrington, C.P., and Manning, G. (1990), Test Statistics and Sample Size Formulae for Comparative Binomial Trials with Null Hypothesis of Non-Zero Risk Difference Or Non-Unity Relative Risk, Statistics in Medicine, 9, Fine, P.E. (1993), Herd Immunity: History, Theory, Practice, Epidemiological Review, 15(2), Greenwood, M., and Yule, U.G. (1915), The Statistics of Antityphoid and Anti-cholera Inoculations and the Interpretation of Such Statistics, Proceedings of the Royal Society of Medicine, 8, Halloran, E. (1998), Chapter 27: Concepts of Infectious Disease in Modern Epidemiology, eds. K. Rothman and S. Greenland, Philadelphia: Lippincott Williams and Wilkins. Halloran, E., Longini, I.M. Jr., Nizam, A., and Yang, Y. (2002), Containing Bioterrorist Smallpox, Science, 298, Halloran, E., Struchiner, C., and Longini, I. Jr. (1997), Study Designs for Evaluating Different Efficacy and Effectiveness Aspects of Vaccines, American Journal of Epidemiology, 146(10), Lekone, P. E., and Finkenstadt B. F. (2006), Statistical Inference in a Stochastic Epidemic SEIR Model with Control Intervention: Ebola as a Case Study, Biometrics, 62, Longini, I. Jr., Halloran, E., and Nizam, A. (2002), Model Based Estimation of Vaccine Effects from Community Vaccine Trials, Statistics in Medicine, 21, 481. Longini, I.M. Jr, Sagatelian, K., Rida, W.N., and Halloran. E. (1998), Optimal Vaccine Trial Design When Estimating Vaccine Efficacy for Susceptibility and Infectiousness from Multiple Populations, Statistics in Medicine, 17, McCormick, A.W., Whitney, C.G., Farley, M.M., et al. (2003), Geographic Diversity and Temporal Trends of Antimicrobial Resistance in Streptococcus pneumoniae in the United States, Nature Medicine, 9, Moulton, L.H., O Brient, K.L., Reid, R., Weatherholtz, R., Santoshan, M., and Siber, G.R. (2006), Evaluation of the Indirect Effects of a Pneumococcal Vaccine in a Community Randomized Study, Journal of Biopharmaceutical Statistics, 16, About the Authors Blake Charvat is Research Associate, Ron Brookmeyer is Professor, and Jay Herson is Senior Associate, Department of Biostatistics, Johns Hopkins Bloomberg School of Public Health, Baltimore, MD ( for correspondence: rbrook@jhsph.edu). 10